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Table 4 Weighting approaches for the simulation study and their application and formulas for the different levels of the hierarchies

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

Weighting approaches School level Student level
No weights
Unscaled weights \({w}_{i}\) \({w}_{ij}\)
Only student weights   \({w}_{ij}\)
Only school weights \({w}_{i}\)  
Scaled weights: Cluster \({w}_{i}\) \({w}_{ij }\frac{{n}_{i}}{\sum_{j=1}^{{n}_{i}}{w}_{ij}}\)
Scaled weights: ECluster \({w}_{i}\) \({w}_{ij }\frac{{{n}^{*}}_{i }}{\sum_{j=1}^{{n}_{i }}{w}_{ij}}\)
Withincluster weights \({{w}_{i }}^{*}\) \({{w}_{j }}^{*}\)
House weights   \({w}_{ij }\frac{n}{\sum_{j=1}^{n}{w}_{ij}}\)
Clustersum \(\sum_{j=1}^{{n}_{i}}{w}_{ij}\) \({w}_{ij }\frac{{n}_{i}}{\sum_{j=1}^{{n}_{i}}{w}_{ij}}\)
  1. Weighting parameters are \({w}_{i}\) = final school weights, \({w}_{ij}\) = final student weights, \({n}_{i}\) = number of sampled students in a school,  \({{n}^{*}}_{i}\) = number of assessed students in a school, \(n\) = number of assessed students from all schools and \({w}_{j}\)= final within school weights