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

Abstract

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 2 responses over time in 5-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 3 possible activity patterns in each interval (inactive, partially active, and completely active), and those patterns, which are observed, induce both nonrandom 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 of activity is exactly one, and the energy expenditure rate is greater than zero and likely to be higher. To address these challenges, we propose a 3-part functional data joint modeling approach. The first part is a continuation-ratio model to reorder the ordinal valued 3 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 3-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 3 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 quasi-likelihood type approximation. We develop an efficient algorithm to fit the model that 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.

A Review of Statistical Analyses on Physical Activity Data Collected from Accelerometers

“…Li et al [33] extend the definition of time intervals from two categories (inactive and active) to three categories (inactive, partially active and active). This extension can cover a wide range of activity combinations.…”

“…The proposed method uses the continuation-ratio model suggested by Molenberghs and Verbeke [37] The model to handle excess zeros can facilitate efficient estimation of the energy expenditure rate for active behaviors, physical activity energy expenditure (PAEE). In the application from Li et al [33], there exists a relationship for Y i (t) + 1.25 = 1.25 × {1− P i (t)}+ P i (t)PAEE i (t), and thus the term Y i (t)/P i (t) represents energy expenditure rate for active behaviors (PAEE− 1.25). A typical research interest is to explore the PAEE in a 5-min interval with active behaviors use more than 2.5 min, which is equivalent to study the conditional expectation E{Y i (t)/P i (t)|C (1) i (t) = 1, P i (t) > 0.5}.…”

Bayesian joint modeling of patient-reported longitudinal data on frequency and duration of migraine

“…In the migraine study, migraine specialists were mainly interested in the co-evolution of migraine frequency and duration over time, as these two outcomes are biologically associated [25]. This medical research question led us to consider joint modeling of these two longitudinal outcomes, as it is known that joint models provide better insights in the analysis of longitudinal multivariate data with increased efficiency due to information exchange between outcomes and allow estimation of the association between outcomes [3,[10][11][12][13]. In this sense, following the novel papers of [8] and [16], a joint model can be constructed as follows: First, separate generalized linear mixed models (GLMMs) can be used to model each longitudinal outcome under an appropriate distribution from the exponential family, and then a bivariate GLMM can be constructed by imposing a bivariate normal distribution on random effects to jointly analyze the longitudinal migraine data with two count outcomes.…”

Multivariate functional additive mixed models

“…Such an approach can become computationally quite expensive when a higher number of functional outcomes are jointly analyzed. Li et al [2018] extend this approach by including an ordinal component to model different distributional assumptions for the bivariate functional outcome depending on the individual's activity pattern. Cao et al [2019] apply a Gaussian copula model to skewed continuous functional data but do not provide an extension to discrete functional responses.…”