Residuals in the Extended Growth Curve Model

Four residuals in the MLNM(A1B1C1 + A2B2C2) are defined. The estimated model which is obtained by using the maximum likelihood theory is considered. The vec operator is applied to the estimated model and it turns out that the estimated model is the projection of the observations on the space generated by the design matrices which is the sum of two tensor product spaces. The estimated model is subtracted from the observations to get the ordinary residuals and the vec operator is then applied. It turns out that the residuals are obtained by projecting the observations on the space orthogonal to the space generated by the design matrices. This space is decomposed into four orthogonal spaces and the residuals are defined by projecting the observation matrix on each of the resulting components. The residuals are interpreted and some remarks are given as to what kind of information they give and how this information might be used to validate the model assumptions. It is shown that the residuals are symmetrically distributed around zero and are uncorrelated with each other. The covariance between the residuals and the estimated model as well as the dispersion matrices for the residuals are also given.