ICCS 2009 Technical Report - IEA

The joint explained variance contribution reflects the proportion of variance explained by morethan one of k sets of predictors. The proportion of variance explained by more than one set ofpredictors JVCj was computed askJVC j = (r 2 n x100) – S UVC jk=1 .Hierarchical linear modelingHierarchical (or multilevel) linear regression models (Raudenbush & Bryk, 2002), wereestimated in order to take school or classroom context effects into account in which studentswere nested within classrooms. In most of the country samples, the classroom level wasequivalent to the school level because typically only one classroom was selected within eachschool. Therefore, as with other IEA studies, it was not possible to separately estimate, in theanalyses presented in the ICCS reports the variance due to the classroom and school levels.A hierarchical regression model with i students nested in j clusters (schools, classrooms) can beestimated asY ij = a j + X n ij b ij +X m j b j +U 0j +e ij ,where Y ij are the criterion variables, X n ij is a vector of student-level variables, with itscorresponding vector of regression coefficients b ij , and X m j is a school- (or classroom-) levelvariable with its corresponding vector of regression coefficients b j . U 0j is the residual term atthe level of the cluster (school or classroom), and e ij is the student-level residual. Both residualterms are assumed to have a mean of 0 and variance that is normally distributed at each level.The explained variance in hierarchical linear models has to be estimated for each levelseparately, with the estimate based on a comparison of each prediction model with the baseline(“null”) model (or ANOVA model) without any predictor variables. Thus:null nullY ij = a j + U 0j + U ij .nullU 0jThe residual termis an estimate of the variance between i students within clusters. The intra-class correlationIC, which reflects the proportion of variance between clusters (in this case, schools), can becomputed from these estimates asnullU 0je ijIC =null nullU 0j + .provides an estimate of the variance in Y ij between j clusters, and e nullijIn order to provide a comparable baseline model for the ICCS multilevel analysis, the “null”model estimated. This model is the one from which students with missing data were excludedafter “missing treatment” procedures had been completed (see section on missing treatmentbelow). The explained variance at the school level EV j was computed asU 0jEV j = (1– ) x 100nullU 0j,and the explained variance at the student level EV ij was computed ase ijEV ij = (1– ) x 100enull ij.Because multilevel modeling takes the hierarchical structure of the cluster sample into account,HLM standard errors that took both sampling and imputation errors into account werereported. Data were weighted (with normalized weights) at both levels.SCALING PROCEDURES FOR ICCS questionnaire ITEMS269