## Implementing Fuzzy Sets in SQL Server, Part 1: Membership Functions and the Fuzzy Taxonomy

**By Steve Bolton**

…………In the first installment of this amateur self-tutorial series on applying fuzzy set theory to SQL Server databases, I discussed how neatly it dovetails with Behavior-Driven Development (BDD) principles and user stories. This is another compelling reason to take notice of fuzzy sets, beyond the advantages of using a set-based language like T-SQL to implement them, which will become obvious as this series progresses. There aren’t any taxing mental gymnastics involved in flagging imprecision in natural language statements like “hot,” “cloudy” or “wide,” which is strikingly similar to the way user stories are handled in BDD. What fuzzy sets bring to the table is the ability to handle imprecise data that resides in the no-man’s land between ordinal and continuous Content types. In addition to flagging imprecision in natural language and domain knowledge that is difficult to pin down, it may be helpful to look for attributes which represent categories that are ranked in some way (which sets them apart from nominal data, which is not ordered on any scale) but which it would be beneficial to express on a continuous numerical scale, even at the cost of inexactness. Thankfully, mathematicians have already hashed out a whole framework for modeling this notoriously tricky class of data, even though it is as underused as the SQL Server Data Mining (SSDM) components I tried to publicize in a previous mistutorial series. It is also fortunate that we already have an ideal tool to implement it with in T-SQL, which can already handle most of the mathematical formulas devised over the last few decades. As I’ll demonstrate in this article, it only takes a few minutes to implement simple membership functions that grade records based on how much they belong to a particular set. It is only when we begin combining different types of imprecision together and assigning more nuanced interpretations to the grading systems that complexity quickly arises and the math becomes challenging. Although I’m still learning the topic as I go – I find it is much easier to absorb this kind of material by writing about it – I hope to reduce the challenge involved by taking a stab at explaining it, which will at a minimum at least help readers avoid repeating my inevitable mistakes.

…………The first challenge to overcome is intimidation, because the underlying concepts don’t even require a college education to grasp; in fact, some DBAs have probably already worked with forerunners of fuzzy sets unwittingly, on occasions where they’ve added columns that rate a row’s inclusion in a particular set. It doesn’t take much mental juggling to start thinking explicitly about such attributes as measures of membership in a particular set. Perhaps the simplest forms of membership functions are single columns filled with data that has been assigned that kind of meaning, which can even be derived from such sources as subjective grades assigned by end users in exactly the same manner as movie or restaurant ratings. The data can even be permanently static. At the next level of complexity, we could of course store such data in the form of computed columns, regardless of whether it is read-only or not.

…………A couple of really simple restrictions are needed to bring this kind of data into line with fuzzy set theory though. First, since the whole object is to treat ordinal data as if it were continuous, we’d normally use T-SQL data types like float, numeric and decimal – which are the closest we can get, considering that our finite computers can’t truly handle infinitesimal scales. Furthermore, it is probably wise to stick with the convention of using a scale between 0 and 1, since this enables us to integrate it seamlessly with evidence theory, stochastics, decision theory, control theory and neural net weights, all of which are also typically bounded in the same range or quite similar ones; some of the theoretical resources I consulted mentioned in an offhand way that it is possible to use other scales, but I haven’t seen a single instance of it yet in the literature. Ordinal categories are often modeled in SQL Server in text data types like nvarchar, tinyint codes or some type of foreign key, which might have to be retained alongside the membership function storage column; in other instances, our membership function may be scoring on the basis of several attributes in a table or view, or perhaps all of them. Of course, in many use cases we won’t need to store the membership function value at all, so it will have to be calculated on the fly. If we’re simply storing a subjective rating or whatever, we might only need some sort of interface to allow end users to enter their own numbers (on a continuous scale of 0 to 1), in which case there is no need for a membership function per se. If a table or view participates in different types of fuzzy sets, it may be necessary to add more of these membership columns for each of them, unless you want to calculate the values as you go. Simply apply the usual rules of data modeling and principles of performance maximization to determine the strategies that fit your use cases best.

**Selecting Membership Functions**

That is all kid stuff for most DBAs. The challenges begin when we try to identify which membership function would be ideal for the use cases at hand. Since the questions being asked of the data vary from one problem to the next, I cannot possibly answer that. I suppose that you could say the general rule of thumb with membership functions is that the sky’s the limit, as long as we stay at an altitude between 0 and 1. Later in this series I’ll demonstrate how to use particular classes of functions called T-norms and T-conorms, since various mathematical theorems demonstrate that they’re ideal for implementing unions and intersections, but even in these cases, there are so many available to us that the difficulty consists chiefly in selecting an appropriate match to the problem you’re trying to solve. There might be more detailed guidelines available from more recent sources, but my favorite reference for the math formulas, George J. Klir and Bo Yuan’s classic *Fuzzy sets and Fuzzy Logic: Theory and Applications*, provides some suggestions. For example, membership values can be derived from sample data through LaGrange interpolation and two methods I have used before, least-squares curve fitting and neural networks.[1] They also discuss how to aggregate the opinions of multiple experts using both direct and indirect methods of collection, in order to ascertain the meaning of fuzzy language terms. The specifics they provide get kind of involved, but it is once again not at all difficult to implement the premises in a basic way; a development team could, for example, reach a definition of the inherently fuzzy term “performance” by scoring their opinions, then weighting them by the authority of their positions.[2] The trick is to pick a mathematical operation that pools them altogether into a single value that stays on a scale of 0 to 1, while still capturing the meaning in a way that is relevant to the problem at hand.

…………Klir and Yuan refer to this as an application of the newborn field of “knowledge engineering,”[3] which has obvious connections to expert systems. Since fuzzy set theory is still a wide-open field there’s a lot of latitude for inventing your own functions; there might be an optimal function that matches the problem at hand, but no one may have discovered it yet. In situations like these, my first choice would be neural nets, since I saw spectacular evidence long ago of how they can be ideal for modeling unknown functions (which pretty much sparked my interest in data mining). Before trying one of these advanced approaches, however, it might be wise to think hard about what mathematical properties you require of your outputs and then consult a calculus book or other math reference to try to find a matching function. While trying to teach myself calculus all over again recently, I was reintroduced to the whole smorgasbord of properties that distinguish mathematical functions from each other, like differentiability, integrability, monotonicity, analycity, concavity, subadditity, superadditivity, discontinuity, splines, super- and subidempotence and the like. You’ll encounter these terms on every other page in fuzzy set math references, which can be differentiated (pun intended?) into broad categories like function magnitude, result, shape and mapping properties. One thing I can help with is to caution that it’s often difficult or even impossible to implement ones (like the popular gamma function) which require calculations of permutations or combinations. It doesn’t matter whether you’re talking about T-SQL, Visual Basic, C# or some computer language implemented outside of the Microsoft ecosystem: it only takes very small input values before you reach the boundaries of the highest data types. This renders certain otherwise useful data mining and statistical algorithms essentially useless in the era of Big Data. An exclamation point in a math formula ought to elicit a groan, because the highest value you might be able to plug into a factorial function in SQL Server is about 170.

**A Trivial Example with Two Membership Functions**

I’ll provide an example here of moderate difficulty, in between the two extremes of advanced techniques like least squares (or God forbid, the gamma function and its relatives) on the one hand and cheesy screenshots of an ordinary table that just happens to have a float column scored between 0 and 1 on the other. As we’ll see in the next few tutorials on fuzzy complements, unions, intersections and the like, when calculating set memberships on the fly we usually end up using a lot of CASE, BETWEEN and MIN/MAX statements in T-SQL, but that won’t be the case in the example below because the values are derived from a stored procedure and stored in two temporary tables. To demonstrate how seamlessly fuzzy set techniques can be integrated with standard outlier detection techniques, I’ll recycle the code from my old tutorial Outlier Detection with SQL Server, part 2.1: Z-Scores and use it as my membership function.

…………There’s a lot of code in Figure 1, but it’s really easy to follow, since all we’re doing is running the Z-Scores procedure on a dataset on the Duchennes form of muscular dystrophy I downloaded from Vanderbilt University’s Department of Biostatistics a couple of tutorial series ago, which now occupies about 9 kilobytes of space in a sham DataMiningProjects database. There’s probably a more efficient way of going about this, but the results are stored in a table variable and the @RescalingMax, @RescalingMin and @RescalingRange variables and the ReversedZScore column are then used to normalize the Z-Score on a range of 0 to 1 (the GroupRank column was needed for the stored procedure definition in the original Z-Scores tutorial, but can be ignored in this context). To illustrate how we can combine fuzzy set approaches together in myriad combinations, I added an identical table that holds Z-Scores for a second column from the same dataset, which is rescaled in exactly the same way. In the subquery SELECT I merely multiply the two membership values together to derive a CombinedMembershipScore. What this essentially does is give us a novel means of multidimensional outlier detection.

** Figure 1: Using Z-Scores for Membership Functions**DECLARE @RescalingMax decimal(38,6), @RescalingMin decimal(38,6), @RescalingRange decimal(38,6)

DECLARE @ZScoreTable1 table

(ID bigint IDENTITY (1,1),

PrimaryKey sql_variant,

Value decimal(38,6),

ZScore decimal(38,6),

ReversedZScore as CAST(1 as decimal(38,6)) – ABS(ZScore),

MembershipScore decimal(38,6),

GroupRank bigint

)

DECLARE @ZScoreTable2 table

(ID bigint IDENTITY (1,1),

PrimaryKey sql_variant,

Value decimal(38,6),

ZScore decimal(38,6),

ReversedZScore as

CAST(1 as decimal(38,6)) – ABS(ZScore),

MembershipScore decimal(38,6),

GroupRank bigint

)

INSERT INTO @ZScoreTable1

(PrimaryKey, Value, ZScore, GroupRank)

EXEC Calculations.ZScoreSP

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’CreatineKinase‘,

@PrimaryKeyName = N’ID’,

@DecimalPrecision = ’38,32′,

@OrderByCode = 8

INSERT INTO @ZScoreTable2

(PrimaryKey, Value, ZScore, GroupRank)

EXEC Calculations.ZScoreSP

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’LactateDehydrogenase‘,

@PrimaryKeyName = N’ID’,

@DecimalPrecision = ’38,32′,

@OrderByCode = 8

— RESCALING FOR COLUMN 1

SELECT @RescalingMax = Max(ReversedZScore), @RescalingMin= Min(ReversedZScore) FROM @ZScoreTable1

SELECT @RescalingRange = @RescalingMax – @RescalingMin

UPDATE @ZScoreTable1

SET MembershipScore = (ReversedZScore – @RescalingMin) / @RescalingRange

— RESCALING FOR COLUMN 2

SELECT @RescalingMax = Max(ReversedZScore), @RescalingMin= Min(ReversedZScore) FROM @ZScoreTable2

SELECT @RescalingRange = @RescalingMax – @RescalingMin

UPDATE @ZScoreTable2

SET MembershipScore = (ReversedZScore – @RescalingMin) / @RescalingRange

SELECT ID, PrimaryKey, Value, ZScore1, ZScore2, MembershipScore1, MembershipScore2, CombinedMembershipScore

FROM (SELECT T1.ID, T1.PrimaryKey, T1.Value, T1.ZScore AS ZScore1, T2.ZScore as ZScore2,

T1.MembershipScore MembershipScore1, T2.MembershipScore AS MembershipScore2, T1.MembershipScore * T2.MembershipScore AS CombinedMembershipScore

FROM @ZScoreTable1 AS T1

INNER JOIN @ZScoreTable2 AS T2

ON T1.ID = T2.ID) AS T3

WHERE CombinedMembershipScore IS NOT NULL

ORDER BY CombinedMembershipScore DESC

— if we want to store the values in the original table, we can use code like this:

UPDATE T4

SET T4.MembershipScore1 = T3.MembershipScore1, T4.MembershipScore2 = T3.MembershipScore2, T4.CombinedMembershipScore =

T3.CombinedMembershipScore

FROM DataMiningProjects.Health.DuchennesTable AS T4

INNER JOIN (SELECT T1.PrimaryKey, T1.MembershipScore AS MembershipScore1, T2.MembershipScore AS MembershipScore2, T1.MembershipScore * T2.MembershipScore AS CombinedMembershipScore

FROM @ZScoreTable1 AS T1

INNER JOIN @ZScoreTable2 AS T2

ON T1.ID = T2.ID) AS T3

ON T4.ID = T3.PrimaryKey

__Figure 2: Sample Results from the Duchennes Practice Data
__

…………Figure 2 gives a glimpse of what the original DuchennesTable might look like if we wanted to store these values rather than calculate them on the fly, which can be accomplished by adding the three float columns on the right to the table definition and executing the UPDATE code at the end of Figure 1. In natural language, we might say that “the first record is 0.941446^{th} of a member in the set around the average Creatine Kinase value” but “the fifth record is only 0.764556^{th} of a member of the set near the mean Lactate Dehydrogenase value.” We could even model deeper levels of imprecision by creating categories like “near” for the high membership values in each column and “outlier” for the lowest ones, then define their boundaries in terms of fuzzy sets. This might be an ideal use for triangular and trapezoidal numbers, which can be worth the expense in extra code, as I’ll explain a few articles from now. We’re also modeling a different type of imprecision in another sense, because we know instinctively that there ought to be some way of gauging whether or not a record’s an outlier when both columns are taken into account; perhaps nobody knows precisely what the rules for constructing such a metric might be, but the CombinedMembershipScore at least allows us to get on the board.

…………Please keep in mind that I’m only using Z-Scores here because it’s familiar to me and is ideal for illustrating how fuzzy sets can be easily adapted to one particular use case, outlier detection. If we needed to make inferences about how well the data fit a gamma or exponential distribution, we might as well have used the corresponding goodness-of-fit tests and applied some rescaling techniques to derive our membership values; if we needed to perform fuzzy clustering, we could have plugged in a Manhattan distance function or one of its relatives. Fuzzy set memberships are often completely unrelated to stochastics and should not be interpreted as probabilities unless you specific intend to model them. The usefulness of fuzzy sets is greatly augmented when we move beyond mere set membership by tweaking the meaning a little, so that they can be interpreted as degrees of evidence, reliability, risk, desirability, or the like, which allow us to plug into various other well-developed mathematical theories. All functions can be differentiated by their return types, number of return and input values, allowable data types and ranges, mathematical properties and the like (not to mention performance costs), but in fuzzy set theory the issue of meaning has a somewhat more prominent role. In some cases, it may even be desirable to use multiple membership functions to determine membership in one fuzzy set, as in my crude example above. These myriad shades of meaning and potential for combinations of them lead to a whole new level of complexity, which may nonetheless be worthwhile to wade through for certain imprecision modeling problems.

**A Taxonomy of Fuzzy Sets (that Doesn’t Tax the Brain)**

I originally figured that I’d have to organize this series according to a taxonomy of different types of fuzzy sets, but it’s actually fairly simply to sketch the outlines of that otherwise advanced topic. Instead of delving into all of the complex math, it’s a lot easier for a layman to dream up all of the combinations of all the places in a set they can apply fuzziness, different means of encoding it and so on. The important thing to keep in mind is that there’s probably a term out there for whatever combination you’re using and that somewhere along the line, mathematicians have probably already figured out most of the logical implications decades ago (thereby saving a lot of the grunt work and reinventing the wheel, assuming that you can interpret their writing and the really thick formulas that often accompany them). The easiest ones to explain are real-valued and interval sets, in which the membership functions are determined on the real number line (which is all we ever encounter in SQL Server) or by a range of values on it.[4] Type-2 Fuzzy Sets illustrate the concept of tacking on further fuzziness perfectly – all we do take an interval-valued set and then assign grades to its boundaries as well. Fuzzy set theorists Yingjie Yang and Chris Hinde state that “A type-2 fuzzy set describes its memberships using type-1 fuzzy sets, but it needs precise crisp values to describe its secondary memberships.”[5] As the levels and number of values needed to define these sets proliferates, the performance costs do as well, so one has to be sure in advance that the extra complexity is useful in modeling real-world data. As Klir and Yuan put it, “Fuzzy sets of type 2 possess a great expressive power, and, hence, are conceptually quite appealing. However, computational demands for dealing with them are even greater than those for dealing with interval-valued sets. This seems to be the primary reason why they have almost never been utilized in any applications.”[6] I’d wager that’s still true, given the fact that the applications of ordinary fuzzy sets to data mining, data warehousing and relational databases have barely been scratched since the mathematicians invented these things years ago.

…………Rough sets also involve fuzzy values on intervals in a sense, but they model approximate distinctions between objects. Say, for example, you classify all of the objects in a child’s bedroom and want to see which qualify as part of a set labeled Toys. A sports car might be considered an adult toy to a certain degree, depending on such factors as whether or not the owner uses it for purposes other than occasional joy rides. The plastic dinosaurs and megafauna in a Prehistoric Playset are certainly toys, as are Fisher Price’s wooden people (well, cheap plastic these days). Medicine definitely wouldn’t belong to the set (at least according to these singing pills). Would one of these classic glow-in-the-dark Godzilla models from the ‘70s qualify? Well, that’s not quite clear, since it’s an object only a child would really appreciate, but they’re unlikely to actually play with it as a toy very often, since it’s designed to stay on display. They could conceivably take them off the shelf and pit them against the Fisher Price people; in this instance, the set membership might be defined by criteria as fuzzy as the whims of a child’s imagination, but we have tools to model it, if a need should arise. The definition of the attribute is in question, not whether a particular row belongs to a set, which is the case with ordinary fuzzy membership functions.

…………In Soft Sets, the characteristics that define the set are themselves fuzzy. I haven’t attempted to model those yet, but I imagine it may require comparisons between tables and views and placing weights on how comparable their different columns are to each other, rather than the rows. Here’s a crude and possibly mistaken example I came up with off the top of my head: in Soft Sets you might have a table with columns for Height, Width and Age and another with columns for Height, Width and Time, in which the first two columns of each are completely related to each and therefore are assigned weights of one, whereas Age and Time are only tangentially related and therefore might be assigned a weight somewhere between 0 and 1. Near sets apparently address a problem tangential to rough and soft sets, by quantifying the quantity and quality of resemblances between objects that might belong to a fuzzy set. Once we’ve been introduced to these concepts, they can obviously be combined together into an endless array of variants, which go by such mouthfuls as “rough intuitionistic Level-2 fuzzy near sets.” Just keep in mind that it is more common to encounter such structures in the real world in everyday language than it is to know the labels and their mathematical properties. It is also easier than it sounds to implement them in practice, if we’re using set-based tools like T-SQL that are ideal for the job.

…………I probably won’t spend much time in this series on even more sophisticated variants that might nonetheless be useful in modeling particular problems. Shadowed sets used multidimensional projections to qualify the lack of knowledge of whether or not a data point belongs to a fuzzy set. Neural nets are a cutting-edge topic I hope to tackle on this blog in a distant future (my interest in data mining was piqued way back in the 1990s when I saw some I cooked up at home do remarkable things) but it is fairly easy to describe Neuro-Fuzzy Sets, in which we’re merely using neural nets to perform some of the functions related to fuzzy sets. The combinations that can be derived from are limited only by one’s imagination; there are already neural nets in use in industry today that use fuzzy functions for activation and fuzzy sets whose membership values are derived from neural nets, and so forth. Undetermined and Neutrosophic Logic are variants of fuzzy logic that can be applied to fuzzy sets if we need to model different types of indeterminacy, which is a topic I’ll take up in a future article on how fuzzy sets can be put to good use in uncertainty management.

…………Blurry sets are a recent innovation designed to incorporate the kind of combinations of fuzziness we’ve just mentioned, but without sacrificing the benefits of normal logic – which might be of great benefit in the long run, since the value of some of recently developed logical systems is at best unproven.[7] Some will probably be substantiated in the long run, but some seem to be motivated by the sort of attention-getting shock value that can make academicians famous overnight these days (some of them seem to be implementations and formal defenses of solipsism, i.e. one of the defining characteristics of schizophrenia). Q-Sets are apparently an even more advanced variants developed for use in the strange world of quantum physics; since making Schrödinger’s cat disappear isn’t among most SQL Server users’ daily duties, I’ll leave that one out for now. I’ll probably also steer away from discussing more advanced types of fuzzy sets that include multiple membership functions, which aren’t referenced often in the literature and apparently are implemented only in rare circumstances.. Intuitionistic Sets have two, one for membership or non-membership, while Vague Sets also use two, except in that case one assesses the truth of the evidence for a record’s membership and the other its falsehood; I presume truth tables and the like are then built from the two values. A novel twist on this theme is the use of multiple membership functions to model the fact that the programmer is uncertain of which membership functions to use in defining fuzzy sets.[8] Multisets are often lumped in with the topic of fuzzy sets, but since they’re just sets that allow duplicate values, I don’t see much benefit in discussing them here. Genuine sets take fuzziness to a new level in an entirely different way, by generalizing the concept of fuzzy set in the same manner that fuzzy sets generalize ordinary “crisp” sets, but I won’t tack on another layer of mathematical complexity at this point, not when the potential for using the established methods of generalization has barely been scratched.

**False Mysticism and the Fuzzy Mystique**

This wide-open field is paradoxically young in terms of mathematical intellectual history, but overripe for implementation, given that many productive uses for it were derived decades ago but haven’t percolated down from academia yet. Taking a long view of the history of math, it seems that new waves of innovation involve the addition of new dimensions to existing objects. Leonhard Euler introduced complex numbers in the 18^{th} Century, then theoreticians like Bernhard Riemann and Charles Hinton contributed the concepts of higher-dimensional space and its curvature in the 19^{th}. Around the same time, Georg Cantor was working out set theory and such mind-blowing structures as high-cardinality infinities and transfinities. More recently, Benoit Mandelbrot elaborated the theory of fractional dimensions, which are now cornerstones in chaos theory and modern art, where they go by the better-known term of fractals. This unifying principle of mathematical innovation stretches back as far as ancient Greece, when concepts like infinity, continuous scales and the like were still controversial; in fact, the concept of zero did not reach the West until it was imbibed from Arab sources in the Middle Ages. Given that zero was accepted so late in history, it is thus not at all surprising that negative numbers were often derided by Western mathematicians as absurdities well into the 18^{th} and 19^{th} Centuries, many centuries after their discovery by Chinese and Indian counterparts.[9] A half-conscious prejudice against the infinite regress of non-repeating digits in pi and Euler’s number is embedded in the moniker they still go by today, “irrational numbers.” The same culprit is behind the term “imaginary number” as well. Each of these incredibly useful innovations was powered by the extension of scales into previously uncharted territory; each was also met by derision and resistance at first, as were fuzzy sets to a certain extent after their development by 20^{th} Century theoreticians like Max Black and Lofti A. Zadeh.

…………Many of these leaps forward were also accompanied by hype and as sort of unbalanced intellectual intoxication, which is the main risk in using these techniques. Fuzzy sets are unique, however, in that some of the pioneers were conscious of the possibility of leveraging the term “fuzzy” for attention; Zadeh openly acknowledges that the term has its uses in terms of publicity power, although he did not originally invent the term for that purpose. The strategy has backfired to a certain extent, however, by drawing the wrong kind of attention. “Fuzzy” is a term that immediately conjures up many alternative images, many of which don’t seem conducive to a high-powered, mission-critical production environment – like teddy bears, static, 1970s cop shows and something out of the back of George Carlin’s fridge.

…………Many of the taxonomic terms listed above also carry a kind of shock value to them; in other branches of academia this usually signifies that the underlying theory is being overstated or is even the product of crackpots with tenure, but in this case there is substantial value once the advertising dross has been stripped away. In fact, I’d wager that if more neutral terms like “graded set” or “continuously-valued set” were used in place of “fuzzy,” these techniques would be commonplace today in a wide variety of industries, perhaps even database management; in this case, the hype has boomeranged by stunting the adoption of an otherwise indispensable set of tools. As McNeill points out, some of the researchers employed in implementing fuzzy sets in various industries (including the development of the space shuttle) back in the early ‘90s had to overcome significant institutional resistance from “higher-ups” who “fretted about image.”[10] They are right to fret within reason, because these tools can certainly be misapplied; in fact, I’ve seen brilliant theorists who grasp the math a lot better than I do abuse it in illogical ways (for the sake of being charitable, I don’t want to call them out by name). Some highly regarded intellectuals don’t recognize *any* boundaries to the theory, for all of reality is fuzzy in their eyes – which is the mark of fanaticism, and certain to stiffen any institutional resistance on the other side. Every mathematical innovation in history has not only been accompanied by knee-jerk opposition from Luddites on one side, but also unwarranted hype and irrational exuberance on the other; fuzzy sets are as susceptible to misuse by bad philosophers and fanatics as higher dimensions, chaos theory and information theory have been for decades, so it is not unwise to tread carefully and maintain intellectual sobriety when integrating fuzzy sets into any development process.

…………Perhaps the best way to overcome this kind of institutional resistance and receive backing for these techniques is to be up front and demonstrate that you recognize the hype factor, plus have clear litmus tests for discerning when and when not to apply fuzzy set theory. Two of these are the aforementioned criteria of searching for data that resides in between ordinal and continuous data in the hierarchy of Content types and sifting through natural language terms for imprecision modeling. It is also imperative to develop clear standards for differentiating between legitimate and illegitimate uses of fuzzy sets, to prevent the main risk: “fuzzifying” data that it is inherently crisp. It is indeed possible to add graded boundaries to any mathematical objects (some of which we’ll explore later in this series), but in many cases, there is no need to bother. Fuzzy logic in the wrong doses and situations can even lead to fallacious conclusions. In fact, applying fuzziness to inherently crisp objects and vice-versa is one of the fundamental strategies human beings have employed since time immemorial to deceive both themselves and others. Here’s a case in point we’ve all seen: you tell your son or daughter they can’t have a snack, but you catch them eating crackers; invariably, their excuse involves taking advantage of the broad interval inherent in the term “snack,” a set which normally, but not always, included crackers. Of course, when people grow up they sometimes only get more skilled at blurring lines through such clever speech (in which case they often rise high in politics, Corporate America and the legal profession). Here’s an important principle to keep in mind: whenever you see a lot of mental energy expended to tamper with the definitions of things, but find the dividing lines *less* clear afterwards, then it’s time to throw a red flag. The whole point of fuzzy sets is not to obscure clear things, but to clear up the parts that remain obscure. Fuzziness is in exactly the same boat as mysticism, which as G.K. Chesterton once said, is only useful when it explains mysteries:

“A verbal accident has confused the mystical with the mysterious. Mysticism is generally felt vaguely to be itself vague—a thing of clouds and curtains, of darkness or concealing vapours, of bewildering conspiracies or impenetrable symbols. Some quacks have indeed dealt in such things: but no true mystic ever loved darkness rather than light. No pure mystic ever loved mere mystery. The mystic does not bring doubts or riddles: the doubts and riddles exist already…The mystic is not the man who makes mysteries but the man who destroys them. The mystic is one who offers an explanation which may be true or false, but which is always comprehensible—by which I mean, not that it is always comprehended, but that it always can be comprehended, because there is always something to comprehend.”[11]

…………Fuzzy sets are not meant to mystify; they’re not nebulous or airy, but designed to squeeze some clarity out of apparently nebulous or airy data and logic. They are akin to spraying Windex on a streaky windshield; if you instead find your vision blocked by streaks of motor oil, it’s time to ask who smeared it there and what their motive was. Fuzziness isn’t an ingredient you add to a numerical recipe to make it better; it’s a quality inherent in the data, which is made clearer by modeling the innate imprecision that results from incomplete measurement, conflicting evidence and many other types of uncertainty. The point is not to make black and white into grey, but to shine a light on it, so that we can distinguish the individual points of black and white that make up grey, which is just a composite of them. These techniques don’t conjure up information; they only ensure that what little information is left over after we’ve defined the obvious crisp sets doesn’t go to waste. Fuzziness can actually arise from a surfeit of detail or thought, rather than a deficit or either; the definition of an object may be incomplete because so many sense impressions, images, stray thoughts, academic theories and whatnot are attached to its meaning that we can neither include them all nor leave any out.

…………As we shall see in future articles on uncertainty management, the manner in which the meaning of set membership can be altered to incorporate evidence theory and the like is indeed empowering, but calls for a lot of mental rigor to resist unconscious drifts in definition. It’s an all-too human problem that can occur to anyone, particular when mind-blowing topics are under discussion; it’s even noticeable at times in the writings of brilliant quantum physicists, who sometimes unconsciously define their terms slightly differently at the beginning of a book than at the end, in ways that nonetheless make all the difference between Schrödinger’s Cat being alive or dead. “Definition drift” also seems to be a Big Problem in Big Analysis for the same reason. It likewise seems to occur in texts on fuzzy sets, where term “fuzz” is often accurately described on one page as a solution to innate imprecision, but on the next, is unconsciously treated as if it were a magic potion that ought to be poured on everything. Another pitfall is getting lost in all of the bewildering combinations of fuzziness I introduced briefly in the taxonomy, but the answer to that is probably to just think of them in terms of ordinary natural language and only use the academic names when sifting through the literature for appropriate membership functions and the like. Above all, avoiding modeling crisp sets that have inherently Boolean yes-or-no membership values as fuzzy sets, because as the saying goes, you can’t be “a little bit pregnant.” Continuous scales can certainly be added to any math object, but if the object being modeled is naturally precise, then it is at best a waste of resources that introduces the risk of fallacious reasoning and at worst, an opening for someone with an axe to grind to pretend a particular scale is much more imprecise than it really is. One dead giveaway is the use of short scales in comparison to the length of the original crisp version. For example, this is the culprit when quibbling erupts over such obviously crisp sets as “dead” and “alive,” on the weak grounds that brain death takes a finite amount of time, albeit just a fraction of a person’s lifespan. It might be possible to develop a Ridiculousness Score by comparing the difference in intervals between those few moments, which occur on an almost infinitesimal scale, against an “alive” state that can span 70-plus years in human beings or the “dead,” which is always infinite. I haven’t seen that done in the literature, but in two weeks, I’ll demonstrate how the complements of fuzzy sets can be used to quantify just how imprecise our fuzzy sets are. The first two installments of this series were lengthy and heavy on text because we needed a solid grounding in the meaning of fuzzy sets before proceeding to lessons in T-SQL code, but the next few articles will be much shorter and immediately beneficial to anyone who wants to put it into action.

[1] pp. 290-293, Klir, George J. and Yuan, Bo, 1995, __Fuzzy Sets and Fuzzy Logic: Theory and Applications__. Prentice Hall: Upper Saddle River, N.J.

[2] *IBID.*, pp. 287-288, 292-293.

[3] *IBID.*, p. 281.

[4] For a quick introduction to the various fuzzy set types, see the __Wikipedia__ article Fuzzy Sets at http://en.wikipedia.org/wiki/Fuzzy_set. I consulted it to make sure that I wasn’t leaving out some of the newer variants that came out since Klir and Yuan and some of the older fuzzy set literature I’ve read, much of which dates from the 1990s. I lost some of the citations to the notes I derived these three paragraphs from (so my apologies go out to anyone I might have inadvertently plagiarized) but nothing I said here can’t be looked up quickly on Wikipedia, Google or any recent reference on fuzzy sets.

[5] Hinde, Chris and Yang, Yingjie, 2000, A New Extension of Fuzzy Sets Using Rough Sets: R-Fuzzy Sets, pp. 354-365 in __Information Sciences__, Vol. 180, No. 3. Available online at the web address https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/13244/3/rough_m13.pdf

[6] p. 17, Klir and Yuan.

[7] Smith, Nicholas J. J., 2004, Vagueness and Blurry Sets, pp 165-235 in __Journal of Philosophical Logic__, April 2004. Vol. 33, No. 2. Multiple online sources are available at http://philpapers.org/rec/SMIVAB

[8] See Pagola, Miguel; Lopez-Molina, Carlos; Fernandez, Javier; Barrenechea, Edurne; Bustince, Humberto , 2013, “Interval Type-2 Fuzzy Sets Constructed From Several Membership Functions: Application to the Fuzzy Thresholding Algorithm,” pp. 230-244 in IEEE Transactions on Fuzzy Systems, April, 2013. Vol. 21, No. 2. I haven’t read the paper yet (I simply can’t afford access to many of these sources) but know of its existence.

[9] See Rogers, Leo, 2014, “The History of Negative Numbers,” published online at the __NRICH.com__ web address http://nrich.maths.org/5961.

[10] pp. 261-262, McNeill.

[11] Chesterton, G.K., 1923, __St Francis of Assisi__. Published online at the __Project Gutenberg__ web address http://gutenberg.net.au/ebooks09/0900611.txt

## Implementing Fuzzy Sets in SQL Server, Part 0: The Buzz About Fuzz

**By Steve Bolton**

…………I originally planned to post a long-delayed series titled Information Measurement with SQL Server next, in which I’d like to cover scores of different metrics for quantifying the data our databases hold – such as how random, chaotic or ordered it might be, or how much information it might provide. I’m putting it off once again, however, because I stumbled onto a neglected topic that could be of immediate benefit to many DBAs: fuzzy sets and their applications in uncertainty management programs and software engineering processes. Since SQL Server is a set-based storage system, I always suspected that the topic would be directly relevant in some way, but never expected to discover just how advantageous they can be. As in my previous series on this blog, I’m posting this mistutorial series in order to introduce myself to the topic, not because I know what I’m talking about; writing about it helps reinforce what I learn along the way, which will hopefully still be of some use to others once all of the inevitable mistakes are overcome. In fact, I guarantee that every DBA and .Net programmer out there has encountered problems which could be more easily and quickly solved through these proven techniques for modeling imprecision, which is precisely what many software engineering and data modeling problems call for. Despite the fact that my own thinking on the topic is still fuzzy (as usual) I’m certain this series can be immediately helpful to many readers, since there’s such an incredible gap between the math, theory and implementations of fuzzy set techniques in other fields one hand, and their slow adoption in the relational and data mining markets on the other.

…………Instead of beating around the bush, I’ll try to encapsulate the purposes and uses cases of fuzzy sets as succinctly as possible: basically, you look for indefinite terms in ordinary speech, then squeeze what little information content you can out of them by assigning grades to records to signify how strongly they belong to a particular set. Most fuzzy set problems are modeled in terms of natural language like this. The overlap with Behavior-Driven Development (BDD) and user stories is quite obvious, but after reading on those hot topics a year prior to learning about fuzzy sets, I was immediately struck by how little these techniques of modeling imprecision are apparently used but how easy it would be to incorporate them into database and application development processes. Uncertainty is a notorious problem in any engineering process, but using sets with graded memberships can even be used to capture it and flesh it out more thoroughly, as part of one of the programs of “uncertainty management” I’ll describe later in this series.

**From Crisp Sets to Membership Functions**

These powerful techniques arise from the quite simple premise that we can assign membership values to records, which some SQL Server users might being doing from time to time insentiently, without realizing that they were approaching the borderlands of fuzzy set theory. Most relational and cube data is in the form of what mathematicians call “crisp sets,” which don’t require membership functions because they’re clear-cut yes-or-no decisions; to theoreticians, these are actually just a special case of a broader class of fuzzy sets, distinguished only by the fact that their membership functions are limited to values of either 0 or 1. In the relational field as it stands today, you either include a row in a set or you don’t, without any in-between. In contrast, most fuzzy membership functions assign continuous values between 0 and 1; although other scales are possible, I have yet to see an example in the literature where any other scale was used. I doubt it is wise to use any other range even if there might be a performance boost of some kind in applying larger-scale float or decimal data types, given that it helps integrate fuzzy sets with the scales used in a lot of other hot techniques I’ll cover later, like Dempster-Shafer evidence theory, possibility theory, decision theory and my personal favorite, neural net weights. That overlap transforms fuzzy sets into an interchangeable part of sorts, in what might be termed modular knowledge discovery.

…………That all sounds very grandiose, but anyone can practice picking out fuzzy sets represented in everyday speech. Artificial intelligence researchers Roger Jang and Enrique Ruspini provide a handy list of obvious ones in a set of slides reprinted by analytics consultant Piero P. Bonissone, including Height, Action Sequences, Hair Color, Sound Intensity, Money, Speed, Distance, Numbers and Decisions. Some corresponding instances of them we encounter routinely might include Tall People, Dangerous Maneuvers, Blonde Individuals, Loud Noises, Large Investments, High Speeds, Close Objects, Large Numbers and Desirable Actions.[i] The literature is replete with such simple examples, of which imprecise weather and height terms like “cloudy,” “hot” and “short” seem to be the most popular. The key things is to look in any BDD or user story implementation are linguistic states where the speech definitely signifies something, but the meaning is not quite clear – particularly when it would still be useful to have a sharper and numerically definable definition, even when we can’t be 100 percent precise.

**Filling a Unique Niche in the Hierarchy of Data Types**

It may be helpful to look at fuzzy sets as a new tool occupying a new rung in the ladder of Content types we already work with routinely, especially in SQL Server Data Mining (SSDM). At the lowest level of data type complexity we have nominal data, which represents categories that are not ranked on any scale; these are basically equivalent to the Discrete Content type in SSDM and are often implemented in text data types or tinyint codes in T-SQL. On the next rung up the ladder we have ordinal data in which categories are assigned some rank, although the gaps may not be defined or even proportional; above that we have continuous data types (or the best approximation we can get, since modern computers can’t handle infinitesimal scales) that are often implemented in T-SQL in the float, numeric and decimal data types. Fuzzy sets represent a new bridge between the top two rungs, by providing more meaningful continuous values to ordinal data that in turn allow us to do productive things we couldn’t do with them before, like performing arithmetic, set operations or calculating stats. Any fuzzy set modeling process ought to focus on looking for data that is ordinal with an underlying scale that is not precisely discernible, but in which it would be useful to work with a continuous scale. That really isn’t much more difficult than picking imprecise terminology out of natural language, which anyone can make a game of. Given their “ability to translate imprecise/vague knowledge of human experts” we might also want to make a habit of flagging instances where we know a rule is operative, but has not yet been articulated.

…………If one were to apply these techniques to database server and other computing terminologies, one of the most obvious examples of imprecise terms would be “performance.” As George J. Klir and Bo Yuan point out in their classic tome *Fuzzy sets and Fuzzy Logic: Theory and Applications*, this is actually an instance of a specific type of fuzzy set called a fuzzy number, which I will introduce later in the series.[ii] Say, for example, that you have a table full of performance data, which you’ve graded the records on scales of 0 to 1 based on whether they fall into categories like “Acceptable,” “Good” and perhaps “Outside Service Level Agreement Boundaries.” That still leaves open the question of what the term “performance “ itself means, so it constitutes another level of fuzziness on top of the membership issue; in fact, it might be necessary to use some of the techniques already hashed out by mathematicians decades for combining the opinions of multiple experts to arrive at a common fuzzy definition of it.

**Modeling Natural Language**

The heavy math in that resource may be too much for some readers to bear, but I highly recommended at least skimming the third section of Chapter 8, where Klir and Yuan identify many different types of fuzziness in ordinary speech. They separate them into four possible combinations of unconditional vs. conditional and unqualified vs. qualified fuzzy propositions, such as the statement “Tina is young is very true,” in which the terms “very” and “true” make it unconditional and qualified.[iii] They also delve into identifying “fuzzy quantifiers” like “about 10, much more than 100, at least about 5,” or “almost all, about half, most,” each of which is modeled by a different type of fuzzy number, which I’ll describe at a later date.[iv] Other distinct types to watch for in natural language include linguistic hedges such as “very, more, less, fairly and extremely” that are used to qualify statements of likelihood or truth and falsehood. These can be chained together in myriad ways, in statements like “Tina is very young is very true,” and the like.[v]

In a moment I’ll describe how chaining together such fuzzy terms and fleshing out other types of imprecision can lead to lesser-known but occasionally invaluable twists on fuzzy sets, but for now I just want to call attention to how quickly it added new layers of complexity to an otherwise simple topic. That is where the highly developed ideas of fuzzy set theory come in handy. The math for implementing all of these natural language concepts has existed for decades, so there’s little reason to reinvent the wheel – yet nor is there a need to overburden readers with all of the equations and jargon, which can look quite daunting on paper. There is a crying need in the data mining field for people willing to act as middlemen of sorts between the end users of the algorithms and their inventors, in the same way that a mechanic fits a need between automotive engineers and drivers; as I’ve pointed out before, it shouldn’t require a doctorate in artificial intelligence to operate data mining software, but the end users are nonetheless routinely buried in equations and formulas they shouldn’t have to decipher. It is imperative for end users to know what such techniques are used for, just as drivers must know how to read a speedometer and operate a brake, but it is not necessary for them to provide lemmas, or even know what a lemma is. While writing these mistutorial series, I’m trying to acquire the skills to do that for the end users by at least removing the bricks from the briefcase, so to speak, which means I’ll keep the equations and jargon down to a minimum and omit mathematical proofs altogether. The jargon is indispensable for helping mathematicians communicate with each other, but is an obstacle to implementing these techniques in practice. It is much easier for end users to think of this topic in terms of natural language, in which they’ve been unwittingly expressing fuzzy sets their whole lives on a daily basis. I can’t simplify this or any other data mining completely, so wall-of-text explanations like this are inevitable – but I’d wager it’s a vast improvement over having to wade through whole textbooks of dry equations, which is sometimes the only alternative. Throughout this series I will have to lean heavily on Klir and Yuan’s aforementioned work for the underlying math, which I will implement in T-SQL. If you want a well-written discussion of the concepts in human language, I’d recommend Dan McNeill’s 1993 book *Fuzzy Logic*.[vi]

**The Low-Hanging Fruits of Fuzzy Set Applications**

These concepts have often proved to be insanely useful whenever they’ve managed to percolate down to various sectors of the economy. The literature is so chock full of them I don’t even know where to begin; the only thing I see in common to the whole smorgasbord is that they seem to seep into industries almost haphazardly, rather than as part of some concerted push. Their “ability to control unstable systems” makes them an ideal choice for many control theory applications.[vii] Klir and Yuan spend several chapters on the myriad implementations already extant when they wrote two decades ago, in fields like robotics,[viii] estimation of longevity of equipment[ix], mechanical and industrial engineering[x], assessing the strength of bridges[xi], traffic scheduling problems[xii] (including the infamous Traveling Salesman) and image sharpening.[xiii] Another example is the field of reliability ratings, where Boolean all-or-nothing rankings like “working” vs. “broken” are often not sufficient to capture in-between states.[xiv] In one detailed example, they demonstrate how to couple weighted matrices of symptoms with fuzzy sets in medical diagnosis.[xv] Klir and Yuan also lament that these techniques are not put to obvious uses in psychology[xvi], where imprecision is rampant, and provide some colorful examples of how to model the imprecision inherent in interpersonal communication, particularly in dating.[xvii] As they point out, some messages are inherently uncertain, on top of any fuzz introduces by the receiver in interpretation; to that we can add the internal imprecision of the speaker, who might not be thinking through their statements thoroughly or selecting their words carefully.

…………Then there is a whole class of applications directly relevant to data mining, such as fuzzy clustering algorithms (like C-Means)[xviii], fuzzy decision trees, neural nets, state sequencing (“fuzzy dynamic systems and automata”)[xix], fuzzified virtual chromosomes in genetic algorithms[xx], fuzzy parameter estimation, pattern recognition[xxi], fuzzy regression procedures and regression on fuzzy data.[xxii] Most of that falls under the rubric of “soft computing,” a catch-all term for bleeding edge topics like artificial intelligence. The one facet of the database server field where fuzzy sets have succeeded in permeating somewhat since Klir and Yuan mentioned the possibility[xxiii] is fuzzy information retrieval, which we can see it in action in SQL Server full-text catalogs.

**The Future of Fuzzy Sets in SQL Server**

Like many of their colleagues, however, they wrote about ongoing research into fuzzy relational databases by researchers like Bill Buckles and F.E. Petry that has not come into widespread use since then.[xxiv] That is where this series comes in. I won’t be following any explicit prescriptions for implementing fuzzy relational databases per se, but will instead leverage the existing capabilities of T-SQL to demonstrate how easy it is to add your own fuzz for imprecision modeling purposes. Researcher Vivek V. Badami pointed out more than two decades ago that fuzz takes more code, but is easier to think about.[xxv] It takes very little experience with fuzzy sets to grasp what he meant by this – especially now that set-based languages like T-SQL that are ideal for this topic are widely used. I wonder if someday it might be possible to extend SQL or systems like SQL Server to incorporate fuzziness more explicitly, for example, by performing the extra operations on membership functions that are required for joins between fuzzy sets, or even more, fuzzy joins between fuzzy sets; later in the series I’ll demonstrate how DBAs can quickly implement DIY versions of these things, but perhaps there are ways to do the dirty work under the hood, in SQL Server internals. Maybe a generation from now we’ll see fuzzy indexes and SQL Server execution plans with Fuzzy Anti-Semi-Join operators – although I wonder how Microsoft could implement the retrieval of only one-seventh of a record and a third of another, using B-trees or *any *other type of internal data structure. In order to determine if a record is worthy of inclusion, it first has to be retrieved and inspected instead of passed over, which could lead to a quandary if SQL Server developers tried to implement fuzzy sets in the internals.

…………The good news is that we don’t have to wait for the theoreticians to hash out how to implement fuzzy relational databases, or for Microsoft and its competition to add the functionality for us. As it stands, T-SQL is already an ideal tool for implementing fuzzy sets. In the next article, I’ll demonstrate some trivial membership functions that any DBA can implement on their own quite easily, so that these concepts don’t seem so daunting. The difficulties can be boiled down chiefly to the fact that the possibilities are almost *too* wide open. Choosing the right membership functions to model the problem at hand is not necessarily straightforward, nor is selecting the right type of fuzzy set to model particular types of imprecision. As in regular data modeling, the wrong choices can sometimes lead not only to the waste of server resources, but also of returning incorrect answers. The greatest risk, in fact, consists of fuzzifying relationships that are inherently crisp and vice-versa, which can lead to fallacious reasoning. Fuzz has become a buzzword of sorts, so it would be wise to come up with a standard to discern its true uses from its abuses. In the next installment, I’ll tackle some criteria we can use to discern the difference, plus provide a crude taxonomy of fuzzy sets and get into some introductory T-SQL samples.

[i] p. 18, Bonissone, Piero P., 1998, “Fuzzy Sets & Expert Systems in Computer Eng. (1).” Available online at http://homepages.rpi.edu/~bonisp/fuzzy-course/99/L1/mot-conc2.pdf

[ii] pp. 101-102, Klir, George J. and Yuan, Bo, 1995, Fuzzy sets and Fuzzy Logic: Theory and Applications. Prentice Hall: Upper Saddle River, N.J.

[iii] *IBID.*, pp. 222-225.

[iv] *IBID.*, pp. 225-226.

[v] *IBID.*, pp. 229-230.

[vi] McNeill, Dan, 1993, __Fuzzy Logic__. Simon & Schuster: New York.

[vii] p. 8, Bonissone.

[viii] Klir and Yuan, p. 440.

[ix] *IBID.*, p. 432.

[x] *IBID.*, pp. 427-432.

[xi] *IBID.*, p. 419.

[xii] *IBID.*, pp. 422-423.

[xiii] *IBID.*, pp. 374-376.

[xiv] *IBID.*, p. 439.

[xv] *IBID.*, pp. 443-450.

[xvi] *IBID.*, pp. 463-464.

[xvii] *IBID.*, pp. 459-461.

[xviii] *IBID.*, pp. 358-364.

[xix] *IBID.*, pp. 349-351.

[xx] *IBID.*, p. 453.

[xxi] *IBID.*, pp. 365-374.

[xxii] *IBID.*, pp. 454-459.

[xxiii] *IBID.*, p. 385.

[xxiv] *IBID.*, pp. 380-381.

[xxv] p. 278, McNeill.

## Goodness-of-Fit Testing with SQL Server Part 7.4: The Cramér–von Mises Criterion

**By Steve Bolton**

…………This last installment of this series of amateur tutorials features a goodness-of-fit metric that is closely related to the Anderson-Darling Test discussed in the last post, with one important caveat: I couldn’t find any published examples to verify my code against. Given that the code is already written and the series is winding down, I’ll post it anyway in the off-chance it may be useful to someone, but my usual disclaimer applies more than ever: I’m writing this series in order to learn about the fields of statistics and data mining, not because I have any real expertise. Apparently, the paucity of information on the Cramér–von Mises Criterion stems from the fact that experience with this particular measure is a lot less common than that of its cousin, the Anderson-Darling Test. They’re both on the high end when it comes to statistical power, so the Cramér–von Mises Criterion might be a good choice when you need to be sure you’re detecting effects with sufficient accuracy.[I]

…………Although it is equivalent to the Anderson-Darling with the weighting function[ii] set to 1, the calculations are more akin to those of the other methods based on the empirical distribution function (EDF) we’ve discussed in this segment of the series. It is in fact a refinement[iii] of the Kolmogorov-Smirnov Test we discussed a few articles ago, one that originated with separate papers published in 1928 by statisticians Harald Cramér and Richard Edler von Mises.[iv] One of the advantages it appears to enjoy over the Anderson-Darling Test is that it seems to perform much better, in the same league as the Kolmogorov-Smirnov, Kuiper’s and Lilliefors Tests. One of the disadvantages is that it that the value of the test statistic might be distorted by Big Data-sized value ranges and counts, which many established statistical tests were never designed to handle. It does appear, however, to suffer from this to a lesser degree than the Anderson-Darling Test, as discussed last time around. Judging from past normality tests I’ve performed on datasets I’m familiar with, it seems to assign higher values to those that were definitely not Gaussian in the proper order, although perhaps not in the correct proportion.

…………That of course assumes that I coded it correctly, which I can’t verify in the case of this particular test. In fact, I had to cut out some of the T-SQL that calculated the Watson Test along with it, since the high negative numbers it routinely returned were obviously wrong. I’ve also omitted the commonly used two-sample version of the test, since sampling occupies a less prominent place in SQL Server use cases than it would in ordinary statistical testing and academic research; one of the benefits of having millions of rows of data in our cubes and relational tables is that we can derive more exact numbers, without depending as much on parameter estimation or inexact methods of random selection. I’ve also omitted the hypothesis testing step that usually accompanies the use of the criterion, for the usual reasons: loss of information content by boiling down the metric to a simple either-or choice, the frequency with which confidence intervals are misinterpreted and most of all, the fact that we’re normally going to be doing exploratory data mining with SQL Server, not narrower tasks like hypothesis testing. One of the things that sets the Cramér–von Mises Criterion apart from other tests I’ve covered in the last two tutorial series is that the test statistic is compared to critical values from the F-distribution rather than the Chi-Squared or Student’s T, but the same limitation still arises: most of the lookup tables have gaps or stop at a few hundred values at best, but calculating them for the millions of degrees of freedom we’d need for such large tables would be computationally costly. Moreover, since I can’t be sure the code below for this less common metric is correct, there is less point in performing those expensive calculations.

…………The bulk of the procedure in Figure 1 is identical to the sample code posted for the Kolmogorov-Smirnov and Lilliefors Tests, which means they can and really ought to be calculated together. The only differences are in the final calculations of the test statistics, which are trivial in comparison to the derivation of the empirical distribution function (EDF) table variable from a dynamic SQL statement. The @Mean and @StDev aggregates are plugged into the Calculations.NormalDistributionSingleCDFFunction I wrote for Goodness-of-Fit Testing with SQL Server, part 2.1: Implementing Probability Plots in Reporting Services. If you want to test other distributions besides the Gaussian or “normal” distribution (i.e. the bell curve), simply substitute a different cumulative distribution function (CDF) here. The final calculation is straightforward: just subtract the CDF from the EDF, square the result and do a SUM over the whole dataset.[v] The two-sample test, which I haven’t included here in order for the sake of brevity and simplicity, merely involves adding together the results for the same calculation across two different samples and making a couple minor corrections. I’ve also included a one-sample version of the test I saw cited at Wikipedia[vi], since it was trivial to calculate. I would’ve liked to include the Watson Test, since “it is useful for distributions on a circle since its value does not depend on the arbitrary point chosen to begin cumulating the probability density and the sample points”[vii] and therefore meets a distinct but important set of use cases related to circular and cyclical data, but my first endeavors were clearly inaccurate, probably due to mistranslations of the equations.

__Figure 1: T-SQL Code for the Cramér–von Mises Criterion__

CREATE PROCEDURE [Calculations].[GoodnessOfFitCramerVonMisesCriterionSP]

@Database1 as nvarchar(128) = NULL, @Schema1 as nvarchar(128), @Table1 as nvarchar(128),@Column1 AS nvarchar(128)

AS

DECLARE @SchemaAndTable1 nvarchar(400),@SQLString nvarchar(max)

SET @SchemaAndTable1 = @Database1 + ‘.’ + @Schema1 + ‘.’ + @Table1

DECLARE @Mean float,

@StDev float,

@Count float

DECLARE @EDFTable table

(ID bigint IDENTITY (1,1),

Value float,

ValueCount bigint,

EDFValue float,

CDFValue decimal(38,37),

EDFCDFDifference decimal(38,37))

DECLARE @ExecSQLString nvarchar(max), @MeanOUT nvarchar(200),@StDevOUT nvarchar(200),@CountOUT nvarchar(200), @ParameterDefinition nvarchar(max)

SET @ParameterDefinition = ‘@MeanOUT nvarchar(200) OUTPUT,@StDevOUT nvarchar(200) OUTPUT,@CountOUT nvarchar(200) OUTPUT ‘

SET @ExecSQLString = ‘SELECT @MeanOUT = CAST(Avg(Value) as float),@StDevOUT = StDev(Value),@CountOUT = CAST(Count(Value) as float)

FROM (SELECT CAST(‘ + @Column1 + ‘ as float) as Value

FROM ‘ + @SchemaAndTable1 + ‘

WHERE ‘ + @Column1 + ‘ IS NOT NULL) AS T1’

EXEC sp_executesql @ExecSQLString,@ParameterDefinition, @MeanOUT = @Mean OUTPUT,@StDevOUT = @StDev OUTPUT,@CountOUT = @Count OUTPUT

SET @SQLString = ‘SELECT Value, ValueCount, SUM(ValueCount) OVER (ORDER BY

Value ASC) / CAST(‘ + CAST(@Count as nvarchar(50)) + ‘AS float) AS EDFValue

FROM (SELECT DISTINCT ‘ + @Column1 + ‘ AS Value, Count(‘ + @Column1 + ‘) OVER (PARTITION BY ‘ + @Column1 + ‘ ORDER BY ‘ + @Column1 + ‘) AS ValueCount

FROM ‘ + @SchemaAndTable1 + ‘

WHERE ‘ + @Column1 + ‘ IS NOT NULL) AS T1’

INSERT INTO @EDFTable

(Value, ValueCount, EDFValue)

EXEC (@SQLString)

UPDATE T1

SET CDFValue = T3.CDFValue, EDFCDFDifference = EDFValue – T3.CDFValue

FROM @EDFTable AS T1

INNER JOIN (SELECT DistinctValue, Calculations.NormalDistributionSingleCDFFunction (DistinctValue, @Mean, @StDev) AS CDFValue

FROM (SELECT DISTINCT Value AS DistinctValue

FROM @EDFTable) AS T2) AS T3

ON T1.Value = T3.DistinctValue

DECLARE @OneDividedByTwelveN As float = 1 / CAST(12 as float)

DECLARE @TwoTimesCount as float = 2 * @Count

DECLARE @ReciprocalOfCount as float = 1 / CAST(@Count as float)

DECLARE @ResultTable table

(CramerVonMisesTest float

)

INSERT INTO @ResultTable

SELECT CramerVonMisesTest

FROM (SELECT @OneDividedByTwelveN

+ Sum(Power(((((2 * ID) – 1) / CAST(@TwoTimesCount as float)) – CDFValue), 2)) AS CramerVonMisesTest

FROM @EDFTable) AS T1

SELECT CramerVonMisesTest

FROM @ResultTable

SELECT ID, Value, ValueCount, EDFValue, CDFValue, EDFCDFDifference

FROM @EDFTable

…………One potential issue with the code above is that I may need to input the EDF rather than the EDFCDFDifference in the final SELECT; in the absence of example data from published journal articles, I can’t be sure of that. Like some of its kin, the criterion can also be adapted “for comparing two empirical distributions”[viii] rather than using the difference between the EDF and the cumulative distribution function (CDF). Most of these concepts have already been covered in the last three articles in this segment of the series; in fact, pretty much all except the last two selects are identical to that of the Kolmogorov-Smirnov, Kuiper’s and Lilliefors procedures. As usual, the parameters allow users to perform the test on any numerical column in any database they have sufficient access to.

** Figure 2: Sample Results from the Duchennes Table**EXEC Calculations.GoodnessOfFitCramerVonMisesCriterionSP

@Database1 = N’DataMiningProjects’,

@Schema1 = N’Health’,

@Table1 = N’DuchennesTable’,

@Column1 = N’PyruvateKinase’

…………I performed queries like the one in Figure 2 against two datasets I’ve used throughout the last two tutorial series, one by on the Duchennes form of muscular dystrophy made publicly available by the Vanderbilt University’s Department of Biostatistics and another on the Higgs Boson provided by the University of California at Irvine’s Machine Learning Repository. Now that I’m familiar with how closely their constituent columns follow the bell curve, I was not surprised to see that the LactateDehydrogenase and PyruvateKinase enzymes scored 0.579157871602764 and 2.25027709042408 respectively, or that the test statistic for the Hemopexin protein was 0.471206505704088. Once again, I can’t guarantee that those figures are accurate in the case of this test, but the values follow the expected order (the same cannot be said of the one-sample Wikipedia version, which varied widely across all of the columns I tested it on). Given that the test statistic is supposed to rise in tandem with the lack of fit, I was likewise not surprised to see that the highly abnormal first float column of the Higgs Boson Dataset scored a 118.555073824395. The second float column obviously follows a bell curve in histograms and had a test statistic of 0.6277795953021942279. Note that the results for the same columns in Goodness-of-Fit Testing with SQL Server Part 7.3: The Anderson-Darling Test were 5.43863473749926, 17.4386371653374, 5.27843535947881, 870424.402686672 and 12987.3380102254 respectively. One of the reasons I posted the code for this week’s test statistic despite by misgivings about its accuracy is that the numbers are easier to read than those for its closest cousin. Unlike the Kolmogorov-Smirnov, Kuiper’s and Lilliefors Tests, the Cramér–von Mises Criterion is not bounded between 0 and 1, but it at least doesn’t reach such inflated sizes as in the Anderson-Darling stat. Furthermore, the vastly higher count in the Higgs Boson Dataset seems to swell the Anderson-Darling results even for clearly Gaussian data like the second float column, which makes it difficult to compare stats across datasets.

** Figure 3: Execution Plan for the Cramér–von Mises Criterion** (click to enlarge)

…………Another advantage the Cramér–von Mises Criterion might enjoy over the Anderson-Darling Test is far better performance. The execution plan in Figure 3 is almost identical to those we saw in Goodness-of-Fit Testing with SQL Server Part 7.1: The Kolmogorov-Smirnov and Kuiper’s Tests and Part 7.2: The Lilliefors Test. Once again there are five queries, only two of which have any real cost. Both of those begin with nonclustered Index Seeks, which is usually a good sign. The only really costly savings we might be able to drudge up are with the Hash Match (Aggregate) operator – but there isn’t much point, since the procedure had the same stellar performance as the Kolmogorov-Smirnov, Kuiper’s and Lilliefors Tests. Given that the procedure is structured so similarly to its high-performing kin, it’s not surprising that it took only 24 seconds to race through the 11 million rows in the Higgs Boson Dataset when processing the first float column and 1:15 for the second; in contrast, the Anderson-Darling took a whopping 9:24 for Column1 and 9:09 for Column2. These times will be several orders of magnitude better when run on a real database server instead of my run-down development machine, but it would probably be wise to go with the better-performing measures anyways, assuming they’re otherwise a good fit for our use cases.

…………I originally intended to include the Akaike, Bayesian and several other metrics with “Information Criterion,” in their name, but decided that these measures of mining model fitness would best be handled in an upcoming series titled Information Measurement with SQL Server. The last tutorial series on outlier detection ended with articles on Cook’s Distance and Mahalanobis Distance, which were both intended to segue into that more advanced series (which I’m entirely unqualified to write), in which we’ll tackle various types of entropy, measures of structure, quantification of order and other mind-blowing topics. We’ll hear from such familiar names as Cramér and Kolmogorov again in that series, but first I must take a detour into a topic that could be of immediate benefit to a wide range of SQL Server users. In the upcoming tutorial series Implementing Fuzzy Sets with SQL Server, I’ll explain how the tragically neglected but techniques used in fuzzy sets can be immediately applied to real-world problems that the SQL Server community encounters routinely, particularly those where modeling data on continuous scales would be preferable but is not currently done because of inexact measurement methods. This article was probably the weakest entry in the Goodness-of-Fit series, which was itself only of narrow interest to certain niches in the SQL Server user base; I only went off on this tangent because I recognized my own weaknesses in this area while writing the Outlier Detection with SQL Server series and sought to rectify it through the school of hard knocks. In the future I may tack a few more articles onto the end of this series, such as sample code for Mardia’s Multivariate Skewness and Kurtosis and other multivariate goodness-of-fit tests, but the bulk of this series is complete. In my next post I’ll introduce some of the basic concepts behind fuzzy sets, before providing code samples that should make this treasure trove of techniques immediately available to a wide range of SQL Server users, in order to solve classes of problems that are not being solved efficiently today. As with much of the research into data mining techniques, the academic study of fuzzy sets is at least two decades ahead of the practice. It’s high time it was brought within reach of non-specialists, many of whom could derive surprising practical benefit from these techniques.

[i] See the Wikipedia webpage “Anderson-Darling Test” at http://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test

[ii] *IBID.*

[iii] See the comment “Several goodness-of-fit tests, such as the Anderson-Darling test and the Cramer Von-Mises test, are refinements of the K-S test,” at National Institute for Standards and Technology, 2014, “1.3.5.16 Kolmogorov-Smirnov Goodness-of-Fit Test,” published in the online edition of the __Engineering Statistics Handbook.__ Available at http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm

[iv] See the __Wikipedia__ page “Cramér–von Mises Criterion” at http://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion

[v] I derived the formula from Anderson, T.W. and Stephens, M.A., 1994, The Modified Cramer-Von Mises Goodness-of-Fit Criterion for Time Series. Technical Report No. 47, Jan. 17, 1994. Published by the Office of Naval Research and the National Science Foundation. Available at the DTIC Online web address http://www.dtic.mil/dtic/tr/fulltext/u2/a275377.pdf Also see Xiao, Yuanhui; Gordon, Alexander and Yakovlev, Andrei, 2006, “A C++ Program for the Cramér-Von Mises Two-Sample Test,” pp. 1-15 in Journal of Statistical Software, January 2007. Vol. 17, No. 8. Available online at http://www.jourlib.org/paper/2885039#.VIXYZP4o4uU All three authors taught at the University of Rochester, which is on the other side of the city from me. I had to dust off my C++ for this one, which brought back interesting memories of typing “Hello World” while precariously balancing a laptop on my knees at my sister’s house ages ago, after having a few beers and a whole box of chocolates on Valentine’s Day.

[vi] See the __Wikipedia__ page “Cramér–von Mises Criterion” at https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion

[vii] p. 57 , Watson, G. S., 1962, “Goodness-of-Fit Tests on a Circle,” p. 57 in __Biometrika,__ Vol. 49, No. 1 and 2. Available online at http://phdtree.org/pdf/33054228-goodness-of-fit-tests-on-a-circle-ii/

[viii] *IBID.*

## Goodness-of-Fit Testing with SQL Server Part 7.3: The Anderson-Darling Test

**By Steve Bolton**

…………As mentioned in previous installments of this series of amateur self-tutorials, goodness-of-fit tests can be differentiated in many ways, including by the data and content types of the inputs and the mathematical properties, data types and cardinality of the outputs, not to mention the performance impact of the internal calculations in between them. Their uses can be further differentiated by the types of probability distributions or regression models they can be applied to and the points within those distributions where their statistical power is highest, such as in the tails of a bell curve or the central point around the median or mean. The Anderson-Darling Test differs from the Kolmogorov-Smirnov Test we recently surveyed and others in its class in a plethora of ways, some of which I was able to glean from sundry comments scattered across the Internet. Unlike many other such tests, it can be applied beyond the usual Gaussian or “normal” distribution to other distributions including the “lognormal, exponential, Weibull, logistic, extreme value type 1” and “Pareto, and logistic.”[1] This is perhaps its major drawing card, since many other such tests are limited to a really narrow range of distributions, usually revolving around the Gaussian.

…………In terms of interpretation of the test statistic, it is “generally valid to compare AD values between distributions and go with the lowest.”[2] When used with the normal distribution it is also “close to optimal” in terms of the Bahadur Slope, one of several methods of assessing the usefulness of the tests statistics produced by goodness-of-fit tests.[3] One of the drawbacks is that “it performs poorly if there are many ties in the data.”[4] Another is that it may be necessary to multiply the final test statistic by specific constants when testing distributions other than the normal[5], but I was unable to find references to any of them in time to include them in this week’s T-SQL stored procedure. This is not true of the Kolmogorov-Smirnov Test we surveyed a few weeks back, which is “distribution-free as the critical values do not depend on whether Gaussianity is being tested or some other form.”[6]

**Parameter Estimation and EDA**

This particular limitation is not as much of an issue in the kind of exploratory data mining that the SQL Server community is more likely to use these tests for, given that we’re normally not performing hypothesis testing; I’ve generally shied away from that topic in this series for many reasons that I’ve belabored in previous articles, like the ease of misinterpretation of confidence intervals and the information loss involved in either-or hypothesis rejections. Don’t get me wrong, hypothesis testing is a valuable and often necessary step when trying to prove a specific point, at the stage of Confirmatory Data Analysis (CDA), but most of our mining use cases revolve around informal Exploratory Data Analysis (EDA), a distinction made in the ‘70s by John W. Tukey, the father of modern data mining.[7] Another issue with hypothesis testing is the fact that most of the lookup tables and approximations weren’t designed with datasets consisting of millions of rows, as DBAs and miners of SQL Server cubes encounter every day.

This size difference has a side benefit, in that we generally don’t have to estimate the means and variances of our datasets, which is a much bigger issue in the kinds of small random samples that hypothesis tests are normally applied to. One of the properties of the Anderson-Darling Test is that parameter estimation is less of an issue with it, whereas the Lilliefors Test, the subject of last week’s article, is designed specifically for cases where the variance is unknown. There are apparently special formulations where different combinations of the mean and standard deviation are unknown, but these aren’t going to be common in our use cases, since the mean and variance are usually trivial to compute in an instant for millions of rows. Another noteworthy property that may be of more use to us is the fact that the Anderson-Darling Test is more sensitive to departures from normality in the tails of distributions in comparison with other popular fitness tests.[8]

**The Perils and Pitfalls of Equation Translation**

It is not surprising that this long and varied list of properties differentiates the Anderson-Darling Test from the Kolmogorov-Smirnov, Kuiper’s and Lilliefors Tests we’ve surveyed in the last few articles, given that there are some marked differences in its internal calculations. The inner workings apparently involve transforming the inputs into a uniform distribution, which is still a bit above my head, because I’m still learning stats and stochastics as I go. The same can be said of some of the equations I had to translate for this week’s article, which contained some major stumbling blocks I wasn’t expecting. Once of these was the fact that the Anderson-Darling Test is usually categorized along with other methods based on the empirical distribution function (EDF), which as explained in recent articles, involves computing the difference between the actual values and the probabilities generated for them by the distribution’s cumulative distribution function (CDF). Nevertheless, the CDF is used twice in the calculation of the test statistic and the EDF is not used at all, which led to quite a bit of confusion on my part.

…………Another issue I ran into is the fact that the term “N +1 – I” in the formula actually requires the calculation of an order statistic of the kind we used in Goodness-of-Fit Testing with SQL Server, part 6.1: The Shapiro-Wilk Test. I won’t recap that topic here, except to say that it is akin to writing all of the values in a dataset on a sheet of paper in order, then folding it in half and adding them up on each side. Prior to that discovery I was mired in trying various combinations of Lead and Lag that just weren’t returning the right outputs. I found an offhand remark after the fact in an academic paper (which I can’t recall in order to give proper credit) to the effect that the identification of this term as an order statistic is missing from most of the literature on the subject for some unknown reason. As I’ve learned over the past few months, the translation of equations[9] is not always as straightforward as I originally thought it would be (even though I already had some experience doing back in fourth and fifth grade, when my father taught college physics classes and I used to read all of his textbooks). Other remaining issues with the code in Figure 1 include the fact that I may be setting the wrong defaults for the LOG operations on the CDFValues when they’re equal to zero and the manner in which I handle ties in the order statistics, which may be incorrect. Some of the literature also refers to plugging the standard normal distribution values of 0 and 1 for the mean and standard deviation. Nevertheless, I verified the output of the procedure on two different sets of examples I found on the Internet, so the code may be correct as is.[10]

** Figure 1: T-SQL Code for the Anderson-Darling Procedure**CREATE PROCEDURE Calculations.GoodnessofFitAndersonDarlingTestSP

@Database1 as nvarchar(128) = NULL, @Schema1 as nvarchar(128), @Table1 as nvarchar(128),@Column1 AS nvarchar(128)

AS

DECLARE @SchemaAndTable1 nvarchar(400),@SQLString nvarchar(max)

SET @SchemaAndTable1 = @Database1 + ‘.’ + @Schema1 + ‘.’ + @Table1

DECLARE @Mean float,

@StDev float,

@Count float

DECLARE @ValueTable table

(Value float)

DECLARE @CDFTable table

(ID bigint IDENTITY (1,1),

Value float,

CDFValue float)

DECLARE @ExecSQLString nvarchar(max), @MeanOUT nvarchar(200),@StDevOUT nvarchar(200),@CountOUT nvarchar(200), @ParameterDefinition nvarchar(max)

SET @ParameterDefinition = ‘@MeanOUT nvarchar(200) OUTPUT,@StDevOUT nvarchar(200) OUTPUT,@CountOUT nvarchar(200) OUTPUT ‘

SET @ExecSQLString = ‘SELECT @MeanOUT = CAST(Avg(‘ + @Column1 + ‘) as float),@StDevOUT = CAST(StDev(‘ + @Column1 + ‘)as float),@CountOUT = CAST(Count(‘ + @Column1 + ‘)as float)

FROM ‘ + @SchemaAndTable1 + ‘

WHERE ‘ + @Column1 + ‘ IS NOT NULL’

EXEC sp_executesql @ExecSQLString,@ParameterDefinition, @MeanOUT = @Mean OUTPUT,@StDevOUT = @StDev OUTPUT,@CountOUT = @Count OUTPUT

SET @SQLString = ‘SELECT ‘ + @Column1 + ‘ AS Value

FROM ‘ + @SchemaAndTable1 + ‘

WHERE ‘ + @Column1 + ‘ IS NOT NULL’

INSERT INTO @ValueTable

(Value)

EXEC (@SQLString)

INSERT INTO @CDFTable

(Value, CDFValue)

SELECT T1.Value, CDFValue

FROM @ValueTable AS T1

INNER JOIN (SELECT DistinctValue, Calculations.NormalDistributionSingleCDFFunction (DistinctValue, @Mean, @StDev) AS CDFValue

FROM (SELECT DISTINCT Value AS DistinctValue

FROM @ValueTable) AS T2) AS T3

ON T1.Value = T3.DistinctValue

SELECT SUM(((1 – (ID * 2)) / @Count) * (AscendingValue + DescendingValue)) – @Count AS AndersonDarlingTestStatistic

FROM (SELECT TOP 9999999999 ID, CASE WHEN CDFValue = 0 THEN 0 ELSE Log(CDFValue) END AS AscendingValue

FROM @CDFTable

ORDER BY ID) AS T1

INNER JOIN (SELECT ROW_NUMBER() OVER (ORDER BY CDFValue DESC) AS RN, CASE WHEN 1 – CDFValue = 0 THEN 0 ELSE Log(1 – CDFValue) END AS DescendingValue

FROM @CDFTable) AS T2

ON T1.ID = T2.RN

— this statement is included merely for convenience and can be eliminated

SELECT Value, CDFValue

FROM @CDFTable

ORDER BY Value

…………The code turned out to be quite short in comparison to the length of time it took to write and the number of mistakes I made along the way. Most of it self-explanatory for readers who are used to the format of the T-SQL procedures I’ve posted in the last two tutorial series. As usual, there is no null-handling, SQL injection or validation code, nor can the parameters handle brackets, which I don’t allow in my own code. The first four allow users to run the procedure on any column in any database they have sufficient access to. The final statement returns the table of CDF values used to calculate the test statistic, since there’s no reason not to now that the costs have already been incurred; as noted above, “this statement is included merely for convenience and can be eliminated.” The joins in the INSERT statement lengthen the code but actually make it more efficient, by enabling the calculation of CDF values just once for each unique column value. The Calculations.NormalDistributionSingleCDFFunction has been introduced in several previous articles, so I won’t rehash it here. In the SELECT where the test statistic is derived, I used an identity value in the join because the ROW_NUMBER operations can be expensive on big table, so I wanted to avoid doing two in one statement.

** Figure 2: Sample Results from the Anderson-Darling Test**EXEC Calculations.GoodnessofFitAndersonDarlingTestSP

@Database1 = N’DataMiningProjects’,

@Schema1 = N’Health’,

@Table1 = N’DuchennesTable’,

@Column1 = N’PyruvateKinase’

…………One of the concerns I had when running queries like the one in Figure 2 against the 209 rows of the Duchennes muscular dystrophy dataset and the 11 million rows of the Higgs Boson dataset (which I downloaded from the Vanderbilt University’s Department of Biostatistics and University of California at Irvine’s Machine Learning Repository and converted into SQL Server tables for use in these tutorial series) is that the values seemed to be influenced by the value ranges and cardinality of the inputs. Unlike the last three tests covered in this series, it is not supposed to be bounded between 0 and 1. As I discovered when verifying the procedure against other people’s examples, it’s not uncommon for the test statistic to get into the single or double digits. In the examples at the NIST webpage, those for the Lognormal and Cauchy distributions were in the double and triple digits respectively, while that of the double exponential distribution were well above 1, so it may not be unusual to get a test statistic this high. It is definitely not bounded between 0 and 1 like the stats returned by other goodness-of-fit tests, but the range might be distribution-dependent. This is exactly what happened with the LactateDehydrogenase, PyruvateKinase and Hemopexin columns, which scored 5.43863473749926, 17.4386371653374 and 5.27843535947881respectively. Now contrast that range with the Higgs Boson results, where the second float column scored a 12987.3380102254 and the first a whopping 870424.402686672. The problem is not with the accuracy of the results, which are about what I’d expect, given that Column2 clearly follows a bell curve in a histogram while Column1 is ridiculously lopsided. The issue is that for very large counts, the test statistic seems to be inflated, so that it can’t be compared across datasets. Furthermore, once a measure gets up to about six or seven digits to the left of the decimal point, it is common for readers to semiconsciously count the digits and interpolate commas, which is a very slow and tedious process. The test statistics are accurate, but suffer from legibility and comparability issues when using Big Data-sized record counts.

__Figure 3: Execution Plan for the Anderson-Darling Procedure
__

…………The procedure also performed poorly on the super-sized Higgs Boson dataset, clocking in at 9:24 for Column1 and 9:09 for Column2; moreover, it gobbled up several gigs of RAM and a lot of space in TempDB, probably as a result of the Table Spool in the execution plan above. Perhaps some optimization could also be performed on the Merge Join, which was accompanied by some expensive Sort operators, by forcing a Hash Match or Nested Loops. The major stumbling block is the number of Table Scans, which I tried to overcome with a series of clustered and non-clustered indexes on the table variables, but this unexpectedly degraded the execution time badly, in tandem with outrageous transaction log growth. I’m sure a T-SQL expert could spot ways to optimize this procedure, but as it stands, the inferior performance means it’s not a good fit for our use cases, unless we’re dealing with small recordsets and need to leverage its specific properties. All told, the Anderson-Darling procedure has some limitations that make it a less attractive option for general-purpose fitness testing than the Kolmogorov-Smirnov Test, at least for our unique uses cases. On the other hand, it has a well-defined set of uses cases based on a well-established properties, which means it could be applied to a wide variety of niche cases. Among these properties is its superior ability “for detecting most departures from normality.”[11] In the last installment of this series, we’ll discuss the Cramér–von Mises Criterion, another EDF-based method that is closely related to the Anderson-Darling Test and enjoys comparable statistical power in detecting non-normality.[12]

[1] See National Institute for Standards and Technology, 2014, “1.3.5.14 Anderson-Darling Test,” published in the online edition of the Engineering Statistics Handbook. Available at http://www.itl.nist.gov/div898/handbook/eda/section3/eda35e.htm

[2] Frost, Jim, 2012, “How to Identify the Distribution of Your Data using Minitab,” published March, 2012 at The Minitab Blog web address http://blog.minitab.com/blog/adventures-in-statistics/how-to-identify-the-distribution-of-your-data-using-minitab

[3] p. 52, No author listed, 1997, __Encyclopaedia of Mathematics, Supplemental Vol. 1__. Reidel: Dordrecht. This particular page was retrieved from Google Books.

[4] No author listed, 2014, “Checking Gaussianity,” published online at the __MedicalBiostatistics.com__ web address http://www.medicalbiostatistics.com/checkinggaussianity.pdf

[5] See the Wikipedia webpage “Anderson-Darling Test” at http://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test

[6] See the aforementioned __MedicalBiostatistics.com __article.

[7] See Tukey, John W., 1977, __Exploratory Data Analysis__. Addison-Wesley: Reading, Mass. I’ve merely heard about the book second-hand and have yet to read it, although I may have encountered a few sections here and there.

[8] *IBID.*

[9] For the most part, I depended on the more legible version in National Institute for Standards and Technology, 2014, “1.3.5.14 Anderson-Darling Test,” published in the online edition of the Engineering Statistics Handbook. Available at http://www.itl.nist.gov/div898/handbook/eda/section3/eda35e.htm All of the sources I consulted though had the same notation, without using the term order statistic.

[10] See Frost, 2012, for the sample data he calculated in Minitab and See Alion System Reliability Center, 2014, “Anderson-Darling: A Goodness of Fit Test for Small Samples Assumptions,” published in __Selected Topics in Assurance Related Technologies__, Vol. 10, No. 5. Available online at the __Alion System Reliability Center __web address http://src.alionscience.com/pdf/A_DTest.pdf These START publications are well-written , so I’m glad I discovered them recently through Freebird2008’s post on Sept. 4, 2008 at the TalkStats thread “The Anderson-Darling Test,” which is available at http://www.talkstats.com/showthread.php/5484-The-Anderson-Darling-Test

[11] See the Wikipedia webpage “Anderson-Darling Test” at http://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test

[12] *IBID.*

## Goodness-of-Fit Testing with SQL Server Part 7.2: The Lilliefors Test

**By Steve Bolton**

…………Since I’m teaching myself as I go in this series of self-tutorials, I often have only a vague idea of the challenges that will arise when trying to implement the next goodness-of-fit test with SQL Server. In retrospect, had I known that the Lilliefors Test was so similar to the Kolmogorov-Smirnov and Kuiper’s Tests, I probably would have combined them into single article. The code for this week’s T-SQL stored procedure is nearly the same, as is the execution plan and the performance. The results are also quite similar right to those of the Kolmogorov-Smirnov Test for some of the practice data I’ve used throughout the series, differing in some cases by just a few decimal places. The slight differences may arise from one of the characteristics of the Lilliefors Test that differentiate it from its more famous cousin, namely that “this test of normality is more powerful than others procedures for a wide range of nonnormal conditions.”[i] Otherwise, they share many mathematical properties in common, like location and scale invariance – i.e., the proportions of the test statistics aren’t altered when using a different starting point or multiplying by a common factor.

…………On the other hand, the test is apparently more restrictive than the Kolmogorov-Smirnov, in that I’ve seen it referred to specifically as a normality test and I haven’t encountered any mention of it being applied to other distributions. Furthermore, its primary use cases seems to be those in which the variance of the data is unknown[ii], which often doesn’t apply in the types of million-row tables the SQL Server community works with daily. The late Hubert Lilliefors (1928-2008), a stats professor at George Washington University, published it in a Journal of the American Statistical Association titled “On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown” back in 1967[iii] – so augmenting its more famous cousin in a few niche scenarios seems to have been the raison d’etre from the beginning. We can always use more statistical tests in our toolbox to meet the never-ending welter of distributions that arise from actual physical processes, but I won’t dwell on the Lilliefors Test for long because its narrower use cases are less suited to our needs than those of the broader Kolmogorov-Smirnov Test.

**Differences from the Kolmogorov-Smirnov**

Another reason for not dwelling on it for too long is that most of the code is identical to that of the stored procedure posted in last week’s article. The Lilliefors Test quantifies the difference between the empirical distribution function (EDF) and cumulative distribution function (CDF) exactly the same as the Kolmogorov-Smirnov and Kuiper’s Tests do; in plain English, it orders the actual values and ranks them on a scale of 0 to 1 and computes the difference from the theoretical probability for the Gaussian “normal” distribution, or bell curve, which is also ranked on a scale of 0 to 1. A couple of notes of caution are in order here, because some of the sources I consulted mentioned inputting Z-Scores into the formula and using the standard normal distribution rather than the actual mean and standard deviation of the dataset, but I verified that the procedure is correct as it stands now against an example at Statd.com.[iv]

…………One of the main characteristics that set it apart from the Kolmogorov-Smirnov Test is that the test statistic is compared against the Lilliefors distribution, which apparently has a better Bahadur Slope[v] (one of many measures of the efficiency of test statistics) than its competitors in certain hypothesis testing scenarios. That is a broad topic I’ve downplayed for several reasons throughout the last two tutorial series. Among the reasons I’ve brought up in the past are the fact that SQL Server users more likely to be using these tests for exploratory data mining, not proving specific points of evidence, as well as the ease of misinterpretation of p-values, critical values and confidence intervals even among professional academic researchers. What we need are continuous measures of *how* closely a dataset follows a particular distribution, not simple Boolean either-or choices of the kind used in hypothesis testing, which reduce the information content of the test statistics as sharply as casting a float data type to a bit would in T-SQL. Furthermore, many of the lookup tables and approximation used in hypothesis testing are only valid up to a few hundred values, not the several million that we would need in Big Data scenarios.

**Abde and Molin’s Approximation**

The Lilliefors distribution was originally derived from Monte Carlo simulations (a broad term encompassing many types of randomized trials) and at least one attempt has been made to approximate it through a set of constants and equations.[vi] I implemented the approximation developed by Hervé Abdi and Paul Molin, but the first couple of SELECTs and the declarations following the comment “code for Molin and Abdi’s approximation” can be safely deleted if you don’t have a need for the P-values the block generates. I verified the P-Values and @A constants used to generate it against the examples given in their undated manuscript “Lilliefors/Van Soest’s Test of Normality,” but as is commonly the case with such workarounds in hypothesis testing, the algorithm is inapplicable when Big Data-sized values and counts are plugged into it.

…………Once @A falls below about 0.74 the approximation begins to return negative P-values and when it climbs above about 5.66 it produces P-values greater than 1, which I believe are invalid under the tenets of probability theory. Most of the practice datasets I plugged into the approximation returned invalid outputs, most of them strongly negative. This is a problem I’ve seen with other approximation techniques when they’re fed values beyond the expected ranges. Nevertheless, since I already coded it, I’ll leave that section intact in case anyone runs into scenarios where they can apply it to smaller datasets.

** Figure 1: T-SQL Code for the Lilliefors Goodness-of-Fit Test**CREATE PROCEDURE [Calculations].[GoodnessOfFitLillieforsTest]

@Database1 as nvarchar(128) = NULL, @Schema1 as nvarchar(128), @Table1 as nvarchar(128),@Column1 AS nvarchar(128), @OrderByCode as tinyint

AS

DECLARE @SchemaAndTable1 nvarchar(400),@SQLString nvarchar(max)

SET @SchemaAndTable1 = @Database1 + ‘.’ + @Schema1 + ‘.’ + @Table1

DECLARE @Mean float,

@StDev float,

@Count float

DECLARE @EDFTable table

(ID bigint IDENTITY (1,1),

Value float,

ValueCount bigint,

EDFValue float,

CDFValue decimal(38,37),

EDFCDFDifference decimal(38,37))

DECLARE @ExecSQLString nvarchar(max), @MeanOUT nvarchar(200),@StDevOUT nvarchar(200),@CountOUT nvarchar(200), @ParameterDefinition nvarchar(max)

SET @ParameterDefinition = ‘@MeanOUT nvarchar(200) UTPUT,@StDevOUT nvarchar(200) OUTPUT,@CountOUT nvarchar(200) OUTPUT ‘

SET @ExecSQLString = ‘SELECT @MeanOUT = CAST(Avg(‘ + @Column1 + ‘) as float),CAST(@StDevOUT = StDev(‘ + @Column1 + ‘) as float),CAST(@CountOUT = Count(‘ + @Column1 + ‘) as float)

FROM ‘ + @SchemaAndTable1 + ‘

WHERE ‘ + @Column1 + ‘ IS NOT NULL’

EXEC sp_executesql @ExecSQLString,@ParameterDefinition, @MeanOUT = @Mean OUTPUT,@StDevOUT = @StDev OUTPUT,@CountOUT = @Count OUTPUT

SET @SQLString = ‘SELECT Value, ValueCount, SUM(ValueCount) OVER (ORDER BY Value ASC) / CAST(‘ + CAST(@Count as nvarchar(50)) + ‘AS float) AS EDFValue

FROM (SELECT DISTINCT ‘ + @Column1 + ‘ AS Value, Count(‘ + @Column1 + ‘) OVER (PARTITION BY ‘ + @Column1 + ‘ ORDER BY ‘ + @Column1 + ‘) AS ValueCount

FROM ‘ + @SchemaAndTable1 + ‘

WHERE ‘ + @Column1 + ‘ IS NOT NULL) AS T1‘

INSERT INTO @EDFTable (Value, ValueCount, EDFValue)

EXEC (@SQLString)

UPDATE T1

SET CDFValue = T3.CDFValue, EDFCDFDifference = EDFValue – T3.CDFValue

FROM @EDFTable AS T1

INNER JOIN (SELECT DistinctValue, Calculations.NormalCalculationsingleCDFFunction (DistinctValue, @Mean, @StDev) AS CDFValue

FROM (SELECT DISTINCT Value AS DistinctValue

FROM @EDFTable) AS T2) AS T3

ON T1.Value = T3.DistinctValue

DECLARE @b0 float = 0.37872256037043,

@b1 float = 1.30748185078790,

@b2 float = 0.08861783849346,

@A float,

@PValue float,

@LillieforsTestStatistic float

SELECT @LillieforsTestStatistic = Max(ABS(EDFCDFDifference))

FROM @EDFTable

— code for Molin and Abdis approximation

— =======================================

SELECT @A = ((-1 * (@b1 + @Count)) + Power(Power((@b1 + @Count), 2) – (4 * @b2 * (@b0 – Power(@LillieforsTestStatistic, –2))), 0.5)) / (2 * @b2)

SELECT @PValue = –0.37782822932809 + (1.67819837908004 * @A)

– (3.02959249450445 * Power(@A, 2))

+ (2.80015798142101 * Power(@A, 3))

– (1.39874347510845 * Power(@A, 4))

+ (0.40466213484419 * Power(@A, 5))

– (0.06353440854207 * Power(@A, 6))

+ (0.00287462087623 * Power(@A, 7))

+ (0.00069650013110 * Power(@A, 8))

– (0.00011872227037 * Power(@A, 9))

+ (0.00000575586834 * Power(@A, 10))

SELECT @LillieforsTestStatistic AS LillieforsTestStatistic, @PValue AS PValueAbdiMollinApproximation

SELECT ID, Value, ValueCount, EDFValue, CDFValue, EDFCDFDifference

FROM @EDFTable

ORDER BY CASE WHEN @OrderByCode = 1 THEN ID END ASC,

CASE WHEN @OrderByCode = 2 THEN ID END DESC,

CASE WHEN @OrderByCode = 3 THEN Value END ASC,

CASE WHEN @OrderByCode = 4 THEN Value END DESC,

CASE WHEN @OrderByCode = 5 THEN ValueCount END ASC,

CASE WHEN @OrderByCode = 6 THEN ValueCount END DESC,

CASE WHEN @OrderByCode = 7 THEN EDFValue END ASC,

CASE WHEN @OrderByCode = 8 THEN EDFValue END DESC,

CASE WHEN @OrderByCode = 9 THEN CDFValue END ASC,

CASE WHEN @OrderByCode = 10 THEN CDFValue END DESC,

CASE WHEN @OrderByCode = 11 THEN EDFCDFDifference END ASC,

CASE WHEN @OrderByCode = 12 THEN EDFCDFDifference END DESC

** Figure 2: Sample Results from the Lilliefors Goodness-of-Fit Test**EXEC Calculations.GoodnessofFitLillieforsTestSP

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’LactateDehydrogenase‘,

@OrderByCode = ‘1’

…………Aside from the approximation section, the code in Figure 1 is almost identical to that of last week’s procedure, so I won’t belabor the point by rehashing the explanation here. As usual, I used queries like the one in Figure 2 to test the procedure against several columns in a 209-row dataset on the Duchennes form of muscular dystrophy and an 11-million-row dataset on the Higgs Boson, which are made publicly available by the Vanderbilt University’s Department of Biostatistics and University of California at Irvine’s Machine Learning Repository respectively. It is not surprising that the results nearly matched the Kolmogorov-Smirnov test statistic for many practice columns. For example, the LactateDehydrogenase enzyme scored 0.128712871287129 here and 0.131875117324784 on the Kolmogorov-Smirnov, while the less abnormal Hemopexin protein scored 0.116783569553499 on the Lilliefors and 0.0607407215998911on the Kolmogorov-Smirnov Test. Likewise, the highly abnormal first float column and Gaussian second column in the Higgs Boson table had test statistics of 0.276267238731715 and 0.0181893798916693 respectively, which were quite close to the results of the Kolmogorov-Smirnov. I cannot say if the departure in the case of Hemopexin was the result of some property of the test itself, like its aforementioned higher statistical power for detecting non-normality, or perhaps a coding error on my part. If so, then it would probably be worth it to calculate the Lilliefors test statistic together with the Kolmogorov-Smirnov and Kuiper’s measures and return them in one batch, to give end users a sharper picture of their data at virtually no computational cost.

** Figure 3: Execution Plan for the Lilliefors Goodness-of-Fit Test** (click to enlarge)

…………There were six queries in the execution plan, just as there were for last week’s tests, but the first accounted for 19 percent and the second 82 percent of the batch total. Both of those began with non-clustered Index Seeks, which is exactly what we want to see. Only the second would provide any worthwhile opportunities for further optimization, perhaps by targeting the only three operators besides the seek that contributed anything worthwhile to the query cost: a Hash Match (Aggregate) at 14 percent, a Stream Aggregate that accounted for 10 percent and two Parallelism (Repartition Streams) that together amount to 53 percent . Optimization might not really be necessary, given that the first float column in the mammoth Higgs Boson dataset returned in just 23 seconds and the second in 27. Your marks are likely to be several orders of magnitude better, considering that the procedure was executed on an antiquated semblance of a workstation that is an adventure to start up, not a real database server. The only other fitness tests in this series this fast were the Kolmogorov-Smirnov and Kuiper’s Tests, which I would have calculated together with this test in a single procedure if I’d known there was so much overlap between them. The Anderson-Darling Test we’ll survey in the next installment of the series is also included in the same category of EDF-based fitness tests, but has less in common with the Lilliefors Test and its aforementioned cousins. Unfortunately, high performance is apparently not among the characteristics the Anderson-Darling Test shares with its fellow EDF-based methods. That’s something of a shame, since it is more widely by real statisticians than many other goodness-of-fit tests.

[i] p. 1, Abdi, Hervé and Molin, Paul, undated manuscript “Lilliefors/Van Soest’s Test of Normality,” published at the __University of Texas at Dallas School of Behavioral and Brain Sciences__ web address https://www.utdallas.edu/~herve/Abdi-Lillie2007-pretty.pdf

[ii] See the __Wikipedia__ page “Lilliefors Test” at http://en.wikipedia.org/wiki/Lilliefors_test

[iii] Lilliefors, Hubert W., 1967, “On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown,” pp. 399-402 in __Journal of the American Statistical Association__, Vol. 62, No. 318. June, 1967.

[iv] See the __Statd.com__ webpage “Lilliefors Normality Test” at http://statltd.com/articles/lilliefors.htm

[v] See Arcones, Miguel A., 2006, “On the Bahadur Slope of the Lilliefors and the Cramér–von Mises Tests of Normality,” pp. 196-206 in the __Institute of Mathematical Statistics Lecture Notes – Monograph Series__. No. 51. Available at the web address https://projecteuclid.org/euclid.lnms/1196284113

[vi] See p. 3, Abdi and Molin and the aforementioned __Wikipedia__ page “Lilliefors Test” at http://en.wikipedia.org/wiki/Lilliefors_test

## Goodness-of-Fit Testing with SQL Server Part 7.1: The Kolmogorov-Smirnov and Kuiper’s Tests

**By Steve Bolton**

…………“The names statisticians use for non-parametric analyses are misnomers too, in my opinion: Kruskal-Wallis tests and Kolmogorov-Smirnov statistics, for example. Good grief! These analyses are simple applications of parametric modeling that belie their intimidating exotic names.”[i]

Apparently even experts like Will G. Hopkins, the author of the plain English online guide to stats A New View of Statistics, perceive just how dry the subject can be. They feel our pain. Sometimes the topics are simply too difficult to express efficiently without brain-taxing equations and really terse writing, but this is not the case with the Kolmogorov-Smirnov Test, the topic of this week’s mistutorial on how to perform goodness-of-fit tests with SQL Server. This particular test got its lengthy moniker from two Russian mathematicians, Vladimir Smirnov (1887-1974) and Andrey Kolmogorov (1903-1987), the latter of whom is well-known in the field but hardly a household name beyond it.[ii] He made many important contributions to information theory, neural nets and other fields directly related to data mining, which I hope to shed some light on in a future tutorial series, Information Measurement with SQL Server. Among them was Kolmogorov Complexity, a fascinating topic that can be used to embed data mining algorithms more firmly into the use of reason, in order to make inferences based on strict logical necessity. Even more importantly, he was apparently sane – unlike most famous mathematicians and physicists, who as I have noted before tend to be not merely eccentric, but often shockingly degenerate or dangerous, or both.[iii] Despite the imposing name, I was actually looking forward to coding this particular test because Kolmogorov’s work always seems to turn out to be quite useful. I wasn’t disappointed. The concepts aren’t half as hard to grasp as the name is to pronounce, because aside from the usual inscrutable equations (which I almost always omit from these articles after translating them into code) the logic behind it is really common sense. Perhaps best of all, the Kolmogorov-Smirnov Test is hands down the fastest and best-performing goodness-of-fit measure we have yet surveyed in this series. The code I provided for the last few articles was some of the weakest I’ve written in all of my tutorial series, which was compounded by the fact that the tests I surveyed aren’t a good match for SQL Server use cases, but all in all, the T-SQL below for the Kolmogorov-Smirnov is some of the best I’ve written to date. After several rewrites, it now executes on an 11-million-row dataset on a beat-up desktop in less than 30 seconds.

**The Benefits of Kolmogorov’s Test**

Several studies comparing the various goodness-of-fit tests often rank the Kolmogorov-Smirnov measure near the bottom along with other popular ones like the Chi-Squared Test) because it has lower statistical power (i.e., the ability to detect an effect on a variable when it is actually present) than rivals like the Shapiro-Wilk. As we have seen in previous articles, however, many of these alternate measures are not as well-suited to the use cases the SQL Server community is likely to encounter – particularly the popular Shapiro-Wilk Test, since it can only be applied to very small datasets. Our scenarios are distinctly different from those encountered in the bulk of academic research, since we’re using recordsets of millions or even billions of rows. These datasets are often big enough to reduce the need for random sampling, since they may represent the full population or something close to it. Furthermore, parameters like averages, counts, standard deviations and variances can be instantly calculated for the entire dataset, thereby obviating the need for the complicated statistical techniques often used to estimate them. This advantage forestalls one of the stumbling blocks otherwise associated with the Kolmogorov-Smirnov Test, i.e. the need to fully specify all of the parameters (typically aggregates) for the distribution being tested.[iv]

…………The ideal goodness-of-fit test for our purposes would be one applicable to the widest number of distributions, but many of them are limited to the Gaussian “normal” distribution or bell curve. That is not true of the Kolmogorov-Smirnov Test, which can be applied to any distribution that would have a continuous Content type in SQL Server Data Mining (SSDM). It is also an exact test whose accuracy is not dependent on the number of data points fed into it.[v] I would also count among its advantages the fact that it has clear bounds, between 0 and 1; other statistical tests sometimes continually increase in tandem with the values and counts fed into them and can be difficult to read, once the number of digits of the decimal place exceeds six or seven, thereby requiring users to waste time counting them. As we shall see, there is a lingering interpretation issue with this test, or at least my amateur implementation of it. The test can also be “more sensitive near the center of the distribution than at the tails,” but this inconvenience is heavily outweighed by its many other advantages.

…………Another plus in its favor is the relative ease with which the inner workings can be grasped. End users should always know how to interpret the numbers returned to them, but there is no reason to burden them with the internal calculations and arcane equations; I think most of the instructional materials on math and stats lose their audiences precisely because they bury non-experts under a mountain of internal details that require a lot of skills they don’t have, nor need. End users are like commuters, who don’t need to give a dissertation in automotive engineering in order to drive to work each day; what they should be able to do is read a speedometer correctly. It is the job of programmers to put the ideas of mathematicians and statisticians into practice and make them accessible to end users, in the same way that mechanics are the middlemen between drivers and automotive engineers. It does help, however, if the internal calculations have the minimum possible level of difficulty, so that programmers and end users alike can interpret and troubleshoot the results better; it’s akin to the way many Americans in rural areas become driveway mechanics on weekends, which isn’t possible if the automotive design becomes too complex for them to work on efficiently.

**Plugging in CDFs and EDFs**

The Kolmogorov-Smirnov Test isn’t trivial to understand, but some end users might find it easier to grasp the inner workings of this particular goodness-of-fit test. The most difficult concept is that of the cumulative distribution function (CDF), which I covered back in Goodness-of-Fit Testing with SQL Server, part 2.1: Implementing Probability Plots in Reporting Services and won’t rehash here. Suffice it to say that all of the probabilities for all of the possible values of a column are arranged so that they accumulate from 0 to 1. The concept is easier to understand than to code, at least for the normal distribution. One of the strongest points of the Kolmogorov-Smirnov Test is that we can plug the CDF of any continuous distribution into it, but I’ll keep things short by simply reusing one of the CDF functions I wrote for the Implementing Probability Plots article.

…………All we have to do to derive the Kolmogorov-Smirnov metric is to add in the concept of the empirical distribution function (EDF), in which we merely put the recordset in the order of the values and assign an EDF value at each point that is equal to the reciprocal of the overall count. Some sources make cautionary statements like this, “Warning: ties should not be present for the Kolmogorov-Smirnov test,”[vi] which would render all of the test based on EDFs useless for our purposes since our billion-row tables are bound to have repeat values. I was fortunate to find a workaround in some undated, uncredited course notes I found at the Penn State University website, which turned out to be the most useful source of info I’ve yet found on implementing EDFs.[vii] To circumvent this issue, all we have to do is use the distinct count for values with ties as the dividend rather than one. Like the CDF, the EDF starts at 0 and accumulates up to a limit of 1, which means they can be easily compared by simply subtracting them at each level. The Kolmogorov-Smirnov test statistic is merely the highest difference between the two.[viii] That’s it. All we’re basically doing is seeing if the order follows the probability we’d expected for each value, if they came from a particular distribution. In fact, we can get two measures for the price of one by using the minimum difference as the test statistic for Kuiper’s Test, which is used sometimes in cases where cyclical variations in the data are an issue.[ix]

__Figure 1: T-SQL Code for the Kolmogorov-Smirnov and Kuiper’s Tests__

CREATE PROCEDURE [Calculations].[GoodnessOfFitKolomgorovSmirnovAndKuipersTestsSP]

@Database1 as nvarchar(128) = NULL, @Schema1 as nvarchar(128), @Table1 as nvarchar(128),@Column1 AS nvarchar(128), @OrderByCode as tinyint

AS

SET @SchemaAndTable1 = @Database1 + ‘.’ + @Schema1 + ‘.’ + @Table1

DECLARE @Mean float,

@StDev float,

@Count float

DECLARE @EDFTable table

(ID bigint IDENTITY (1,1),

Value float,

ValueCount bigint,

EDFValue float,

CDFValue decimal(38,37),

EDFCDFDifference decimal(38,37))

DECLARE @ExecSQLString nvarchar(max), @MeanOUT nvarchar(200),@StDevOUT nvarchar(200),@CountOUT nvarchar(200), @ParameterDefinition nvarchar(max)

SET @ParameterDefinition = ‘@MeanOUT nvarchar(200) OUTPUT,@StDevOUT nvarchar(200) OUTPUT,@CountOUT nvarchar(200) OUTPUT ‘

SET @ExecSQLString = ‘SELECT @MeanOUT = Avg(‘ + @Column1 + ‘),@StDevOUT = StDev(‘ + @Column1 + ‘),@CountOUT = Count(‘ + @Column1 + ‘)

FROM ‘ + @SchemaAndTable1 + ‘

WHERE ‘ + @Column1 + ‘ IS NOT NULL’

EXEC sp_executesql @ExecSQLString,@ParameterDefinition, @MeanOUT = @Mean OUTPUT,@StDevOUT = @StDev OUTPUT,@CountOUT = @Count OUTPUT

SET @SQLString = ‘SELECT Value, ValueCount, SUM(ValueCount) OVER (ORDER BY Value ASC) / CAST(‘ +

CAST(@Count as nvarchar(50)) + ‘AS float) AS EDFValue

FROM (SELECT DISTINCT ‘ + @Column1 + ‘ AS Value, Count(‘ + @Column1 + ‘) OVER (PARTITION BY ‘ + @Column1 + ‘ ORDER BY ‘ + @Column1 + ‘) AS ValueCount

FROM ‘ + @SchemaAndTable1 + ‘

WHERE ‘ + @Column1 + ‘ IS NOT NULL) AS T1‘

INSERT INTO @EDFTable

(Value, ValueCount, EDFValue)

EXEC (@SQLString)

UPDATE T1

SET CDFValue = T3.CDFValue, EDFCDFDifference = EDFValue – T3.CDFValue

FROM @EDFTable AS T1

INNER JOIN (SELECT DistinctValue, Calculations.NormalDistributionSingleCDFFunction (DistinctValue, @Mean, @StDev) AS CDFValue

FROM (SELECT DISTINCT Value AS DistinctValue

FROM @EDFTable) AS T2) AS T3

ON T1.Value = T3.DistinctValue

SELECT KolomgorovSmirnovSupremum AS KolomgorovSmirnovTest, KolomgorovSmirnovSupremum – KolomgorovSmirnovMinimum AS KuipersTest

FROM (SELECT Max(ABS(EDFValue – CDFValue)) AS KolomgorovSmirnovSupremum, — the supremum i.e. max

Min(ABS(EDFValue – CDFValue)) AS KolomgorovSmirnovMinimum

FROM @EDFTable

WHERE EDFCDFDifference > 0) AS T3

SELECT ID, Value, ValueCount, EDFValue, CDFValue, EDFCDFDifference

FROM @EDFTable

ORDER BY CASE WHEN @OrderByCode = 1 THEN ID END ASC,

CASE WHEN @OrderByCode = 2 THEN ID END DESC,

CASE WHEN @OrderByCode = 3 THEN Value END ASC,

CASE WHEN @OrderByCode = 4 THEN Value END DESC,

CASE WHEN @OrderByCode = 5 THEN ValueCount END ASC,

CASE WHEN @OrderByCode = 6 THEN ValueCount END DESC,

CASE WHEN @OrderByCode = 7 THEN EDFValue END ASC,

CASE WHEN @OrderByCode = 8 THEN EDFValue END DESC,

CASE WHEN @OrderByCode = 9 THEN CDFValue END ASC,

CASE WHEN @OrderByCode = 10 THEN CDFValue END DESC,

CASE WHEN @OrderByCode = 11 THEN EDFCDFDifference END ASC,

CASE WHEN @OrderByCode = 12 THEN EDFCDFDifference END DESC

…………The code above is actually a lot simpler than it looks, given that the last 12 lines are dedicated to implementing the @OrderByCode parameter, which I’ve occasionally provided as an affordance over the course of the last two tutorial series. It’s particularly useful in this test when the column values, distinct counts EDF and CDF results in the @EDFTable are of interest in addition to the test statistic; ordinarily, this would be taken care of in an app’s presentation layer, so the ordering code can be safely deleted if you’re not using SQL Server Management Studio (SSMS). In this instance, 1 orders the results by ID ASC, 2 is by ID Desc, 3 is by Value ASC, 4 is by Value DESC, 5 is by ValueCount ASC, 6 is by ValueCount DESC, 7 is by EDFValue ASC, 9 is by EDFValue Desc, 9 is by CDFValue ASC, 10 is by CDFValue DESC, 11 is by EDFCDFDifference ASC and 12 is by EDFCDFDifference DESC. The rest of the parameters and first couple of line of dynamic SQL allow users to perform the tests against any column in any database they have sufficient access to. As usual, you’ll have to add in your own validation, null handling and SQL injection protection code. Two dynamic SQL statements are necessary because separate count, mean and standard deviation have to be extracted from the original base table. The retrieval of those aggregates needed for subsequent calculations occurs shortly after the declarations section.

…………Note that this procedure was markedly faster after substituting the sp_executesql statement for a dynamic INSERT EXEC on the base table (which had been used to populate the @EDFTable in an inefficient way). One quirk I should point out though is the use of the DISTINCT clause in the UPDATE subquery, which is needed to prevent unnecessary repetitive calls to the somewhat expensive Calculations.NormalDistributionSingleCDFFunction in the case of duplicate values. This somewhat convoluted method actually save a big performance hit on large tables with lots of duplicates. In the final query, I bet that the outer subquery would be less expensive than retrieving the max twice in a single query. One of the few concerns I have about the procedure is the use of the actual mean and standard deviation in calculating the CDF values. Some sources recommended using the standard normal, but this typically resulted in ridiculous distortions for most of the recordsets I tested them against. On the other hand, I verified the correctness of the calculations as they stand now by working through the example in the Alion System Reliability Center’s Selected Topics in Assurance Related Technologies, a series of publications on stats I recently discovered and now can’t live without.[x]

** Figure 2: Sample Results from the Kolmogorov-Smirnov and Kuiper’s Tests**EXEC Calculations.GoodnessOfFitKolomgorovSmirnovAndKuipersTestsSP

@Database1 = N’DataMiningProjects’,

@Schema1 = N’Health’,

@Table1 = N’DuchennesTable’,

@Column1 = N’LactateDehydrogenase’,

@OrderByCode = 1

…………Even more good news: when I tested it on the 209 rows of the tiny 9-kilobyte set of data on the Duchennes muscular dystrophy and the 11 million rows and near 6 gigs of data in the Higgs Boson dataset (which I downloaded from the Vanderbilt University’s Department of Biostatistics and University of California at Irvine’s Machine Learning Repository respectively) I got pretty much the results I expected. After using the same datasets for the last dozen articles or so, I know which ones follow the Gaussian distribution and which do not, and the Kolmogorov-Smirnov Test consistently returned lower figures for the ones that followed a bell curve and higher ones for those that do not. For example, the query in Figure 2 returned a value of 0.131875117324784 for the LactateDehydrogenase enzyme, while the less abnormal Hemopexin scored a 0.0607407215998911. On the other hand, the highly abnormal, really lopsided first float column in the Higgs Boson dataset scored a whopping 0.276266847552121, while the second float column scored just a 0.0181892303151281 probably because it clearly follows a bell curve in a histogram.

…………Other programmers may also want to consider adding in their own logic to implement confidence intervals and the like, which I typically omit for reasons of simplicity, the difficulty of deriving lookup values on a Big Data scale and philosophical concerns about their applicability, not to mention the widespread concern among many professional statisticians about the rampant misuse and misinterpretation of hypothesis testing methods. Suffice it to say that my own interval of confidence in them is steadily narrowing, at least for the unique use cases the SQL Server community faces. The good news is that if you decide to use standard hypothesis testing methods, then the Kolmogorov-Smirnov test statistic doesn’t require modifications before plugging it into lookup tables, unlike the popular Shapiro-Wilk and Anderson-Darling tests.[xi]

** Figure 3: Execution Plan for the Kolmogorov-Smirnov and Kuiper’s Procedure** (click to enlarge)

…………When all is said and done, the Kolmogorov-Smirnov Test is the closest thing to the ideal goodness-of-fit measure for our use cases. It may have low statistical power, but it can handle big datasets and a wide range of distributions. The internals are a shorter to code and moderately easier to explain to end users than those of some other procedures and the final test statistic is easy to read because it has clear bounds. It also comes with some freebies, like the ability to simultaneously calculate Kuiper’s Test at virtually no extra cost. For most columns I tested there wasn’t much of a difference between the Kolmogorov-Smirnov and Kuiper’s Test results till we got down to the second through fifth decimal places, but there’s no reason not to calculate it if the costs are dwarfed by those incurred by the rest of the procedure. Note that I also return the full @EDFTable, including the ValueCount for each distinct Value, since there’s no point in discarding all that information once the burden of computing it all has been borne. One of the few remaining concerns I have about the test is that much of this information may be wasted in the final test statistics, since merely taking minimums and maximums is often an inefficient way of making inferences about a dataset. This means that more useful, expanded versions of the tests might be possible by calculating more sophisticated measures on the same EDF and CDF data.

…………Best of all, the test outperforms any of the others we’ve used in the last two tutorial series. After eliminating most of the dynamic SQL I overused in previous articles, the execution time actually worsened, till I experimented with some different execution plans. On the first float column in the 11-million-row, 6-gig Higgs Boson dataset, the procedure return in just 24 seconds, but for the equally-sized second float column, in returned in an average of just 29. That’s not shabby at all for such a useful statistical test on such a huge dataset, on a clunker of a desktop that’s held together with duct tape. I can’t account for that difference, given that the execution plans were identical and the two columns share the same data type and count; the only significant difference I know of is that one is highly abnormal and the other follows a bell curve. For smaller datasets of a few thousand rows the test was almost instantaneous. I don’t think the execution plan in Figure 3 can be improved upon much, given that just two of the five queries account for practically all of the cost and both of them begin with Index Seeks. In the case of the first, that initial Seek accounts for 92 percent of the cost. The second ought to be the target of any optimization efforts, since it accounts for 85 percent of the batch; within it, however, the only operators that might be worth experimenting with are the Hash Match (Aggregate) and the Sort. Besides, the procedure already performs well enough as it is and should be practically instantly on a real database server. In the next installment, we’ll see whether the Lilliefors Test, another measure based on the EDF, can compete with the Kolmogorov-Smirnov Test, which is thus far the most promising measure of fit we’ve yet covered in the series.

[i] See Hopkins, 2014, “Rank Transformation: Non-Parametric Models,” published at the __A New View of Statistics __webpage http://www.sportsci.org/resource/stats/nonparms.html

[ii] See the Wikipedia pages “Andrey Kolmogorov” and “Vladimir Smirnov” at http://en.wikipedia.org/wiki/Andrey_Kolmogorov and http://en.wikipedia.org/wiki/Vladimir_Smirnov_(mathematician) respectively.

[iii] I’m slowly compiling a list of the crazy ones and their bizarre antics for a future editorial or whatever – which will include such cases as Rene Descartes’ charming habit of carrying a dummy of his dead sister around Europe and carrying on conversations with it in public. I’m sure there’ll also be room for Kurt Gödel, who had a bizarre fear of being poisoned – so he forced his wife to serve as his food-taster. Nothing says romance like putting the love of your life in harm’s way when you think people are out to get you. When she was hospitalized, he ended up starving to death. Such tales are the norm among the great names in these fields, which is why I’m glad I deliberately decided back in fifth grade not to pursue my fascination with particle physics.

[iv] See National Institute for Standards and Technology, 2014, “1.3.5.16 Kolmogorov-Smirnov Goodness-of-Fit Test,” published in the online edition of the __Engineering Statistics Handbook.__ Available at http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm

[v] *IBID.*

[vi] p. 14, Hofmann, Heike, 2013, “Nonparametric Inference and Bootstrap { Q-Q plots; Kolmogorov Test,” lecture notes published Oct. 11, 2013 at the Iowa State University web address http://www.public.iastate.edu/~hofmann/stat415/lectures/07-qqplots.pdf

[vii] Penn State University, 2014, “Empirical Distribution Functions,” undated course notes posted at the __Penn State University __website and retrieved Nov. 5, 2014 from the web address https://onlinecourses.science.psu.edu/stat414/node/333

[viii] I also consulted the Wikipedia page “Kolmogorov-Smirnov Test” at http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test for some of these calculations.

[ix] See the Wikipedia article “Kuiper’s Test” at http://en.wikipedia.org/wiki/Kuiper’s_test

[x] See Alion System Reliability Center, 2014, “Kolmogorov-Simirnov: A Goodness of Fit Test for Small Samples,” published in __Selected Topics in Assurance Related Technologies__, Vol. 10, No. 6. Available online at the __Alion System Reliability Center __web address

https://src.alionscience.com/pdf/K_STest.pdf

[xi] “Critical value beyond which the hypothesis is rejected in Anderson-Darling test is different when Gaussian pattern is being tested than when another distribution such a lognormal is being tested. Shapiro-Wilk critical value also depends on the distribution under test. But Kolmogorov-Smirnov test is distribution-free as the critical values do not depend on whether Gaussianity is being tested or some other form. No author listed, 2014, “Checking Gaussianity,” published online at the MedicalBiostatistics.com web address http://www.medicalbiostatistics.com/checkinggaussianity.pdf

## Goodness-of-Fit Testing with SQL Server Part 6.2: The Ryan-Joiner Test

**By Steve Bolton**

…………In the last installment of this amateur series of self-tutorials, we saw how the Shapiro-Wilk Test might probably prove less useful to SQL Server users, despite the fact that it is one of the most popular goodness-of-fit tests among statisticians and researchers. Its impressive statistical power is rendered impotent by the fact that the logic of its internal calculations limits it to inputs of just 50 rows (or up to 2,000 when certain revisions are applied), which is chump change when we’re talking about SQL Server tables that often number in the millions of rows. Thankfully, a little-known rival in the same category[1] is available that shares few of the drawbacks that make Shapiro-Wilk such a disappointing choice for our particular use cases. In fact, the Ryan-Joiner Test “is very highly correlated with that of Shapiro and Wilk, so either test may be used and will produce very similar resultspWhen wading through various head-to-head comparisons of the goodness-of-fit tests published on the Internet, I noticed that on the occasions when the Ryan-Joiner Test was mentioned, it received favorable marks like this review by Jim Colton at the Minitab Blog:

“I should note that the three scenarios evaluated in this blog are not designed to assess the validity of the Normality assumption for tests that benefit from the Central Limit Theorem, such as such as 1-sample, 2-sample, and paired t-tests. Our focus here is detecting Non-Normality when using a distribution to estimate the probability of manufacturing defective (out-of-spec) unit.

…………“In scenario 1, the Ryan-Joiner test was a clear winner. The simulation results are below…

…………“…The Anderson-Darling test was never the worst test, but it was not nearly as effective as the RJ test at detecting a 4-sigma outlier. If you’re analyzing data from a manufacturing process tends to produce individual outliers, the Ryan-Joiner test is the most appropriate…”

…………“…The RJ test performed very well in two of the scenarios, but was poor at detecting Non-Normality when there was a shift in the data. If you’re analyzing data from a manufacturing process that tends to shift due to unexpected changes, the AD test is the most appropriate.”[3]

…………The most striking drawback is the paucity of public information available on the test, which doesn’t even have a Wikipedia page, thereby forcing me to resort to even less professional sources like Answers.com for matter-of-fact explanations like this: “The Ryan-Joiner test is implemented in the Minitab software package but not widely elsewhere.”[4] It was apparently the brainchild of Brian Joiner and Barbara Ryan, “the founder of Minitab,” but I was unable to find a publicly available copy of the original academic paper they published on the test back in 1976 until after I’d already written most of the code below.[5] Publication of this kind signifies that it is not a proprietary algorithm exclusively owned by Minitab, so we are free to implement it ourselves – provided we can find adequate detail on its inner workings, which turned out to be a tall order. The main drawback of the Ryan-Joiner Test is the difficulty in finding information that can be applied to implementation and testing, which is certainly a consequence of its close association with Minitab, a stats package that competes only tangentially with SQL Server Data Mining (SSDM) as I addressed in Integrating Other Data Mining Tools with SQL Server, Part 2.1: The Minuscule Hassles of Minitab and Integrating Other Data Mining Tools with SQL Server, Part 2.2: Minitab vs. SSDM and Reporting Services. This makes it somewhat opaque, but it I was able to overcome this inscrutability enough to get a T-SQL version of it up and running.

…………The underlying mechanisms are still somewhat unclear, but this brief introduction in the LinkedIn discussion group Lean Sigma Six Group Brazil is adequate enough for our purposes: “This test assesses normality by calculating the correlation between your data and the normal scores of your data. If the correlation coefficient is near 1, the population is likely to be normal. The Ryan-Joiner statistic assesses the strength of this correlation; if it falls below the appropriate critical value, you will reject the null hypothesis of population normality.”[6] As usual, I’ll be omitting those critical values, because of the numerous issues with hypothesis testing I’ve pointed out in previous blog posts. Apparently my misgivings are widely shared by professional statisticians and mathematicians who actually know what they’re talking about, particularly when it comes to the ease and frequency with which all of the caveats and context that statements of statistical significance are carelessly dispensed with. It is not that significance level stats aren’t useful, but that the either-or nature of standard hypothesis testing techniques discards an awful lot of information by effectively shrinking our hard-won calculations down to simple Boolean either-or choices; not only is this equivalent to casting a float or decimal value down to a SQL Server bit data type, but can also easily lead to errors in interpretation. For this reason and concerns about brevity and simplicity, I’ll leave out the critical values, which can be easily tacked on to my code by anyone with a need for them.

**Interpreting the Results in Terms of the Bell Curve**

Aside from that, the final test statistic isn’t that hard to interpret: the closer we get to 1, the more closely the data follows the Gaussian or “normal” distribution, i.e. the bell curve. So far, my test results have all remained within the range of 0 to 1 as expected, but I cannot rule out the possibility that in some situations an undiscovered error will cause them to exceed these bounds. When writing the T-SQL code in Figure 1 I had to make use of just two incomplete sources[7], before finally finding the original paper by Ryan and Joiner at the Minitab website late in the game.[8] This find was invaluable because it pointed out that the Z-Scores (a basic topic I explained way back in Outlier Detection with SQL Server, part 2.1: Z-Scores) in the internal calculations should be done against the standard normal distribution, not the data points.

…………My standard disclaimer that I still a novice in the fields of data mining and statistics and that my sample code has not yet been thoroughly tested ought not be glossed over, given the number of other mistakes I caught myself making when writing the code below. At one point I accidentally used a minus sign rather than an asterisk in the top divisor; I tested it once against the wrong online calculator, for the normal probability density function (PDF) rather than the cumulative density function (CDF); later, I realized I should have used the standard normal inverse CDF rather than the CDF or PDF; I also used several different improper step values for the RangeCTE, including one that was based on minimum and maximum values rather than the count and another based on fractions. Worst of all, I garbled my code at the last minute by accidentally (and not for the first time) using the All Open Documents option with Quick Replace in SQL Server Management Studio (SSMS). Once I figured out my mistakes, the procedure ended up being a lot shorter and easier to follow than I ever expected. Keep in mind, however, that I didn’t have any published examples to test it against, so there may be other reliability issues lurking within.

** Figure 1: T-SQL Code for the Ryan-Joiner Test Procedure**CREATE PROCEDURE [Calculations].[NormalityTestRyanJoinerTestSP]

@Database1 as nvarchar(128) = NULL, @Schema1 as nvarchar(128), @Table1 as nvarchar(128),@Column1 AS nvarchar(128)

AS

SET @SchemaAndTable1 = @Database1 + ‘.’ + @Schema1 + ‘.’ + @Table1

DECLARE @ValueTable table

(ID bigint IDENTITY (1,1),

Value float)

SET @SQLString = ‘SELECT ‘ + @Column1 + ‘ AS Value

FROM ‘ + @SchemaAndTable1 +

‘ WHERE ‘ + @Column1 + ‘ IS NOT NULL’

INSERT INTO @ValueTable

(Value)

EXEC (@SQLString)

DECLARE @Var AS decimal(38,11),

@Count bigint,

@ConstantBasedOnCount decimal(5,4),

@Mean AS decimal(38,11)

SELECT @Count = Count(*), @Var = Var(Value)

FROM @ValueTable

— the NOT NULL clause is not necessary here because that’s taken care of the in @SQLString

SELECT @ConstantBasedOnCount = CASE WHEN @Count > 10 THEN 0.5 ELSE 0.375 END

; WITH RangeCTE(RangeNumber) as

(

SELECT 1 as RangeNumber

UNION ALL

SELECT RangeNumber + 1

FROM RangeCTE

WHERE RangeNumber < @Count

)

SELECT SUM(Value * RankitApproximation) / Power((@Var * (@Count – 1) * SUM(Power(CAST(RankitApproximation AS float), 2))), 0.5) AS RyanJoinerTestStatistic

FROM (SELECT RN, Value, Calculations.NormalDistributionInverseCDFFunction((RN – @ConstantBasedOnCount) / (@Count + 1 – (2 * @ConstantBasedOnCount))) AS RankitApproximation

FROM (SELECT ROW_NUMBER() OVER (ORDER BY Value) AS RN, Value

FROM @ValueTable AS T0) AS T1

INNER JOIN RangeCTE AS T2

ON T1.RN = T2.RangeNumber) AS T3

OPTION (MAXRECURSION 0)

…………Much of the code above is easy to follow if you’ve seen procedures I’ve posted over the last two tutorial series. As usual, the parameters and first couple of lines in the body allow users to perform the test on any column in any database they have sufficient access to, as well as to adjust the precision of the calculations to avoid arithmetic overflows. Starting with my last article, I began using a lot less dynamic SQL to code procedures like these, by instead caching the original values in a @ValueTable table variable. A couple of simple declarations and assignments needed for the rest of the computations follow this. The RangeCTE generates a set of integers that is fed to the Calculations.NormalDistributionInverseCDFFunction I introduced in Goodness-of-Fit Testing with SQL Server, part 2: Implementing Probability Plots in Reporting Services.

…………In lieu of making this article any more verbose and dry than it absolutely has to be, I’ll omit a rehash of that topic and simply point users back to the code from that previous article. Once those numbers are derived, the calculations are actually quite simple in comparison to some of the more complex procedures I’ve posted in the past. As usual, I’ll avoid a precise explanation of how I translated the mathematical formulas into code, for the same reason that driver’s ed classes don’t require a graduate degree in automotive engineering: end users need to be able to interpret the final test statistic accurately – which is why I’m not including the easily misunderstood critical values – but shouldn’t be bothered with the internals. I’ve supplied just enough detail so that the T-SQL equivalent of a mechanic can fix my shoddy engineering, if need be. It may be worth noting though that I can’t simply use the standard deviation in place of the root of the variance as I normally do, because the square root in the Ryan-Joiner equations is calculated after the variance has been multiplied by other terms.

** Figure 2: Sample Results from Ryan-Joiner Test Procedure**EXEC Calculations.NormalityTestRyanJoinerTestSP

@DatabaseName = N’DataMiningProjects’,

@SchemaName = N’Health’,

@TableName = N’DuchennesTable’,

@ColumnName = N’Hemopexin’

…………The sample query in Figure 2 was run against a column of data on the Hemopexin protein contained within a dataset on the Duchennes form of muscular dystrophy, which I downloaded long ago from the Vanderbilt University’s Department of Biostatistics and converted to a SQL Server table. Since this table only has 209 rows and occupies just 9 kilobytes, I customarily stress test the procedures I post in these tutorials against an 11-million-row table of data on the Higgs Boson, which is made freely available by the University of California at Irvine’s Machine Learning Repository and now occupies nearly 6 gigabytes in the same database.

…………I also tested it against columns in other tables I’m quite familiar with and discovered a pattern that is both comforting and disconcerting at the same time: the test statistic is indeed closer to 0 than 1 on columns I already know to be abnormal, but there may be a scaling issue in the internal calculations because the values are still unexpectedly high for all columns. I know from previous goodness-of-fit and outlier detection tests that the Hemopexin column is more abnormal than some of the other Duchennes columns and as expected, it had a lower Ryan-Joiner statistic; the problem is that it was still fairly close to 1. Likewise, the histograms I posted way back in Outlier Detection with SQL Server, part 6.1: Visual Outlier Detection with Reporting Services clearly show that the first float column in the Higgs Boson dataset is hopelessly lopsided and therefore can’t come from a normal distribution, while the second follows a clear bell curve. It is not surprising then that the former scored a 0.909093348747035 and the latter a 0.996126753961487 in the Ryan-Joiner Test. The order of the values always seems to correctly match the degree of normality for every practice dataset I use, which is a good sign, but the gaps between them may not be proportionally correct. In the absence of example data to verify my procedure against, I can’t tell for sure if this is a problem or not.

**Next Up: Kolmogorov and EDF-Based Tests**

Either way, the test is useful as-is, because it at least assigns test statistic values that are in the expected order, regardless of whether or not they are scaled correctly. These results come at a moderate, tolerable performance cost, clocking in at 6:43 for the first float column and 6:14 for the second. As usual, your results will probably be several orders of magnitude better than mine, given that I’m using a clunker of a development machine, not a real database server. The execution plans consist of two queries, the second of which accounts for 97 percent of the cost of the whole batch; out of the 24 operators in that query, a single Sort accounts for 96 percent of the cost. It occurs prior to a Merge Join, so there may be some way to optimize the procedure with join hints or recoding with optimization in mind. We’re unlikely to get much benefit out of analyzing the execution plan further, because it consists almost entirely of spools and Compute Scalar operators with infinitesimal costs, plus two Index Seeks, which is what we want to see.

…………The Ryan-Joiner Test performs well enough that DBA and data miners might find it a more useful addition to their toolbelt than the far better-known Shapiro-Wilk Test, which is simply inapplicable to most Big Data scenarios because of its fatal limitations on input sizes. There may be some lingering concerns about it reliability, but this can be rectified through a more diligent search of the available literature for examples that we can test it against; if we really need this particular statistic, then conferring with a professional statistician for ten minutes to verify the correctness of the results might also get the job done. If misgivings about its reliability are a real concern, then we can always turn to the alternatives we’ll cover in the next segment of this series, like the Kolmogorov-Smirnov (my personal favorite, which was also invented by my favorite mathematician), Anderson-Darling Kuiper’s and Lilliefors Tests, as well as the Cramér–von Mises Criterion. Judging from the fact that experts seem to divide the various goodness-of-fit tests into categories along the same lines[9], I was right to segregate the Jarque-Bera and D’Agostino’s K-Squared Test into a separate segment at the beginning of this series for measures based on kurtosis and skewness. The Shapiro-Wilk and Ryan-Joiner Tests likewise have a separate set of internal mechanism in commons, based on measures of correlation. In the next five articles, we’ll cover a set of goodness-of-fit measures that rely on a different type of internal mechanism, the empirical distribution function (EDF), which is a lot easier to calculate and explain than the long-winded name would suggest.

[1] These authors say it is “similar to the SW test”: p. 2142, Yap, B. W. and Sim, C. H., 2011, “Comparisons of Various Types of Normality Tests,” pp. 2141-2155 in __Journal of Statistical Computation and Simulation__, Vol. 81, No. 12. Also see the remark to the effect that “This test is similar to the Shapiro-Wilk normality test” at Gilberto, S. 2013, “Which Normality Test May I Use?” published in the__ Lean Sigma Six Group Brazil__ discussion group, at the LinkedIn web address http://www.linkedin.com/groups/Which-normality-test-may-I-3713927.S.51120536

[2] See the __Answers.com__ webpage “What is Ryan Joiner Test” at http://www.answers.com/Q/What_is_Ryan_joiner_test

[3] Colton, Jim, 2013, “Anderson-Darling, Ryan-Joiner, or Kolmogorov-Smirnov: Which Normality Test Is the Best?” published Oct. 10, 2013 at __The Minitab Blog__ web address http://blog.minitab.com/blog/the-statistical-mentor/anderson-darling-ryan-joiner-or-kolmogorov-smirnov-which-normality-test-is-the-best

[4] See the aforementioned __Answers.com__ webpage.

[5] See the comment by the user named Mikel on Jan. 23, 2008 in the__ iSixSigma__ thread “Ryan-Joiner Test” at http://www.isixsigma.com/topic/ryan-joiner-test/

[6] Gilberto, S. 2013, “Which Normality Test May I Use?” published in the__ Lean Sigma Six Group Brazil__ discussion group, at the LinkedIn web address http://www.linkedin.com/groups/Which-normality-test-may-I-3713927.S.51120536

[7] No author listed, 2014, “7.5 – Tests for Error Normality,” published at the __Penn State University__ web address https://onlinecourses.science.psu.edu/stat501/node/366 .This source has several other goodness-of-test formulas arranged in a convenient format. Also see Uaieshafizh, 2011, “Normality Test Dengan Menggunakan Uji Ryan-Joiner,” published Nov. 1, 2011 at the __Coretan Uaies Hafizh__ web address http://uaieshafizh.wordpress.com/2011/11/01/uji-ryan-joiner/ . Translated from Indonesian by Google Translate.

[8] Ryan, Jr., Thomas A. and Joiner, Brian L., 1976, “Normal Probability Plots and Tests for Normality,” Technical Report, published by the __Pennsylvania State University Statistics Department__. Available online at the Minitab web address http://www.minitab.com/uploadedFiles/Content/News/Published_Articles/normal_probability_plots.pdf

[9] For an example, see p. 2143, Yap, B. W. and Sim, C. H., 2011, “Comparisons of Various Types of Normality Tests,” pp. 2141-2155 in __Journal of Statistical Computation and Simulation__, Vol. 81, No. 12. Available online at the web address http://www.tandfonline.com/doi/pdf/10.1080/00949655.2010.520163 “Normality tests can be classified into tests based on regression and correlation (SW, Shapiro–Francia and Ryan–Joiner tests), CSQ test, empirical distribution test (such as KS, LL, AD andCVM), moment tests (skewness test, kurtosis test, D’Agostino test, JB test), spacings test (Rao’s test, Greenwood test) and other special tests.” I have yet to see the latter two tests mentioned anywhere else, so I’ll omit them from the series for now on the grounds that sufficient information will likely be even harder to find than it was for the Ryan-Joiner Test.

## Goodness-of-Fit Testing with SQL Server Part 6.1: The Shapiro-Wilk Test

**By Steve Bolton**

…………Just as a good garage mechanic will fill his or her Craftsman with tools designed to fix specific problems, it is obviously wise for data miners to stockpile a wide range of algorithms, statistical tools, software packages and the like to deal with a wide variety of user scenarios. Some of the tests and algorithms I’ve covered in this amateur self-tutorial series and the previous one on outlier detection are applicable to a broad range of problems, while others are tailor-made to address specific issues; what works in one instance may be entirely inappropriate in a different context. For example, some fitness tests are specifically applicable only to linear regression and other to logistic regression, as explained in Goodness-of-Fit Testing with SQL Server, part 4.1::R2, RMSE and Regression-Related Routines and Goodness-of-Fit Testing with SQL Server part 4.2:: The Hosmer-Lemeshow Test. Other measures we’ve surveyed recently, like the Chi-Squared, Jarque-Bera and D’Agostino-Pearson Tests, can only be applied to particular probability distributions or are calculated in ways that can be a drag on performance, when run against the wrong type of dataset. The metric I’ll be discussing this week stands out as one of the most popular goodness-of-fit tests, in large part because it is has better “statistical power,” which is a numerical measure of how often the actual effects of a variable are detected by a particular test.

…………The Shapiro-Wilk Test is also apparently flexible enough to be extended to other distribution beyond the “normal” Gaussian (i.e. the bell curve), such as the uniform, the exponential, and to a certain extent “to any symmetric distribution.”[1] Its flexibility is augmented by scale and origin invariance, two properties that statisticians prefer to endow their metrics with because multiplying the terms by a common factor or choosing a different starting point doesn’t lead to incomparable values.[2] For these reasons it is widely implemented in statistical software that competes in a tangential way with SQL Server Data Mining (SSDM), most notably “R, Stata, SPSS and SAS.”[3] As we shall see, however, there is less incentive to implement it in SQL Server than in these dedicated stats packages, because of the specific nature of the datasets we work with.

**The Fatal Flaw of Shapiro-Wilk for Big Data**

The usefulness of the Shapiro-Wilk Test is severely constrained by a number of drawbacks, such as sensitivity to outliers and the fact that its authors envisioned it as an adjunct to the kind of visualizations we covered in Goodness-of-Fit Testing with SQL Server, part 2: Implementing Probability Plots in Reporting Services, not as a replacement for them.[4] The fatal flaw, however, is that the Shapiro-Wilk Test can only handle datasets up to 50 rows in size; approximations have been developed by statisticians like Patrick Royston that can extend it to at least 2,000 rows, but that is still a drop in the bucket compared to the millions of rows found in SQL Server tables. As I’ve pointed out in previous articles, one of the great strengths of the “Big Data” era is that we can now plumb the depths of such huge treasures troves in order to derive information of greater detail, which is an advantage we shouldn’t have to sacrifice merely to accommodate metrics that were designed generations ago with entirely different contexts in mind. Furthermore, the test is normally used in hypothesis testing on random samples when the means and variances are unknown, which as I have explained in the past, are not user scenarios that the SQL Server community will encounter often.[5] The means and variances of particular columns are trivial to calculate with built-in T-SQL functions. Moreover, random sampling is not as necessary in our field because we have access to such huge repositories of information, which are often equivalent to the full population, depending on what questions we choose to ask about our data.

…………I’ll have to implement the T-SQL code for this article against a small sample of our available practice data, simply because of the built-in limitation on row counts. In order to accommodate larger datasets, we’d have to find a different way of performing the internal calculations, which are subject to combinatorial explosion. The main sticking point it a constant in the Shapiro-Wilk equations which must be derived through covariance matrices, which are too large to calculate for large datasets, regardless of the performance costs. As Royston notes, deriving the constant for a 1,500-row table would require the storage of 1,126,500 reals, given that the covariance matrix requires a number of comparisons equivalent to the count of the table multiplied by one less than itself. That exponential growth isn’t ameliorated much by the fact that those results are then halved; I’m still learning the subject of computational complexity classes so I can’t identify which this calculation might fit into, but it certainly isn’t one of those that are easily computable in polynomial time.

**Workarounds for Combinatorial Explosion**

My math may be off, but I calculated that stress-testing the Shapiro-Wilk procedure against the first float column in the 11-million-row Higgs Boson Dataset, which I downloaded from the University of California at Irvine’s Machine Learning Repository and converted into a SQL Server table of about 6 gigabytes) would require about 1.2 trillion float values and 67 terabytes of storage space. I have the sneaking suspicion that no one in the SQL Server community has that much free space in their TempDB. And that is before factoring in such further performance hits as the matrix inversion and other such transforms.

…………While writing a recent article on Mahalanobis Distance, combinatorial explosion of matrix determinants forced me to scrap my sample code for a type of covariance matrix that compared the global variance values for each column against one another; even that was a cheap workaround for calculating what amounts to a cross product against each set of local values. In this case, we’re only talking about a bivariate comparison, so inserting the easily calculable global variance value would leave us with a covariance matrix of just one entry, which isn’t going to fly.[6] We can’t fudge the covariance matrix in this way, but it might be possible to use one of Royston’s approximations to derive that pesky constant in a more efficient way. Alas, I was only able to read a couple of pages in his 1991 academic journal article on the topic, since Springer.com charges an arm and a leg for full access. I had the distinct sense, however, that it would still not scale to the size of datasets typically associated with the Big Data buzzword. Furthermore, a lot of it was still over my head, as was the original 1965 paper by Samuel S. Shapiro and Martin B. Wilk (although not as far as such topics used to be, which is precisely why I am using exercises like these in order to acquire the skills I lack). Thankfully, that article in *Biometrika* provides an easily adaptable table of lookup values for that constant[7], as well as a legible example that I was able to verify my results against. Figure 1 below provides DDL for creating a lookup table to hold those values, which you’ll have to copy yourself from one of the many publicly available sources on the Internet, including the original paper.[8]

** Figure 1: DDL for the Shapiro-Wilk Lookup Table**CREATE TABLE [Calculations].[ShapiroWilkLookupTable](

[ID] [smallint] IDENTITY(1,1) NOT NULL,

[ICount] bigint NULL,

[NCount] bigint NULL,

[Coefficient] [decimal](5, 4) NULL,

CONSTRAINT [PK_ShapiroWilkLookupTable] PRIMARY KEY CLUSTERED ([ID] ASC)WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE

= OFF, IGNORE_DUP_KEY = OFF, ALLOW_ROW_LOCKS = ON, ALLOW_PAGE_LOCKS = ON) ON [PRIMARY]

) ON [PRIMARY]

** Figure 2: T-SQL Code for the Shapiro-Wilk Test**CREATE PROCEDURE [Calculations].[GoodnessOfFitShapiroWilkTestSP]

@DatabaseName as nvarchar(128) = NULL, @SchemaName as nvarchar(128), @TableName as nvarchar(128),@ColumnName AS nvarchar(128),@DecimalPrecision AS nvarchar(50)

AS

DECLARE @SchemaAndTableName nvarchar(400),@SQLString nvarchar(max)

SET @SchemaAndTableName = @DatabaseName + ‘.’ + @SchemaName + ‘.’ + @TableName

DECLARE @ValueTable table

(ID bigint IDENTITY (1,1),

Value float)

SET @SQLString = ‘SELECT ‘ + @Column1 + ‘ AS Value

FROM ‘ + @SchemaAndTable1 + ‘

WHERE ‘ + @Column1 + ‘ IS NOT NULL’

INSERT INTO @ValueTable

(Value)

EXEC (@SQLString)

DECLARE @Count bigint,

@CountPlusOneQuarter decimal(38,2),

@CountIsOdd bit = 0,

@CountDivisor float,

@S2 float,

@ShapiroWilkTestStatistic float,

@One float = 1

SELECT @Count = Count(*)

FROM @ValueTable

SELECT @CountPlusOneQuarter = @Count + 0.25

SELECT @CountIsOdd = CASE WHEN @Count % 2 = 1 THEN 1 ELSE 0 END

SELECT @CountDivisor = CASE WHEN @CountIsOdd = 1 THEN (@Count / CAST(2 as float)) + 1 ELSE (@Count / CAST(2 as float)) END

SELECT TOP 1 @S2 = Sum(Power(Value, 2)) OVER (ORDER BY Value) – (Power(Sum(Value) OVER (ORDER BY Value), 2) * (@One / CAST(@Count as float)))

FROM @ValueTable

ORDER BY Value DESC

SELECT @ShapiroWilkTestStatistic = Power(CoefficientSum, 2) / @S2

FROM (SELECT TOP 1 SUM(FactorByShapiroWilkLookup * Coefficient) OVER (ORDER BY Coefficient DESC) AS CoefficientSum

FROM (SELECT T1.RN AS RN, T2.Value – T1.Value AS FactorByShapiroWilkLookup

FROM (SELECT TOP 99999999999 Value, ROW_NUMBER () OVER (ORDER BY Value ASC) AS RN

FROM @ValueTable

WHERE Value IS NOT NULL

ORDER BY Value ASC) AS T1

INNER JOIN (SELECT TOP 99999999999 Value, ROW_NUMBER () OVER (ORDER BY Value DESC) AS RN

FROM @ValueTable

WHERE Value IS NOT NULL

ORDER BY Value DESC) AS T2

ON T1.RN = T2.RN

WHERE T1.RN <= @CountDivisor) AS T3

INNER JOIN OutlierDetection.LookupShapiroWilkTable

ON RN = ICount AND NCount = @Count

ORDER BY RN DESC) AS T4

SELECT @ShapiroWilkTestStatistic AS ShapiroWilkTestStatistic

…………The use of the lookup table removes the need for the complex matrix logic, which might have made the T-SQL in Figure 2 even longer than the matrix code I originally wrote for Outlier Detection with SQL Server, part 8: A T-SQL Hack for Mahalanobis Distance (which might have set a record for the lengthiest T-SQL samples ever posted in a blog, if I hadn’t found a workaround at the last minute). Longtime readers may notice a big change in the format of my SQL; gone is the @DecimalPrecision parameter, which enabled users to set their own precision and scale, but which made the code a lot less legible by requiring much bigger blocks of dynamic SQL. From now on, I’ll be using short dynamic SQL statements like the one included in @SQLString and performing a lot of the math operations on a table variable that holds the results. I ought to have done this sooner, but one of the disadvantages of working in isolation is that you’re missing the feedback that would ferret out bad coding habits more quickly. As usual, the parameters and first couple of lines within the body enable users to perform the test on any table column in any database they have sufficient access to.

…………Most of the internal variables and constants we’ll need for our computations are declared near the top, followed by the some simple assignments of values based on the record count. The @S2 assignment requires a little more code. It is then employed in a simple division operation in the last block, which is a series of subqueries and windowing operations that retrieve the appropriate lookup value, which depends on the record count. It also sorts the dataset by value, then derives order statistics by essentially folding the table in half, so that the first and last values are compared, the second from the beginning and second from the end, etc. etc. right up to the midpoint. The final calculations on the lookup values and these order statistics are actually quite simple. For this part, I also consulted the National Institute for Standards and Technology’s Engineering Statistics Handbook, which is one of the most succinctly written sources of information I’ve found to date on the topic of statistics.[9] Because I’m still a novice, the reasons why these particular calculations are used is still a mystery to me, although I’ve frequently seen Shapiro and Wilk mentioned in connection with Analysis of Variance (ANOVA), which is a simpler topic to grasp if not to implement. If a float would do in place of variable precision then this code could be simplified, by inserting the results of a query on the @SchemaAndTableName into a table variable and then performing all the math on it outside of the dynamic SQL block.

** Figure 3: Sample Results from the Shapiro-Wilk Test**EXEC Calculations.GoodnessOfFitShapiroWilkTestSP

@DatabaseName = N’DataMiningProjects’,

@SchemaName = N’Health’,

@TableName = N’First50RowsPyruvateKinaseView’,

@ColumnName = N’PyruvateKinase‘

…………In Figure 3, I ran the procedure against a view created on the first 50 non-null values of the Pyruvate Kinase enzyme, derived from the 209-row table of Duchennes muscular dystrophy data I downloaded from the Vanderbilt University’s Department of Biostatistics. Given that we can’t yet calculate this on more than 50 rows at this point, doing the traditional performance test of the procedure on the HiggsBosonTable is basically pointless. Only if the lookup table could be extended somehow with new coefficients it might pay to look at the execution plan. When run against the trivial 7-row example in the Shapiro-Wilk paper, it had a couple of Clustered Index Scans that could probably be turned into Seeks with proper indexing on both the lookup table and the table being tested. It also had a couple of expensive Sort operators and a Hash Match that might warrant further inspection if the procedure could somehow be extended to datasets big enough to affect performance.

…………Interpretation of the final test statistics is straightforward in one sense, yet tricky in another. The closer the statistic is to 1, the more closely the data approaches a normal distribution. It is common to assign confidence intervals, P-values and the like with the Shapiro-Wilk Test, but I am omitting this step out of growing concern about the applicability of hypothesis testing to our use cases. I’ve often questioned the wisdom of reducing high-precision test statistics down to simple Boolean, yes-no answers about whether a particular column is normally distributed, or a particular value is an outlier; not only is it akin to taking a float column in a table and casting it to a bit, but it prevents us from asking more sophisticated questions of our hard-won computations like, “*How* normally distributed is my data?”

**More Misgivings About Hypothesis Testing-Style Metrics**

The more I read by professional statisticians and data miners who really know what they’re talking about, the less at ease I feel. Doubts about the utility of hypothesis tests of normality are routinely expressed in the literature; for some easily accessible examples that pertain directly to today’s metric, see the StackOverflow threads “Seeing if Data is Normally Distributed in R”[10] and “Perform a Shapiro-Wilk Normality Test”.[11] Some of the books I’ve read recently in my crash course in stats have not just echoed the same sentiments, but added dozens of different potential pitfalls in interpretation.[12] Hypothesis testing encompasses a set of techniques that are routinely wielded without the precision and skill required to derive useful information from them, as many professional statisticians lament. Worse still, the inherent difficulties are greatly magnified by Big Data, which comes with a unique set of use cases. The SQL Server user community might find bona fide niches for applying hypothesis testing, but for the foreseeable future I’ll forego that step and simply use the test statistics as measures in their own right, which still gives end users the freedom to implement confidence intervals and the like if they find a need.

…………The Shapiro-Wilk Test in its current form is likewise not as likely to be as useful to us as it is to researchers in other fields, in large part because of the severe limitations on input sizes. As a rule, DBAs and data miners are going to be more interested in exploratory data mining rather than hypothesis testing, using very large datasets where the means and variances are often easily discernible and sampling is less necessary. Perhaps the Shapiro-Wilk Test could be adapted to accommodate much larger datasets, as Royston apparently attempted to do by using quintic regression coefficients to approximate that constant the Shapiro-Wilk equations depend upon.[13] In fact, given that I’m still learning about the field of statistics, it is entirely possible that a better workaround is already available. I’ve already toyed with the idea of breaking up entire datasets into random samples of no more than 50 rows, but I’m not qualified to say if averaging the test statistics together would be a logically valid measure. I suspect that the measure would be incorrectly scaled because of the higher record counts.

…………Until some kind of enhancement becomes available, it is unlikely that the Shapiro-Wilk Test will occupy a prominent place in any DBA’s fitness testing toolbox. There might be niches where small random sampling and hypothesis testing might make it a good choice, but for now it is simply too small to accommodate the sheer size of the data we’re working with. I looked into another potential workaround in the form of the Shapiro-Francia Test, but since it is calculated in a similar way and is “asymptotically equivalent”[14] to the Shapiro-Wilk (i.e., they basically converge and become equal for all intents and purposes) I chose to skip that alternative for the time being. In next week’s article we’ll instead discuss the Ryan-Joiner Test, which is often lumped in the same category with the Shapiro-Wilk. After that, we’ll survey a set of loosely related techniques that are likely to be of more use to the SQL Server community, encompassing the Kolmogorov-Smirnov, Anderson-Darling, Kuiper’s and Lilliefors Tests, as well as the Cramér–von Mises Criterion

[1] Royston, Patrick, 1991, “Approximating the Shapiro-Wilk W-Test for Non-Normality,” pp. 117-119 in __Statistics and Computing__, September, 1992. Vol. 2, No. 3. Available online at http://link.springer.com/article/10.1007/BF01891203#page-1

[2] p. 591, Shapiro, Samuel S. and Wilk, Martin B., 1965, “An Analysis of Variance Test for Normality (Complete Samples),” pp. 591-611 in __Biometrika__, December 1965. Vol. 52, Nos. 3-4.

[3] See the Wikipedia page “Shapiro-Wilk Test” at http://en.wikipedia.org/wiki/Shapiro%E2%80%93Wilk_test

[4] p. 610, Shapiro and Wilk, 1965.

[5] p. 593, Shapiro, Samuel S. and Wilk, Martin B., 1965, “An Analysis of Variance Test for Normality (Complete Samples),” pp. 591-611 in __Biometrika__, December 1965. Vol. 52, Nos. 3-4.

[6] Apparently there is another competing definition of the term, in which values are compared within a particular column, not against across columns. See the Wikipedia page “Covariance Matrix” at http://en.wikipedia.org/wiki/Covariance_matrix#Conflicting_nomenclatures_and_notations

[7] pp. 603-604, Shapiro, Samuel S. and Wilk, Martin B., 1965, “An Analysis of Variance Test for Normality (Complete Samples),” pp. 591-611 in __Biometrika__, December 1965. Vol. 52, Nos. 3-4.

[8] Another source of the Shapiro-Wilk coefficient is Zaiontz, Charles, 2014, “Shapiro-Wilk Tables,” posted at the Real Statistics Using Excel blog web address http://www.real-statistics.com/statistics-tables/shapiro-wilk-table/

[9] For this part, I also consulted the National Institute for Standards and Technology, 2014, “7.2.1.3 Anderson-Darling and Shapiro-Wilk Tests,” published in the online edition of the __Engineering Statistics Handbook__. Available at http://www.itl.nist.gov/div898/handbook/prc/section2/prc213.htm

[10] See especially the comment by Ian Fellows on Oct. 17, 2011:

“Normality tests don’t do what most think they do. Shapiro’s test, Anderson Darling, and others are null hypothesis tests AGAINST the assumption of normality. These should not be used to determine whether to use normal theory statistical procedures. In fact they are of virtually no value to the data analyst. Under what conditions are we interested in rejecting the null hypothesis that the data are normally distributed? I have never come across a situation where a normal test is the right thing to do. When the sample size is small, even big departures from normality are not detected, and when your sample size is large, even the smallest deviation from normality will lead to a rejected null…”

…………“…So, in both these cases (binomial and lognormal variates) the p-value is > 0.05 causing a failure to reject the null (that the data are normal). Does this mean we are to conclude that the data are normal? (hint: the answer is no). Failure to reject is not the same thing as accepting. This is hypothesis testing 101.”

…………“But what about larger sample sizes? Let’s take the case where there the distribution is very nearly normal.”

…………“Here we are using a t-distribution with 200 degrees of freedom. The qq-plot shows the distribution is closer to normal than any distribution you are likely to see in the real world, but the test rejects normality with a very high degree of confidence.”

…………“Does the significant test against normality mean that we should not use normal theory statistics in this case? (another hint: the answer is no )”

[11] Note these helpful comments by Paul Hiemstra on March 15, 2013:

“An additional issue with the Shapiro-Wilks test is that when you feed it more data, the chances of the null hypothesis being rejected becomes larger. So what happens is that for large amounts of data even veeeery small deviations from normality can be detected, leading to rejection of the null hypothesis even though for practical purposes the data is more than normal enough…”

…………“…In practice, if an analysis assumes normality, e.g. lm, I would not do this Shapiro-Wilks test, but do the analysis and look at diagnostic plots of the outcome of the analysis to judge whether any assumptions of the analysis where violated too much. For linear regression using lm this is done by looking at some of the diagnostic plots you get using plot (lm()). Statistics is not a series of steps that cough up a few numbers (hey p < 0.05!) but requires a lot of experience and skill in judging how to analysis your data correctly.”

[12] A case in point with an entire chapter devoted to the shortcomings of hypothesis testing methods is Kault, David, 2003, __Statistics with Common Sense__. Greenwood Press: Westport, Connecticut.

[13] His approximation method is also based on Weisberg, Sanford and Bingham, Christopher, 1975, “An Approximate Analysis of Variance Test for Non-Normality Suitable for Machine Calculation,” pp 133-134 in __Technometrics__, Vol. 17.

[14] p. 117, Royston.

## Goodness-of-Fit Testing with SQL Server Part 5: The Chi-Squared Test

**By Steve Bolton**

…………As I’ve cautioned before, I’m writing this series of amateur self-tutorials in order to learn how to use SQL Server to perform goodness-of-fit testing on probability distributions and regression lines, not because I already know the topic well. Along the way, one of the things I’ve absorbed is that the use cases for the various goodness-of-fit tests are more sharply delineated than the statistical tests for the outlier detection, which was the topic of my last tutorial series. I covered some of the more general measures in Goodness-of-Fit Testing with SQL Server, part 1: The Simplest Methods, but even some of these – like the 3-Sigma Rule – are limited only to the Gaussian or “normal” distribution, i.e. the bell curve. Many of the other metrics we’ll survey later in this series are likewise limited to specific data types, such as the popular Kolmogorov-Smirnov and Anderson-Darling Tests, which cannot be applied to nominal data (i.e. corresponding to the Discrete Content type in SQL Server Data Mining).[1] For that task, you need a metric like the Chi-Squared Test (or 𝜒²), which can handle nominal data types as well as continuous ones, which are measured in infinitesimal gradations; cases in point would include the decimal and float types in SQL Server.

…………In addition to the bell curve, the 𝜒²can handle such other popular distributions as the Poisson, log normal, Weibull, exponential, binomial and logistic, plus any others that have an associated cumulative distribution function (CDF).[2] Ralph B. D’Agostino, one of the inventors of the metric we discussed in Goodness-of-Fit Testing with SQL Server, part 3.2: D’Agostino’s K-Squared Test, cautions though that analyses of the 𝜒² Test indicate this flexibility comes at the cost of decreased statistical power; as he and some of his colleagues put it in a 1990 academic paper, “The extensive power studies just mentioned have also demonstrated convincingly that the old warhorses, the chi-squared test and the Kolmogorov test (1933), have poor power properties and should not be used when testing for normality.”[3] Some experts consider this flaw to be almost fatal, to the point where one writes, “If you want to test normality, a chi-squared test is a really bad way to do it. Why not, say, a Shapiro-Francia test or say an Anderson-Darling adjusted for estimation? You’ll have far more power.”[4] As we shall see in a few weeks, the Anderson-Darling Test has other limitations beyond its inability to handle nominal columns, whereas I believe the Shapiro-Francia Test is based on the Shapiro-Wilk, which is computationally expensive and limited to what the SQL Server community would consider very small sample sizes. Each test has its own unique set of strengths and weaknesses, which ought to strongly influence a data miner’s choices.

**More “Gotchas” with the ****𝜒****² Test (and Its Inventor)**

A further caveat of the 𝜒² Test is that the population ought to be ten times more numerous than the sample[5], but one of the strengths of Big Data-era analysis is that we can use modern set-based methods to traverse gigantic datasets, rather than taking dinky slices of the kind normally seen in hypothesis testing. As discussed over the course of the last two tutorial series, I’m shying away from the whole field of hypothesis testing because it is not well-suited to our use cases, which may involve hundreds of millions of rows that might represent a full population rather than 50 or 100 from a sample that rarely does; furthermore, the act of applying the usual confidence and significance levels and the like reduces such tests down to a simple Boolean, yes-no answer. This represents a substantial reduction in the information provided by the test statistics, akin to truncating a float or decimal column down to a SQL Server bit data type; by retaining the full statistic, we can measure *how* normal or exponential or uniform a particular dataset may be.[6]

That is why in the last article, I skipped the usual step of plugging the Hosmer-Lemeshow Test results into a 𝜒² Test, to derive confidence levels and the like based on how well they approximate a 𝜒² distribution.[7] In fact, such comparisons to the 𝜒² distribution seem to be as common in hypothesis testing as those to Student’s T-distribution, or the F-distribution in the case of Analysis of Variance (ANOVA). Further adding to the confusion is the fact that there is also a 𝜒² Test of Independence, in which contingency tables are used to establish relationships between multiple variables. There is some overlap in the underlying concepts, but the two 𝜒² Tests are not identical.[8] The goodness-of-fit version we’re speaking of here was the developed by Karl Pearson, one of the most brilliant statisticians of the 19^{th} Century – but also one of the most twisted. As I’ve pointed out several times since beginning my series on A Rickety Stairway to SQL Server Data Mining, ordinary mathematicians might be stable people, but a frightening share of the rare geniuses among them have been not just eccentric, but Norman Bates-level crazy. Pearson was a blatant racist and Social Darwinist who sought to extinguish the “unfit” through such schemes as eugenics[9], and thereby helped feed the intellectual current in Europe that eventually brought Hitler to power. We can still use his statistical tests, just as we use the rockets devised by Werner von Braun and the quantum mechanics pioneered by Werner Heisenberg – provided that we put them to better purposes.

**Deriving the ****𝜒****² from CDFs**

You don’t have to be a proverbial rocket scientist in order to calculate the 𝜒² Test, nor do you need to do the kind of mental gymnastics required for the Heisenberg Uncertainty Principle. The equation is actually quite simple, especially since it follows a form similar to that of many of the other test statistics surveyed in the last two tutorial series. Like Z-Scores and so many other metrics, the 𝜒² Test involves subtracting one value from another for each row, squaring the result and then summing them across the entire dataset, all of which can be easily implemented with T-SQL windowing functions. The difference is that in this case, we’re putting the data in ascending order, then subtracting probabilities generated by the CDF of the distribution we’re testing from the actual value.

Some CDFs are trivial to calculate, but as I mentioned in Goodness-of-Fit Testing with SQL Server, part 2: Implementing Probability Plots in Reporting Services, I had a hell of a time deriving the correct values for the normal distribution’s CDF – as do many novices, in large part because there is no closed-form solution to that particular formula. Rather than rehash that whole topic of how to use approximations to derive the normal CDF, I’ll simply reuse most of the code from that article to implement that part of the 𝜒² equation. I had to tweak a little so that I could calculate only the handful of CDF values we actually need, rather than every single probability in a defined range; this called for passing it a table parameter of the type shown below, which is populated in the middle of Figure 1. Keep in mind that this Gaussian CDF is based on the simplest approximation I could find, so once you get about five or six places right of the decimal point, some inaccuracy creeps in, which might be magnified in certain cases by the use of float rather than decimal in the type definition.

__Figure 1: DDL for the ____𝜒__** ²Test**CREATE TYPE [Calculations].[SimpleFloatValueTableParameter] AS TABLE(

[RN] [bigint] NULL,

[Value] float(53) NULL)

CREATE PROCEDURE [Calculations].[NormalDistributionCDFSupplyTableParameterSP]

@Mean decimal(38,21), @StDev decimal(38,21), @InputTableParameter AS [Calculations].[SimpleFloatValueTableParameter] READONLY

AS

DECLARE @StDevTimesSquareRootOf2 as decimal(38,21), @One as decimal(38,37) = 1, @Two as decimal(38,37) = 2, @EulersConstant decimal(38,37) = 2.7182818284590452353602874713526624977

SELECT @StDevTimesSquareRootOf2 = @StDev * Power(@Two, 0.5)

SELECT ColumnValue, CASE WHEN ColumnValue >= @Mean THEN CDFValue ELSE 1 – CDFValue END AS CDFValue

FROM (SELECT Value AS ColumnValue, 0.5 + (0.5 * Power(@One – Power(@EulersConstant, ((-0.147 * Power(ErrorFunctionInput, 4)) – (1.27324 * Power(ErrorFunctionInput, 2))) / (@One + (0.147 * Power(ErrorFunctionInput, 2)))), 0.5)) AS CDFValue

FROM ( SELECT Value, (Value – @Mean) / @StDevTimesSquareRootOf2 AS ErrorFunctionInput

FROM @InputTableParameter

WHERE Value IS NOT NULL) AS T1) AS T2

…………As annoying as it might be to create these extra objects just to run the procedure in Figure 2, it saves us from having to calculate zillions of CDF values on large tables, when we only need the minimum and maximum values for each band. The 𝜒² Test is applied to distributions rather than regression lines as the Hosmer-Lemeshow Test is, but they have at least one thing in common: the division of the dataset into probability bands, which are then graded on how close the expected values match the actual observations. The criteria for membership in these bands is up to you, but in my implementation, I’m simply using the NTILE windowing function to break up a dataset into subsets of almost equal size, in order of the values of the column being tested. Several sources caution that the type of banding can have a strong effect on the final test statistic. As the National Institute for Standards and Technology’s Engineering Statistics Handbook (one of the best online resources for anyone learning statistics) puts it, “This test is sensitive to the choice of bins. There is no optimal choice for the bin width (since the optimal bin width depends on the distribution). Most reasonable choices should produce similar, but not identical, results… The chi-square goodness-of-fit test is applied to binned data (i.e., data put into classes). This is actually not a restriction since for non-binned data you can simply calculate a histogram or frequency table before generating the chi-square test. However, the value of the chi-square test statistic are dependent on how the data is binned.”[10]

They’re not kidding, as I observed first-hand. I set the @NumberOfBands parameter to a default of 10, but you’re probably going to want to run several trials and experiment with higher and lower values, especially when it’s calculated against large tables, because it can dramatically affect the test statistic. Many sources mention that the count of records in each bucket ought to be more than 5, so you don’t want to set the @NumberOfBands so high that the bucket size falls below this threshold. I found it helpful to look at the output of the @FrequencyTable to make sure there weren’t too many bands with identical bounds, which will happen if the @NumberOfBounds is too high. Use some common sense: if you’re operating on nominals that can only be assigned integer values between 0 and 5, then a bin count of 6 might be a wise starting point.

**An Explanation of the Sample Code**

Most of the rest of the code is self-explanatory, to those who have slogged their way through one of my procedures before. As usual, the first four parameters allow you to run it against any numerical column in any database you have adequate access to, while the first couple of lines in the body help implement this. The rest is all dynamic SQL, beginning with the usual declaration sections and assignments of the aggregate values we’ll need for other calculations. After that I declare a couple of table variables and a table parameter to hold the final results as well as some intermediate steps. Most of the work occurs in the first INSERT, which divides the dataset into bands; a few statements later, the distinct minimum and maximum values that are inserted in this step are fed to the CDF procedure to derive probability values. Note that it can be drastically simplified if the flexible variable ranges that the @DecimalPrecision parameter implements are not needed; in that case, simply return the results from @SchemaAndTableName into a table variable and perform all the math on it outside the dynamic SQL block.

…………If you receive NULL values for your CDFs in the final results, it’s a clue that you probably need to try a @DecimalPrecision parameter (which I normally provide to help end users avoid arithmetic overflows) with a smaller scale; it signifies that the procedure can’t match values in the joins properly due to rounding somewhere. For a distribution other than the normal, simply plug in a different CDF and adjust the degrees of freedom to account for additional parameters, such as the shape parameter used in the Weibull. There might be a more efficient way to do the updates to the @FrequencyTable that follow, but the costs of these statements compared to the rest of the batch are inconsequential, plus the procedure is easier to follow this way. The two cumulative frequency counts are provided just as a convenience and can be safely eliminated if you don’t need them. After that, I return the full @FrequencyTable to the user (since the costs of calculating it have already been incurred) and compute the final test statistic in a single line in the last SELECT.

As mentioned in previous articles, many of these older tests were not designed for datasets of the size found in modern relational databases and data warehouse, so there are no checks built in to keep the final test statistic from being grossly inflated by the accumulation of millions of values. For that reason, I’m using a variant known as “Reduced 𝜒²” that simply divides by the count of records to scale the results back down to a user-friendly, easily readable stat. Note that in previous articles, I misidentified Euler’s Number in my variable names as Euler’s Constant, for obvious reasons. Adding to the confusion is the fact that the former is sometimes also known as Napier’s Constant or the Exponential Constant, while the latter is also referred to as the Euler-Mascheroni Constant, which I originally thought to be distinct from Euler’s Constant. I used the correct constant and high-precision value for it, but applied the wrong name in my variable declarations.

__Figure 2: T-SQL for the ____𝜒__** ² Goodness-of-Fit Test**CREATE PROCEDURE [Calculations].[GoodnessOfFitChiSquaredTestSP]

@DatabaseName as nvarchar(128) = NULL, @SchemaName as nvarchar(128), @TableName as nvarchar(128),@ColumnName AS nvarchar(128), @DecimalPrecision AS nvarchar(50), @NumberOfBands as bigint = 10

AS

DECLARE @SchemaAndTableName nvarchar(400),@SQLString nvarchar(max)

SET @SchemaAndTableName = @DatabaseName + ‘.’ + @SchemaName + ‘.’ + @TableName

SET @SQLString = ‘DECLARE @Mean decimal(‘ + @DecimalPrecision + ‘), @StDev decimal(‘ + @DecimalPrecision + ‘), @Count decimal(‘ + @DecimalPrecision + ‘),

@EulersNumber decimal(38,37) = 2.7182818284590452353602874713526624977

SELECT @Count=Count(CAST(‘ + @ColumnName + ‘ AS Decimal(‘ + @DecimalPrecision + ‘))), @Mean = Avg(CAST(‘ + @ColumnName + ‘ AS Decimal(‘ + @DecimalPrecision + ‘))), @StDev = StDev(CAST(‘ + @ColumnName + ‘ AS Decimal(‘ + @DecimalPrecision + ‘)))

FROM ‘ + @SchemaAndTableName + ‘ WHERE ‘ + @ColumnName + ‘ IS NOT NULL

DECLARE @CDFTable table

(ID bigint IDENTITY (1,1),

Value decimal(‘ + @DecimalPrecision + ‘),

CDFValue decimal(‘ + @DecimalPrecision + ‘))

DECLARE @FrequencyTable table

(ID bigint,

MinValue decimal(‘ + @DecimalPrecision + ‘),

MaxValue decimal(‘ + @DecimalPrecision + ‘),

LowerCDFValue decimal(‘ + @DecimalPrecision + ‘),

UpperCDFValue decimal(‘ + @DecimalPrecision + ‘),

ActualFrequencyCount bigint,

ExpectedFrequencyCount decimal(‘ + @DecimalPrecision + ‘),

CumulativeActualFrequencyCount decimal(‘ + @DecimalPrecision + ‘),

CumulativeExpectedFrequencyCount decimal(‘ + @DecimalPrecision + ‘)

)

INSERT INTO @FrequencyTable

(ID, MinValue, MaxValue, ActualFrequencyCount)

SELECT DISTINCT BandNumber, Min(CAST(‘ + @ColumnName + ‘ AS decimal(‘ + @DecimalPrecision + ‘))) OVER (PARTITION BY BandNumber ORDER BY ‘ + @ColumnName + ‘) AS BandMin,

Max(CAST(‘ + @ColumnName + ‘ AS decimal(‘ + @DecimalPrecision + ‘))) OVER (PARTITION BY BandNumber ORDER BY ‘ + @ColumnName + ‘ DESC) AS BandMax, — note the DESC to go in the opposite order

Count(*) OVER (PARTITION BY BandNumber) AS BandFrequencyCount

FROM (SELECT ‘ + @ColumnName + ‘, NTile(‘ + CAST(@NumberOfBands as nvarchar(128)) + ‘) OVER (ORDER BY ‘ + @ColumnName + ‘) AS BandNumber

FROM ‘ + @SchemaAndTableName + ‘

WHERE ‘ + @ColumnName + ‘ IS NOT NULL) AS T1

DECLARE @InputTableParameter AS Calculations.SimpleFloatValueTableParameter

INSERT INTO @InputTableParameter

(Value)

SELECT DISTINCT Value FROM (SELECT MinValue AS Value

FROM @FrequencyTable

UNION

SELECT MaxValue AS Value

FROM @FrequencyTable)

AS T1

INSERT INTO @CDFTable

(Value, CDFValue)

EXEC Calculations.NormalDistributionCDFSupplyTableParameterSP @Mean, @StDev, @InputTableParameter

UPDATE T1

SET LowerCDFValue = T2.CDFValue

FROM @FrequencyTable AS T1

INNER JOIN @CDFTable AS T2

ON T1.MinValue = Value

UPDATE T1

SET UpperCDFValue = T2.CDFValue

FROM @FrequencyTable AS T1

INNER JOIN @CDFTable AS T2

ON T1.MaxValue = T2.Value

UPDATE @FrequencyTable

SET ExpectedFrequencyCount = (UpperCDFValue – LowerCDFValue) * @Count

— the Cumulatives are just for convenience and can be safely eliminated from the table if you don’t need them

UPDATE T1

SET T1.CumulativeActualFrequencyCount = T2.CumulativeActualFrequencyCount,

T1.CumulativeExpectedFrequencyCount = T2.CumulativeExpectedFrequencyCount

FROM @FrequencyTable AS T1

INNER JOIN (SELECT ID, Sum(ActualFrequencyCount) OVER (ORDER BY ID) AS CumulativeActualFrequencyCount, Sum(ExpectedFrequencyCount)

OVER (ORDER BY ID) AS CumulativeExpectedFrequencyCount

FROM @FrequencyTable) AS T2

ON T1.ID = T2.ID

— return all of the results

SELECT ID, MinValue, MaxValue, LowerCDFValue, UpperCDFValue, ActualFrequencyCount, ExpectedFrequencyCount, CumulativeActualFrequencyCount, CumulativeExpectedFrequencyCount

FROM @FrequencyTable

ORDER BY ID

— this is an alternate version of the test called “reduced chi squared” in which the degrees of freedom are taken into account to scale the results back down

SELECT Sum(ExpectedFrequencyCountSum) / Count(*) AS ChiSquaredTestReduced,

@Count AS FullPopulationCount, @Mean AS Mean, @StDev AS StDev

FROM (SELECT CASE WHEN ExpectedFrequencyCount = 0 THEN 0 ELSE Power(ActualFrequencyCount – ExpectedFrequencyCount, 2) / ExpectedFrequencyCount END AS ExpectedFrequencyCountSum

FROM @FrequencyTable) AS T1‘

–SELECT @SQLString — uncomment this to debug dynamic SQL errors

EXEC (@SQLString)

…………As has become standard fare over the past two tutorial series, I first tested the results against a tiny 9-kilobyte table of data on the Duchennes form of muscular dystrophy from the Vanderbilt University’s Department of Biostatistics. Then I stress-tested it against the 11 million rows in the Higgs Boson Dataset I downloaded from the University of California at Irvine’s Machine Learning Repository and converted into a nearly 6-gigabyte SQL Server table. The query in Figure 3 on the Hemopexin protein produced the first resultset below it, while the much longer resultset was the product of a similar query on the first float column in the HiggsBosonTable. An unreasonable selection of bands can also apparently affect performance; on my first trial on the HiggsBosonTable, I forgot to set the number well above 7, which may be why it took 7:26. Subsequent trials with values around 100 took between 5:46 and 5:52; the results depicted here are only for the first 22 out of 110 bands.

…………I’m not surprised that the final test statistic has six digits to the left of the decimal points, given that I know from previous outlier detection and goodness-of-fit tests that Column 1 is highly abnormal. Column 2 follows an obvious bell curve when displayed in a histogram, so it is likewise not surprising that its 𝜒² Test result was only 1,326, or less a hundredth of Column1. I have the feeling that the sheer size of the dataset can distort the final test statistic, thereby making it difficult to compare them across datasets, but probably not as severely as in other measures, particularly the Jarque-Bera and K² Tests. The query on the second float column likewise took 5:45 on my beat-up development machine – which more closely resembles the Bluesmobile than a real server, so your mileage will probably be a lot better. It’s not as quick as the procedure I wrote in Goodness-of-Fit Testing with SQL Server Part 4.1: R^{2}, RMSE and Regression-Related Routines, but certainly faster than many others I’ve done in past articles.

** Figure 3: Sample Results from the Duchennes and Higgs Boson Datasets**EXEC @return_value = [Calculations].[GoodnessOfFitChiSquaredTestSP]

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’Hemopexin‘,

@DecimalPrecision = N’38,17′,

@NumberOfBands = 7

…………The full execution plan it too large to depict here, but suffice it to say that it consists of 11 separate queries – with one of them, the insert into the @FrequencyTable, accounting for 99 percent of the computational cost of the entire batch. I’m not sure at this point how to go about optimizing that particular query, given that it starts with an Index Seek, which is normally what we want to see; there are also a couple of Nested Loops operators and a Hash Match within that query, but together they only account for about 12 percent of its internal costs. Almost all of the performance hit comes on two Sort operators, which a better-trained T-SQL aficionado might be able to dispose of with a few optimizations.

Efficiency is something I’ll sorely need for next week’s article, in which I tackle the Shapiro-Wilk Test. Many sources I’ve stumbled upon while researching this series indicate that it has better statistical power than most of the competing goodness-of-fit tests, but it has many limitations which severely crimp its usability, at least for our purposes. First, it can apparently only be calculated on just 50 values, although I’ve seen figures as high as a couple of hundred. Either way, that’s about a few hundred million rows short; the sheer sizes of datasets available to DBAs and data miners today are one of their strengths, and we shouldn’t have to sacrifice that hard-won advantage by taking Lilliputian slices of it. Worst of all, the calculations are dogged by a form of combinatorial explosion, which can be the kiss of death for Big Analysis. I have learned to fear the dreaded factorial symbol n! and the more insidious menace posed by calculations upon infinitesimal reciprocals, of the kind that afflicted the Hosmer-Lemeshow Test in last week’s article. My implementation of the Shapiro-Wilk Test will sink or swim depending on whether or not I can find a reasonable workaround for the covariance matrices, which are calculated based on a cross product of rows. In a table of a million rows, that means 1 trillion calculations just to derive an intermediary statistic. A workaround might be worthwhile, however, given the greater accuracy most sources ascribe to the Shapiro-Wilk Test.

[1] See National Institute for Standards and Technology, 2014, “1.3.5.15 Chi-Square Goodness-of-Fit Test,” published in the online edition of the Engineering Statistics Handbook. Available at http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm

[2] *IBID.*

[3] p. 316 D’Agostino, Ralph B.; Belanger, Albert and D’Agostino Jr., Ralph B, 1990, “A Suggestion for Using Powerful and Informative Tests of Normality,” pp. 316–321 in __The American Statistician__. Vol. 44, No. 4. Available online at http://www.ohio.edu/plantbio/staff/mccarthy/quantmet/D’Agostino.pdf

[4] See the reply by the user named Glen_b in the __CrossValidated__ thread “How to Get the Expected Counts When Computing a Chi-Squared Test?” dated March 14, 2013, which is available at the web address

[5] See the StatTrek webpage titled “When to Use the Chi-Square Goodness of Fit Test” at http://stattrek.com/chi-square-test/goodness-of-fit.aspx

[6] I may be a novice, but am apparently not alone in my reluctance to use tests that enforce either-or choices. See the reply by the user named Glen_b in the __CrossValidated__ thread “How to Get the Expected Counts When Computing a Chi-Squared Test?” dated March 14, 2013, which is available at the web address http://stats.stackexchange.com/questions/52209/how-to-get-the-expected-counts-when-computing-a-chi-squared-test as well as the reply by the same user to the thread “What Tests Do I Use to Confirm That Residuals are Normally Distributed?” posted Sept. 13, 2013 at the CrossValidated forum web address http://stats.stackexchange.com/questions/36212/what-tests-do-i-use-to-confirm-that-residuals-are-normally-distributed/36220#36220 He makes several very good points about goodness-of-fit testing that are worth quoting here. In the first, he says that

“No test will prove your data is normally distributed. In fact I bet that it isn’t. (Why would any distribution be exactly normal? Can you name anything that actually is?)

2) When considering the distributional form, usually, hypothesis tests answer the wrong question

What’s a good reason to use a hypothesis test for checking normality?

I can think of a few cases where it makes some sense to formally test a distribution. One common use is in testing some random number generating algorithm for generating a uniform or a normal.

In the second thread, he similarly points out that:

“1.No test will tell you your residuals are normally distributed. In fact, you can reliably bet that they are not.”

“2.Hypothesis tests are not generally a good idea as checks on your assumptions. The effect of non-normality on your inference is not generally a function of sample size*, but the result of a significance test is. A small deviation from normality will be obvious at a large sample size even though the answer to the question of actual interest (‘to what extent did this impact my inference?’) may be ‘hardly at all’. Correspondingly, a large deviation from normality at a small sample size may not approach significance…”

“…If you must use a test, Shapiro-Wilk is probably as good as anything else. (But it’s answering a question you already know the answer to – and every time you fail to reject, giving an answer you can be sure is wrong.)”

[7] Just a side note on terminology: I see both the tests and the distribution referred to as “Chi-Squared” with a final D as often as I do “Chi-Square” without one, which are sometimes mixed together in the same sources. I’ll stick with a closing D for the sake of consistency, even if it turns out to be semantically incorrect.

[8] For a readable explanation of the independence test, see Hopkins, Will G., 2001, “Contingency Table (Chi-Squared Test),” published at the __A New View of Statistics__ website address http://www.sportsci.org/resource/stats/continge.html

[9] For a quick introduction to this sordid tale, see the Wikipedia page “Karl Pearson” at http://en.wikipedia.org/wiki/Karl_Pearson

[10] See National Institute for Standards and Technology, 2014, “1.3.5.15 Chi-Square Goodness-of-Fit Test,” published in the online edition of the Engineering Statistics Handbook. Available at http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm The formula for the goodness-of-fit test is widely available, but I depended mostly on this NIST webpage when writing my code because their equation was more legible.

## Goodness-of-Fit Testing with SQL Server Part 4.2: The Hosmer–Lemeshow Test with Logistic Regression

**By Steve Bolton**

…………The last installment of this amateur series of self-tutorials was the beginning of a short detour into using SQL Server to perform goodness-of-fit testing on regression lines, rather than on probability distributions. These are actually quite simple concepts; any college freshman ought to be able to grasp the idea of a single regression line, since all they do is graph the values of one variable as it changes in tandem with another, whereas distributions merely summarize the distinct counts for each value of a variable, as the famous bell curve does. In the first case we’re dealing with how well a regression line models the relationship between two variables – assuming one exists – and in the second, we’re assessing whether the distribution of a single variable matches one of many distributions that constantly recur in nature, like the Gaussian “normal” distribution, or other common ones like the Poisson, binomial, uniform and exponential distributions.

…………These topics can of course become quite complex rather quickly once we factor in modifications like multivariate cases, but I’ll avoid these topics for the sake of simplicity, plus the fact that there are an endless number of variants of them that could keep us busy to the end of time. I really can’t end this segment though without addressing a particular variant, since logistic regression is a widely used algorithm that addresses a distinctive set of use cases. There is likewise an endless array of alternative takes on the basic logistic regression algorithm, such as adaptations for multinomial cases, but I’ll stick to the old maxim, KISS: Keep It Simple, Stupid. The Hosmer-Lemeshow Test that is commonly used for fitness testing on logistic regression may not be applicable in some of these more advanced cases, but it is indispensable for the bare bones version of the algorithm.

**Adapting Regression Lines to Use the Logistic Function**

This regression algorithm is a topic I became somewhat familiar with while writing A Rickety Stairway to SQL Server Data Mining, Algorithm 4: Logistic Regression a while back, in a different tutorial series. Suffice it to say that the algorithm is ideal for situations in which you need to place bounds on the outputs that a regression line can generate; most commonly this is a Boolean yes-no choice that ranges between 0 and 1, but it can be adapted to other scales, such as 0 to 100. A linear regression line would produce nonsensical values in these instances, since the values would be free to go off the charts, but a logistic regression is guaranteed to stay within its assigned bounds. This is accomplished by using the logistic function, which is really not much more difficult to implement than linear regression (with one critical limitation). The formula is widely available on the Internet, so when writing the code for this article I retrieved it from the most convenient source as usual: Wikipedia.[1]

…………In many ways, the logistic function behaves like the cumulative distribution functions (CDFs) I mentioned in Goodness-of-Fit Testing with SQL Server, Part 2: Implementing Probability Plots in Reporting Services, in that the probabilities assigned to the lowest value begin at 0 and accumulate up to 1 by the time we reach the final value in an ordered dataset. It also behaves in many ways like a weighted function, in that the pressure on it to conform to the bounds increases as it nears them; I think of it in terms of the way quarks are inextricably bound together by the strong force within hadrons, which increases as they come closer to breaking free. In between the upper and lower bounds the regression takes on the appearance of an S-curve rather than the lines seen in normal regression.

…………Another easily readable and insightful commentary on logistic regression can be found at University of Toronto Prof. Saed Sayad’s website[2], in which he provides a succinct explanation of the logistic function equation and some alternative measures for the accuracy of the mining models it generates. Three of these are subspecies of the R^{2} measure we discussed last week in connection with linear regression, which are discussed together under the rubric of Pseudo R^{2}. Fitness testing of this kind is as necessary on regression lines as it is for distribution matching, because they rarely model the relationships between variables perfectly; as statistician George Box (one of the pioneers of Time Series) once put it so colorfully, “All models are wrong, but some are useful.”[3] Sayad also mentions alternative methods like Likelihood Ratio tests and the Wald Test. The measure of fit I’ve seen mentioned most often in connection with logistic regression goes by the memorable moniker of the Hosmer-Lemeshow Test. It apparently has its limitations, as we shall see, but it is not terribly difficult to implement – with some important caveats.

**Banding Issues with Coding the Hosmer-Lemeshow Test**

In fact, the first three steps in the dynamic SQL in Figure 1 are almost identical to those used to calculate regression lines in last week’s article and a tutorial from a different series, Outlier Detection with SQL Server Part 7: Cook’s Distance, which I won’t waste time recapping here.[4] The fourth step just applies the logistic function to the results, in a single, simple line of T-SQL. After this, I simply insert the logistic regression values into a table variable for later retrieval, including returning it to the user towards the end of the procedure; there’s really no reason not to return the correlation, covariance, slope, intercept and standard deviations for both variables as well, given that we’ve already calculated them. Step 5 is where the meat and potatoes of the Hosmer-Lemeshow Test can be found. Its strategy is essentially to divide the values into bands, which are often referred to as “deciles of risk” when the test is employed in one of its most common applications, risk analysis.[5] The bands are then compared to the values we’d expect for them based on probabilistic calculations and the gap between them is quantified and summarized.

…………It is now time for my usual disclaimer: I am writing this series in order to familiarize myself with these data mining tools, not because I have expertise in using them, so it would be wise to check my code thoroughly before putting it to use (or even, God forbid, a production environment). Normally I check my code against the examples provided in various sources, especially the original academic papers whenever possible. In this case, however, I couldn’t find any at a juncture where they would have come in handy, given that I am not quite certain that I am splitting the data into bands on the right axis. I am still learning how to decipher the equations that underpin algorithms of this kind and some of them differ significantly in notation and nomenclature, so it may be that I ought to be counting the observed and expected values differently, which affects how they are split; from the wording in a book inventors David W. Hosmer Jr. and Stanley Lemeshow wrote on *Applied Logistic Regression*, it seems that the counts between the bands are supposed to vary much more significantly[6] than they do in my version, which could be a side effect of incorrect banding. There are apparently many different ways of doing banding[7], however, including my method below, in which the @BandCount parameter is plugged into the NTILE windowing function in Step 1.

…………I’ve seen two general rules of thumb mentioned in the literature for setting the @BandCount to an optimal level: using groups of fewer than five members leads to incorrect results, while using fewer than six groups “almost always” leads to passing the fitness test.[8] Averages for both the X and Y axes are calculated for each band, then the regression line is derived through the usual methods in Steps 3 through 5, with one crucial difference: some of the aggregates have to be derived from the bands rather than the original table, otherwise we’d end up with an apples vs. oranges comparison. This is one of several points where the procedure can go wrong, since the banding obscures the detail of the original table and can lead to a substantially different regression line. Keep in mind though that the literature mentions several alternative methods of banding, so there may be better ways of accomplishing this.

** Figure 1: T-SQL Code for the Hosmer–Lemeshow Test Procedure**CREATE PROCEDURE [Calculations].[GoodnessOfFitLogisticRegressionHosmerLemsehowTestSP]

@DatabaseName as nvarchar(128) = NULL, @SchemaName as nvarchar(128), @TableName as nvarchar(128),@ColumnName1 AS nvarchar(128),@ColumnName2 AS nvarchar(128), @BandCount As bigint

AS

DECLARE @SchemaAndTableName nvarchar(400),@SQLString nvarchar(max)

SET @SchemaAndTableName = @DatabaseName + ‘.’ + @SchemaName + ‘.’ + @TableName

SET @SQLString = ‘DECLARE @MeanX float,@MeanY float, @StDevX float, @StDevY float,

@Count float, @Correlation float, @Covariance float, @Slope float, @Intercept float,

@ValueRange float, @HosmerLemeshowTest float

— STEP #1 — GET THE RANGE OF VALUES FOR THE Y COLUMN

SELECT @ValueRange = CASE WHEN RecordCount % 2 = 0 THEN ValueRange + 1 ELSE ValueRange END

FROM (SELECT Max(‘ + @ColumnName2 + ‘) – Min(‘ + @ColumnName2 + ‘) AS ValueRange, Count(*) AS RecordCount

FROM ‘ + @SchemaAndTableName + ‘

WHERE ‘ + @ColumnName2 + ‘ IS NOT NULL) AS T1

— STEP #2 — CREATE THE BANDS

DECLARE @LogisticRegressionTable table

(ID bigint, — ID is the decile identifier

CountForGroup bigint,

AverageX float,

AverageY float,

RescaledY float,

LogisticResult float

)

INSERT INTO @LogisticRegressionTable

(ID, CountForGroup, AverageX, AverageY)

SELECT DISTINCT DecileNumber, COUNT(*) OVER (PARTITION BY DecileNumber) AS CountForGroup , Avg(CAST(‘ + @ColumnName1 + ‘ AS float)) OVER

(PARTITION BY DecileNumber ORDER BY DecileNumber) AS AverageX, Avg(CAST(‘ + @ColumnName2 + ‘ AS float)) OVER (PARTITION BY DecileNumber

ORDER BY DecileNumber) AS AverageY

FROM (SELECT ‘ + @ColumnName1 + ‘, ‘ + @ColumnName2 + ‘, NTILE(‘ + CAST(@BandCount AS nvarchar(128)) + ‘) OVER (ORDER BY ‘ + @ColumnName1 + ‘) AS DecileNumber

FROM ‘ + @SchemaAndTableName + ‘) AS T1

UPDATE @LogisticRegressionTable — this could be done in one step, but Im leaving it this way for legibility purposes

SET RescaledY = AverageY / @ValueRange

— STEP #3 – RETRIEVE THE GLOBAL AGGREGATES NEEDED FOR

OTHER CALCULATIONS

— note that we cant operate on the original table here, otherwise the stats would be different from those of the bands

SELECT @MeanX = Avg(CAST(X AS float)), @MeanY = Avg(CAST(Y AS float)), @StDevX = StDev(CAST(X AS float)), @StDevY = StDev(CAST(Y AS float))

FROM (SELECT ‘ + @ColumnName1 + ‘ AS X, ‘ + @ColumnName2 + ‘ AS Y

FROM ‘ + @SchemaAndTableName + ‘

WHERE ‘ + @ColumnName1 + ‘ IS NOT NULL AND ‘ + @ColumnName2 + ‘ IS NOT NULL) AS T1

— STEP #4 – CALCULATE THE CORRELATION (BY FIRST GETTING THE COVARIANCE)

SELECT @Covariance = SUM((AverageX – @MeanX) * (AverageY – @MeanY)) / (‘ + CAST(@BandCount AS nvarchar(128)) + ‘ – 1)

FROM @LogisticRegressionTable

SELECT @Correlation = @Covariance / (@StDevX * @StDevY)

— STEP #5 – CALCULATE THE SLOPE AND INTERCEPT

SELECT @Slope = @Correlation * (@StDevY / @StDevX)

SELECT @Intercept = @MeanY – (@Slope * @MeanX)

— STEP #6 – CALCULATE THE LOGISTIC FUNCTION

UPDATE T1

SET LogisticResult = 1 / (1 + EXP(-1 * (@Intercept + (@Slope * AverageX))))

FROM @LogisticRegressionTable AS T1

— RETURN THE RESULTS

SELECT @HosmerLemeshowTest = SUM(Power((RescaledY – LogisticResult), 2) / (CountForGroup * LogisticResult * (1- LogisticResult)))

FROM @LogisticRegressionTable AS T1

SELECT * FROM @LogisticRegressionTable

SELECT @HosmerLemeshowTest AS HosmerLemeshowTest, @Covariance AS Covariance, @Correlation AS Correlation, @MeanX As MeanX, @MeanY As MeanY, @StDevX as StDevX, @StDevY AS StDevY, @Intercept AS Intercept, @Slope AS Slope‘

–SELECT @SQLString — uncomment this to debug dynamic SQL errors

EXEC (@SQLString)

…………Another potential problem arises when we derive the expected values of column Y for each band, using the logistic function in Step 6.[9] This is one of those maddening scaling issues that continually seem to arise whenever these older statistical tests are applied to Big Data-sized recordsets. This very simple and well-known function is implemented in the formula 1 / (1 + EXP(-1 * (@Intercept + (@Slope * AverageX)))), but when the result of the exponent operation is an infinitesimally small value, it gets truncated when adding the 1. This often occurs when the values for the @Intercept are high, particularly above 100. When 1 is divided by the resulting 1, the logistic function result is 1, which leads to division by zero errors when calculating the test statistic in the last SELECT assignment. There might be a way for a more mathematically adept programmer to rescale the @Intercept, @Slope and other variables so that this truncation doesn’t occur, but I’m not going to attempt to implement a workaround unless I’m sure it won’t lead to incorrect results in some unforeseen way.

…………Yet another issue is that my implementation allows the second column to be Continuous rather than the usual Boolean either-or choice seen in simple logistic regression. That requires rescaling to the range of permissible values, but the way I’ve implemented it through the RescaledY table variable column and @ValueRange variable may be incorrect. The SELECT that assigns the value to @HosmerLemeshowTest can also probably be done a little more concisely in fewer steps, but I want to highlight the internal logic so that it is easier to follow and debug. The rest of the code follows much the same format as usual, in which null handling, SQL injection protection, bracket handling and validation code are all omitted for the sake of legibility and simplicity. Most of the parameters are designed to allow the user to perform the regression on any two columns in the same table or view, in any database they have sufficient access to. The next-to-last line allows programmers to debug the dynamic SQL, which will probably be necessary before putting this procedure to professional use. In the last two statements I return all of the bands in the regression table plus the regression stats, since the costs of calculating them have already been incurred. It would be child’s play for us to also calculate the Mean Squared Error from these figures with virtually no computational cost, but I’m not yet sure if it enjoys the same validity and significance with logistic regression as it does with linear.

** Figure 2: Sample Results from Duchennes and Higgs Boson Datasets**EXEC Calculations.GoodnessOfFitLogisticRegressionHosmerLemsehowTestSP

@DatabaseName = N’DataMiningProjects’,

@SchemaName = N’Physics’,

@TableName = N’HiggsBosonTable’,

@ColumnName1 = N’Column1′,

@ColumnName2 = N’Column2′,

@BandCount = 12

…………I’ve tested most of the procedures for the last two tutorial series against a 9-kilobyte dataset on the Duchennes form of muscular dystrophy, which is made publicly available by Vanderbilt University’s Department of Biostatistics. For this week’s article, however, the intercepts were too high for most combinations of comparisons between the PyruvateKinase, CreatineKinase, LactateDehydrogenase and Hemopexin columns, resulting in the aforementioned divide-by-zero errors. For the few combinations that worked, the test statistic was ridiculously inflated; for other databases I’m familiar with, it returned results in the single and double digits (which is apparently permissible, since I’ve seen professionals post Hosmer-Lemeshow results online that fall in that range) but for whatever reason, this was not the case with Duchennes dataset.

…………That is why I derived the sample results in Figure 2 from the first two float columns of the Higgs Boson Dataset I downloaded from University of California at Irvine’s Machine Learning Repository, which I normally use for stress-testing because its 11 million rows occupy nearly 6 gigabytes in the SQL Server table I converted it into. Given that the first column is obviously abnormal and the second clearly follows a bell curve, I expected the results to indicate a serious lack of fit, but the test statistic was only a minuscule 1.30909988070185E-05. In fact, the values seem to shrink in tandem with the record count, which makes me wonder if another, more familiar scaling issue is operative here. As we’ve seen through the last two tutorial series, many common statistical measures were not designed with today’s Big Data table sizes in mind and thus end up distorted when we try to cram too much data into them. Given that there are so many other issues with my implementation, it is hard to tell if that is an additional problem or some inaccuracy in my code. Substituting the AverageY value for the RescaledY I used in the test statistic only seems to introduce new problems, without solving this one.

**The Case Against Hosmer-Lemeshow**

…………Regardless of the quality of my implementation, the usefulness of the Hosmer-Lemeshow Test is apparently still questionable, given that even professionals publish articles with inspiring titles like “Why I Don’t Trust the Hosmer-Lemeshow Test for Logistic Regression.”[10] University of Pennsylvania Sociology Prof. Paul Allison lists several drawbacks to the test even when it is correctly implemented, including the most common one mentioned on the Internet, the weaknesses inherent in dividing the dataset into groups. This can even lead to significantly different results when different stats software is run against the same data.[11] Hosmer and Lemeshow themselves point out that the choice of boundaries (“cut points”) for the groups can lead to significantly different results.[12] Furthermore, as Frank Harrell puts it in a thread in the CrossValidated forum, “The Hosmer-Lemeshow test is to some extent obsolete because it requires arbitrary binning of predicted probabilities and does not possess excellent power to detect lack of calibration. It also does not fully penalize for extreme overfitting of the model. …More importantly, this kind of assessment just addresses overall model calibration (agreement between predicted and observed) and does not address lack of fit such as improperly transforming a predictor.”[13] He recommends alternatives like a “generalized R^{2},” while Allison gets into “daunting” alternatives like “standardized Pearson, unweighted sum of squared residuals, Stukel’s test, and the information matrix test.”[14]

…………Nevertheless, despite these well-known shortcomings, the Hosmer-Lemeshow Test remains perhaps the best-known goodness-of-fit measure for logistic regression. Fitness tests seem to have well-defined use cases in comparison to the outlier detection methods we covered in my last tutorial series, with the Hosmer-Lemeshow among those that occupy a very distinct niche. The other methods mentioned by these authors seem to be more advanced and less well-known, so I thought it still worthwhile to post the code, even if there are certain to be problems with it. On the other hand, it is not worth it at this point to optimize the procedures much until the accuracy issues can either be fixed or debunked. It performed well in comparison to other procedures in this series anyways, with a time of 3:08 for the trial run in Figure 2. Only 1 of the 8 queries accounted for 89 percent of the cost of the whole execution plan, and that one contained two expensive Nested Loops operators, which might mean there’s room for further optimization if and when the accuracy can be verified.

…………Given the number of issues with my code as well as the inherent issues with the test itself, it might be fitting to write a rebuttal to my mistutorial titled titled like Allison article, such as “Why I Don’t Trust Steve Bolton’s Version of the Hosmer-Lemeshow Test for Logistic Regression.” It may still be useful in the absence of any other measures, but I’d assign it a lot less trust than some of the other code I’ve posted in the last two series. On the other hand, this also gives me an opportunity to jump into my usual spiel about my own lack of trust in hypothesis testing methods. I have a lack of confidence in confidence intervals and the like, at least as far as our use cases go, for multiple reasons. First and foremost, plugging test statistics into distributions just to derive a simple Boolean pass/fail measure sharply reduces the information content. Another critical problem is that most of the lookup tables for the tests and distributions they’re plugged into stop at just a few hundred values and are often full of gaps; furthermore, calculating the missing values yourself for millions of degrees of freedom can be prohibitively expensive. Once again, these measures were designed long before Big Data became a buzz word, in an era when most statistical tests were done against a few dozen or few hundred records at best. For that reason I have often omitted the hypothesis testing stage that accompanies many of the goodness-of-fit measures in this series, including the Hosmer-Lemeshow Test, which is normally plugged into a Chi-Squared distribution.[15] On the other hand, we can make use of the separate Chi-Squared goodness-of-fit measure, which as we shall see next week, is a versatile metric that can be adapted to assess the fit of a wide variety of probability distributions – with a much higher degree of confidence than we can assign to the Hosmer-Lemeshow Test results on logistic regression.

[1] See the Wikipedia article “Logistic Regression” at http://en.wikipedia.org/wiki/Logistic_regression

[2] See Sayad, Saed, 2014, “Logistic Regression,” published at the __SaedSayad.com__ web address http://www.saedsayad.com/logistic_regression.htm

[3] See the undated publication “Goodness of Fit in Linear Regression” retrieved from Lawrence Joseph’s course notes on Bayesian Statistics on Oct. 30, 2014, which are published at the website of the __McGill University Faculty of Medicine__. Available at the web address http://www.medicine.mcgill.ca/epidemiology/joseph/courses/EPIB-621/fit.pdf. No author is listed but I presume that Prof. Joseph wrote it.

[4] I derived some of the code for regular linear regression routine long ago from the Dummies.com webpage “How to Calculate a Regression Line” at http://www.dummies.com/how-to/content/how-to-calculate-a-regression-line.html

[5] See the Wikipedia article Hosmer-Lemeshow Test at the web address http://en.wikipedia.org/wiki/Hosmer%E2%80%93Lemeshow_test

[6] p. 160, Hosmer Jr., David W.; Lemeshow, Stanley and Sturdivan, Rodney X., 2013, __Applied Logistic Regression__. John Wiley & Sons: Hoboken, New Jersey.

[7] *IBID.*, pp. 160-163.

[8] *IBID.*, p. 161 for the second comment.

[9] I was initially confused about the assignment of the expected values (as well as the use of mean scores), but they are definitely derived from the logistic function, according to p. 2 of the undated manuscript, “Logistic Regression,” published at the Portland State University web address http://www.upa.pdx.edu/IOA/newsom/da2/ho_logistic.pdf . It is part of the instructional materials for one of Prof. Jason Newsom’s classes so I assume he wrote it, but cannot be sure.

[10] Allison, Paul, 2013, “Why I Don’t Trust the Hosmer-Lemeshow Test for Logistic Regression,” published March 5, 2013 at the __Statistical Horizons__ web address http://www.statisticalhorizons.com/hosmer-lemeshow

[11] Allison, Paul, 2014, untitled article published in March, 2014 at the __Statistical Horizons__ web address http://www.statisticalhorizons.com/2014/04

[12] pp. 965-966, 968, Hosmer, D.W.; T. Hosmer; Le Cessie, S. and Lemeshow, S., 1997, “A Comparison of Goodness-of-Fit Tests for the Logistic Regression Model,” pp. 965-980 in __Statistics in Medicine__. Vol. 16.

[13] Harrell, Frank, 2011, “Hosmer-Lemeshow vs AIC for Logistic Regression,” published Nov. 22, 2011 at the __CrossValidated__ web address http://stats.stackexchange.com/questions/18750/hosmer-lemeshow-vs-aic-for-logistic-regression

[14] See Allison, 2014.

[15] For more on the usual implementation involving the Chi-Squared distribution, see p. 977. Hosmer et al., 1997 and p. 158, Hosmer et al. 2013.