The Analysis of Multivariate Longitudinal Data: An Instructive Application of the Longitudinal Three-Mode Model

Structural equation modeling is a common technique to assess change in longitudinal designs. However, these models can become of unmanageable size with many measurement occasions. One solution is the imposition of Kronecker product restrictions to model the multivariate longitudinal structure of the data. The resulting longitudinal three-mode models (L3MMs) are very parsimonious and have attractive interpretation. This paper provides an instructive description of L3MMs. The models are applied to health-related quality of life (HRQL) data obtained from 682 patients with painful bone metastasis, with eight measurements at 13 occasions; before and every week after treatment with radiotherapy. We explain (1) how the imposition of Kronecker product restrictions can be used to model the multivariate longitudinal structure of the data, (2) how to interpret the Kronecker product restrictions and the resulting L3MM parameters, and (3) how to test substantive hypotheses in L3MMs. In addition, we discuss the challenges for the evaluation of (differences in) fit of these complex and parsimonious models. The L3MM restrictions lead to parsimonious models and provide insight in the change patterns of relationships between variables in addition to the general patterns of change. The L3MM thus provides a convenient model for multivariate longitudinal data, as it not only facilitates the analysis of complex longitudinal data but also the substantive interpretation of the dynamics of change.