Model-assisted analyses of longitudinal, ordinal outcomes with absorbing states

Abstract

Studies of critically ill, hospitalized patients often follow participants and characterize daily health status using an ordinal outcome variable. Statistically, longitudinal proportional odds models are a natural choice in these settings since such models can parsimoniously summarize differences across patient groups and over time. However, when one or more of the outcome states is absorbing, the proportional odds assumption for the follow-up time parameter will likely be violated, and more flexible longitudinal models are needed. Motivated by the VIOLET Study (Ginde et al), a parallel-arm, randomized clinical trial of Vitamin $D_3$ in critically ill patients, we discuss and contrast several treatment effect estimands based on time-dependent odds ratio parameters, and we detail contemporary modeling approaches. In VIOLET, the outcome is a four-level ordinal variable where the lowest "not alive" state is absorbing and the highest "at-home" state is nearly absorbing. We discuss flexible extensions of the proportional odds model for longitudinal data that can be used for either model-based inference, where the odds ratio estimator is taken directly from the model fit, or for model-assisted inferences, where heterogeneity across cumulative log odds dichotomizations is modeled and results are summarized to obtain an overall odds ratio estimator. We focus on direct estimation of cumulative probability model (CPM) parameters using likelihood-based analysis procedures that naturally handle absorbing states. We illustrate the modeling procedures, the relative precision of model-based and model-assisted estimators, and the possible differences in the values for which the estimators are consistent through simulations and analysis of the VIOLET Study data.

Keywords: absorbing state; longitudinal data; marginalized models; ordinal responses; partial proportional odds; proportional odds; randomized clinical trial.

FIGURE 1

VIOLET Study data: Empirical cumulative log odds across 27 follow-up days

FIGURE 2

Dichotomization-specific functional form of the intervention association [i.e., log odds ratio (LOR)] across no $\left({\beta }_{xt,2}^{ppo2},{\beta }_{xt,3}^{ppo2}\right)=(0,0)$, moderate $\left({\beta }_{xt,2}^{ppo2},{\beta }_{xt,3}^{ppo2}\right)=(-0.1,-0.2)$, and severe $\left({\beta }_{xt,2}^{ppo2},{\beta }_{xt,3}^{ppo2}\right)=(-0.2,-0.4)$ violations of the PO assumption for the simulation studies, where subscripts 2 and 3 indicate contrasts for k = 2 and k = 3 versus k = 1.

FIGURE 3

Model-based and model-assisted summaries: For high (a) and moderate (b) response dependence settings, we report averages of the t_ij–specific estimates of intervention effects using a global log odds ratio [LOR(t_ij)] and the empirical standard errors (ESE) across 1000 replicates. We includes scenarios with no violation $\left[\left({\beta }_{xt,2}^{ppo\text{2 }},{\beta }_{xt,3}^{ppo\text{2 }}\right)=(0,0)\right]$, moderate violation $\left[\left({\beta }_{xt,2}^{ppo2},{\beta }_{xt,3}^{ppo2}\right)=(-0.1,-0.2)\right]$, and severe violation $\left[\left({\beta }_{xt,2}^{ppo2},{\beta }_{xt,3}^{ppo2}\right)=(-0.2,-0.4)\right]$ of the proportional odds assumption for the intervention effect.

FIGURE 4

Calibrations plots for the cumulative pr[Y_i(t_ij) ≤ k | X_i(t_ij)] over the 27 follow-up days. For models ppo1 and ppo2 we display the predicted minus the observed probabilities of falling at or below state k ∈ {1, 2, 3}.

FIGURE 5

VIOLET Study results: Intervention effect estimates and confidence interval widths for model-based, model-assisted, and cross-sectional model fits