HLM

The Levels of Hierarchy

HLM allows users to model data sets that have two to four levels of hierarchy. For each level, there is a submodel with its own structural and random components. The structural component represents, at that level, the relations among variables, and the random component denotes the residual variability. In a school effects study in which the sample consists of students clustered within schools, for example, there are two levels of nesting and subsequently two submodels. With i = 1,…,nj students (Level-1 units) nested within j = 1,…, J schools (Level-2 units), the first sub- or Level-1 model for the achievement of student i in school j, Achij can be represented as:

$$Ac{h}_{ij}={\beta }_{0j}+{\displaystyle \sum _{q=1}^Q{\beta }_{q\text{}j}{X}_{qij}}+r_{ij}$$

where the structural component consists of the intercept, β0j, and the Level-1 coefficients and predictors, βqj and Xqij(q = 1,…,Q). The random component is denoted by rij, which is assumed to be normally distributed with a variance of σ2. Some Level-1 predictor examples are family socioeconomic status (SES), gender, and prior achievement. A given βqj relates the qth predictor to the achievement outcome. Letting the qth predictor be SES, the coefficient βqj will thus index the strength and direction of the student SES-achievement association for school j.

The second sub- or Level-2 model predicts the intercept, β0j, and the Level-1 coefficients βqj. For the intercept and the qth Level-1 coefficient, the model can generally be represented as:

$$\begin{array}{c}{\beta }_{0\text{}j}={\gamma }_{00}+{\displaystyle \sum _{s=1}^{S_0}{\gamma }_{0s}{W}_{s\text{}j}}+u_{0j}\\ {\beta }_{q\text{}j}={\gamma }_{q0}+{\displaystyle \sum _{s=1}^{S_q}{\gamma }_{qs}{W}_{s\text{}j}}+u_{q\text{}j},\end{array}$$

where the structural component consists of the intercepts, γ00 and γq0, and the Level-2 coefficients and predictors, γ0s, γqs, and Wsj (s = 1,…,S0 or q). The random component is denoted by u0j and uqj, which are assumed to be distributed as bivariate normal with dispersion T. The matrix T contains the variance and covariance components τ00, τqq, and τq0. Some Level-2 predictor examples are school type (private vs. public), school SES, and curricular policies. The parameter γ0s and a given γqs relate the sth school-level predictor to the Level-1 intercept and the qth Level-1 coefficient, respectively. Letting sth school-level predictor be school type, a researcher can use it to model the Level-1 intercept, β0j, and assess how it relates to school achievement. The relationship will be captured by γ0s. The same predictor can also be used to model a given βqj, for example, the coefficient for SES, to see if the SES-achievement associations vary between the two types of schools or, equivalently, whether school type moderates the relationships between student SES and achievement. The estimate of γqs captures this moderating effect.

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