Statistical Issues in the Analysis of Correlated Data.

Abstract

We first extend the original classification and regression trees (CART) paradigm (Breiman et al. 1984) to clustered binary outcomes, where individuals within a cluster are correlated. We propose to generate tree models using residuals from a null generalized linear mixed model (with fixed and random intercepts only) as the outcome, which circumvents modeling the correlation structure explicitly while still accounting for the cluster-correlated design, thereby allowing us to adopt the original CART machinery in tree growing, pruning and cross-validation. Based on simulations, we find that our residual-based tree is more appropriate for analyzing clustered binary data, and provides more accurate classification predictions than the standard CART that ignores the clustering. We also illustrated our method using data from clinical studies, and residual-based trees identified clinically meaningful subgroups.

Clinical attachment level (CAL) is a tooth-level measure that quantifies the severity of periodontal disease. The within-mouth correlation of tooth-level CAL is difficult to model because it must reflect the three-dimensional spatial geography of teeth and their functional similarity. In the second project, we propose two linear mixed effects (LME) models with random effects that quantify the within-mouth correlation of teeth and their shared functionality. Via simulations, we demonstrate that our mixed models give consistent and more efficient estimates than a t-test and generalized estimating equations that fail to model the within-mouth correlation accurately. We also evaluate the performance of the approaches when data are missing under different biologically plausible missing data mechanisms.

Inference for the fixed effects in an LME model is dependent upon the correlation structure implied by the random effects included in the LME model. However, limited methods are available for making inference about the fit of the assumed covariance structure in the LME model. In the third project, we propose three permutation tests, all of which are based on comparing the estimated assumed covariance matrix to the covariance matrix of the marginal residuals. Cholesky residuals, which are exchangeable both within and among subjects, are employed in the permutations. Through simulations, we show that two of our tests have valid size and comparable power in testing different covariance structure assumptions.