Approximations to Distributions of Test Statistics in Complex Mixed Linear Models Using SAS Proc MIXED

The MIXED procedure of SAS® has made use of the linear mixed model accessible to researchers. However, a sticky problem for the procedure has been the specification of appropriate denominator degrees of freedom for test statistics for fixed effects in both balanced designs with simple covariance structures and complex designs involving complicated covariance structures, unbalanced data and/or small sample sizes. This paper compares the denominator degrees-of-freedom options in Proc MIXED. INTRODUCTION It is no exaggeration to claim that statistical practice has been changed by the development of the MIXED procedure (‘Proc MIXED’) of SAS (Littell et al. 1996). Because of its generality and relative ease of use, Proc MIXED has made linear mixed model technology accessible to researchers in a wide variety of fields and with a wide range of statistical training. Recent books discuss linear mixed models in general (McCulloch and Searle 2001), as well as the Proc MIXED implementation of mixed model calculations (Littell et al. 1996, Verbeke and Molenberghs 1997, Brown and Prescott 1999). With a little instruction and experience, researchers can use Proc MIXED to intelligently model and analyze data involving multiple random terms, heterogeneous variances, and correlations resulting from clustered and serially measured units. Although there are sometimes modifications to be made when the data are unbalanced, Proc MIXED in general is equally easy to use for balanced or unbalanced (even seriously so) data. To use Proc MIXED, the linear model for the means or fixed effects is specified in one statement (the MODEL statement) and the variance-correlation structure is specified in one or more separate statements (RANDOM and REPEATED statements). Proc MIXED uses either maximum likelihood or residual maximum likelihood together with estimated generalized least squares to estimate the variance-covariance and mean parameters. Both procedures are statistically defensible, and have well-known optimality properties. Proc MIXED also provides sensible test statistics for linear hypotheses involving fixed effects. Proc MIXED and linear mixed models in general have a weakness, however. Null distributions of the test statistics are often unknown, and p-values cannot be computed exactly. This is not invariably true because in many balanced data situations with simple covariance structures, test statistics are in fact known to follow F distributions with specific denominator degrees of freedom (ddf). Nonetheless, in a great many situations the distributions are unknown. For both estimation methods, test statistics asymptotically follow the chi-square distribution; but this fact is of little help because small sample distributions are commonly encountered in applications of linear mixed models and can be very different from chi-square. Some progress can be made by assuming that test statistics approximately follow an F-distribution with a carefully calculated ddf. This paper reviews methods of calculating the ddf in Proc MIXED of SAS® version 8. Methods appropriate for balanced designs are reviewed, and two recent general methods, the FaiCornelius (1996) and the Kenward-Roger (1997) methods are described in detail. Past simulation studies of these methods are reviewed, as are results of a new study involving the effects of covariance structures, imbalance, and sample size on adequacy of the Proc MIXED implementation of these methods. EXAMPLE In a recent study of differential olfactory responses of male and female lady beetles, Hamilton et al. (1999) examined the antennae of a small sample of male and female lady beetles using electron microscopy. For a sample of 3 males and 3 females, they counted the number of ‘sensilla’ (small hair-like attachments) on each of 10 segments of an antenna. Although the design was balanced, it was a small sample repeated measures design with an apparently heterogeneous-variance correlation structure for the repeated measurements. Particular attention was focused on the last segment, and its relationship to the other segments and sex. The Proc MIXED statements used in the analysis were: proc mixed data=beetle covtest; class sex seg indiv last; model sensilla = last seg(last) sex last*sex / ddfm= ; random indiv(sex); repeated seg / subject=indiv type=arh(1); Even though the design was balanced, the complex covariance structure dictated that the test statistics follow an unknown null distribution. The output below indicates how the tests of fixed effects varied depending on the ddf options available in Proc MIXED. The default option in this case was containment. Num Den Effect DF DF F Value Pr > F ddfm=residual last 1 48 9713.06 <.0001 seg(last) 8 48 61.52 <.0001 sex 1 48 7.26 0.0097 sex*last 1 48 6.77 0.0123 ddfm=contain last 1 44 9713.06 <.0001 seg(last) 8 44 61.52 <.0001