R 2 as measures of external and internal consistency in the linear mixed model

Several statistics have been proposed for linear mixed models (LMMs) to assess adequacy of fit. However, Orelien and Edwards (2007) showed that many of these statistics performed poorly in that they showed little variation when important variables related to the outcome were missing from the model. It was shown that statistics that can be classified as marginal are more useful than conditional statistics in selecting fixed effect covariates. In this chapter, we review the theoretical framework of the different approaches that can be used or have been used for statistics in the LMM. Limitations of each of these approaches are discussed. We then propose new statistics based on approaches that have not been considered thus far. Two of the statistics that we propose have the advantage that they can be easily interpreted. One of the statistics measures what we denote as “external consistency” (how well the model performs compared to a null model) while the other measures “internal consistency” (how much of the variation in the outcome is explained by the model at hand, assuming that it is the true one). This latter statistic has a corresponding population parameter assuming that the model is fully specified. Simulation results show that these statistics can be used to assess the goodness of the fit of a model or compare the fixed effects of alternative models. In assessing the ability of the statistics proposed to compare fixed effect covariates of competing models, the comparison is limited to models having the same random effects.