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

Mang, Julia; Küchenhoff, Helmut; Meinck, Sabine; Prenzel, Manfred

doi:10.1186/s40536-021-00099-0

Table 7 Coverage rates of high 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—high variances simulated data—Model 1
SAS	No weights	0.00			0.93	0.97
	Unscaled weights	0.98			0.52	0.99
	Only school weights	0.99			0.94	0.99
	Only student weights	0.00			0.32	0.97
	Withincluster weights	0.99			0.66	0.99
	Scaled weights: cluster	0.99			0.94	0.99
	Scaled weights: ECluster	0.99			0.94	0.99
	Clustersum	0.00			0.94	0.96
	House weights	0.00			0.93	0.96
Mplus	No weights	0.00			0.94	0.90
	Unscaled weights	0.99			0.94	0.97
	Only school weights	0.99			0.94	0.97
	Only student weights	0.00			0.94	0.90
	Withincluster weights	0.99			0.94	0.97
	Scaled weights: cluster	0.99			0.95	0.98
	Scaled Weights: ECluster	0.99			0.95	0.98
	Clustersum	0.00			0.95	0.78
	House weights	0.00			0.94	0.82
B: Coverage rates—high variances simulated data—Model 2
SAS	No weights	0.00	0.84		0.94	0.02
	Unscaled weights	0.99	0.89		0.54	0.06
	Only school weights	0.99	0.86		0.94	0.07
	Only student weights	0.00	0.82		0.35	0.02
	Withincluster weights	0.99	0.88		0.71	0.06
	Scaled weights: cluster	0.99	0.86		0.95	0.07
	Scaled weights: ECluster	0.99	0.86		0.95	0.07
	Clustersum	0.00	0.89		0.94	0.96
	House weights	0.00	0.84		0.95	0.02
Mplus	No weights	0.00	0.88		0.94	0.97
	Unscaled weights	0.98	0.89		0.93	0.94
	Only school weights	0.98	0.90		0.94	0.94
	Only student weights	0.00	0.88		0.94	0.97
	Withincluster weights	0.98	0.89		0.93	0.94
	Scaled weights: cluster	0.99	0.88		0.94	0.95
	Scaled weights: ECluster	0.99	0.89		0.94	0.94
	Clustersum	0.00	0.89		0.94	0.96
	House weights	0.00	0.87		0.95	0.96
C: Coverage rates—high variances simulated data—Model 3
SAS	No weights	0.12	0.90	0.96	0.94	0.96
	Unscaled weights	0.96	0.89	0.93	0.56	0.99
	Only school weights	0.96	0.91	0.94	0.93	0.99
	Only student weights	0.13	0.90	0.96	0.37	0.95
	Withincluster weights	0.96	0.90	0.94	0.71	0.99
	Scaled weights: cluster	0.98	0.92	0.94	0.94	0.98
	Scaled weights: ECluster	0.97	0.92	0.94	0.94	0.98
	Clustersum	0.10	0.89	0.95	0.94	0.94
	House weights	0.13	0.88	0.95	0.94	0.93
Mplus	No weights	0.11	0.90	0.96	0.94	0.96
	Unscaled weights	0.96	0.91	0.94	0.93	0.92
	Only school weights	0.96	0.90	0.94	0.93	0.92
	Only student weights	0.11	0.91	0.96	0.94	0.96
	Withincluster weights	0.96	0.91	0.94	0.93	0.92
	Scaled weights: cluster	0.97	0.91	0.94	0.94	0.92
	Scaled Weights: ECluster	0.97	0.91	0.94	0.94	0.92
	Clustersum	0.10	0.89	0.94	0.94	0.94
	House weights	0.14	0.89	0.95	0.94	0.94

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 \({y}_{ij}={\beta }_{0}+ {\beta }_{1}*{x}_{ij}+{\beta }_{2}*{x}_{i}+ {\tau }_{i}+ {\varepsilon }_{ij}\). High 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

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