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In this paper, we discuss the theoretical validity of the observed partial likelihood (OPL) constructed in a Coxtype model under incomplete data with two class possibilities, such as missing binary covariates, a cure-mixture model or doubly censored data. A main result is establishing the asymptotic convergence of the OPL. To reach this result, as it is difficult to apply some standard tools in the survival analysis, we develop tools for weak convergence based on partial-sum processes. The result of the asymptotic convergence shown here indicates that a suitable order of the number of Monte Carlo trials is less than the square of the sample size. In addition, using numerical examples, we investigate how the asymptotic properties discussed here behave in a finite sample.

Although the Cox model [

In this paper, we discuss the theoretical validity of the OPL which appears in a Cox-type model under incomplete data beyond the right-censored form. In this area, one advantage of the OPL is that the baseline hazard function as a nuisance included in a Cox-type model is eliminated completely in the inferential likelihood. This yields a more stable computational system for optimization than that of the SPFL. For example, in a Cox-cure model, a computational process based the EM algorithm to obtain the SPFL easily fails to converge if a suitable starting value is not provided (e.g., see [

Generally, it is difficult to investigate computationally to what extent the MC approximations of the OPL are valid, since the exact computation requires a huge number of summands, as the sample size and incomplete information of data are increasing. For this reason, it is worth studying the OPL theoretically. However, it is not easy to complete such a study in one go, because standard tools to study asymptotic properties of Cox’s partial likelihood or the SPFL cannot be applied directly to an objective of the OPL. Therefore in this paper, for the sake of simplicity, we focus on the OPL constructed in incomplete data composed of unobserved two class labels. Typical cases of this type occur in a Cox-type model with incomplete data, such as missing binary covariates, a cure-mixture model or doubly censored data. As a main result, we establish the asymptotic convergence of the OPL and derive a limit form of the OPL. This result is a foundation or precondition for applying an infinite-dimensional Laplace approximation for integral on the baseline hazard. Such a Laplace approximation method will yield the other limit form of the OPL [

A matter of interest in practice concerns MC approximations of the OPL. One other significant point is that the result for the convergence of the exact OPL can be easily tailored to the context of the MC approximations. Based on such an argument, we show that a suitable order of the number of MC trials is less than O(n^{2}) Further, in Section 4 we investigate how the asymptotic properties discussed here behave in a finite sample.

In Section 2 we formulate the OPL in incomplete data with two class possibilities, providing several examples of interest; in Section 3 we develop the tools to obtain the main result and show the convergence of the OPL, and in Section 4 we discuss the performances of MC approximations.

Let

In the case that

where

coefficient vector

In this paper, we consider incomplete data where some of the

Each of these is used to construct the likelihoods. Further, if the event of

where

Let

bilities, the observed full likelihood (OFL) can be generally written as

with the elements such that

and

is the survival function of the

In the following three examples, we show how the form of the OFL is related to the representative cases. Hereafter, we will often omit

Example: Missing Binary Covariates. Let us assume that

first covariate is binary and may be missing,

where

Using the binomial expansion, this can be rewritten as

Example: Cox Cure-Mixture Model. The Cox cure-mixture model [10,12-15] is presumed to hold the proportional hazards model for uncured individuals and to be zero-hazard for cured ones. That is, we assume

We observe that

note here that

Example: Doubly Censored Data. In doubly censored data [16,17], left-censored data may be included. Let

In the phenomenal meaning, the common model is assumed regardless of the type of observations, but we do not define

where note that

Let

where

Let

and

Using these expressions, let

where

be the n-dimensional version of Minkowski’s measure

where

and

which is

where

Remark 1. In (2.5), even if we consider a difference or quotient between

We will now discuss how the mean of the log OPL converges to a deterministic function and provide Theorem 1 of the main result. The following conditions are assumed for these discussions.

Conditions A. Let

A1:

A2:

A4:

Condition A2 means

right-censoring time under a given

Theorem 1. Suppose that Conditions A are satisfied. Then, as

Theorem 1 is proved in Section 3.3. We prepare useful tools for such a proof in Sections 3.1 and 3.2 below. In Section 3.1, we discuss a relation needed to show that two OPL’s converge to the same limit, determining a plan (Lemmas 1 and 2) to obtain Theorem 1. In Section 3.2, following the plan, we provide a tool (Lemma 3) to give a weak convergence of all possible partial-sum processes.

Note that the OPL is constructed by an integral on

be functions that exist around

Lemma 1. Suppose that

with probability 1. Then, as

where

(Proof of Lemma 1). Using Taylor expansions of

the difference between

where, with some

Let us assume that

are shown by

Applying (3.1) to the above inequality, this lemma is proved.

Using Lemma 1, for several patterns of

Lemma 2. Suppose that

where

Remark 2. For simplicity, letting

denote

as the area of

because it is always satisfied that

Thus, we show that the operators of

(Proof of Lemma 2). Note that

because

This shows

Therefore, condition (3.2) gives (3.1).

Note that, by the portmanteau theorem, (3.2) is equivalent to, as

where

For simplicity, let

tions of more essential terms to consider in the convergence on

where examples of

can be regarded as

which is the conditional expectation of

So, letting

Example: Missing Binary Covariates. For simplicity, we assume that

while the expectations of terms which may form all possible partial-sums in

In these calculi, note that the Bayes rule is used, such as

Example: Cox Cure-Mixture Model. In this model,

which may form all possible partial-sums are

where

On Weak Convergence. Let

Remark 3. For example, if

However, a result of interest here is whether

Incidentally, we cannot obtain the almost sure convergence in this problem, since

is always apart from zero.

Lemma 3. Let

(i) The class of functions

(ii)

(iii)

then, as

Lemma 3 is proved in Appendix A.1. The following examples show that the conditions needed in Lemma 3 are

satisfied for

Example 1. Let

Example 2. Let

Consider

i.e.

and

by Conditions A1 and A4, similar to the standard Cox model (see [

is obtained as

using Lemma 3 in Example 1 and applying the strong law of large numbers (SLLN) to

by applying the SLLN on

using Lemma 3 in Example 2. Hence,

using Lemma 3 in Example 2 and the continuous mapping theorem about log-function. For the latter application, note that

converges in probability to zero as

Applying the above three results to Lemmas 1 and 2, therefore, we obtain

respectively, so that we conclude

Although (3.4) shows that the limit of

In discussing a convergence about the form

Let

at arbitrary point

note that

and then

Lemma 4.

uniformly on

A proof of Lemma 4 is provided briefly in Appendix A.2 since it is similar to Lemma 3. Now, applying Lemma

4 to

Let

nation of this result and (3.4) yields

On a Limit Form. The result of (3.5) shows only that the limit of

Here we discuss a limit form of

sets of

Then,

Because of

so that, via the general binomial theorem, we can show that

Also,

In addition, (3.6) is derived in the case of

A limit function of

Corollary 1. If Theorem 1 holds, then a limit expression to which

It usually takes a long time for the exact computation of the OPL. So, another subject of interest is the performance of its MC approximations. Let

We assign a point

Given fixed data

using

By the standard asymptotic theory, as

provided

To evaluate the quantity of

similar to the discussion for (3.6). As this result means that

using the delta method. Now consider the other aspect of (4.2) under

where

We will investigate two circumstances in the finite samples using the Cox cure-mixture model. One is how a relation such as (3.6) obtained as

Ovarian Cancer Data: For the first purpose, we use survival data of ovarian cancer patients [

Here, let

and

Simulated Data: For the second purpose, we prepare simulated data with

We use these to observe a better estimation performance than

computed at

Further, even if the approximations were reduced to

we set

computed under these settings. Although

A main result of this paper was to show the almost sure convergence of the OPL constructed in incomplete data with two class possibilities. To obtain this result, we discussed the principle of formulating this type of structure of the OPL, and then developed the tools based on a partial-sum processes argument. The limit function of the OPL resulting finally (Corollary 1) is the essential supremum of partial likelihoods obtained based on all the forms of complete data included in incomplete data, which is similar to

Unfortunately, it will be difficult to show consistency and asymptotic normality of the maximum OPL estimator (MOPLE) using the limit function of the OPL provided in Corollary 1. However, if the consistency is

achieved (as almost expected), the global essential maximum will be accomplished around true complete data under a true regression parameter. On the other hand, for the purpose of showing the consistency of the MOPLE, there will be other convenient limit expressions, although not discussed in this paper. A future paper on this topic is based on an infinite-dimensional Laplace approximation for integral on the baseline hazard function [

The results on the convergence of the exact OPL could easily suit the context of MC approximations. For example, at the end of Section 4.1 we show that, by applying Theorem 1 and Corollary 1, the size of the MC error is less than

In future study, it is important to derive the other expression of the limit function based on an infinite-dimen- sional Laplace approximation for integral on the baseline hazard and then to discuss the consistency and asymptotic normality of the MOPLE, since the asymptotic convergence of the OPL is given in this paper. Further, it is an interesting issue how the discussion of the OPL of the binary class as considered here could be extended to that under continuous class possibilities, such as the Cox frailty model.

The author is grateful to anonymous referees for their careful reading. This work is financially supported by JSPS KAKENHI grant number 23700336.