Linear models for multivariate repeated measures data with block exchangeable covariance structure

The popularity of the classical general linear model (CGLM) is attributable mostly to its ease of fitting and validating; however, the CGLM is inappropriate for correlated observations. In this paper we explore linear models for correlated observations with an exchangeable structure (Arnold in J Am Stat Assoc 74:194–199, 1979). For the case of $$N>1$$ repeated measures observations having site-dependent or site-independent covariates, the maximum likelihood estimates (MLEs) of the model’s parameters are derived, likelihood ratio tests are obtained for relevant model building hypotheses, and some Monte Carlo simulation studies are performed to illuminate important aspects of the models and their tests of hypotheses. For the case of site-independent covariates, closed-form solutions exist for the MLEs and exact tests can be constructed for the model building hypotheses. Simulations revealed that these exact tests remain robust in the presence of moderate skewness or outliers. However, these fortuitous closed-form occurrences vanish for the case of site-dependent covariates. In order to ameliorate this deficiency, some Monte Carlo simulations are performed to estimate the bias of these MLEs, the probability of a multimodal likelihood, and the suitability of the limiting chi-squared approximation to the model building hypotheses. These simulations reveal that the estimated biases of the slope parameters are negligible for sample size combinations (n, N) as small as (2, 6). Likewise, this sample size combination resulted in only an approximate 1% estimated probability of a multimodal likelihood, which drastically decreased with the increase of either n or N. Moreover, the limiting $$\chi ^2$$ distributional assumption appears to hold reasonably well for a sample size of $$N=100$$ , regardless of the value of n. Finally, we provide examples of fitting our model and conducting tests of hypotheses using two medical datasets.