Parametric modelling of prevalent cohort data with uncertainty in the measurement of the initial onset date

Abstract

In prevalent cohort studies with follow-up, if disease duration is the focus, the date of onset must be obtained retrospectively. For some diseases, such as Alzheimer’s disease, the very notion of a date of onset is unclear, and it can be assumed that the reported date of onset acts only as a proxy for the unknown true date of onset. When adjusting for onset dates reported with error, the features of left-truncation and potential right-censoring of the failure times must be modeled appropriately. Under the assumptions of a classical measurement error model for the onset times and an underlying parametric failure time model, we propose a maximum likelihood estimator for the failure time distribution parameters which requires only the observed backward recurrence times. Costly and time-consuming follow-up may therefore be avoided. We validate the maximum likelihood estimator on simulated datasets under varying parameter combinations and apply the proposed method to the Canadian Study of Health and Aging dataset.

Appendix

We derive the likelihood given in (3) by calculating the approximate observation specific probability contributions. Let $\varDelta t_i$ denote an arbitrarily small positive number. Then,

$$\begin{aligned}&{\mathbb {P}}(T_i^{\text {obs}} \in [t_i, t_i + \varDelta t_i) | X_i> T_i^{\text {brt}}) \\&\quad = \frac{{\mathbb {P}}(T_i^{\text {obs}} \in [t_i, t_i + \varDelta t_i), X_i> T_i^{\text {brt}})}{{\mathbb {P}}(X_i> T_i^{\text {brt}})} \\&\quad = \frac{\int _{0}^{\infty } {\mathbb {P}}(T_i^{\text {obs}} \in [t_i, t_i + \varDelta t_i), X_i> T_i^{\text {brt}} | T_i^{\text {brt}} = u) f_{T_i^{\text {brt}}}(u) du}{\int _{0}^{\infty } {\mathbb {P}}(X_i > T_i^{\text {brt}} | T_i^{\text {brt}} = v)f_{T_i^{\text {brt}}}(v) dv} \end{aligned}$$

which, through the assumption that $T_i^{\text {obs}}$ is independent of $X_i$ conditional on $T_i^{\text {brt}}$, for all i,

$$\begin{aligned} = \frac{\int _{0}^{\infty } {\mathbb {P}}(T_i^{\text {obs}} \in [t_i, t_i + \varDelta t_i) | T_i^{\text {brt}} = u) S(u) f_{T_i^{\text {brt}}}(u) du}{\int _{0}^{\infty } S(u) f_{T_i^{\text {brt}}}(u) du} \end{aligned}$$

Dividing by $\varDelta t_i$ and letting $\varDelta t_i$ tend to zero, under the assumption of a stationary onset process, the equation above corresponds to the i-th term given in (3).