ABSTRACT

Received October 21, 2013; revised November 21, 2013; accepted November 28, 2013

In this paper, we discuss the theoretical validity of the observed partial likelihood (OPL) constructed in a Cox- type 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.

Keywords:

Cox’s Regression Model; Logistic Regression Model; Incomplete Binary Data; Partial Likelihood; Partial-Sum Processes; Profile Likelihood

1. Introduction

Although the Cox model [1] is a standard tool for the analysis of time-to-event data, in practice analysts are often confronted with some problems in handling incomplete data beyond the right-censored form, such as interval-censored data [2], missing covariates [3-5] or (statistical) structural modelling. The inference for the Cox model under such cases of incomplete data can usually be performed based on the semiparametric profile likelihood (SPFL) [6] as a generalization of Cox’s partial likelihood [7]. On the other hand, as a substitute for the SPFL method, one can analyse the same data using the imputation method, which yields a sum of partial likelihoods. By describing the sum of all possible partial likelihoods more exactly, we can formulate the marginal of partial likelihoods [8], that is, the observed partial likelihood (OPL). In this area, the theory for the SPFL has been studied by many authors (e.g., [6,9]). However, to the best of our knowledge, there has not been much development in mathematical theory for the OPL.

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 [10]). The main disadvantages of the OPL are, for instance, that a great length of time is required for the exact computation and it is not clear how much the amount of computation can be reduced by the Monte Carlo (MC) method. However, even if the feasible number of MC trials is smaller than desirable to approximate the OPL, and hence the MC approximation is quite rough, it may be sufficient for a starting value in the computational process of the SPFL.

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 [11], which is useful in discussing the consistency and asymptotic normality of the estimators. However, the method is not convenient for showing the convergence of the OPL. For these reasons, it is also valuable to discuss the convergence of the OPL using the arguments employed in this paper.

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(n²) 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.

2. Observed Full and Partial Likelihoods

2.1. Notations and Motivated Examples

Let and be the observed survival time and right-censoring indicator of the individual, where are continuous random variables independent of and is the indicator function. Suppose that the individuals possess some difference between models or observations identified by the two classes. We define such a class variable by

In the case that expresses the difference between models, assume that the distribution of follows the proportional hazards model formulated as

where is the baseline hazard function, is the function given by

is the covariate vector from the population of the class, and is the regression

coefficient vector As usual, the information on can be re-expressed using the counting processes and at-risk processes

In this paper, we consider incomplete data where some of the’s are treated as missing. Let

Each of these is used to construct the likelihoods. Further, if the event of can be expressed by a pro- bability, we use the following notation and assumption

where is the covariate vector related to and is the regression coefficient vector For simplicity, we will write hereafter.

Let be the collection of true’s. In many cases of incomplete data with two class possi-

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

(2.1)

with the elements such that

and where the space denotes the collection of all the vectors composed of 0 or 1 with length n, expresses one element of, in which there exists one true element,

is the survival function of the individual belonging to the class is the cumulative baseline hazard function, and is given by either of or in advance.

In the following three examples, we show how the form of the OFL is related to the representative cases. Hereafter, we will often omit when it is clear that a function depends on e.g.

and so on.

Example: Missing Binary Covariates. Let us assume that but For example, the

first covariate is binary and may be missing, and Then, we can write

where In this case, the OFL is

Using the binomial expansion, this can be rewritten as

(2.2)

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 but so that we can write

We observe that if and is missing otherwise. The OFL is usually

(2.3)

note here that for all Then, (2.3) can be rewritten in the same form as (2.2).

Example: Doubly Censored Data. In doubly censored data [16,17], left-censored data may be included. Let indicate whether the observation is left-censored or not, the OFL is then

(2.4)

In the phenomenal meaning, the common model is assumed regardless of the type of observations, but we do not define as Here we use the rule and such that designates the type of model rather than an observation. Under this rule, because we have for all (2.4) can be expressed as

where note that is defined as missing data (i.e.) if the observation is left-censored, and as complete data of (i.e.) otherwise.

2.2. Observed Partial Likelihood

Let be the integral operator proposed by [18] to derive the partial likelihood in the Cox model without time-dependent covariates. Without loss of generality, let us suppose that there are no ties. By operating to the OFL we have the OPL

where

Let To discuss an asymptotic form of we will prepare some convenient expressions. First, to pack the expression of into and, we define

and

Using these expressions, let

where is an empirical version of the theoretical expectation. Further, as an important definition, let

be the n-dimensional version of Minkowski’s measure Then, using these notations, can be written as

(2.5)

where

is the greatest follow-up time and The quantity of is the total number that the same’s are repeated on In addition, we define

and

which is in which’s are replaced by the true intensity

where and are the true and In the case of

such as Cox cure-mixture model or doubly censored data, we have

Remark 1. In (2.5), even if we consider a difference or quotient between and we cannot remove the potential increasing factor n in the summands because of the existence of the operator Thus, it may potentially be difficult to apply some of the standard tools in the survival analysis to the asymptotic discussion. For these reasons, our strategy to obtain a limit of is to regard all the summands of as a process on We will then derive the result of a weak convergence on

3. Convergence of the Observed Partial Likelihood

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 be a compact set of which includes where and are the true and The true baseline function is continuous and non-decreasing on with

A1: are i.i.d. vectors from the population of the class.

A2: A3:

A4: A5:

Condition A2 means for all because of

where and and is the i-th survival function of

right-censoring time under a given By the compact condition of we have

on as a matter of course. However, in the case that there are no as in the example of doubly censored data, such conditions on are omitted because always.

Theorem 1. Suppose that Conditions A are satisfied. Then, as converges almost surely to a deterministic function uniformly on

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.

3.1. Relations between Two Observed Partial Likelihoods

Note that the OPL is constructed by an integral on with the measure Thus, to give two OPL’s with the same limit, it is predicted that the difference between the integrands of two OPL’s should converge weakly to zeros on for example, by analogy of the dominated convergence theorem. Let

be functions that exist around where for simplicity. Then, we have the following lemma about some and

Lemma 1. Suppose that

(3.1)

with probability 1. Then, as

where denotes almost sure convergence.

(Proof of Lemma 1). Using Taylor expansions of

and,

the difference between and is derived as

where, with some on and

Let us assume that without loss of generality. Then, and

are bounded on by some finite values and In fact,

and

are shown by Therefore, we have

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

Using Lemma 1, for several patterns of and we can investigate whether they converge to the same limit. The important problem is how to show the condition (3.1). For this purpose, we make the use of meaning that a convergence in implicated in (3.1), since is a probability measure on We have the following lemma to establish the condition (3.1).

Lemma 2. Suppose that

(3.2)

where denotes convergence in probability. Then, (3.1) is established.

Remark 2. For simplicity, letting

denote

as the area of. We can immediately show that (3.2) provides a version of convergence in probability of (3.1), i.e.

(3.3)

because it is always satisfied that

Thus, we show that the operators of and are mutually exchangeable in (3.3) in a proof of Lemma 2.

(Proof of Lemma 2). Note that

because is independent of due to the n-dimensional projection to from From condition (3.2), limits of are zeros almost everywhere on which can be eventually written as independently of The dominated convergence theorem provides

This shows via Markov’s inequality such that

Therefore, condition (3.2) gives (3.1).

3.2. All Possible Partial-Sum Processes

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

where denotes convergence in distribution. In this section, we develop a tool to show such a weak conver- gence on

For simplicity, let and An important key to obtaining (3.2) or a limit of

is a convergence result of on As representa-

tions of more essential terms to consider in the convergence on we denote

where examples of are and so on. Then,

can be regarded as, which is the empirical version of the conditional expectation on a unique

but is calculated letting be fixed. Let

which is the conditional expectation of on given Although is treated as unknown in many incomplete problems, this is not the case for terms included in the observed partial likelihood. Hence, for a fixed

the conditional expectation of on is

So, letting and be means to centralize and, then

Example: Missing Binary Covariates. For simplicity, we assume that or 0 occurs independently of

Letting, then

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

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

Example: Cox Cure-Mixture Model. In this model, is usually assumed. The expectations of terms

which may form all possible partial-sums are and

where Similarly, we have

On Weak Convergence. Let In our application, note that centred

are zero-means and mutually independent but are not sampled from an identical distribution. We will therefore discuss the partial-sum processes about sampled from two populations. Lemma 3 shows that converges in probability to zero uniformly on; recall that an n-dimensional element means marginal collection of elements of In advance, let

Remark 3. For example, if by Chebyshev’s inequality, we immediately have

However, a result of interest here is whether and can be exchanged, that is, about

.

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

is always apart from zero.

Lemma 3. Let be random elements on sampled from one of two distributions (populations) with at every where the population to which the element belongs is known and indexed by or 1. Suppose that are mutually independent and have at every If the following three conditions are satisfied,

(i) The class of functions is Glivenko-Cantelli,

(ii) and

(iii) for every

then, as

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

satisfied for and

Example 1. Let From Condition A5, we have Condition (ii). Since is independent of, Conditions (i) and (iii) are clearly satisfied.

Example 2. Let Conditions (i) and (ii) are shown by Conditions A1 and A4. Arbitrary satisfies by Condition A4 and

by Condition A3, where means the Euclidean norm for vectors. Hence, Condition (iii) is satisfied.

3.3. Proof of Theorem 1

Consider in which the random quantities are reduced less than that of where and are in which is replaced by and,

i.e. and

: It is satisfied that

and

by Conditions A1 and A4, similar to the standard Cox model (see [19]). Thus,

is obtained as

: We have

using Lemma 3 in Example 1 and applying the strong law of large numbers (SLLN) to and by Conditions A1 and A5. For the latter application, note that be- cause of Also, it is shown that

by applying the SLLN on (see [1]) from Conditions A1 and A4. In addition, we have

using Lemma 3 in Example 2. Hence, converges in probability to zero as.

: It satisfies that

using Lemma 3 in Example 2 and the continuous mapping theorem about log-function. For the latter application, note that is bounded away from zero on by Condition A2. Hence,

converges in probability to zero as

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

respectively, so that we conclude

(3.4)

Although (3.4) shows that the limit of is equivalent to that of, still depends on n and’s. We will therefore investigate the limit form of further.

In discussing a convergence about the form and included in and of the partial sums, note that they can be written as

Let Similarly to Lemma 3 (proof of s2), we show that

at arbitrary point In particular, because of

note that

and then. We have the following lemma.

Lemma 4. converge in probability to

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 we obtain their limits as

Let be, in which is replaced by

: We obtain

by Lemma 4 and the continuous mapping theorem about log-function. Therefore, converges in probability to zero as. Hence, using Lemmas 1 and 2, we can show so that a triangle combi-

nation of this result and (3.4) yields

(3.5)

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

Here we discuss a limit form of To consider the case of, let and be the sub-

sets of such that if and if For simplicity, let

Then,

Because of, we have

,

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

Also, Therefore, because of and similar to the derivation of the on the Banach space which results in the essential supremum, we conclude

(3.6)

In addition, (3.6) is derived in the case of. Results (3.5) and (3.6) show that Theorem 1 is complete.

A limit function of is concretely provided by (3.6), which is summarized as follows.

Corollary 1. If Theorem 1 holds, then a limit expression to which converges almost surely as is

4. Additional Considerations

4.1. Monte Carlo (MC) Approximations

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 be all the elements of labelled in order such that

We assign a point to using and let denote the distribution Using these notations, we redefine as

Given fixed data let be random elements from

where and is either 1 or 0 with an equal probability 0.5 if and

if An MC approximation of is

using and the corresponding empirical measure

By the standard asymptotic theory, as it follows that

(4.1)

provided exists and where

To evaluate the quantity of in the case of, consider, then, as

similar to the discussion for (3.6). As this result means that may increase exponentially according to n, direct use of (4.1) is not particularly productive. Therefore, although (4.1) is the rationale in this context, it will be modified, as to

(4.2)

using the delta method. Now consider the other aspect of (4.2) under Applying Theorem 1 and Corollary 1 to such a problem, we obtain the following results

where This means from the point of view of the k-asymp- totic variance in the second line of (4.2). Hence, we can show that the order of is less than in (4.2) under That is, using the MC method, a computational load of needed in the exact computation can be reduced to one of at most

4.2. Numerical Examples

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 is located in the finite samples. The other is to observe numerically the practical size of the error in MC approximations, which was shown to be less than in the previous section.

Ovarian Cancer Data: For the first purpose, we use survival data of ovarian cancer patients [20]. We set the covariates as and where Treat is the type of chemotherapy (0 = single, 1 = combined), Age is the age of the patient (in years) and Rdisease is the extent of residual disease (0 = complete, 1 = incomplete). The maximum of the OPL is achieved approximately at

Here, let Figure 1 shows plots of

and at where and the y-axis is drawn in exponential scale. Although the total number of is in fact’s of are sorted on This data are small in size However, circumstances close to the relation in (3.6) are observed at least at and

Simulated Data: For the second purpose, we prepare simulated data with and where follows the standard uniform distribution. The latent distribution of is standard exponential and the censoring follows a uniform distribution Under these settings, the simulated means of cure and censored rate are about 48% and 58%, respectively. We generate 100 pairs of simulated data of size n. We perform m MC approximations for each simulated data set. Let be the element of m’s. For each simulated data set, we estimate and by and, where

We use these to observe a better estimation performance than

Figure 2 shows simulated averages and standard errors (SEs) of 100 pairs of

computed at in simulated data of under and where Although is considerably smaller than the needed in the exact method, approximates well enough.

Further, even if the approximations were reduced to the simulated average of would still yield sufficiently good approximations of Based on these empirical findings,

we set and Figure 3 shows

computed under these settings. Although increases over an initial domain of n, Figure 3 shows that the rate of such an increase is smaller as n increases. This provides a conjecture that may be bounded by some order smaller than such as for a sufficiently large We leave further investigation of this to future research.

5. Concluding Remarks

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 on a Banach space. In Section 4.2, we showed numerically how an essential supremum approximates the OPL in real data for the Cox cure-mixture model.

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 [11]. However, in applying such a Laplace approximation to the OPL, a precondition that the OPL converges to a deterministic function is necessary. Hence, in order to obtain this precondition and for the reason that it is generally difficult to show the convergence result directly using the Laplace approximation, it is meaningful to discuss the asymptotic convergence of the OPL using the argument employed in this paper.

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 This suggests that the MC method, for which the number is at most, achieves an appropriate approximation and can reduce the vast computational load of implied by the exact method up to a feasible level. Further, in Section 4.2 we performed numerical experiments to investigate the practical size of the error in MC approximations using the Cox cure-mixture model. These experiments indicate that the exact OPL may be sufficiently approximated with the number of MC trials smaller than, such as, as is larger.

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.

Acknowledgements

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

References