Estimation

Several estimation techniques are used in hierarchical linear modelling since the model comprises different types of parameters. Specifically, the level 1 coefficients, fij, can be fixed (that is, equal to a constant: Pu of model 3 in Table I), non-randomly varying (that is, vary across level 2 units, for example, physicians, but solely as a function of a level 2 predictor Wy. j of model 4 in Table I) or random (that is, vary across level 2 units, for example, physicians: fioj of models 1-6, of models 5 and 6 in Table I). The level 2 coefficients, y, are considered fixed effects and the level 1 and 2 variances and covariances (<r2, t00, Toi and tu) are called the covariance components. The estimation techniques for each type of paramter are outlined below. More theoretical details are available in Littell et al.2 and in Searle et al.10

4.1.1. Estimating Fixed Effects (y)

Weighted least squares (WLS) or generalized least squares (GLS) is used to estimate y as shown below:

where

A is the N x 4 design matrix with N = Y,j= x «j (see (9)), and V is V with G and -R replaced by their maximum likelihood estimates. The elements of G and R (that is, t00, x01, :u and a2) are called the variance-covariance components and are estimated by maximum likelihood (ML) or restricted maximum likelihood (REML) as described in Section 4.1.2. The variance of the estimator y (11) is estimated by vâr (f) = (ATP~1A)-1.

Liang and Zeger11 recommended the following as an alternative to (13), particularly in the case when the variances of Y are not homogeneous across level 2 units (for example, physicians):

vâr(y) = (Ar V ~1 A)~1AT V ~1 (Y — Ay) (Y — Ay)TV~1A(ATV~1A)~1

4.1.2. Estimating Covariance Components (R and G)

If the design is perfectly balanced (that is, nj all equal and the distribution of level 1 predictors within each level 2 unit, (for example, within each physician practice, is the same) there are closed-form formulae for estimating the variance-covariance parameters.10 When the design is unbalanced, iterative numerical procedures are used to obtain the estimates. Usually these procedures are based on maximum likelihood estimation techniques.

Maximum likelihood (ML) estimates of G and R are found by maximizing the following log-likelihood function (see also Littell et al.2 and Searle et al,10):

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