where F is the mean item intercorrelation at the school level, which can be estimated using the variances in the intercept-only model by
7 = alupii/(aluPi, + aL) • The relationship between
Equation (8) and Equation (9), based on the Spearman-Brown formula, is explained by Raudenbush et al. (1991).
Equation (9) shows that the internal consistency reliability can be improved not only by including more items in the scale, but also by sampling a larger number of pupils in each school. Raudenbush et al. (1991) demonstrated that increasing the number of pupils making judgments per school increases the school-level reliability faster than increasing the number of items in the scale. Even with a low interitem correlation and a low intra-class correlation, increasing the number of pupils to infinity will in the end produce a reliability equal to one, whereas increasing the number of items to infinity generally will not.
If we want to predict the evaluation scores of the school principal using school-level variables (e.g., the experience or gender of the school principal or type of school), we can simply include these variables as explanatory variables in the multilevel model. We can also estimate the school principals' evaluation scores using the school-level residuals. We can add pupil-level explanatory variables to the model, which would lead to evaluation scores that are conditional on the pupil-level variables. This can be used to correct the evaluation scores for inequalities in the composition of the pupil population across schools.
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