Positive Regression Dependence And PRDS

Why PRDS Matters

Genomic tests are often dependent: genes can share biological pathways, samples, and technical sources of variation. Dependence matters when interpreting procedures that control the false discovery rate (FDR). A useful way to describe one favorable dependence pattern is positive regression dependence on a subset, abbreviated PRDS. The subset is the set of true-null tests, not necessarily the whole family.

PRDS is a condition on the joint distribution of the p-values. It is stronger than saying that each true-null p-value is valid on its own, and it is not the same as independence. Its direction can seem counterintuitive: when one true-null p-value is allowed to be larger, the conditional probability of an increasing event in the full p-value vector should not decrease. Because small p-values correspond to large test statistics in a one-sided test, positive dependence in test statistics can produce this pattern.

Definition and Direction

Definition: Let \(P=(P_1,\ldots,P_m)\) be the vector of p-values for a family of \(m\) tests, and let \(I_0\subseteq\{1,\ldots,m\}\) index the true-null hypotheses. A set \(D\subseteq[0,1]^m\) is increasing if \(p\in D\) and \(p'_j\ge p_j\) for every coordinate \(j\) imply \(p'\in D\). The vector \(P\) is PRDS on \(I_0\) if, for every increasing measurable set \(D\) and every \(i\in I_0\), \(P(P\in D\mid P_i\le t)\) is nondecreasing in \(t\) wherever the conditioning event has positive probability.

The phrase “on \(I_0\)” matters. The requirement is imposed when conditioning on a p-value from a true-null test. It does not require every p-value to be null, nor does it require the p-values to be exchangeable. In a continuous test, \(P_i\le t\) is a tail event, so increasing \(t\) makes the conditioning event less restrictive.

An increasing set describes a property that remains true if any p-value is made larger. For example, \(D=\{p:p_2\ge 0.5\}\) is increasing. The set \(\{p:p_2\le 0.05\}\), which describes a small p-value, is not increasing. This distinction helps keep the direction of PRDS straight.

For a true null, marginal validity means \(P(P_i\le t)\le t\) for every \(t\in[0,1]\); an exactly calibrated continuous null p-value is uniform on \([0,1]\). These are marginal statements. PRDS instead concerns conditional probabilities involving the joint vector. Valid marginal p-values do not, by themselves, guarantee PRDS.

Worked Example: A Positively Correlated Pair of Null Tests

Suppose two hypothetical one-sided test statistics satisfy \((Z_1,Z_2)\sim N((0,0),\Sigma)\), where both marginal variances are 1 and their correlation is \(0.6\). Both tests are true nulls, and \(P_i=1-\Phi(Z_i)\), where \(\Phi\) is the standard normal cumulative distribution function. The p-values are marginally uniform, and the covariance is positive.

Take the increasing set \(D=\{p:p_2\ge 0.5\}\). Given \(Z_1=z\), the conditional distribution is \(Z_2\mid Z_1=z\sim N(0.6z,0.64)\). Since \(P_2\ge0.5\) is equivalent to \(Z_2\le0\),

P(P_2\ge 0.5\mid Z_1=z) =\Phi\left(\frac{0-0.6z}{0.8}\right) =\Phi(-0.75z).

At \(z=0\), this probability is \(0.500\); at \(z=1\), it is \(\Phi(-0.75)\approx0.2266\); and at \(z=2\), it is \(\Phi(-1.50)\approx0.0668\). Larger \(z\) means smaller \(P_1\). Thus the probability of the increasing event rises as \(P_1\) rises, in the direction required by PRDS. These conditional values illustrate the direction for this event; the Gaussian justification below establishes the condition for all increasing sets under the stated sufficient assumptions.

A Gaussian Sufficient Condition

A particularly useful case for genomic analyses is a vector of jointly Gaussian, standardized test statistics with nonnegative correlations and one-sided p-values. For each coordinate, define \(P_j=1-\Phi(Z_j)\). For a true-null coordinate \(i\), assume \(Z_i\sim N(0,1)\), as required for its p-value to be exactly uniform. Other coordinates may have nonzero means, as can occur under alternatives. If the correlations between \(Z_i\) and every other coordinate are nonnegative, the conditional behavior has the PRDS direction. If this holds for each \(i\in I_0\), it gives PRDS on \(I_0\).

Here is the justification. For jointly Gaussian statistics, the conditional distribution of the remaining coordinates given \(Z_i=z\) has a covariance matrix that does not depend on \(z\). For each \(j\ne i\), its conditional mean is

E(Z_j\mid Z_i=z)=\mu_j+\rho_{ji}\frac{\sigma_j}{\sigma_i}(z-\mu_i),

where \(\mu_j\) and \(\sigma_j\) are the marginal mean and standard deviation of \(Z_j\), and \(\rho_{ji}\) is the correlation between \(Z_j\) and \(Z_i\). When \(\rho_{ji}\ge0\), increasing \(z\) shifts each conditional mean weakly upward. Because the conditional covariance is fixed, the conditional vectors at two values \(z_1\le z_2\) can be coupled using the same Gaussian residuals; every coordinate at \(z_2\) is then at least as large as its counterpart at \(z_1\).

For one-sided p-values, a larger test statistic gives a smaller p-value. Consequently, conditional on a larger \(Z_i\), the other p-values shift weakly downward. Meanwhile \(P_i=1-\Phi(Z_i)\) itself decreases as \(Z_i\) increases. Therefore, conditional on \(P_i=u\), increasing \(u\) shifts the full p-value vector in the coordinatewise increasing direction. The conditional probability \(g(u)=P(P\in D\mid P_i=u)\) is nondecreasing for every increasing set \(D\).

To connect this statement to the definition, let \(U=P_i\). For \(t_1<t_2\), the distribution of \(U\mid U\le t_2\) is a mixture of \(U\mid U\le t_1\) and values in \((t_1,t_2]\), when both parts have positive probability. Since \(g\) is nondecreasing, its average over the added values is at least its average over \(U\le t_1\). Thus \(E[g(U)\mid U\le t_2]\ge E[g(U)\mid U\le t_1]\), which is the PRDS condition.

This is a sufficient condition, not a definition or a necessary condition. PRDS can hold in other distributions, and nonnegative correlations alone do not establish it for every kind of p-value construction. The direction of the test and the transformation from statistic to p-value matter.

Worked Example: A Genomic Factor Model

Consider three hypothetical standardized gene-level statistics with a common pairwise correlation of \(0.4\), so their covariance matrix has 1 on the diagonal and \(0.4\) off the diagonal. Let their means be \(0,0,1.2\): genes 1 and 2 are true nulls, while gene 3 has a positive mean under an alternative. Use one-sided p-values \(P_j=1-\Phi(Z_j)\). The covariance matrix is positive definite: its eigenvalues are \(0.6,0.6,1.8\). Each true-null statistic has a standard normal marginal distribution, and its p-value is valid.

Conditioning on the true-null statistic \(Z_1=z\), the conditional mean of \(Z_2\) is \(0.4z\), and its conditional variance is \(1-0.4^2=0.84\). For the increasing event \(D=\{p:p_2\ge0.5\}\),

P(P_2\ge0.5\mid Z_1=z) =\Phi\left(\frac{-0.4z}{\sqrt{0.84}}\right).

At \(z=0\), this is \(0.500\). At \(z=1.645\), it is \(\Phi(-0.718)\approx0.236\). The larger statistic corresponds to a smaller \(P_1\), and the event becomes less likely, as expected when \(P_1\) is smaller. The nonnegative correlations between the true-null statistics and all coordinates meet the Gaussian sufficient condition for PRDS on \(I_0=\{1,2\}\). The nonzero mean for gene 3 does not change the conditional covariance or the direction of its conditional mean shift.

Worked Example: Negative Correlation Can Reverse the Condition

Now suppose two hypothetical true-null statistics are standard bivariate normal with correlation \(-0.5\), again using \(P_i=1-\Phi(Z_i)\). The marginal p-values are still uniform. Consider the same increasing event \(D=\{p:p_2\ge0.5\}\). The conditional distribution is \(Z_2\mid Z_1=z\sim N(-0.5z,0.75)\), so

P(P_2\ge0.5\mid Z_1=z) =\Phi\left(\frac{0.5z}{\sqrt{0.75}}\right).

As \(z\) increases, this probability increases, even though \(P_1=1-\Phi(z)\) decreases. More directly, as \(t\) approaches zero, conditioning on \(P_1\le t\) forces \(Z_1\) far into its upper tail, and the conditional probability of \(P_2\ge0.5\) approaches 1. At \(t=1\), there is no effective conditioning and \(P(P_2\ge0.5)=0.5\). The required probability therefore decreases between these limits, contradicting the nondecreasing condition. Marginal validity alone does not prevent this failure.

How to Use the Condition Carefully

PRDS is useful because dependence need not be absent for a multiple-testing argument to apply. It describes a structured form of dependence involving the true-null p-values and all increasing events, rather than a single correlation coefficient. The following distinctions are important when assessing whether the condition is plausible:

Independence is stronger in a different way. Independent p-values do not change their joint conditional behavior when one coordinate is conditioned on. PRDS allows dependence but constrains its direction for increasing sets.
Pairwise correlations are not the definition. A correlation matrix summarizes linear association. PRDS constrains conditional probabilities for every increasing measurable set. Nonnegative Gaussian correlations provide a useful sufficient condition for one-sided p-values, not a universal shortcut.
Two-sided p-values require separate care. For two-sided tests, p-values depend on the magnitude of the statistic, not monotonically on the signed statistic. The Gaussian argument above therefore does not automatically establish PRDS for them.
Marginal calibration is not joint calibration. Uniform true-null p-values establish their marginal validity, but say nothing by themselves about the conditional probabilities in the PRDS definition.

In a genomic analysis, knowledge of shared samples, common factors, or correlated expression can motivate a dependence model, but it does not prove PRDS. A diagnostic review should check the p-value construction, the direction of each test, and the relevant joint model; it should also avoid treating a handful of estimated pairwise correlations as a verification of a condition defined over all increasing sets. PRDS is a mathematical assumption about the joint distribution, not a visual pattern that can be confirmed from one plot.

Common Mistake

Common mistake: “All true-null p-values are uniform, and most observed correlations are positive, so PRDS holds.” Uniformity checks each null p-value separately, while PRDS is a joint conditional requirement for every increasing set. Positive observed correlations may support a model-based argument, but do not establish PRDS without appropriate assumptions about the statistics and p-value transformation.

Another frequent error is to reverse the order: a smaller \(P_i\) means more evidence against its null, but the PRDS definition asks about conditioning on \(P_i\le t\) as \(t\) increases. With one-sided Gaussian statistics and positive correlations, making \(P_i\) smaller tends to make other p-values smaller too. For an increasing event such as “another p-value is at least 0.5,” that makes the event less likely, not more likely.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

Is \(D=\{p:p_1+p_2\ge1\}\) increasing? Explain.
For one-sided p-values \(P_j=1-\Phi(Z_j)\), what happens to the other test statistics’ conditional means as \(Z_i\) increases when their correlations with \(Z_i\) are nonnegative?
Can uniform marginal distributions for every true-null p-value establish PRDS? Why or why not?
Why does the Gaussian sufficient-condition argument not automatically apply to two-sided p-values?

Solutions. (1) Yes. Increasing either coordinate cannot reduce the sum, so any point in the set remains in it when coordinates increase. (2) Each conditional mean shifts weakly upward, because its slope in \(Z_i\) is the nonnegative correlation times a positive standard-deviation ratio. (3) No. Uniformity is a marginal property; PRDS imposes conditional requirements on the joint vector for every increasing set. (4) A two-sided p-value depends on the magnitude of a statistic, so it is not a monotone decreasing transformation of the signed statistic. The one-sided monotonicity argument therefore does not establish the required ordering.