On Some Procedures Based on Fisher’s Inverse Chi-Square Statistic

Abstract

We present approximations to the distribution of the weighted combination of independent and dependent p - values, . In case that independence of Pi 's is not assumed, it is argued that the quantity A is implic-itly dominated by positive definite quadratic forms that induce a chi-square distribution. This gives way to the approximation of the associated degrees of freedom using Satterthwaite (1946) or Patnaik (1949) method. An approximation by Brown (1975) is used to estimate the covariance between the log transformed P - values. The performance of the approximations is compared using simulations. For both the independent and dependent cases, the approximations are shown to yield probability values close to the nominal level, even for arbitrary weights, ωi ’s.