Regularized reduced rank growth curve models

The growth curve model (GCM), also known as GMANOVA, is a useful technique for investigating patterns of change in repeated measurement data over time and examining the effects of predictor variables on temporal trajectories. The reduced rank feature had been introduced previously to GCM for capturing redundant information in the criterion variables in a parsimonious way. In this paper, a ridge type of regularization was incorporated to obtain better estimates of parameters. Separate ridge parameters were allowed in column and row regressions, and the generalized singular value decomposition (GSVD) was applied for rank reduction. It was shown that the regularized estimates of parameters could be obtained in closed form for fixed values of ridge parameters. Permutation tests were used to identify the best dimensionality in the solution, and the K-fold cross validation method was used to choose optimal values of the ridge parameters. A bootstrap method was used to assess the reliability of parameter estimates. The proposed model was further extended to a mixture of GMANOVA and MANOVA. Illustrative examples were given to demonstrate the usefulness of the proposed method.