# A Simulation Study

The same unbalanced datasets that were used in Section 15.2.1 are considered here. Multiple imputation (MI) was used to introduce balance in terms of cluster size and treatment allocation (Z). Proper multivariate imputations were conducted using the default Markov chain Monte Carlo method (Schafer, 1997) in SAS with a non-informative prior (Jeffreys). This imputation model included *S*, *T,* and *Z.* The imputation model was run “by cluster.” A total of 200 burn-in iterations were used, and the number of iterations equaled 100.

The “balanced” data were subsequently analyzed by fitting Model (15.1) using the UN and FA0(4) parameterizations for the *D* matrix (see Section 15.2.1). The outcome of interest was again model convergence. In addition, the bias, efficiency (standard deviation of the estimate), and Mean Squared Errors of the estimates of the Rriai and R?_{ndiv} metrics were evaluated.

*Results*

Table 15.3 shows the convergence rates for the MI UN and MI FA0(4) models. Compared to the case for the models in the unbalanced non-MI scenarios (see Table 15.1), the rates of proper convergence were substantially higher in both the MI UN and MI FA0(4) scenarios — and this was particularly so when N was small. The use of MI to make the unbalanced data “balanced” was thus a successful strategy to improve proper convergence.

Tables 15.4 and 15.5 show the bias, efficiency, and MSE of the estimates of Rndiv and R_{rial}, respectively, in the non-MI and MI settings that properly converged. As expected, the bias, efficiency, and MSE in the estimation of ^{both R}Liv and *R^2 _{rial}* improved when the number of clusters increased. With respect to the estimation of R

^{2}ndiv, the bias was low in all scenarios but the efficiency and MSE were poorer in the MI scenarios compared to the non- MI scenarios (see Table 15.4). In contrast to R?

_{ndiv}, the bias, efficiency and MSE in R2

_{rial}were of similar magnitude in the MI and non-MI scenarios (see Table 15.5). Only when N = 5 did the bias in the estimation of R2

_{rial}tend to be substantially higher in the MI scenario compared to the non-MI scenario. Note that the bias was negative in all scenarios, indicating that the true R2

_{rial }tends to be underestimated. Further, the bias and MSE in the estimation of Rndiv was smaller compared to what was observed for R2

_{rial}, and the efficiency somewhat lower, because there is less replication than for the individual-level quantity.