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Multivariate ordinal and quantitative longitudinal data measuring the same latent construct are frequently collected in psychology. We propose an approach to describe change over time of the latent process underlying multiple longitudinal outcomes of different types (binary, ordinal, quantitative). By relying on random-effect models, this approach handles individually varying and outcome-specific measurement times. A linear mixed model describes the latent process trajectory while equations of observation combine outcome-specific threshold models for binary or ordinal outcomes and models based on flexible parameterized non-linear families of transformations for Gaussian and non-Gaussian quantitative outcomes. As models assuming continuous distributions may be also used with discrete outcomes, we propose likelihood and information criteria for discrete data to compare the goodness of fit of models assuming either a continuous or a discrete distribution for discrete data. Two analyses of the repeated measures of the Mini-Mental State Examination, a 20-item psychometric test, illustrate the method. First, we highlight the usefulness of parameterized non-linear transformations by comparing different flexible families of transformation for modelling the test as a sum score. Then, change over time of the latent construct underlying directly the 20 items is described using two-parameter longitudinal item response models that are specific cases of the approach.