# FDR correction when tests are correlated

I have a data set with small number of samples and large number of variables. I did hypothesis testing (T test) on each of the variable and got a number of p values. However, the variables are correlated to each other and FDR correction (Benjamini–Hochberg procedure) assumes that tests are independent or positively regression dependent.

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From the paper of BY (2001), http://projecteuclid.org/euclid.aos/1013699998, it seems to me that BY proves that BH procedure also work well in data set where variables are independent or positively regression dependent to each other. But they also mentioned that there may other form of dependency that BH procedure won't work very well. From http://www.math.tau.ac.il/~yekutiel/papers/JSPI%20--%20Dani.pdf, Y extents the BH procedure to BY procedure to meet the situation of non-positive regression dependency. My question is that what is positive regression dependency and what is non-positive regression dependency? An few examples would be greatly helpful!

• FDR does not assume independence; see my answers here and here for example. There are several more specialized procedures for individual types of dependence, though you would have to provide more detail as to what kind of dependence you have in your dataset. – Chris C Apr 5 '16 at 21:50
• @ChrisC， Thank you for the reference. Can you explain to me what is positive regression dependency? In the paper of BY (2001), PRDS is just a extension or a special case of positive regression dependency. Also, this paper, math.tau.ac.il/~yekutiel/papers/JSPI%20--%20Dani.pdf, there is a non-positive regression dependency. I have no idea what these two things are. Can you help? – WCMC Apr 5 '16 at 23:16
• To answer your question, a positive correlation between two random variables means that when the one has a certain property, the other tends to have the same. At the opposite there is a negative correlation when two random variables tend to not share the same property more often than if they were independent. I did not read the paper of BY, but my guess is that the cases where BH do not apply correctly are very special so BH should be fine in most of the cases. – Jean Paul Sep 29 '17 at 13:36
• @JeanPaul, Please be more precise in your descriptions. Which property are you referring to? The question was about positive regression dependency. Also guessing is not really an answer. – Knarpie Sep 29 '17 at 15:13

You're looking for the Benjamini-Yekutieli procedure:

Benjamini, Yoav; Yekutieli, Daniel. The control of the false discovery rate in multiple testing under dependency. Ann. Statist. 29 (2001), no. 4, 1165--1188. doi:10.1214/aos/1013699998. http://projecteuclid.org/euclid.aos/1013699998

The procedure is available in R using the method = "BY" option in p.adjust(). For more info, try ?p.adjust.