Bivariate joint models for survival and change of cognitive function

Abstract

Changes in cognitive function over time are of interest in ageing research. A joint model is constructed to investigate. Generally, cognitive function is measured through more than one test, and the test scores are integers. The aim is to investigate two test scores and use an extension of a bivariate binomial distribution to define a new joint model. This bivariate distribution model the correlation between the two test scores. To deal with attrition due to death, the Weibull hazard model and the Gompertz hazard model are used. A shared random-effects model is constructed, and the random effects are assumed to follow a bivariate normal distribution. It is shown how to incorporate random effects that link the bivariate longitudinal model and the survival model. The joint model is applied to the English Longitudinal Study of Ageing data.

Keywords: Joint model; bivariate binomial distribution; cognitive function; shared random-effects model; survival analysis.

Figure 1.

Bias and corresponding Monte Carlo 95% confidence interval. Circles represent biases, and geometry bars represent Monte Carlo 95% confidence interval.

Figure 2.

Frequency distribution of recalled words at baseline interview. $Y_1$ represents immediate-recall words; $Y_2$ represents delayed-recall words.

Figure 3.

Recalled words trajectories. Left-hand side: immediate-recall words $Y_1$ trajectories. Right-hand side: delayed-recall words $Y_2$ trajectories. Individuals are represented in the same colour in both plots.

Figure 4.

Fitted bivariate binomial distribution for immediate recall ( $Y_1$) and delayed recall ( $Y_2$). Left hand side: conditional on age 50; Right hand side: conditional on age 80.

Figure 5.

Comparison of K-M survival curves: predicted survival curves (grey lines), the mean of those survival curves (blue line) and the median of survival curves (red line).