Maximum likelihood estimation for length-biased and interval-censored data with a nonsusceptible fraction

Abstract

Left-truncated data are often encountered in epidemiological cohort studies, where individuals are recruited according to a certain cross-sectional sampling criterion. Length-biased data, a special case of left-truncated data, assume that the incidence of the initial event follows a homogeneous Poisson process. In this article, we consider an analysis of length-biased and interval-censored data with a nonsusceptible fraction. We first point out the importance of a well-defined target population, which depends on the prior knowledge for the support of the failure times of susceptible individuals. Given the target population, we proceed with a length-biased sampling and draw valid inferences from a length-biased sample. When there is no covariate, we show that it suffices to consider a discrete version of the survival function for the susceptible individuals with jump points at the left endpoints of the censoring intervals when maximizing the full likelihood function, and propose an EM algorithm to obtain the nonparametric maximum likelihood estimates of nonsusceptible rate and the survival function of the susceptible individuals. We also develop a novel graphical method for assessing the stationarity assumption. When covariates are present, we consider the Cox proportional hazards model for the survival time of the susceptible individuals and the logistic regression model for the probability of being susceptible. We construct the full likelihood function and obtain the nonparametric maximum likelihood estimates of the regression parameters by employing the EM algorithm. The large sample properties of the estimates are established. The performance of the method is assessed by simulations. The proposed model and method are applied to data from an early-onset diabetes mellitus study.

Appendix: Determine the possible jump points of G(t)

The major issue here is to determine the possible jump points of G(t), which implies the possible jump points of F(t). We could show that it suffices to consider the discrete distribution function G(t) with jumps points at $l_i, i=1,\ldots ,n$. The arguments are as follows. From (2.3), for any pair $(\pi ^*,F^*)$, the likelihood function can be reparameterized as

$$\begin{aligned}&{\mathcal {L}}(\pi ^{*}, G^{*})\\&\qquad =\prod _{i=1}^n \left( {\tilde{\pi }}^{*} \int _{l_i}^{u_i} \frac{1}{t} dG^{*}(t) \right) ^{\delta _i}\left( \frac{1-{\tilde{\pi }}^{*}}{\tau }\right. \\&\left. \qquad \quad +{\tilde{\pi }}^{*} \int _{l_i}^{\infty } \frac{1}{t} dG^{*}(t) \right) ^{(1-\delta _i)I(\omega _i=0)} \left( \frac{1-{\tilde{\pi }}^{*}}{\tau }\right) ^{(1-\delta _i)I(\omega _i=1)}, \end{aligned}$$

where ${\tilde{\pi }}^{*}=\pi ^{*} \mu ^{*}/(\tau (1-\pi ^{*})+\pi ^{*}\mu ^{*})$ and $\mu ^{*}=1/(\int t^{-1} dG^{*}(t))$. If $G^*$ is a step function which assigns positive probability to the points between $l_i$ and $u_i$ for some i, then we can obtain greater value for $\int _{l_i}^{u_i} t^{-1} dG(t)$ by shifting the mass to $l_i$ since the integrand 1/t is a decreasing function of t. Similarly, if $G^*$ puts mass at a point on the left side of the smallest of $l_i$’s, a greater likelihood value can be obtained by shifting that point to the smallest of $l_i$’s since none of the integration in (2) contains that point. Although $\mu =1/(\int t^{-1} dG(t))$ is smaller than $\mu ^*=1/(\int t^{-1} dG^{*}(t))$, we can choose $\pi =\tau {{\tilde{\pi }}}^{*}/((1-{{\tilde{\pi }}}^{*})\mu +\tau {{\tilde{\pi }}}^{*})$ such that ${\tilde{\pi }}={\tilde{\pi }}^{*}$. Then, ${\mathcal {L}}(\pi , G)>{\mathcal {L}}(\pi ^*, G^*)$. The proof is complete.