Consistency of the Semi-Parametric MLE under the Piecewise Proportional Hazards Models with Interval-Censored Data

We consider the piecewise proportional hazards (PWPH) model with interval censored (IC) relapse times under the distribution-free set-up. The partial likelihood approach is not applicable for IC data, and the generalized likelihood approach is studied by Wong et al. [1]. It turns out that under the PWPH model with IC data, the semi-parametric MLE(SMLE) of the covariate effect under the standard generalized likelihood may not be unique and may not be consistent. In fact, the parameter under the PWPH model with IC data is not identifiable unless the Identifiability assumption is imposed. They proposed a modification to the likelihood function so that its SMLE is unique. Under certain regularity conditions, we show that the SMLE is consistent and is asymptotically normally distributed.


Introduction
We establish the consistency of the semi-parametric MLE under the piecewise proportional hazards (PWPH) model, with interval-censored (IC) continuous survival time Y. The proportional hazards (PH) model specifies that a covariate vector Z has a proportional effect on the hazard function of Y. It is a common regression model for survival analysis. The PWPH model is a special PH model.  The observable  random vector is   0 , , ,   [10]. Wong et al. [11] applied the PWPH model to analyze their cancer research data. In a cancer research data set, Yi is the relapse time of a cancer patient after surgery, Zi is a vector with numerical or categorical coordinates, containing information about the age, tumor size at surgery, nodal number, bone marrow micro metastasis (bmm) or other information about the i-th patient. One is interested in the conditional survival function | SY z instead of SY. For instance, Wong et al. [11] considered a problem of studying the relation between the covariate bmm with IC relapse time Y of a breast cancer patient after the surgery. The covariate bmm is a categorical variable taking two values, say 1 (bmm positive) and 0 (otherwise). Some medical doctors suspected that the bmm effect might depend on time T. Then a PWPH model is as follows. The semi-parametric problem under the PWPH model with IC data was studied by Wong et al. [1]. It turns out that under PWPH model(1) with IC data, the parameter β is not identifiable unless further assumptions are imposed (see Example 1). Moreover, in general, the SMLE of β under the likelihood function (1.4) may not be unique. Both phenomena do not occur if the covariates are time-independent . They specified the Identifiability condition for such problems and studied the estimation problem of deriving the SMLE. Their simulation results suggest that the SMLEs of So and β are consistent under the mixed case IC model [2]. We give the proof of the consistency and asymptotic normality of the SMLE in this paper.

The Main Results
We study consistency of the SMLE under the PWPH model with one cut point assuming Y is continuous in this paper. In particular, we consider the model where Z is a time-independent covariate vector (2.1). Y is subject to interval censoring under the mixed case IC model with the following up times ki C and the random number of follow-up times K. We first present some preliminary results [13].

Proposition 1
Without loss of generality (WLOG), we can assume that the covariates p i Z R ∈ and take at least p linearly independent values.

Given a random variable, say Y , let
SFL and SFR are defined in a similar manner.
with the parameter ( ) to assumption (2.2). If assumption (2.2) is violated, β is not identifiable, as is the case in the next example. I.
. Moreover, assume the Case 2 model, that is, the observable random vector and So be absolutely continuous, where Then is not identifiable. The proof is given in the Appendix.
The likelihood function with IC data is given by (1.4), i.e., . For the PH model, there are two differences between right censoring and interval censoring: (a) One can show that the SMLE is unique and is consistent under the standard RC model but may not be so under the standard interval censorship model, unless further assumptions are imposed (due to Identifiability). Proof. We shall give the proof in 4 steps. Abusing notation, write Let Ω be the sample space.
Step 2: It can be verified that Consequently, ( ) Step 3: Now we assign weight pn,i to each interval is the GMLE of ( ) ( ). Step 4 (Conclusion). Let ˆn Q denote the empirical estimator of Q, the distribution of (L,R,Z) and For simplicity in notation we shall assume that ( ) ( ) In order to prove the Lemma 3, we will introduce the Fatou's Lemma with varying measures.