Positive Regression Dependence on a Subset

The statistical dependence condition Positive Regression Dependence on a Subset (PRDS) often appears in the context of multiple hypothesis testing.

The formal definition is abstract and unintuitive:

Definition (PRDS).

A random vector $X=(X_1,\dots,X_m)$ is positively regression dependent on a subset $I_0\subseteq{1,\dots,m}$ if, for every $i\in I_0$ and every increasing measurable set $D\subseteq\mathbb{R}^m$,

$$x\mapsto \Pr(X\in D\mid X_i=x)$$

is nondecreasing.

Here, $D$ is increasing if $x\in D$ and $y\ge x$ coordinatewise imply $y\in D$.

Intuitively, assuming larger test statistics mean stronger evidence against the null, conditioning on one test statistic being larger makes significance-favoring joint outcomes no less likely.

In other words, PRDS describes a form of positive dependence in the joint hypothesis-testing space: significance tends to cluster with significance.

In simpler terms:

Picture the test statistics as gauges on a control panel, one for each hypothesis. Higher readings mean stronger evidence against the respective null hypothesis.

PRDS says that, for the gauges corresponding to true null hypotheses, a higher reading never makes jointly high readings elsewhere on the panel less likely.

In other words, if a true-null test statistic is higher, the other test statistics are, if anything, more likely to be higher too.

Why would anyone need to name this property at all?

The reason is that in multiple testing, dependence between tests can change the probability of getting many small p-values together, and that directly affects false discovery rate control.

Take the Benjamini–Hochberg procedure: Its decision for one hypothesis does not depend only on that hypothesis’s p-value. It depends on the whole collection of p-values because they are ranked together. So whether a true-null hypothesis gets falsely rejected partly depends on what all the other tests are doing.

Under independence the problem does not exist. Under dependence, one test looking significant can change the distribution of all the others.

For example, one true-null statistic could become more extreme, making the rejection threshold harmfully permissive and undermining error-control validity.

PRDS prevents this.

In general, the Benjamini–Hochberg procedure says: the more convincing p-values you have, the more hypotheses you may reject.

If some other tests become more significant, BH will generally keep the same number of rejections or reject more. It will not suddenly become more conservative.

PRDS says: If one true-null test happens to look more significant, the other tests do not suddenly become systematically less significant.

Stronger evidence generally leads to the same number or more rejections via the Benjamini–Hochberg procedure, and PRDS ensures that dependence does not systematically shift the remaining evidence in the opposite direction.

Both work in the same monotone direction. PRDS is the condition that prevents dependence among tests from interacting with BH in the wrong direction.

PRDS is the antidote to dependence breaking BH’s FDR control.