## ABSTRACT

The direct approach for obtaining maximum likelihood (ML) or restricted maximum likelihood (REML) estimates of the B matrix and the parameters of the W_{
i
} matrix in the Laird-Ware model, (2.1), is to use initial guesses of the parameters to calculate the likelihood in equation (2.9) or the modified likelihood in equation (2.23). A nonlinear optimization program is used to search over the values of the unknown parameters to find the values that maximize the likelihood. The direct approach requires calculations with matrices of size n_{i}
× n_{i}
, where n_{i}
is the number of observations on subject i. These calculations are no problem when n_{i}
is not large, and the use of the Cholesky factorization for calculating the likelihood is very stable numerically. This chapter considers an alternate method of calculating likelihoods using state space representations and the Kalman filter. State space methods use recursive calculations, entering the observations one at a time, and require calculations with small matrices the size of which do not depend on how many observations are available on a subject. Also, state space representations are very powerful in their own right. When a problem can be set up in state space form, the methodology exists for obtaining estimates of the unknown parameters. In Chapter 5 the Laird-Ware model for longitudinal data is given in state space form so that the Kalman Filter can be used to calculate exact likelihoods for Gaussian errors.