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2 Some theory 172.5 Inference: Random effectsTests of hypotheses: variance parametersInference concerning variance parameters of a linear mixed effects model usuallyrelies on approximate distributions for the (RE)ML estimates derived fromasymptotic results.It can be shown that the approximate variance matrix for the REML estimates isgiven by the inverse of the expected information matrix (Cox and Hinkley, 1974,section 4.8). Since this matrix is not available in ASReml we replace the expectedinformation matrix by the AI matrix. Furthermore the REML estimates are consistentand asymptotically normal, though in small samples this approximationappears to be unreliable (see later).A general method for comparing the fit of nested models fitted by REML is theREML likelihood ratio test, or REMLRT. The REMLRT is only valid if the fixedeffects are the same for both models. In ASReml this requires not only the samefixed effects model, but also the same parameterisation.If l R2 is the REML log-likelihood of the more general model and l R1 is the REMLlog-likelihood of the restricted model (that is, the REML log-likelihood under thenull hypothesis), then the REMLRT is given byD = 2 log(l R2 /l R1 ) = 2 [log(l R2 ) − log(l R1 )] (2.14)which is strictly positive. If r i is the number of parameters estimated in modeli, then the asymptotic distribution of the REMLRT, under the restricted modelis χ 2 r 2 −r 1.The REMLRT is implicitly two-sided, and must be adjusted when the test involvesan hypothesis with the parameter on the boundary of the parameter space. Infact, theoretically it can be shown that for a single variance component, say,the asymptotic distribution of the REMLRT is a mixture of χ 2 variates, wherethe mixing probabilities are 0.5, one with 0 degrees of freedom (spike at 0) andthe other with 1 degree of freedom. The approximate P-value for the REMLRTstatistic (D), is 0.5(1-Pr(χ 2 1 ≤ d)) where d is the observed value of D. Thedistribution of the REMLRT for the test that k variance components are zero, ortests involved in random regressions, which involve both variance and covariancecomponents, involves a mixture of χ 2 variates from 0 to k degrees of freedom.See Self and Liang (1987) for details.Tests concerning variance components in generally balanced designs, such as the