series : "Statistics in Python"
This is a follow-up to my previous posts, here and this post, which are on software development, and multiple comparisons which looks at a specific case of pairwise mean comparisons.
In the following, I provide a brief introduction to the statsmodels functions for p-value asjustements to correct for multiple testing problems and then illustrate some properties using several Monte Carlo experiments.
subtitle: "The earth is round after all"
series : "Adventures with Statistics in Python"
If you run an experiment and it shows that the earth is not round, then you better check your experiment, your instrument, and don't forget to look up the definition of "round"
Statsmodels has 11 methods for correcting p-values to take account of multiple testing (or it will have after I merge my latest changes).
The following mainly describes how it took me some time to figure out what the interpretation of the results of a Monte Carlo run is. I wrote the Monte Carlo to verify that the multiple testing p-value corrections make sense. I will provide some additional explanations of the multiple testing function in statsmodels in a follow-up post.
Let's start with the earth that doesn't look round.
The first results
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b s sh hs h hommel fdr_i fdr_n fdr_tsbky fdr_tsbh fdr_gbs
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reject 9.6118 9.619 9.7178 9.7274 9.7178 9.72 10.3128 9.8724 10.5152 10.5474 10.5328
r_>k 0.0236 0.0246 0.0374 0.0384 0.0374 0.0376 0.2908 0.0736 0.3962 0.4118 0.4022
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The headers are shortcuts for the p-value correction method. In the first line, reject, are the average number of rejections across Monte Carlo iterations. The second line, r_>k, are the fraction of cases where we reject more than 10 hypothesis. The average number of rejections is large because the alternative in the simulation is far away from the null hypothesis, and the corresponding p-values are small. So all methods are able to reject most of the false hypotheses.
The last three methods estimate, as part of the algorithm, the number of null hypotheses that are correct. All three of those methods reject a true null hypothesis in roughly 40% of all cases. All methods are supposed to limit the false discovery rate (FDR) to alpha which is 5% in this simulations. I expected the fraction in the last line to be below 0.05. So what's wrong?
It looks obvious, after the fact, but it had me puzzled for 3 days.
Changing the experiment: The above data are based on p-values that are the outcome of 30 independent t-tests, which is already my second version for generating random p-values. For my third version, I changed to a data generating process similar to Benjamini, Krieger and Yekutieli 2006, which is the article on which fdr_tsbky is based. None of the changes makes a qualitative difference to the results.
Checking the instrument: All p-values corrections except fdr_tsbky and fdr_gbs are tested against R. For the case at hand, the p-values for fdr_tsbh are tested against R's multtest package. However, the first step is a discrete estimate (number of true null hypothesis) and since it is discrete, the tests will not find differences that show up only in borderline cases. I checked a few more cases which also verify against R. Also, most methods have a double implementation, separately for the p-value correction and for the rejection boolean. Since they all give identical or similar answers, I start to doubt that there is a problem with the instrument.
Is the earth really round? I try to read through the proof that these adaptive methods limit the FDR to alpha, to see if I missed some assumptions, but give up quickly. These are famous authors, and papers that have long been accepted and been widely used. I also don't find any assumption besides independence of the p-values, which I have in my Monte Carlo. However, looking a bit more closely at the proofs shows that I don't really understand FDR. When I implemented these functions, I focused on the algorithms and only skimmed the interpretation.
What is the False Discovery Rate? Got it. I should not rely on vague memories of definitions that I read two years ago. What I was looking at, is not the FDR.
One of my new results (with a different data generating process in the Monte Carlo, but still 10 out of 30 hypotheses are false)
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b s sh hs h hommel fdr_i fdr_n fdr_tsbky fdr_tsbh fdr_gbs
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reject 5.2924 5.3264 5.5316 5.5576 5.5272 5.5818 8.1904 5.8318 8.5982 8.692 8.633
rejecta 5.2596 5.2926 5.492 5.5176 5.488 5.5408 7.876 5.7744 8.162 8.23 8.1804
reject0 0.0328 0.0338 0.0396 0.04 0.0392 0.041 0.3144 0.0574 0.4362 0.462 0.4526
r_>k 0.0002 0.0002 0.0006 0.0006 0.0006 0.0006 0.0636 0.0016 0.1224 0.1344 0.1308
fdr 0.0057 0.0058 0.0065 0.0065 0.0064 0.0067 0.0336 0.0081 0.0438 0.046 0.0451
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reject : average number of rejections
rejecta : average number of rejections for cases where null hypotheses is false (10)
rejecta : average number of rejections for cases where null hypotheses is true (20)
r_>k : fraction of Monte Carlo iterations where we reject more than 10 hypotheses
fdr : average of the fraction of rejections when null is true out of all rejections
The last numbers look much better, the numbers are below alpha=0.05 as required, including the fdr for the last three methods.
"Consider the problem of testing m null hypotheses h1, ..., hm simultaneously, of which m0 are true nulls. The proportion of true null hypotheses is denoted by mu0 = m0/m. Benjamini and Hochberg(1995) used R and V to denote, respectively, the total number of rejections and the number of false rejections, and this notation has persisted in the literature. <...> The FDR was loosely defined by Benjamini and Hochberg(1995) as E(V/R) where V/R is interpreted as zero if R = 0." Benjamini, Krieger and Yekutieli 2006, page 2127
Some additional explanations are in this Wikipedia page
What I had in mind when I wrote the code for my Monte Carlo results, was the family wise error rate, FWER,
"The FWER is the probability of making even one type I error in the family, FWER = Pr(V >= 1)" Wikipedia
Although, I did not look up that definition either. What I actually used, is Pr(R > k) where k is the number of false hypothesis in the data generating process. Although, I had chosen my initial cases so Pr(R > k) is close to Pr(V > 0).
In the follow-up post I will go over the new Monte Carlo results, which now look all pretty good.
Reference
Benjamini, Yoav, Abba M. Krieger, and Daniel Yekutieli. 2006. “Adaptive Linear Step-up Procedures That Control the False Discovery Rate.” Biometrika 93 (3) (September 1): 491–507. doi:10.1093/biomet/93.3.491.
Statistical tests are often grouped into one-sample, two-sample and k-sample tests, depending on how many samples are involved in the test. In k-sample tests the usual Null hypothesis is that a statistic, for example the mean as in a one-way ANOVA, or the distribution in goodness-of-fit tests, is the same in all groups or samples. The common test is the joint test that all samples have the same value, against the alternative that at least one sample or group has a different value.
However, often we are not just interested in the joint hypothesis if all samples are the same, but we would also like to know for which pairs of samples the hypothesis of equal values is rejected. In this case we conduct several tests at the same time, one test for each pair of samples.
This results, as a consequence, in a multiple testing problem and we should correct our test distribution or p-values to account for this.
I mentioned some of the one- and two sample test in statsmodels before. Today, I just want to look at pairwise comparison of means. We have k samples and we want to test for each pair whether the mean is the same, or not.
Instead of adding more explanations here, I just want to point to R tutorial and also the brief description on Wikipedia. A search for "Tukey HSD" or multiple comparison on the internet will find many tutorials and explanations.
The following are examples in statsmodels and R interspersed with a few explanatory comments.
