Steve Bolton is one of my favorite long-form analytics bloggers, and his ongoing goodness of fit series is a testament as to why.
Goodness-of-fit tests are also sometimes applicable to regression models, which I introduced in posts like A Rickety Stairway to SQL Server Data Mining, Algorithm 2: Linear Regression and A Rickety Stairway to SQL Server Data Mining, Algorithm 4: Logistic Regression. I won’t rehash the explanations here for the sake of brevity; suffice it to say that regressions can be differentiated from probability distributions by looking at them as line charts which point towards the predicted values of one or more variables, whereas distributions are more often represented as histograms representing the full range of a variable’s actual or potential values. I will deal with methods more applicable to regression later in this series, but in this article I’ll explain some simple methods for implementing the more difficult concept of a probability distribution.
As I found out the hard way, the difficult part with implementing these visual aids is not in representing the data in Reporting Services, but in calculating the deceptively short formulas in T-SQL. For P-P Plots, we need to compare two cumulative distribution functions (CDFs). That may be a mouthful, but one that is not particularly difficult to swallow once we understand how to calculate probability distribution functions. PDFs are easily depicted in histograms, where we can plot the probability of the occurrence of each particular value in a distribution from left to right to derive such familiar shapes as the bell curve. Since probabilities in stochastic theory always start at 0 and sum to 1, we can plot them a different way, by summing them in succession for each associated value until we reach that ceiling. Q-Q Plots are a tad more difficult because they involve comparing the inverse of the CDFs, using what is alternately known as quantile or percent point functions, but not terribly so. Apparently the raison d’etre for these operations is to distill distributions like the Gaussian down to the uniform distribution, i.e. a flat line in which all outcomes are equally likely, for easier comparison.
The most well-known extension of these somewhat forgotten stats is the Jarque-Bera Test, which only dates back to the 1970s despite being one of earliest examples of normality testing. All of these measures have fallen out of favor with statisticians to some extent, for reasons that will be apparent shortly, but one of the side effects of this is that it is a little more difficult to find variations on them that are more suited to the unique needs of the SQL Server community. One of the strengths of data mining on database servers like SQL Server is that you typically have such an enormous number of records to draw from that you can actually perform calculations on the full population, or a proportion close to it. In ordinary statistics, however, you’re often limited to making inferences based on small samples of just a few dozen or a few hundred rows, out of a much larger population that is often of unknown size; the results can still be logically valid, but often only if other preconditions are met on the data (including normality tests, which are often not performed). For that reason, I usually prefer to leverage SQL Server’s fast set-based retrieval methods to quickly calculate statistics on full populations whenever possible, especially when there are simpler versions of the mathematical formulas available for the full dataset.
Steve doesn’t post very frequently, but if you have a few hours on a lazy Friday, check him out.