Use of R2 statistics for assessing goodness-of-fit and model selection in the Linear Mixed Model for longitudinal data

In the Linear Mixed Model (LMM), several R 2 statistics have been proposed for assessing goodness-of-fit. However, the performance of these statistics has not been demonstrated. In this dissertation research, first we show that many of the R2 statistics that have been proposed in the statistical literature are not appropriate to assess adequacy of the fixed effect terms because they are unable to detect when important covariates are missing from the model. A distinction is made between R2 statistics that can be classified as marginal and those that can be classified as conditional. We show through simulations that only marginal R2 statistics are appropriate for assessing the adequacy of the fixed effects in the LMM. To remedy the shortcoming of R2 statistics that have been proposed, we introduce new R2 statistics that measure the extent to which the model at hand is better than a null model and statistics that measure how much of the variation in the outcome is explained by the model at hand assuming that the model is adequate. Results from simulations show that our proposed R 2 statistics perform well in assessing adequacy of model fit for the fixed effects or selection of the fixed effects covariates. For selecting the random effects, our proposed R2 statistics are able to distinguish between a model that includes a time covariate and one that doesn't (such as a model with only a random intercept). However, these statistics were unable to discriminate between a full and a reduced model in the random effects that both included a time covariate such as a full model with an intercept, linear and quadratic component for time and a reduced model with an intercept and linear component for time. We found that even when the true model of the random effects involves variables (polynomial components) beyond the linear term, the reduced model with an intercept and a linear term for time may be as good as the full model.