Hierarchical Linear Modeling

Model A: Random-Intercepts Model

The random-intercepts model is the simplest model in which only group membership (here, schools) affects the level of achievement. In HLM, separate sets of regression equations are written at the individual (level 1) and group (level 2) levels of analysis.

Level 1 (Student-Level) Model

where i represents each student and j represents each school. Note that no predictors are included in Equation 1. β0j is the mean MathAch score for school j: eij is the within-school residual that captures the difference between individual MathAch score and the school mean MathAch. eij is assumed to be normally distributed, and the variance of eij is assumed to be homogeneous across schools [i.e., eij∼N(0, σ2) for all 160 schools]. As presented in Table 1 (Model A), the variance of eij is equal to σ2 = 39.15.

Level 2 (School-Level) Model

The level 2 model partitions each school's mean MathAch score into two parts

Here, γ00 = 12.64 is the overall mean MathAch score, averaging over the 160 school means. U0j captures the residual difference between individual school mean MathAch and the overall mean MathAch. τ00 = Var(U0j) = 8.55 is the variance of the residuals at level 2.

The random-intercept model not only provides an important baseline for model comparison, but it also allows the computation of the intraclass correlation (ICC), which is the proportion of the between variance to the sum of the between and within variance.

In this example, the ICC is

This ICC generally ranges from 0 to 1 based on Equation 3, with higher values indicating greater clustering. As the product of the ICC and average cluster size increases, the Type I error rate for the study quickly increases to unacceptable levels if ignoring the clustering and treating all observations as independent from each other. For example, with an average of approximately 45 students per school (i.e., 7,185 students/160 schools) and an ICC of approximately .20, a t test that compared the mean of two different school types (i.e., high minority enrollment vs. low minority enrollment), but ignored the clustering, would have a Type I error rate of more than .50 compared with the nominal level of α = .05. Similarly, the estimated 95% confidence interval is approximately .33 of the width of the correct confidence interval. In contrast, use of HLM procedures can maintain the Type I error rate at the nominal α = .05 level.

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