Tests of Linear Hypotheses using Indirect Information
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In multigroup data settings with small within-group sample sizes, standard F-tests of group-specific linear hypotheses can have low power, particularly if the within-group sample sizes are not large relative to the number of explanatory variables. To remedy this situation, in this article we derive alternative test statistics based on information-sharing across groups. Each group-specific test has potentially much larger power than the standard F-test, while still exactly maintaining a target type I error rate if the hypothesis for the group is true. The proposed test for a given group uses a statistic that has optimal marginal power under a prior distribution derived from the data of the other groups. This statistic approaches the usual F-statistic as the prior distribution becomes more diffuse, but approaches a limiting "cone" test statistic as the prior distribution becomes extremely concentrated. We compare the power and p-values of the cone test to that of the F-test in some high-dimensional asymptotic scenarios. An analysis of educational outcome data is provided, demonstrating empirically that the proposed test is more powerful than the F-test.
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