The Power Function of Conditional Log-Linear Model Tests

原文摘要

In fitting a log-linear model to data, it is common to examine the P values associated with the conditional likelihood-ratio tests corresponding to a nested sequence of candidate log-linear models. For an appropriate Pitman sequence of models these test statistics are asymptotically independent and chi-squared in distribution. This article is an analytic study of the noncentrality parameters of these limiting distributions, pointing out factors that determine the powers of these tests. The conditional likelihood-ratio tests corresponding to a given nested sequence of log-linear models depend only on the models, but not on their parameterizations. The noncentrality parameters of their limiting chi-squared distributions depend on the pair of models compared, the true model, and the limiting model, but not on their parameterizations. Often a nested sequence of log-linear models is specified by a particular parameterization with adjacent models in the sequence differing only in the presence or absence of a particular set of parameters. In general, the test associated with a pair of adjacent models is not sensitive only to the presence of that parameter set; rather it is sensitive to that parameter set and parameter sets farther out in the sequence of models as well. Analysis of this test's noncentrality parameter shows that it may not be an indicator of the presence or absence of this parameter set: The noncentrality parameter may be large when the parameter set is null, or the parameter set may be nonnull even though the noncentrality parameter equals 0. In this case, the given parameterization of the nested sequence of models is not well suited to identifying to what the likelihood-ratio test statistic is sensitive. Another parameterization is well suited to this identification; it depends on the limiting model of the Pitman sequence. We present numerical calculations and simulation results illustrating general results in particular cases, and we show that the asymptotic effects occur in a simple setting at moderate sample sizes.

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