An essential issue in the estimation of a covariance matrix in longitudinal data is its positive definiteness. This constraint creates a major obstacle and subsequently several alternative techniques are introduced in the literature to take over the problem. In this thesis a robust technique is addressed to develop the correlation matrix of the longitudinal data . This is constructed upon the Cholesky decomposition of the underlying covariance matrix. The technique is shown to be effective in the analysis of longitudinal data in the sense that the positive-definiteness of the estimated covariance is guaranteed. It has also the unique distinction of providing an unconstrained and statistically meaningful re-parameterization of the covariance matrix, but at the expense of imposing an order among the underlying random variables . The decomposition involves a positive-definite diagonal matrix proportional to the square root of diagonal entries of the covariance matrix and a unit lower-triangular matrix to enable constructing uniquely the correlation matrix. Consequently , such factorization amounts to directly assembling the covariance matrix but in a manner that its estimate does not depend on the quality of formulating and estimating the innovation variances . In other words , the estimation of the correlation matrix is robust to the miecification of models for the innovation variances since their components are shared by the use of alternative Cholesky decomposition . Then, the first robustness is provided with respect to the model miecification for the innovation variances and the second is to the effects of outliers in the data . The latter is handled using the heavy-tailed multivariate t - distribution with unknown degrees of freedom . We note that, the assumption of multivariate normality is commonly made for the vector of repeated measures on a subject may be implausible in many practical cases when outliers exist or the underlying longitudinal data exhibit heavy-tails . We implement a single method to robust estimating topics on a variety of real data sets . The proposed approach replaces the normal distribution by the t distribution in fitting statistical models. There is no closed-form solution for the maximum likelihood estimates of the alternative Cholesky decomposition models , and thus a Fisher scoring algorithm is developed to compute the MLEs of model parameters. The algorithm is used such as to take into account the non-redundant and unconstrained entries of the diagonal matrix and the unit lower triangular matrix. The computation method and the entries form of the Fisher information matrix are slightly different for the alternative Cholesky decomposition. Thus, a modified Cholesky decomposition is employed. A comparative study is done for the usual and modified Cholesky decomposition methods by the use of conducting some simulations and analyzing a real data set .