Estimation and statistical inferences of variance components in the analysis of single-case experimental design using multilevel modeling

Multilevel models (MLMs) can be used to examine treatment heterogeneity in single-case experimental designs (SCEDs). With small sample sizes, common issues for estimating between-case variance components in MLMs include nonpositive definite matrix, biased estimates, misspecification of covariance structures, and invalid Wald tests for variance components with bounded distributions. To address these issues, unconstrained optimization, model selection procedure based on parametric bootstrap, and restricted likelihood ratio test (RLRT)-based procedure are introduced. Using simulation studies, we compared the performance of two types of optimization methods (constrained vs. unconstrained) when the covariance structures are correctly specified or misspecified. We also examined the performance of a model selection procedure to obtain the optimal covariance structure. The results showed that the unconstrained optimization can avoid nonpositive definite issues to a great extent without a compromise in model convergence. The misspecification of covariance structures would cause biased estimates, especially with small between case variance components. However, the model selection procedure was found to attenuate the magnitude of bias. A practical guideline was generated for empirical researchers in SCEDs, providing conditions under which trustworthy point and interval estimates can be obtained for between-case variance components in MLMs, as well as the conditions under which the RLRT-based procedure can produce acceptable empirical type I error rate and power.