In statistical hypothesis testing, specifically multiple hypothesis testing, the q-value in the Storey procedure provides a means to estimate the positive false discovery rate (pFDR). Just as the p-value gives the expected false positive rate obtained by rejecting the null hypothesis for any result with an equal or smaller p-value, the q-value gives the expected pFDR obtained by rejecting the null hypothesis for any result with an equal or smaller q-value.
History In statistics, testing multiple hypotheses simultaneously using methods appropriate for testing single hypotheses tends to yield many false positives: the so-called multiple comparisons problem. For example, assume that one were to test 1,000 null hypotheses, all of which are true, and (as is conventional in single hypothesis testing) to reject null hypotheses with a significance level of 0.05; due to random chance, one would expect 5% of the results to appear significant (P < 0.05), yielding 50 false positives (rejections of the null hypothesis). Since the 1950s, statisticians had been developing methods for multiple comparisons that reduced the number of false positives, such as controlling the family-wise error rate (FWER) using the Bonferroni correction, but these methods also increased the number of false negatives (i.e. reduced the statistical power). In 1995, Yoav Benjamini and Yosef Hochberg proposed controlling the false discovery rate (FDR) as a more statistically powerful alternative to controlling the FWER in multiple hypothesis testing. The pFDR and the q-value were introduced by John D. Storey in 2002.
Definition Let there be a null hypothesis H 0 {\displaystyle H_{0}} and an alternative hypothesis H 1 {\displaystyle H_{1}} . Perform m {\displaystyle m} hypothesis tests; let the test statistics be i.i.d. random variables T 1 , … , T m {\displaystyle T_{1},\ldots ,T_{m}} such that T i ∣ D i ∼ ( 1 − D i ) ⋅ F 0 + D i ⋅ F 1 {\displaystyle T_{i}\mid D_{i}\sim (1-D_{i})\cdot F_{0}+D_{i}\cdot F_{1}} . That is, if H 0 {\displaystyle H_{0}} is true for test i {\displaystyle i} ( D i = 0 {\displaystyle D_{i}=0} ), then T i {\displaystyle T_{i}} follows the null distribution F 0 {\displaystyle F_{0}} ; while if H 1 {\displaystyle H_{1}} is true ( D i = 1 {\displaystyle D_{i}=1} ), then T i {\displaystyle T_{i}} follows the alternative distribution F 1 {\displaystyle F_{1}} . Let D i ∼ Bernoulli ( π 1 ) {\displaystyle D_{i}\sim \operatorname {Bernoulli} (\pi _{1})} , that is, for each test, H 1 {\displaystyle H_{1}} is true with probability π 1 {\displaystyle \pi _{1}} and H 0 {\displaystyle H_{0}} is true with probability π 0 = 1 − π 1 {\displaystyle \pi _{0}=1-\pi _{1}} . Denote the critical region (the values of T i {\displaystyle T_{i}} for which H 0 {\displaystyle H_{0}} is rejected) at significance level α {\displaystyle \alpha } by Γ α {\displaystyle \Gamma _{\alpha }} . Let an experiment yield a value t {\displaystyle t} for the test statistic. The q-value of t {\displaystyle t} is formally defined as
inf { Γ α : t ∈ Γ α } pFDR ( Γ α ) {\displaystyle \inf _{\{\Gamma _{\alpha }:t\in \Gamma _{\alpha }\}}\operatorname {pFDR} (\Gamma _{\alpha })}
… excerpt ends here. Continue reading the full article.
