Combining Independent Chi-Squared Tests
Abstract Classes of Bayes tests for combining n independent noncentral chi-squared statistics Ti ∼ x2 ki(θi) are derived, including the simple sum test based on Σ Ti , and are compared in power to the common “omnibus” procedures such as Fisher's based on II Pi , the product of the attained significance levels. Linear Bayes statistics Σ biTi with appropriate weights bi are found to yield more powerful tests against prespecified alternatives (θ1, …, θ n ) than weighted Fisher procedures advocated by others, provided each ki , > 2. Over the range of alternatives considered, the test based on II Pi minimizes the maximum shortcoming in power relative to the other tests studied when each ki ≥ 2, while the sum test has this property when each ki = 1.