SUMMARY Use of the maximum likelihood method for parameter estimation in a general mixed effects model requires initial estimates for all the model parameters. When the model is complicated, it may not be easy to obtain such estimates, especially estimates of the variances and covariances. A computational strategy is given for obtaining initial estimates. It is based on a general formulation of the method of moments for mixed effects models which allows the variances and covariances to be estimated for fixed values of the regression parameters. A number of numerical algorithms has been proposed for obtaining maximum, or restricted maximum, likelihood estimates of the parameters of a linear mixed effects model (Hemmerle & Hartley, 1973; Jennrich & Sampson, 1976; Corbeil & Searle, 1976; Laird & Ware, 1982). The maximum likelihood method can also be used with a model in which some fixed effects enter nonlinearly, while all random effects enter linearly; see ? 2. With any of the algorithms it is important to have good initial estimates for all the model parameters. One computational strategy sometimes used for finding initial estimates is to first maximize the full likelihood or log likelihood function, referred to here as the objective function, over a mesh, or grid, of points covering the parameter space and then to use the mesh point corresponding to the maximum as the initial estimate. However, with a complicated mixed effects model the dimension of the parameter space can be large enough to make this strategy impractical. Also, this strategy implies that practical bounds can be placed on all variances and covariances of the model, and though this may be possible, it may not be easy to do, especially when the model is complicated. This paper presents a modification of this strategy. It makes use of a general formulation of the method of moments for mixed effects models which is applicable to obtaining estimates of the variances and covariances of such models for fixed values of the regression parameters. Thus initial estimates of the variances-covariances are obtained without recourse to a grid search over at least those dimensions of the parameter space corresponding to the variances and covariances. The method of moments formulation permits estimates to be obtained under various constraints which might be imposed on the variances and covariances. It is also applicable to multivariate mixed effects models, and