**2.4 Multilevel model estimation**

The estimation of a multilevel model is complex because, in addition to the residuals at the individual level in the model, there are more residual terms of random intercepts and/or slopes of higher levels. Simultaneously, three types of parameters need to be estimated: the fixed effects, the random effects, and the residual variance/ covariance components in matrices *D*, *G*, and *R*. Statistical theory and estimation algorithms for multilevel modeling are beyond the scope of this chapter, but some ideas are given.

When matrices *D*, *G*, and *R* are known, they can be used to estimate the combined model using generalized least square (GLS). The variance of *y*, given that the matrices *D* and *G* are known, is

$$
\hat{V} = \textbf{W}\textbf{D}\textbf{W}' + \textbf{Z}\textbf{G}\textbf{Z}' + \textbf{R} \tag{22}
$$

The inverse of the *V*^ matrix can be used as a weight; the regression coefficients of the model can be estimated using GLS. However, the matrices *D* and *G* are unknown.

The maximum likelihood estimation method is the most used for estimating multilevel models. It consists of maximizing the likelihood function that generally involves an iterative process that takes the parameter estimates as the initial parameter values for the next iteration of parameter estimation. This process is repeated until the parameter estimates have stabilized from one iteration to the next. The default *tolerance number*, which is sometimes defined by the users, is usually a sufficiently small number, for example, 10�8. The model converges if the tolerance number is reached between two consecutive iterations. However, sometimes this does not happen. If the limit of specified iterations is reached and the tolerance number between two consecutive iterations has not been reached, the method is said to not converge, and this fact may indicate model specification problems or a small sample size.

Other estimation methods used in multilevel models are *generalized estimating equations*, *bootstrap methods,* and *Bayesian methods* [3]. When the assumptions of the multilevel models (Section 2.3) are not met, these methods are adequate.
