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Table 6 Coverage rates of low variances simulated data

From: Sampling weights in multilevel modelling: an investigation using PISA sampling structures

Software Weighting approach CR \(\widehat{{\beta }_{0}}\) CR \(\widehat{{\beta }_{1}}\) CR \(\widehat{{\beta }_{2}}\) CR \(\widehat{{\sigma }_{\varepsilon }^{2}}\) CR \(\widehat{{\sigma }_{\tau }^{2}}\)
A: Coverage rates—low variances simulated data—Model 1
 SAS No weights 0.94    0.94 0.87
Unscaled weights 0.93    0.62 0.74
Only school weights 0.94    0.94 0.87
Only student weights 0.91    0.43 0.70
Withincluster weights 0.94    0.77 0.94
Scaled weights: cluster 0.94    0.94 0.87
Scaled weights: ECluster 0.94    0.94 0.87
Clustersum 0.94    0.95 0.87
House weights 0.94    0.95 0.90
 Mplus No weights 0.94    0.94 0.92
Unscaled weights 0.95    0.94 0.90
Only school weights 0.94    0.94 0.90
Only student weights 0.94    0.94 0.91
Withincluster weights 0.94    0.94 0.91
Scaled weights: cluster 0.95    0.94 0.90
Scaled weights: ECluster 0.95    0.94 0.90
Clustersum 0.94    0.94 0.92
House weights 0.94    0.94 0.91
B: Coverage rates—low variances simulated data—Model 2
 SAS No weights 0.89 0.90   0.94 0.91
Unscaled weights 0.91 0.92   0.62 0.06
Only school weights 0.91 0.91   0.96 0.91
Only student weights 0.87 0.91   0.41 0.72
Withincluster weights 0.91 0.93   0.78 0.52
Scaled weights: cluster 0.91 0.92   0.96 0.91
Scaled weights: ECluster 0.91 0.92   0.96 0.91
Clustersum 0.89 0.91   0.95 0.92
House weights 0.90 0.91   0.95 0.93
 Mplus No weights 0.89 0.90   0.95 0.92
Unscaled weights 0.91 0.92   0.95 0.92
Only school weights 0.91 0.92   0.95 0.92
Only student weights 0.89 0.90   0.95 0.93
Withincluster weights 0.91 0.92   0.95 0.92
Scaled weights: cluster 0.91 0.92   0.95 0.92
Scaled weights: ECluster 0.91 0.92   0.95 0.92
Clustersum 0.89 0.91   0.95 0.92
House weights 0.89 0.90   0.95 0.93
C: Coverage rates—low variances simulated data—Model 3
 SAS No weights 0.87 0.93 0.95 0.95 0.91
Unscaled weights 0.91 0.93 0.93 0.62 0.03
Only school weights 0.90 0.94 0.95 0.96 0.90
Only student weights 0.86 0.93 0.94 0.40 0.52
Withincluster weights 0.90 0.93 0.95 0.74 0.43
Scaled weights: cluster 0.90 0.94 0.95 0.96 0.91
Scaled weights: ECluster 0.90 0.94 0.95 0.96 0.90
Clustersum 0.87 0.93 0.94 0.95 0.91
House weights 0.88 0.93 0.95 0.95 0.92
 Mplus No weights 0.87 0.93 0.95 0.95 0.93
Unscaled weights 0.90 0.94 0.95 0.95 0.92
Only school weights 0.90 0.94 0.95 0.95 0.91
Only student weights 0.87 0.93 0.95 0.95 0.93
Withincluster weights 0.90 0.94 0.95 0.95 0.92
Scaled weights: Cluster 0.90 0.94 0.95 0.95 0.92
Scaled weights: ECluster 0.90 0.94 0.95 0.95 0.92
Clustersum 0.87 0.93 0.94 0.95 0.93
House weights 0.87 0.93 0.95 0.95 0.93
  1. The CR represents the compliance rate of the estimators within its 95% confidence interval of three hierarchical models. Model 1 is declared as \({y}_{ij}={\beta }_{0}+ {\tau }_{i}+ {\varepsilon }_{ij}\), Model 2 as \({y}_{ij}={\beta }_{0}+ {\beta }_{1}*{x}_{ij}+ {\tau }_{i}+ {\varepsilon }_{ij}\) and Model 3 as \({y}_{ij}={\beta }_{0}+ {\beta }_{1}*{x}_{ij}+{\beta }_{2}*{x}_{i}+ {\tau }_{i}+ {\varepsilon }_{ij}\). Low variances between schools simulated data serves as scenario template. Simulation variation is displayed with the different weighting approaches combined with different estimation algorithms implemented in the two examined software packages