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Table 1 Variable definitions for hierarchical models used in this paper

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

\({y}_{ij}\) Student achievement, i.e., PISA competence (Math, Reading or Science)
\({x}_{ij}\) Student socio-economic status, i.e., the PISA Economic, Social and Cultural Index (ESCS)
\({x}_{i}\) The school’s socio-economic Index
\({\beta }_{0}\) Grand (i.e., overall) mean, intercept of the model
\({\beta }_{1}\) Fixed effect on student level
\({\beta }_{2}\) Fixed effect on school level
\({\varepsilon }_{ij}\) Residual
\({\tau }_{i}\) School random effect