A general model and an associated method of data analysis are presented. It is assumed that observations on a set of response variables have a multivariate normal distribution with a general parametric form of the mean vector and the variance-covariance matrix. Any parameter of the model may be fixed, free or constrained to be equal to other parameters. The free and constrained parameters are estimated by the maximum likelihood method. Approximate standard errors and confidence intervals for the estimated parameters may be obtained by computing the inverse of the information matrix. The adequacy of any specific model contained in the general model may be tested by the likelihood ratio technique, yielding a large sample chi-square test of goodness of fit. Great generality and flexibility are obtained, in that a wide range of models is contained in the general model by imposing various specifications on the parametric structure of the general model. Part II of this paper (RB-69-47) deals with applications to various problems mainly from the field of psychology. In part I the general model is dealt with purely formally without reference to any particular specialization or application of it. Expressions for first-order derivatives and expected values of second-order derivatives of the likelihood function are derived, and the method of maximizing the likelihood function is described.