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

We illustrate the application of the proposed Bayesian piecewise modeling approach by using it to investigate the divergence in child-to adult trajectories of BMI between participants who do and do not develop adult T2DM in a well-studied ongoing population-based prospective cohort, the Cardiovascular Risk in YFS [15]. Details on study design and on the collection of cardiovascular risk factors between 1980 and 2011 are published elsewhere [98] and summarized in Additional file 1.

In a previously published work on the YFS cohort, elevated BMI in children between 9 and 18 years was associated with an increased risk of developing T2DM in adulthood [87]. Additionally, a sex- and insulin-adjusted mixed model incorporating participants ages as a categorical variable, suggested that differences in average BMI values between those who do and those who do not develop adult T2DM tended to emerge during adolescence, becoming marginally significant from the age of 15 years onwards. In this approach, the between-group difference at each age groups was assessed by pairwise comparisons of the predicted marginal means (i.e. LS-means), and did not incorporate BMI trajectory information at the individual or population level. In contrast, the proposed hierarchical piecewise regression approach considers and makes full use of individual trajectory information to test for group-differences at specific stages of BMI development from childhood to adulthood. Unlike categorical approaches, the proposed growth model provides a clearer representation of the underlying pathological BMI development among those who develop T2DM in adulthood.

In our illustration, we include 2540 YFS participants (1401 females and 1139 males) followed-up a maximum of six times between 1980 and 2011 (Additional file 2: Table S1). Information on adult T2DM status was collected on participants at their latest individual adult follow-up (i.e. dichotomous outcome coded 0 for participants without T2DM, and 1 for those with T2DM in 2001, 2007, or 2011). Included participants had at least one BMI measure available in childhood (i.e. in 1980, 1983 or 1986 between age 3 and 18 years). Participants had on average 4.98 repeated measures of BMI over the study period, with 90.7% of participants having 4 or more BMI measures (Additional file 2: Figure S1). 88 included participants (3.5%, 44 females and 44 males) had T2DM in adulthood. We excluded BMI observations made among those aged 3 years in 1980 so that the ages of participants considered in the trajectory analysis ranged from 6 to 49 years. 3 year olds were not included in the analysis since only 3 participants in this birth cohort developed T2DM in adulthood. Furthermore, the lack of BMI measures between 3 and 6 years, prevented modeling the downwards slope from infancy peak, nor the age at adiposity rebound, which usually occurs before age 6 years in normal weight children [99, 100]. Using BMI data collected on participants aged 6 years and over, we expect that most participants had reached this important childhood milestone, and that a linear trend was thus an appropriate functional form to approximate childhood BMI growth from that age (Additional file 3: Figure S1).

Since sex differences in childhood growth and pubertal timing have been demonstrated [101, 102], subsequent BMI trajectory modeling between age 6 and 49 years was conducted among males and females separately [103]. BMI, especially in adulthood, is slightly right skewed, but using log10 transformed BMI in the modeling approach presented below did not alter our conclusions. For ease of interpretation, we thus present results using untransformed BMI only.

Visual inspection of the sex-specific smoothed BMI trajectories confirms the presence of a divergence between the two groups in adolescence (Additional file 3: Figure S2). Compared to participants who remain healthy, those who develop T2DM seem to have greater average BMI levels by the time they are young adults, although it is unclear whether this divergence results from a group-difference in the timing at which the transition to a slower BMI growth rate happens (Type II divergence) from a group-difference in rate itself after puberty (Type I divergence), or from both (Type III divergence).

Although the distal outcome of ‘adult T2DM’ is the grouping factor of interest in our illustrative trajectory divergence analyses, we also demonstrate how the same modeling approach can be used to investigate potential inter-cohort variation in childhood to adulthood BMI trajectories by considering models with ‘year of birth’ as a categorical level 2 predictor of each of the 4 trajectory parameters. Individual age- and sex-specific BMI Z-scores at the first clinic (in 1980) were also included as level 2 predictors of each BMI trajectory parameters to investigate if systematic deviation from participants of comparable age and sex at baseline had any influence on the development of BMI trajectories later in life. All continuous covariates used in the analyses were standardized in order to stabilize the variance, improve normality of errors and linearity of the mean.

Specific values for the hyperparameters used in our illustrative analyses are given in Additional file 4. While in principle φ can be unstructured, in our application, convergence for some parameters could not be reached when considering an unrestricted covariance structure between all four random effects in the unconditional change point model (equations 1.1 and 1.2), probably due to over parameterisation. Because initial analyses suggested a correlation between the slopes before the change point (b _1i ) and the difference in slopes after the change point (b _2i ), we constrained the model by including a non-zero correlation between these two random effects but setting independence for all other random effects, leading to a block diagonal structure of φ (Additional file 4). Based on DIC, this covariance structure was preferred over mutually independent random effects for both males and females (Additional file 4), and used when expending the trajectory models with level 2 predictors. In our application, we investigated prior sensitivity by fitting the unconditional BMI trajectory model using three sets of priors for the hyperparameters (Additional file 4). Because we found that the choice of hyperparameters had a minor influence on the marginal posterior distributions, for subsequent (conditional) analyses, we chose to report posterior estimates of parameters estimated from the set of priors that yielded the lowest DIC in the sensitivity analyses (Additional file 4). In this set the priors for the means of the change points were based on the sex-specific estimates that maximized the profile log likelihood for the fixed (population-average) breakpoints in the unconditional model (estimated at 16 years for females and 22 for males, see estimation method in Additional file 5). Using these priors for the change point means also kept computation running times reasonable.

For the other participants varying variable included in the analysis, sex- and age-specific BMI z-scores at the first visit and birth cohort, priors were set to N ~ (0,0.001) for all corresponding parameters (i.e. all β _cohort priors and β _{initialBMI − z − score} ). To remain consistent with previous analyses of this data set [87], time-varying measures of fasting insulin were log-transformed and standardized before being included as a level-1 predictor in the Bayesian hierarchical models to improve right skewedness and to linearize its relationship with BMI About 17% of the insulin measures were not available in the data. The missing data mechanism for ‘insulin’ was considered non-informative, as we have no reason to believe that the probability of an individual insulin measure being missing depends on the true value of this missing insulin observation (although it may be related to other observed variables for that individual). We thus consider that insulin is missing at random (MAR), and we specify a prior for this covariate [104]. Since log (insulin) is approximately normally distributed, we specify a N ~ (μ _log(insulin), τ _log(insulin) ) likelihood for log(insulin) _i and place a vague prior on its variance (i.e. τ _{log(insulin) ~ Gamma(0.001,0,001)} ). Under this parametrization, the posterior predictive distribution for μ _log(insulin) and τ _log(insulin) will be informed by the observed part of the data only. Although individual insulin measurements change at each data collection point, by adding log(insulin) as a level 1 covariate in the multilevel model, the estimated relationship between insulin and BMI development remains constant across time [45]. This is a reasonable assumption in our application, since data exploration did not suggest any systematic patterns of change in insulin levels at the intra-individual level as people age. That is, the age smoother estimate obtained by fitting a generalized additive mixed model had an estimated degree of freedom (edf) close to 1 and was not significant (p-value >0.3), which did not suggest a non-linear relationship between log(insulin) and age [105].

Approximate posterior distributions of the parameters of the models considered throughout the analyses are obtained via MCMC simulations. Each model ran with 4 independent parallel chains of the Gibbs sampler (see Appendix 3 for an example of code). For each model, the first 50000 iterations were discarded in a burn-in run, and the draws from the posterior were thinned by a factor of 10 to reduce serial correlation of the chains. The following 20000 iterations were used to obtain posterior distributions of model parameters by mixing the 4 sequences. Model convergence was assessed through MCMC iterations traceplots and Gelman-Rubin diagnostic [92], and residual errors were plotted to confirm they approximately followed a normal distribution.

Following the modeling approach presented in the Methods and the priors and their corresponding hyperparameters (Additional file 4: Table S1) we fitted the following set of conditional Bayesian hierarchical piecewise models for each sex: unconditional (Model A), adult T2DM status adjusted intercept (Model B), adult T2DM status adjusted childhood slope (Model C), adult T2DM status adjusted adult slope (Model D), adult T2DM status adjusted change point (CP) (Model E), adult T2DM status adjusted CP and adult slope (Model F), adult T2DM status adjusted change point, childhood and adult slopes (Model G), adult T2DM status adjusted intercept, and change point (Model G), and a model with all four parameters adjusted for adult T2DM status (Model H). As mentioned above, previous research on this data set suggested BMI levels were not significantly different between the two groups in childhood [87]. Models C (i.e. group difference in childhood slopes) and B (i.e. BMI response consistently higher in one group across the life course) were thus fitted to demonstrate our modeling approach. An annotated extract showing the RJAGS code syntax used to fit Model E is available in Additional file 6.

For both sex, the lowest DIC was obtained when fitting model E, which was also the best fitting model with PP p-values close to 0.5 (Table 1). This supported the type II divergence mechanism where a difference in BMI levels emerged between the two groups due to a group difference in the change point timing. BMI growth rate in adulthood for both sexes was decreased by two-thirds compared to childhood (i.e. 0.67- vs. 0.18 -, and 0.61- vs. 0.15 kg/m² per year in childhood and adulthood for females and males respectively), and participants who developed T2DM had similar BMI yearly rates in adulthood compared to those who remained healthy (β_2T2DM effect not significant in model F for both sex Table 2). However, females who developed T2DM reached their developmental transition in BMI rate on average 12.37 years later (Table 2).

Similarly for males, estimated BMI growth rates were not markedly different between the two T2DM groups in childhood or in adulthood, and comparable to those estimated in females (Table 2). But again, compared with healthy adults, those who developed T2DM reached their slower BMI growth rate on average 6.47 years later.

The effect of the time-varying covariate of insulin at level 1 was significant for both males and females, with a 1-sd increase in log(Insulin) resulting in a BMI observation increased by 2.6 and 2.8 kg/m² respectively (i.e. exp(β _log(insulin)), Table 2). To assess potential differences in the magnitude of the insulin effect as a function of between-person characteristics, we expanded model E by including an interaction between ‘adult T2DM status’ and log(insulin). For each sex, the estimated parameters were not significant (95% CI included 0), suggesting that the effect of insulin on BMI was homogenous between the two groups and across genders.

The estimates of the variance-covariance parameters of model E showed that the correlation between an individual’s BMI growth rate in childhood and adulthood is equal to 0.61 for females and 0.47 for males, suggesting that children who have greater yearly BMI increase rates also have greater adult rates of increase (correlation estimated as: \( \raisebox{1ex}{${\sigma}_{\beta 1\beta 2}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{\sigma_{\beta 1}^2*{\sigma}_{\beta 2}^2}$}\right. \), Table 2). The between-participant variation around the change point σ _CP was comparable between males and females (Table 2).

Figure 2 shows the estimated population-average prototypical trajectories for each sex and T2DM group obtained from the estimated parameters for Model E, along with 100 trajectories predicted for each sex and T2DM group from Model E by Monte Carlo simulation. This illustrates a range of credible individual profiles generated under this model (see Appendix for code). For each sex and adult T2DM status group, Fig. 3. shows a box and whiskers plot of the estimated individual BMI slopes obtained from Model E after the average change point in the healthy group and before the T2DM groups reach their average CP (i.e. slopes between 16.02 and 28.4 years in females, and slopes between 21.62 and 28.09 years in males). Figure 3 illustrates that individual rates of change after puberty provides better discrimination of participants who went on to develop T2DM from those who did not, compared to punctual individual BMI levels at age 15 or 18 for females, and ages 21 and 24 for males (Additional file 2: Figure S2). While the distribution of BMI levels at age groups surrounding the age at divergence overlaps considerably (Additional file 2: Figure S2), individual slopes allow to differentiate participants who have switched to a rate consistent with a normal slowing down of BMI development after puberty, from those who are still on the trajectory of increasing BMI development consistent with the rate from childhood.

A short series of simulations was conducted to compare difference in estimates of the age at which the groups diverge when using the proposed Bayesian piecewise growth modeling approach compared to a more traditional approach based on pairwise comparison of LS-means estimated from a categorical mixed model. We simulated repeated measure data from a Type II divergence model (i.e. group-difference in the change point timing only), using the posterior estimates of mean growth parameters for the model fitted for females (average parameters are set to: β _O = 26.5, β ₁ = 0.67, CP = 16.02, = β _GroupCP =12.37, β ₂ = −0.49, matching Model E posterior estimates for females in Table 2), and both a participant-level random effect (σ _error ²  = 2.77) and an observation-level residual error (σ _error ²  = 2.47). Under this model, “CP”, the change point for the first group to depart from the population-average childhood slope represents the age at which the two groups of participants diverge in their outcome trajectories (i.e. the second group maintains this rate of change for 12.30 years longer). To resemble the YFS BMI data, we randomly sampled baseline ages from the YFS cohort subtracted by 25 years as ages at the first visit for each participant, with 6 non-missing repeated measures 3, 6, 9, 21, 27 and 31 years later for each participant. We considered 3 scenarios of sample sizes for the number of participants in each group (group 1/group 2): (1) 100/100, (2) 50/100, and (3) 30/100. For each of the three scenarios, we simulated 100 datasets and fitted both a mixed model with age as a categorical variable and the Type II divergence Bayesian Hierarchical piecewise model using the set of priors defined in Additional file 4. For each piecewise model, we recorded the posterior estimate for the “CP” parameter, and for each fitted categorical mixed model, we applied pairwise comparison of the least-square means (LS-means) with Tukey adjustment for multiplicity to retrieve: (1) the earliest age at which the group-difference in means was found significant (p < 0.05), and (2) the midway point between two consecutive ages that had a minimum number of non-significant differences in means before and significant differences in means after the “midway point” method (2) is a potential alternative definition of age at which the group-difference appears in the LS means. Compared to the “earliest age with p-val <0.05” method (1), the “midway” point definition minimises the impact of simulations where some tests show significance at a young age, even though tests for the surrounding ages are not. For each scenario, estimated ages at divergence using the 3 methods were averaged across the 100 simulations. Figure 4 presents the simulation results in term of the quartiles distribution and means of these estimates of age at divergence across the 100 simulations. When sample size decreases for one group of participants, the pairwise LS mean comparison method will tend to overestimate the age at divergence, with significant variability in the estimates arising due to random variation, especially when age at significance is determined using the first age at which a p-value <0.05 occurs (Fig. 4.). In contrast, the hierarchical Bayesian piecewise model was less sensitive to sample size, and the true age at divergence was consistently within the estimated interquartile range of the produced estimates, indicating that the Bayesian trajectory divergence model outperforms the LS mean method in both accuracy and precision, regardless of the way “age at divergence” is defined from the model output.