# True meaning of $\pi_0 = 1$ in false discovery rate

I am not a statistician so apologies if my terminology is wrong

I have a dataset of >14 million p values derived from Fisher's exact testing on genome scale sequencing data.

Benjamini-Hochberg correction of these p-values turns out only ~1500 significant p-values so I have been attempting to use qvalue R package to relax the FDR and therefore gain a greater number of significant p-values.

Applying the qvalue package with default options returns a $\pi_0$ of 1

qobj<-qvalue(p.fish.unadj)
qsummary(qobj)
pi0:    1


I presume this is a "good thing" and means that:

1. I have good power in my dataset to detect true null results and therefore true alternative results also
2. $\pi_0=1$ is an approximation because my true alternatives i.e. significant tests is such a small number compared to the overall size of the dataset

Am I right?

• "...so I have been attempting to use qvalue R package to relax the FDR and therefore gain a greater number of significant p-values." Danger, Will Robinson! What is the point of gaining a greater number of significant $p$-values here? – cardinal Feb 6 '13 at 12:53
• @cardinal - let me revise my language. The Fisher test is used in my work to assess significance of a difference in a score relating to individual bases in a genome. If groups of certain scores occur together this may be biologically significant. Although this could be looked at using a Stouffer-Liptak-Kechris correction I wondered if 'relaxation' of the FDR could be used to inform about any spatial correlation of significant p-values not necessarily significant at original FDR – plumb_r Feb 6 '13 at 13:12

## 1 Answer

It is actually $\pi_{0}(\lambda)=1$ (i.e. a function of $\lambda$ which controls the fraction of p-values used for the null-distribution). Recall that when the null-hypothesis is true, p-values will have a uniform distribution.

In other words, $\pi_{0}(0)=1$ "...is usually going to be much too conservative in genomewide data sets, where a sizable proportion of features are expected to be truly alternative. However, as we set $\lambda$ closer to 1, the variance of $\pi_{0}$ increases..."

I suggest you re-run your qvalue() estimation with a fixed lambda of 0.5 to indeed be confident your analysis.

Storey, John D., and Robert Tibshirani. "Statistical significance for genomewide studies." Proceedings of the National Academy of Sciences 100.16 (2003): 9440-9445.