Multilevel Longitudinal Functional Principal Component Model

Abstract

Sensor devices, such as accelerometers, are widely used for measuring physical activity (PA). These devices provide outputs at fine granularity (e.g., 10-100 Hz or minute-level), which while providing rich data on activity patterns, also pose computational challenges with multilevel densely sampled data, resulting in PA records that are measured continuously across multiple days and visits. On the other hand, a scalar health outcome (e.g., BMI) is usually observed only at the individual or visit level. This leads to a discrepancy in numbers of nested levels between the predictors (PA) and outcomes, raising analytic challenges. To address this issue, we proposed a multilevel longitudinal functional principal component analysis (mLFPCA) model to directly model multilevel functional PA inputs in a longitudinal study, and then implemented a longitudinal functional principal component regression (FPCR) to explore the association between PA and obesity-related health outcomes. Additionally, we conducted a comprehensive simulation study to examine the impact of imbalanced multilevel data on both mLFPCA and FPCR performance and offer guidelines for selecting optimal methods.

Keywords: functional principal component analysis; functional regression; unbalanced study design.

FIGURE 1 ∣

An example of the first 600 min (x-axis) of daily physical activity (PA) magnitude counts on 3 days (denoted by varying colors) from minute-level accelerometer count data for one subject across three visits (baseline, 6, and 12 months). The daily records were realigned to have a “common” starting time of device wear denoted as “0” on the x-axis. Values on y-axis provide the vector magnitude of PA measured from accelerometers.

FIGURE 2 ∣

MENU study application: The first three estimated principal components (rows) for the random intercept (1st column), random slope (2nd column), visit-specific process (3rd column) and day-specific process (4th column). The y-axis of the plots give the overall mean value curve $\mu (t)$ (black) with addition (red) or subtraction (blue) of 2 square root of eigenvalues multiplying first, second or third level principal component curves. The %s in the bottom left of each graph are the percent of variation explained by that component. The x-axis is time (min) representing 10 h (600 min) of activity.

FIGURE 3 ∣

An example of PA magnitude counts (y-axis) with raw count inputs (thin solid), smoothed curves (thick solid) and model-recovered curves (thick dashed) at baseline (top), 6 months (middle), and 12 months (bottom), during 600 min (x-axis) of each day, starting from when the participants began wearing the device. The daily records were realigned to have a “common” starting time of device wear denoted as “0” on the x-axis. Different colors of the line represent the day of the measurement.

FIGURE 4 ∣

MENU study application: Estimated functional coefficients curve with 95% pointwise confidence intervals (shaded grey) when implementing the longitudinal FPCR model on log(Insulin) (top), BMI (middle) and HOMA (bottom), with $U$ (Level 1) and $V$ (Level 2) processes reconstructed from the fitted multilevel longitudinal FPCA (mLFPCA) model as functional predictors, after adjusting for age, ethnicity, smoking status, and visit >1; x-axis is time in minutes.