Balanced Data with 10 Time Points

Tables 4.4 - 4.6 presents the weighted and unweighted results for the unconditional model. Each of the three models give identical results for both the weighted and un- weighted fixed effects estimates, however the standard error estimates differ marginally when level two sample sizes are 50 (Table 4.4). The standard error estimates in the weighted case are generally underestimating variability as measured by thestdr

ˆ

θr

, but improve with increases in level two sample size. The opposite is true for the unweighted

standard error estimates where the larger sample sizes result in underestimation of the standard deviation of the estimates.

The unweighted and weighted results reveal that the intercept and slope parame- ters are markedly biased due to the unequal selection of level two observations. The unweighted intercept is estimated around 15 compared to the parameter value of 20 and the unweighted slope is estimated around 3 compared to the parameter value of 2. The use of time invariant weights in this case results in essentially unbiased estimates of the population parameter values for the fixed intercept and slope. The weighted and unweighted random effects estimates presented in Table 4.5 show that the selection mechanism used here does not affect the random effects parameter estimates because the unweighted estimates are unbiased. The weighted estimates for the larger sample sizes are also converging on the population parameter values. Comparison of the standard deviations of the weighted random effects estimates to the unweighted random effects estimates reveals that the weighted estimates are much more volatile due to the weights and hence require the larger sample size for asymptotic properties to be fulfilled. Finally, coverage rates given in Table 4.6 reveal that there is zero coverage for the unweighted fixed effects estimates However, coverage rates for the generally unbiased random effects estimates are much superior in the unweighted analysis. Coverage rates for the weighted random intercepts and random slopes for a sample size of 500 approaches the coverage rates for the comparable unweighted estimates with a sample size of 50 indicating that the UWE for random effects parameters is very large and far greater than the UWE for a mean as presented in Table 4.3 for the time invariant weight.

Results for the conditional model with complete and balanced level one data with 10 data points per level two observation using the time invariant weights are presented in Tables 4.7 through 4.9. For the larger conditional growth model, larger sample sizes are required for convergence of the weighted fixed intercept and slope estimates on the parameter values (Table 4.7). The unweighted estimates reveal that the selection of level

two observations results in intercept estimates close to 6 compared to the population parameter value of 5. As well, the expected value of the unweighted fixed slope estimates are nearly two points higher than the parameter value of 4. Estimates for the effects of covariates (Table 4.7) are good for both the unweighted and weighted analysis since the selection design did not affect these relationships. Again, the mixed model and marginal models with independent and uniform correlation matrix produce the same results within sample size and weighting method indicating that unequal selection of level two observations only may be addressed equally in the various estimation types. Standard error estimates are again lower than the standard deviations for the fixed effects estimates (Table 4.8. - covariate effects exhibit the same pattern but are not shown in this table). This is true for all of the fixed effects in the weighted analysis, but the estimates improve with level two sample size. The standard error estimates for the unweighted fixed effects are unbiased except for the unweighted slope estimates, which are themselves biased. Random effects results for the conditional model are similar to those of the unconditional model (Table 4.9), though the unweighted analysis reveals that there may be a slight underestimation of the random slopes indicating that unequal selection may have decreased the variance of the slopes. In general, the weighted estimates require much larger sample sizes for convergence on the random effects population parameters. Coverage rates (Table 4.10) for the weighted estimates in the conditional model are poorer for biased parameters such as the fixed slope than the coverage rates for similar parameters in the unconditional model. For example, the fixed slope estimate coverage reaches only 0.78 with a level two sample of size of 500. As expected, the coverage rates for the biased parameters, the fixed intercept and slope, are very poor for the unweighted estimates. However, the coverage for unbiased fixed and random effects is better in the unweighted analysis than the weighted analysis. The coverage rate for the slightly biased random slope of the unweighted analysis gets worse with increased sample size probably due to smaller standard error estimates.

Results for the mixed and marginal models for the case where there is unequal selection of level two observations but no unequal selection of time points, reveal that using the time invariant weight corrects for biases due to the selection design. This is true for both the mixed and marginal models with either correlation matrix, all of which give very similar results within weighting approach and level two sample size. The weighted analysis requires fairly large samples sizes to obtain decent coverage rates, particularly for random effects estimates measured in a mixed model. Next, we turn to analysis of data with both unequal inclusion of level two and level one observations. In this situation, there are many more weighting options and the type of model and estimation techniques have more variable results.