A joint modeling and estimation method for multivariate longitudinal data with mixed types of responses to analyze physical activity data generated by accelerometers

Abstract

A mixed effect model is proposed to jointly analyze multivariate longitudinal data with continuous, proportion, count, and binary responses. The association of the variables is modeled through the correlation of random effects. We use a quasi-likelihood type approximation for nonlinear variables and transform the proposed model into a multivariate linear mixed model framework for estimation and inference. Via an extension to the EM approach, an efficient algorithm is developed to fit the model. The method is applied to physical activity data, which uses a wearable accelerometer device to measure daily movement and energy expenditure information. Our approach is also evaluated by a simulation study.

Keywords: accelerometers; longitudinal data; mixed effects model; multivariate longitudinal data; penalized quasi-likelihood.

Figure 1

Sample data from two subjects on weeks 0, 3, 6, 9 and 12. Solid lines and “X” labels display the observations from individual with ID 4. Dashed lines and ”O” labels represent the outcomes from individual with ID 5. (a) Y⁽¹⁾: continuous variable for daily sedentary hours; (b) Y⁽²⁾: continuous variable for energy expenditure; (c) Y⁽³⁾: proportion of sedentary time greater than 20 minutes, (d) Y⁽⁴⁾: proportion of active time greater than 5 minutes; (e) Y⁽⁵⁾: count number of daily standing up behaviors; (f) Y⁽⁶⁾: count number of daily steps; (g) Y⁽⁷⁾: binary variable for whether daily moderate to vigorous physical activity (MVPA) time is greater than one hour; (h) Y⁽⁸⁾: binary variable for whether the highest energy expenditure rate measured by metabolic equivalents (METs) in 10 minutes is greater than 3.

Figure 2

Simulation results for the conditional expectations for ℓ = 1, 2 defined in Section 4. (a)(b) moderate sample size scenario with n = 200 and J_i = 5, (c)(d) large sample size scenario with n = 400 and J_i = 9. Dotted lines denote the true conditional expectation values. Solid lines represent the averaged values of the estimates from our JOINT-PQL2 method. Shadowed areas display the 10% to 90% quantiles of the estimated values in 500 simulation runs. Thick and thin dashed lines represent the averaged estimates by the NAIVE1 and NAIVE2 methods, respectively.

Figure 3

The estimates of conditional expectations across five weeks for daily sedentary hours $(Y_{ij}^{(1)})$ and energy expenditure levels $(Y_{ij}^{(2)})$ defined in Section 5. The individual who reaches the criteria in Section 5 at Week 0 is defined as active participant, while those who do not meet any term at Week 0 is defined as inactive participant. Thick and thin lines represent the estimates of active and inactive participants, respectively. Solid and dotted lines display the exercise treatment group and the control group, respectively.