Parametric Models of Covariance Matrices for Repeated Measures: A Simulation Study on Fit, Error and Statistical Power in Mixed Linear Models

Abstract

The premise in experiments with repeated measures is that observations taken in the same experimental unit are correlated and that correlations decrease proportionally to the increase in the distance between measurements in time or space. Nevertheless, these experiments are often analyzed as if the correlations between the repeated measures were constant or using methods that only consider correlations different, which may impact on the rejection rate of the null hypothesis, and ultimately type I error rate and statistical power. In this context, this study investigated the application of mixed linear models with different assumptions about the covariance matrix in data sets from simulated experiments with repeated measures. 84 scenarios that varied in terms of the covariance matrix pattern (14 structures), number of repeated measurements (4 and 8) and sample size (4, 8 and 12) were evaluated. 10,000 data sets were simulated for each scenario based on a multivariate normal distribution and were subsequently analyzed using mixed linear models. Type I error rate and statistical power for the hypothesis test of the interaction between treatments and repeated measures were estimated as the proportion of p values less than or equal to 0.01 or 0.05 out of a total of 10,000 tests for each scenario. The models were also evaluated for their ability to fit the data using Bayesian Information Criteria (BIC). Thus, the frequency with which the covariance structures were chosen by the selection criteria was computed. Results indicate that the assumption chosen most frequently by the information criteria resulted from the specified covariance structure that corresponded to the empirical covariance structure of the analyzed data sets, particularly for those with larger number of repeated measures and sample sizes. Results also indicate that the use of covariance models that do not recognize heterogeneous correlations between repeated measures can inflate type I error or reduce it to very conservative levels, which may affect the conclusion of agricultural experiments. For a 5% significance level, type I error bias was greater than 2α, while for 1% significance level, bias was over 5α. In addition, the statistical power was reduced when the assumption about the covariance matrix of the data sets did not correspond to the empirical covariance structure, particularly for those data sets with a smaller sample size.