# Wilcoxon test with multiple testing: which correction for p values?

Please, I'm not very confident in statistics, and I'm trying to respond to a reviewer for a paper on the following issues:

In my experiment I observed 15 babies during a test where they were free to play with an experimental toy, for 10 minutes

• each baby was tested individually
• age ranges from 8 months to 36 months
• during the test I recorder the durations in seconds of 12 selected behaviours (i.e. how long the baby smile? how long the baby explore the toy? and so on...)

In order to see if there were differences due to the age, I split the group in two samples (threshold: 24 months) with N=7 and N=8.

I then run a Wilcoxon rank sum test to compare, for each behaviour, the averages of durations, obtaining 12 p values, some of which are significant (values lower than alpha=0.05 )

The reviewer says that I need to correct alpha with Bonferroni, as I'm performing a multiple testing.

This leads alpha to be very low:

alpha corrected = 0.05/12 = 0.004

with the consequence that all significant results disappear.

Now, googling a little bit, I found that Bonferroni is not a good method when comparisons are more that 3 or 4, as it is too conservative, and False Discovery Rate (FDR) is proposed instead.

Do you agree on this?

• I wonder how the reviewer would feel if there were a multivariate omnibus test before the individual Wilcoxon tests... For example, in the coin package in R, there's an example using the independence_test function to conduct a test analogous to the Kruskal-Wallis test with multiple dependent variables. Sep 6, 2018 at 14:33

Suppose you have a collection of hypotheses $H_1, \dots, H_s$ that is under consideration.

While Bonferroni correction controls the Family-wise Error Rate (FWER), its ability to detect cases when a hypothesis $H_i$, $i=1,\dots,s$ is false is low since the Bonferroni condition $\alpha/s$ is quite stringent. In other words, what you're observing is the result of testing against a much smaller level than the conventional $\alpha$ level.

However, if you still want to use a Bonferroni-like procedure, the Holm procedure (or any stepdown procedure for that matter) will control the FWER while individual tests are increased over the $\alpha/s$ level of the Bonferroni correction.

The False Discovery Rate (FDR) is definitely a weakening of FWER. In general $FDR\le FWER$, so the FDR is more liberal (more rejections) than the FWER.

A final word, I wouldn't go into saying that " Bonferroni is not a good method when comparisons are more that 3 or 4, as it is too conservative" as your search concluded. In fact, Lehmann and Romano (2005) state that "when the number of tests is in the tens or hundreds of thousands, control of the FWER at conventional levels becomes so stringent that individual departures from the hypothesis have little chance of being detected."

I hope this helps.

Here's my take as a non-statistician.

In this situation, I would not apply a p-value correction. I think hypotheses about different dependent variables (DV) aren't a family of hypotheses, though you can see that some of the definitions below can apply.

That being said, if you are testing hypotheses about multiple DV, you have to concede that the likelihood of making a type I error across these hypotheses exceeds your nominal alpha value (typically 0.05).

To unpack things a bit:

The first question is, Does it make sense to treat your collection of hypotheses as a family of hypotheses?

For a definition of "family", I'll use some quotes from the Wikipedia article on familywise error rate.

• Hochberg & Tamhane defined "family" in 1987 as "any collection of inferences for which it is meaningful to take into account some combined measure of error".[1][page needed]

• According to Cox in 1982, a set of inferences should be regarded a family:[citation needed]

1) To take into account the selection effect due to data dredging

2) To ensure simultaneous correctness of a set of inferences as to guarantee a correct overall decision

To summarize, a family could best be defined by the potential selective inference that is being faced: A family is the smallest set of items of inference in an analysis, interchangeable about their meaning for the goal of research, from which selection of results for action, presentation or highlighting could be made (Yoav Benjamini).[citation needed]

Second, even if you are dealing with a family of hypotheses, Does it meet the goals of your analysis to use a p-value correction?

Using a p-value correction controls for either the familywise error rate (FWER) or false discovery rate (FDR). However, it is up to the discretion of the analyst whether or it is desirable or not to control the type I error rate at the expense of missing identifying potentially significant hypotheses. Choosing to err on the side of identifying more hypotheses is valid, as long as it acknowledged that the nominal type I error rate is likely exceeded, and that this makes sense for the goals of the analysis.

• In this particular application, which appears to be a bit of an underpowered exploration with a potential for correlated responses and all the attendant problems, I agree with the reviewer's request to correct the p-values (and would mistrust all the p-values without such an analysis). Nevertheless your post is well-reasoned well-supported, useful, and thoughtful (+1).
– whuber
Sep 6, 2018 at 13:46