Joint Bayesian longitudinal models for mixed outcome types and associated model selection techniques

Abstract

Motivated by data measuring progression of leishmaniosis in a cohort of US dogs, we develop a Bayesian longitudinal model with autoregressive errors to jointly analyze ordinal and continuous outcomes. Multivariate methods can borrow strength across responses and may produce improved longitudinal forecasts of disease progression over univariate methods. We explore the performance of our proposed model under simulation, and demonstrate that it has improved prediction accuracy over traditional Bayesian hierarchical models. We further identify an appropriate model selection criterion. We show that our method holds promise for use in the clinical setting, particularly when ordinal outcomes are measured alongside other variables types that may aid clinical decision making. This approach is particularly applicable when multiple, imperfect measures of disease progression are available.

Keywords: Bayesian; Longitudinal data analysis; MCMC; Ordinal regression.

Fig. 1

(Left) LeishVet score across time by subject. LeishVet score was not recorded at the first follow-up for all subjects. (Right) Log anti-SLA across time by subject. The linear regression line (blue) indicates a small positive trend in log anti-SLA over time

Fig. 2

(Left) Comparison of prior densities for ${\Sigma }_{\alpha 11}$ using the proposed IW and SIW implementations while $\mathrm{J}=2$ (the number of outcomes). (Right) Histogram of the implied prior on the correlations between the patient specific effects. The plot is based on $\frac{{\Sigma }_{\alpha 31}}{\sqrt{{\Sigma }_{\alpha 11}{\Sigma }_{\alpha 33}}}$, but all off diagonal terms have a similar result

Fig. 3

Trace and density plots for the cutpoint parameters corresponding to the latent outcome. Data comes from the randomly selected replicate dataset

Fig. 4

Trace and density plots residual variance and subject specific intercept variance corresponding to the latent outcome

Fig. 5

1–4 step ahead mean based point estimates for 9 sample canine patients, where lines are colored according to the model. Black circles indicate observations that were used to fit the model, while the black triangles were used to evaluate predictions

Fig. 6

Log anti-SLA plotted against LeishVet score. This image does not account for the dependent nature of the responses. LeishVet score is equal to 4 for only a single observation, thus presenting zero variability

Fig. 7

Marginal associations between the four covariates and baseline values of the dependent variables. The $p$-value between log SLA and log DPP is based on pearson correlation, while $p$-values for LeishVet score used fishers exact test

Fig. 8

leishmaniosis application: Trace and density plots for the one free cutpoint and variance term corresponding to the latent outcome

Fig. 9

leishmaniosis application: Trace and density plots subject specific variance terms corresponding to the latent outcome

Fig. 10

Posterior predictive checks for the continuous outcome for the leishmaniosis dataset. Checks are based on replicated the final outcomes for each subject using the posterior draws and their previous $n_i-1$ responses. Plots were made using the ppc_stat function from the bayesplot package (Gabry and Mahr 2021)

Fig. 11

LeishVet score probability forecasts using a bmrarm. Black dots indicate observed outcomes, which have probability of 1 for their true value

Fig. 12

Log anti-SLA forecasts using a bmrarm. Red dots indicate observed responses, while blue indicate the mean and 95% credible intervals of the predicted outcomes