Modeling semi-competing risks data as a longitudinal bivariate process

Abstract

As individuals age, death is a competing risk for Alzheimer's disease (AD) but the reverse is not the case. As such, studies of AD can be placed within the semi-competing risks framework. Central to semi-competing risks, and in contrast to standard competing risks , is that one can learn about the dependence structure between the two events. To-date, however, most methods for semi-competing risks treat dependence as a nuisance and not a potential source of new clinical knowledge. We propose a novel regression-based framework that views the two time-to-event outcomes through the lens of a longitudinal bivariate process on a partition of the time scales of the two events. A key innovation of the framework is that dependence is represented in two distinct forms, local and global dependence, both of which have intuitive clinical interpretations. Estimation and inference are performed via penalized maximum likelihood, and can accommodate right censoring, left truncation, and time-varying covariates. An important consequence of the partitioning of the time scale is that an ambiguity regarding the specific form of the likelihood contribution may arise; a strategy for sensitivity analyses regarding this issue is described. The framework is then used to investigate the role of gender and having ≥1 apolipoprotein E (APOE) ε4 allele on the joint risk of AD and death using data from the Adult Changes in Thought study.

Keywords: alzheimer's disease; b-splines; discrete-time survival; longitudinal modeling; penalized maximum likelihood.

FIGURE 1

Graphical representation of the interplay between standard notation for semi-competing risks outcome data and the proposed bivariate longitudinal framework; see Sections 2 and 3.2 for details.

FIGURE 2

Comparison in the ACT data between the three approaches to overcome within-interval censoring and study entry times. The figures present the estimated baseline time trends, $\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{i}\mathrm{t}\left({\widehat{\mathit{\alpha }}}_1\right),\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{i}\mathrm{t}\left({\widehat{\mathit{\alpha }}}_2\right)$, and $\mathrm{e}\mathrm{x}\mathrm{p}\left({\widehat{\mathit{\alpha }}}_{\theta }\right)$, under the unstructured specification and the B-spline specification with $\lambda \in \{0.0,2.5\}$ and under the two partitions ${\mathit{\tau }}^{2.5}$ (top panel) and ${\mathit{\tau }}^{5.0}$ (bottom panel). Note that the $y$-axis has been truncated at 0.8 for $\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{i}\mathrm{t}\left({\widehat{\mathit{\alpha }}}_1\right)$ and at 10 for $\mathrm{e}\mathrm{x}\mathrm{p}\left({\widehat{\mathit{\alpha }}}_{\theta }\right)$.

FIGURE 3

Estimated baseline time trends, $\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{i}\mathrm{t}\left({\widehat{\mathit{\alpha }}}_1\right),\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{i}\mathrm{t}\left({\widehat{\mathit{\alpha }}}_2\right)$, and $\mathrm{e}\mathrm{x}\mathrm{p}\left({\widehat{\mathit{\alpha }}}_{\theta }\right)$ from a series of analyses to the ACT data, under the unstructured specification and the B-spline specification with $\lambda \in \{0.0,2.5\}$ and under the two partitions ${\mathit{\tau }}^{2.5}$ (top panel) and ${\mathit{\tau }}^{5.0}$ (bottom panel). See Section 6.1 for details. Also shown are 95% confidence intervals. Note that the $y$-axis has been truncated at 0.8 for $\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{i}\mathrm{t}\left({\widehat{\mathit{\alpha }}}_1\right)$ and at 10 for $\mathrm{e}\mathrm{x}\mathrm{p}\left({\widehat{\mathit{\alpha }}}_{\theta }\right)$. For results under more $\lambda$ values, see Supplementary Figures A.10 and A.11. Results presented under the nearest neighbor strategy