Nonparametric estimation of the lifetime and disease onset distributions for a survival-sacrifice model

Abstract

In carcinogenicity experiments with animals where the tumor is not palpable it is common to observe only the time of death of the animal, the cause of death (the tumor or another independent cause, as sacrifice) and whether the tumor was present at the time of death. These last two indicator variables are evaluated after an autopsy. Defining the non-negative variables $T_{1}$ (time of tumor onset), $T_{2}$ (time of death from the tumor) and $C$ (time of death from an unrelated cause), we observe $(Y,\Delta_{1},\Delta_{2})$, where $Y=\min\left\{T_{2},C\right\}$, $\Delta_{1}=1_{\left\{T_{1}\leq C\right\}}$, and $\Delta_{2}=1_{\left\{T_{2}\leq C\right\}}$. The random variables $T_{1}$ and $T_{2}$ are independent of $C$ and have a joint distribution such that $P(T_{1}\leq T_{2})=1$. Some authors call this model a “survival-sacrifice model”.

[20] (generally to be denoted by LJP (1997)) proposed a Weighted Least Squares estimator for $F_{1}$ (the marginal distribution function of $T_{1}$), using the Kaplan-Meier estimator of $F_{2}$ (the marginal distribution function of $T_{2}$). The authors claimed that their estimator is more efficient than the MLE (maximum likelihood estimator) of $F_{1}$ and that the Kaplan-Meier estimator is more efficient than the MLE of $F_{2}$. However, we show that the MLE of $F_{1}$ was not computed correctly, and that the (claimed) MLE estimate of $F_{1}$ is even undefined in the case of active constraints.

In our simulation study we used a primal-dual interior point algorithm to obtain the true MLE of $F_{1}$. The results showed a better performance of the MLE of $F_{1}$ over the weighted least squares estimator in LJP (1997) for points where $F_{1}$ is close to $F_{2}$. Moreover, application to the model, used in the simulation study of LJP (1997), showed smaller variances of the MLE estimators of the first and second moments for both $F_{1}$ and $F_{2}$, and sample sizes from 100 up to 5000, in comparison to the estimates, based on the weighted least squares estimator for $F_{1}$, proposed in LJP (1997), and the Kaplan-Meier estimator for $F_{2}$. R scripts are provided for computing the estimates either with the primal-dual interior point method or by the EM algorithm.

In spite of the long history of the model in the biometrics literature (since about 1982), basic properties of the real maximum likelihood estimator (MLE) were still unknown. We give necessary and sufficient conditions for the MLE (Theorem 3.1), as an element of a cone, where the number of generators of the cone increases quadratically with sample size. From this and a self-consistency equation, turned into a Volterra integral equation, we derive the consistency of the MLE (Theorem 4.1). We conjecture that (under some natural conditions) one can extend the methods, used to prove consistency, to proving that the MLE is $\sqrt{n}$ consistent for $F_{2}$ and cube root $n$ convergent for $F_{1}$, but this has presently not yet been proved.