Three-part joint modeling methods for complex functional data mixed with zero-and-one inflated proportions and zero inflated continuous outcomes with skewness

摘要

We take a functional data approach to longitudinal studies with complex bivariate outcomes. This work is motivated by data from a physical activity study that measured two responses over time in five-minute intervals. One response is the proportion of time active in each interval, a continuous proportions with excess zeros and ones. The other response, energy expenditure rate in the interval, is a continuous variable with excess zeros and skewness. This outcome is complex because there are three possible activity patterns in each interval (inactive, partially active and completely active), and those patterns, which are observed, induce both non-random and random associations between the responses. More specifically, the inactive pattern requires a zero value in both the proportion for active behavior and the energy expenditure rate; a partially active pattern means that the proportion of activity is strictly between zero and one and that the energy expenditure rate is greater than zero and likely to be moderate, and the completely active pattern means that the proportion active is exactly one, and the energy expenditure rate is greater than zero and likely to be higher. To address these challenges, we propose a three-part functional data joint modeling approach. The first part is a continuation-ratio model to reorder the ordinal valued three activity patterns. The second part models the proportions when they are in interval (0, 1). The last component specifies the skewed continuous energy expenditure rate with Box-Cox transformations when they are greater than zero. In this three-part model, the regression structures are specified as smooth curves measured at various time-points with random effects that have a correlation structure. The smoothed random curves for each variable are summarized using a few important principal components, and the association of the three longitudinal components is modeled through the association of the principal component scores. The difficulties in handling the ordinal and proportional variables are addressed using a quasilikelihood type approximation. We develop an efficient algorithm to fit the model which also involves the selection of the number of principal components. The method is applied to physical activity data and is evaluated empirically by a simulation study.