Estimating a time-to-event distribution from right-truncated data in an epidemic: A review of methods

Abstract

Time-to-event data are right-truncated if only individuals who have experienced the event by a certain time can be included in the sample. For example, we may be interested in estimating the distribution of time from onset of disease symptoms to death and only have data on individuals who have died. This may be the case, for example, at the beginning of an epidemic. Right truncation causes the distribution of times to event in the sample to be biased towards shorter times compared to the population distribution, and appropriate statistical methods should be used to account for this bias. This article is a review of such methods, particularly in the context of an infectious disease epidemic, like COVID-19. We consider methods for estimating the marginal time-to-event distribution, and compare their efficiencies. (Non-)identifiability of the distribution is an important issue with right-truncated data, particularly at the beginning of an epidemic, and this is discussed in detail. We also review methods for estimating the effects of covariates on the time to event. An illustration of the application of many of these methods is provided, using data on individuals who had died with coronavirus disease by 5 April 2020.

Keywords: Coronavirus disease; Cox regression; failure time; identifiability; relative efficiency; right-truncation; survival analysis.

Figure 1.

For each of five values of $\lambda$ and of four values of ${\theta }_1$ , graph shows the asymptotic relative efficiency (ARE) of the estimator of $E(T)$ based on (i) $L_1^{\mathrm{kwn}}$ (solid line), (ii) $L_3$ (broken line) and (iii) $L_4$ (dot-dash line) all compared to (iv) $L_1^{\mathrm{est}}$ . The $x$ -axis of each graph is $\tau$ and the $y$ -axis is the ARE.

Figure 2.

Distribution of symptom onset times in the sample of 304 individuals.

Figure 3.

Estimated survival curves from the gamma model (left) and log normal model (right), obtained using likelihoods $L_1^{\mathrm{est}}$ , $L_1^{\mathrm{kwn}}$ , $L_3$ and $L_4$ . Dotted lines represent 95% confidence intervals (Estimates using $L_1^{\mathrm{kwn}}$ and $L_4$ are so close they may be hard to distinguish.).

Figure 4.

Comparison of non-parametric estimate (step function) of survival conditional on delay being less than 31 days with corresponding estimates from the gamma model (left) and log normal model (right). Dotted lines represent 95% confidence intervals for the non-parametric estimate. Estimates using $L_1^{\mathrm{kwn}}$ and $L_4$ are so close that they are shown by a single line, and estimates using $L_1^{\mathrm{est}}$ and $L_3$ are so close that they may be difficult to distinguish.

Figure 5.

Estimate of log hazard ratio associated with sex=female as a function of $P(T\ge \tau \mid \text{sex=male})$ . Dotted lines indicate 95% confidence limits calculated by bootstrap; these may be unreliable when $P(T\ge \tau \mid \text{sex=male})$ is large (see text).