Cauchy combination of p-values — acat

Combines several p-values into one using the Cauchy combination test of Liu and Xie (2020). The p-values may be arbitrarily dependent on one another.

Usage

acat_combine(
  p = NULL,
  weights = NULL,
  log_p = NULL,
  log_1mp = NULL,
  log.p = FALSE
)

Arguments

p: Numeric vector of p-values in $[0, 1]$. Supply this or the pair log_p and log_1mp.
weights: Numeric vector of nonnegative weights, the same length as the p-values. Rescaled to sum to 1. Defaults to equal weights.
log_p: Numeric vector of $\log(p_k)$.
log_1mp: Numeric vector of $\log(1 - p_k)$.
log.p: If TRUE, return the log of the combined p-value rather than the p-value itself. Use this when the answer is small enough to round to 0.

Value

A single p-value, or its log when log.p is TRUE. A component at exactly 0 gives 0 and one at exactly 1 gives 1, since either decides the combination on its own. Components at both endpoints at once raise an error, because the two disagree infinitely; supply the log-scale pair instead, which usually shows neither was really at an endpoint.

Details

The statistic is a weighted sum of Cauchy-transformed p-values, $$T = \sum_k w_k \tan\{(0.5 - p_k)\pi\},$$ referred to a standard Cauchy distribution. The transform is what makes this work: when $p_k$ is uniform on $(0,1)$ under the null, as a p-value must be, $\tan\{(0.5 - p_k)\pi\}$ is standard Cauchy, and the Cauchy family is closed under averaging. The p-value supplied must therefore be the ordinary two-sided p-value on $(0,1)$, not a one-sided p-value on $(0, 0.5)$. If you hold a one-sided p-value $q$, double it before passing it here.

Liu and Xie prove that the tail of the null distribution of $T$ is well approximated by a standard Cauchy under any correlation structure among the p-values, with the ratio of the true size to the nominal level tending to 1 as the level tends to 0. Under independence, and under perfect dependence where every $p_k$ is equal, the null distribution is exactly standard Cauchy, so the combination returns a uniform p-value.

The transform has poles at $p = 0$ and $p = 1$, so a component that arrives already rounded to either endpoint carries infinite weight. Supply log_p and log_1mp instead of p whenever you can compute them, which for a test with a known reference distribution means asking for both tails on the log scale rather than subtracting from 1. The arithmetic below never forms $\tan()$ of a saturated number and never overflows, however extreme the components are.

References

Liu, Y. and Xie, J. (2020). Cauchy Combination Test: A Powerful Test With Analytic p-Value Calculation Under Arbitrary Dependency Structures. Journal of the American Statistical Association 115(529), 393–402.