Table 2 Hierarchical linear models used to answer research questions

Model 1: Unconditional model

 Level-1: \({Y}_{ij} = {\upbeta }_{0j}+ {r}_{ij}\)

 Level-2: \({\upbeta }_{0j} = {\upgamma}_{00}+ {u}_{0j}\)

Model 2: Model including SES at both levels

 Level-1: \({Y}_{ij} = {\upbeta }_{0j}+{\upbeta }_{1j}{\left(\mathrm{ses}\right)}_{j}+ {r}_{ij}\)

 Level-2: \({\upbeta }_{0j} = {{\upgamma}}_{00}+{{\upgamma}}_{01}{\left(\mathrm{m}\_\mathrm{ses}\right)}_{j}+ {u}_{0j}\)

 \({\upbeta }_{1j} = {{\upgamma}}_{10}+ {u}_{1j}\)

Model 3: Model including student and school variables, without SES

 Level-1: \({Y}_{ij} = {\upbeta }_{0j}+{\upbeta }_{1j}{\left(\mathrm{engage}\right)}_{j}+{\upbeta }_{2j}{\left(\mathrm{litnum}\right)}_{j}+ {\upbeta }_{3j}{\left(\mathrm{liking}\right)}_{j}+ {\upbeta }_{4j}{\left(\mathrm{preschool}\right)}_{j}+ {r}_{ij}\)

 Level-2: \({\upbeta }_{0j} = {{\upgamma}}_{00}+{{\upgamma}}_{01}{\left(\mathrm{instruction}\right)}_{j}+{\upgamma}{}_{02}{\left(\mathrm{readiness}\right)}_{j}+{{\upgamma}}_{03}{\left(\mathrm{condition}\right)}_{j}+\)

                \({{\upgamma}}_{04}{(\mathrm{emphasis})}_{j}+{{\upgamma}}_{05}{(\mathrm{safe})}_{j}+ {u}_{0j}\)

                \({\upbeta }_{1j} = {{\upgamma}}_{10}+ {u}_{1j}\)

                \({\upbeta }_{2j}= {{\upgamma}}_{20}+ {u}_{2j}\)

                \({\upbeta }_{3j} = {{\upgamma}}_{30}+ {u}_{3j}\)

                \({\upbeta }_{4j} = {{\upgamma}}_{40}\)

Model 4: Model including all variables

 Level-1: \({Y}_{ij} = {\upbeta }_{0j}+{\upbeta }_{1j}{\left(\mathrm{ses}\right)}_{j}+ {\upbeta }_{2j}{\left(\mathrm{engage}\right)}_{j}+{\upbeta }_{3j}{\left(\mathrm{litnum}\right)}_{j}+ {\upbeta }_{4j}{\left(\mathrm{liking}\right)}_{j}+{\upbeta }_{5j}{\left(\mathrm{preschool}\right)}_{j}+ {r}_{ij}\)

 Level-2: \({\upbeta }_{0j} = {{\upgamma}}_{00}+{{\upgamma}}_{01}{\left(\mathrm{m}\_\mathrm{ses}\right)}_{j}+ {{\upgamma}}_{02}{\left(\mathrm{instruction}\right)}_{j}+{{\upgamma}}_{03}{\left(\mathrm{readiness}\right)}_{j}+{{\upgamma}}_{04}{(\mathrm{condition})}_{j}+{{\upgamma}}_{05}{(\mathrm{emphasis})}_{j}+{{\upgamma}}_{06}{(\mathrm{safe})}_{j}+ {u}_{0j}\)

                \({\upbeta }_{1j} = {{\upgamma}}_{10}+ {u}_{1j}\)

                \({\upbeta }_{2j} = {{\upgamma}}_{20}+ {u}_{2j}\)

                \({\upbeta }_{3j} = {{\upgamma}}_{30}+ {u}_{3j}\)

                \({\upbeta }_{4j}= {{\upgamma}}_{40}+ {\mathrm{u}}_{4j}\)

                \({\upbeta }_{5j} = {{\upgamma}}_{50}\)