Hierarchical Linear Models: Estimation of Effects ; In estimating the level-1 and level-2 models discussed above, a distinction is made between fixed effects, random coefficients, and variance components Fixed effects are parameter estimates that do not vary across groups, for example, the [Gamma]'s from equations 2 and 3. Alternatively, random coefficients are parameter estimates that are allowed to vary across groups such as the level-1 regression coefficients (e.g., [[Beta].sub.0j] and [[Beta].sub.1j]). In addition to these level-1 and level-2 regression coefficients, hierarchical linear models also include estimates of the variance components which include: (1) the variance in the level-1 residual (i.e., [r.sub.ij] referred to as ([[Sigma].sup.2]), (2) the variance in the level-2 residuals (i.e., [U.sub.0j] and [U.sub.ij]), and (3) the covariance of the level-2 residuals [ie, cov ([U.sub.0j], [U.sub.1j])]. The variance-covariance matrix of the level-2 residuals is referred to in the hierarchical linear modeling literature as [Tau], therefore, element [[Tau].sub.00] represents the variance in [U.sub.0j], element [[Tau].sub.11] represents the variance in [U.sub.1j] and element [[Tau].sub.10] represents the covariance between [U.sub.0j] and [U.sub.ij]. Obviously, the number of elements in the x matrix will depend on the number of level-2 equations estimated.
The [Gamma]'s in equations 2 and 3 represent fixed effects in hierarchical linear models Although these level-2 regression weights could be estimated using an Ordinary Least squares (OLS) regression approach, this is not appropriate given that the precision of the level-1 parameters will most likely vary across groups. Given this varying precision, an OLS approach is not appropriate due to the violation of the homoscedasticity assumption Hierarchical linear models use a Generalized Least squares (GLS) estimate for the level-2 parameters which provide a weighted level-2 regression such that the groups with more precise level-1 estimates (i.e., more precise estimates of the dependent variable; that is, intercepts and slopes) receive more weight in the level-2 regression equation.
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