Order‐restricted inference for multivariate longitudinal data with applications to the natural history of hearing loss

Multivariate outcomes are often measured longitudinally. For example, in hearing loss studies, hearing thresholds for each subject are measured repeatedly over time at several frequencies. Thus, each patient is associated with a multivariate longitudinal outcome. The multivariate mixed-effects model is a useful tool for the analysis of such data. There are situations in which the parameters of the model are subject to some restrictions or constraints. For example, it is known that hearing thresholds, at every frequency, increase with age. Moreover, this age-related threshold elevation is monotone in frequency, that is, the higher the frequency, the higher, on average, is the rate of threshold elevation. This means that there is a natural ordering among the different frequencies in the rate of hearing loss. In practice, this amounts to imposing a set of constraints on the different frequencies' regression coefficients modeling the mean effect of time and age at entry to the study on hearing thresholds. The aforementioned constraints should be accounted for in the analysis. The result is a multivariate longitudinal model with restricted parameters. We propose estimation and testing procedures for such models. We show that ignoring the constraints may lead to misleading inferences regarding the direction and the magnitude of various effects. Moreover, simulations show that incorporating the constraints substantially improves the mean squared error of the estimates and the power of the tests. We used this methodology to analyze a real hearing loss study.