Power for a Generalization of the Glmm 1with Fixed and Random Predictors

Multivariate models with random predictors are often considered in public health. Prospective power analysis for such models must take into account all possible stochastic realizations of the predictors. An extended definition is given for General Linear Multivariate Models with both random and fixed predictors. A classification scheme for hypotheses is introduced. Those that concern both fixed and Gaussian predictors are called GLH(F, G), those about Gaussian predictors only are called GLH(G), and those about fixed predictors are called GLH(F). Although there are some asymptotic power approximations for tests of the independence of sets of Gaussian random variables, (a special case of a GLMM(G), GLH(G)), power for more complicated models has not been widely studied. In the GLMM(F), the power for some multivariate tests and for the univariate approach to repeated measures (UNIREP) can be approximated by using noncentral F statistics (Muller et al., 1992). These results are conditional powers since they depend on the observed values of the predictors. Unconditional power is the expected value of conditional power with respect to the density of the random noncentrality matrix. The law of total probability has been used similarly to calculate unconditional power for general linear univariate models with random Gaussian predictors (Sampson, 1974; Gatsonis and Sampson, 1989). For a GLMM(F, G) with one random predictor and a GLH(F, G) with one degree of freedom, small sample unconditional power results are derived for the HotellingLawley and Pillai-Bartlett traces, Wilks' Lambda and the UNIREP tests. For the GLMM(F, G) and the GLH(G), in the linear case, small sample unconditional power results are derived for the tests just listed. When the noncentrality has more than one non-zero eigenvalue, results are given for the Hotelling-Lawley and UNIREP tests. Asymptotic limits are examined under Pitman local alternatives. Numerical algorithms are developed for some unconditional power results. Comparisons with simulations allow concluding that the unconditional power results are very accurate for many cases, and that the deviations from expected values are due to approximations for conditional power. Unconditional and conditional power results are compared in simulations, and by application to a clinical trial on bone density.