Combining Independent Chi-Squared Tests

Abstract

Classes of Bayes tests for combining n independent noncentral chi-squared statistics T i ∼ x 2 ki (θ i ) are derived, including the simple sum test based on Σ T i , and are compared in power to the common “omnibus” procedures such as Fisher's based on II P i , the product of the attained significance levels. Linear Bayes statistics Σ b i T i with appropriate weights b i are found to yield more powerful tests against prespecified alternatives (θ 1 , …, θ n ) than weighted Fisher procedures advocated by others, provided each k i , > 2. Over the range of alternatives considered, the test based on II P i minimizes the maximum shortcoming in power relative to the other tests studied when each k i ≥ 2, while the sum test has this property when each k i = 1.