Bayesian hierarchical piecewise regression models: a tool to detect trajectory divergence between groups in long-term observational studies

Abstract

Background: Bayesian hierarchical piecewise regression (BHPR) modeling has not been previously formulated to detect and characterise the mechanism of trajectory divergence between groups of participants that have longitudinal responses with distinct developmental phases. These models are useful when participants in a prospective cohort study are grouped according to a distal dichotomous health outcome. Indeed, a refined understanding of how deleterious risk factor profiles develop across the life-course may help inform early-life interventions. Previous techniques to determine between-group differences in risk factors at each age may result in biased estimate of the age at divergence.

Methods: We demonstrate the use of Bayesian hierarchical piecewise regression (BHPR) to generate a point estimate and credible interval for the age at which trajectories diverge between groups for continuous outcome measures that exhibit non-linear within-person response profiles over time. We illustrate our approach by modeling the divergence in childhood-to-adulthood body mass index (BMI) trajectories between two groups of adults with/without type 2 diabetes mellitus (T2DM) in the Cardiovascular Risk in Young Finns Study (YFS).

Results: Using the proposed BHPR approach, we estimated the BMI profiles of participants with T2DM diverged from healthy participants at age 16 years for males (95% credible interval (CI):13.5-18 years) and 21 years for females (95% CI: 19.5-23 years). These data suggest that a critical window for weight management intervention in preventing T2DM might exist before the age when BMI growth rate is naturally expected to decrease. Simulation showed that when using pairwise comparison of least-square means from categorical mixed models, smaller sample sizes tended to conclude a later age of divergence. In contrast, the point estimate of the divergence time is not biased by sample size when using the proposed BHPR method.

Conclusions: BHPR is a powerful analytic tool to model long-term non-linear longitudinal outcomes, enabling the identification of the age at which risk factor trajectories diverge between groups of participants. The method is suitable for the analysis of unbalanced longitudinal data, with only a limited number of repeated measures per participants and where the time-related outcome is typically marked by transitional changes or by distinct phases of change over time.

Keywords: Accelerated longitudinal design; Cohort effect; Group divergence; Hierarchical regression; Non-linear trajectory model; Piecewise model.

Fig. 1

Three hypothetical models of between-group divergence in curvilinear response trajectories over time. Red and black solid lines indicate the average response curve of participants belonging to one or the other group; dashed lines show the position and age at change point(s) for the two groups of participants, or the age at which trajectories diverge between the two groups. Graph obtained using simulated data

Fig. 2

Sex-specific population average prototypical BMI trajectories for healthy and T2DM adults in the YFS cohort (solid blue and solid red lines, respectively) and prediction of 200 individual trajectories for each sex (100 per T2DM status group). The dashed trajectories were obtained by MCMC simulation using sex-specific posterior estimates of mean and variance of growth parameters for the best fitting models (Model E). In these predictions, time varying measures of log(insulin) were set to the average log(Insulin) observed in the cohort

Fig. 3

Box and whiskers plot of fitted individuals random slopes between 16.02 and 28.4 years for females (a) and between 21.62 and 28.09 years for males (b). Individual random slopes are estimated from the Bayesian hierarchical random change point model E. Solid lines in the boxplot indicate the group-specific median for the slopes (equivalent to the 50th percentiles of the posterior distribution)

Fig. 4

Boxplots and mean “Age at divergence” (x) estimated across 100 simulations using the three methods. Bottom and top of the boxes are the lower (Q1) and upper quartiles (Q3), respectively; the bands near the middle of the boxes are the medians, the lengths of the boxes represent the interquartile range (IQR = Q3-Q1); the upper whiskers are defined as min(max(x), Q3 + 1.5 * IQR) and the lower whiskers as max(min(x), Q1 – 1.5 * IQR). Means of age at divergence across the 100 simulations for each scenario are indicated with empty circles. The horizontal dashed line indicates the true age at divergence set in the simulations (16.02 years old)