Optimal scaling of a longitudinal choice variable with time-varying representation of individuals

A new method of homogeneity analysis is proposed for analysing the repeated measurements of n individuals' choices among m categories. In this method, time-varying scores are assigned to each individual, while time-invariant ones are assigned to each category. For obtaining such scores, a requirement of homogeneity for time-varying individual scores is introduced in addition to the individual-category homogeneity requirement underlying the existing homogeneity analysis. The loss function based on these two requirements is minimized under a suitable normalization condition. This minimization is explicitly attained through eigen-value decomposition, and the graphical representation of the resulting scores allows us to easily find the longitudinal changes in individuals and the relationships among categories.