Multilevel Modeling

The intraclass correlation (ICC) is the correlation between observations in the same cluster; equivalently, the proportion of total variance attributable to clustering. ICC = between-cluster variance / total variance. ICC = 0 means no clustering (all variation within clusters); ICC = 1 means complete clustering (all variation between clusters). ICC > 0.05 typically justifies multilevel analysis. Small ICC (0.01–0.05) means clusters are similar, but clustering still inflates standard errors if ignored. Rules of thumb: ICC > 0.10 strong clustering, 0.05–0.10 moderate, < 0.05 weak. Calculate ICC after fitting unconditional model (no predictors) to assess baseline clustering.

Fixed effects are average relationships across clusters (e.g., the average effect of age on outcome across schools). Random effects are cluster-specific deviations from the average (e.g., how each school's age effect differs from the grand mean). A random intercept allows the baseline outcome to differ by cluster (each school has its own intercept). A random slope allows the effect of a predictor to differ by cluster (each school's effect of age differs). Include random effects if clusters differ substantively in the effect; include fixed effects if the effect is conceptually the same across clusters. Random effect variances quantify heterogeneity; large variance indicates effects differ substantially.

REML (Restricted Maximum Likelihood) is preferred for comparing random effects models (e.g., comparing random slopes models); it produces unbiased variance component estimates. ML (Maximum Likelihood) is preferred for comparing fixed effects (especially when comparing non-nested models) and for likelihood ratio tests of fixed effects. With large samples, REML and ML estimates converge. With small samples and few level 2 clusters, REML is more reliable. Always report which method you used; results can differ, especially with small samples.

Centering transforms predictors to improve interpretation and numerical stability. Grand-mean centering (subtract overall mean) centers level 1 predictors around the global mean; the intercept represents outcome when the predictor equals the grand mean. Group-mean centering (subtract cluster mean) centers level 1 predictors around each cluster's mean; separates within-cluster and between-cluster effects. For example, centering student-level socioeconomic status (SES) at school mean isolates the within-school effect of SES on achievement; the cluster mean (school-level SES) becomes a level 2 predictor, representing school composition effect. Grand-mean centering is default; group-mean centering used when distinguishing within vs between effects theoretically important.