Open Journal of Statistics
Vol.04 No.02(2014), Article ID:43291,18 pages
10.4236/ojs.2014.42012
On the Convergence of Observed Partial Likelihood under Incomplete Data with Two Class Possibilities
Tomoyuki Sugimoto
Department of Mathematical Sciences, Hirosaki University, Hirosaki, Japan
Email: tomoyuki@cc.hirosaki-u.ac.jp
Copyright © 2014 Tomoyuki Sugimoto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the owner of the intellectual property Tomoyuki Sugimoto. All Copyright © 2014 are guarded by law and by SCIRP as a guardian.
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,
Figure 1. Plots of
on
(solid curves) and
(horizontal dotted lines) at
, in ovarian cancer data (
: exponential scale).
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
Figure 2. Plots of averages (polygonal lines) and SEs (horizontal whiskers) of,
and
at
,
obtained from 100 simulated data sets of
(
and
).
Figure 3. Plots of averages of
at
obtained from 100 simulated data sets of n = 30, 100, 300, 500 and 1000 (
and
).
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
- 1. D. R. Cox, “Regression Models and Life Tables (with Discussion),” Journal of the Royal Statistical Society, Series B, Vol. 34, No. 2, 1972, pp. 187-220.
- 2. J. S. Kim, “Maximum Likelihood Estimation for the Proportional Hazards Model with Partly Interval-Censored Data,” Journal of the Royal Statistical Society, Series B, Vol. 65, No. 2, 2003, pp. 489-502.
http://dx.doi.org/10.1111/1467-9868.00398 - 3. M. C. Paik and W.-Y. Tsai, “On Using the Cox Proportional Hazards Model with Missing Covariates,” Biometrika, Vol. 84, No. 3, 1997, pp. 579-593. http://dx.doi.org/10.1093/biomet/84.3.579
- 4. H. Y. Chen and R. J. A. Little, “Proportional Hazards Regression with Missing Covariates,” Journal of the American Statistical Association, Vol. 94, No. 447, 1999, pp. 896-908.
http://dx.doi.org/10.1080/01621459.1999.10474195 - 5. A. H. Herring and J. G. Ibrahim, “Likelihood-Based Methods for Missing Covariates in the Cox Proportional Hazards Model,” Journal of the American Statistical Association, Vol. 96, No. 453, 2001, pp. 292-302.
http://dx.doi.org/10.1198/016214501750332866 - 6. S. A. Murphy and A. W. van der Vaart, “On Profile Likelihood (with Discussion),” Journal of the American Statistical Association, Vol. 95, No. 450, 2000, pp. 449-465.
http://dx.doi.org/10.1080/01621459.2000.10474219 - 7. D. R. Cox, “Partial Likelihood,” Biometrika, Vol. 62, No. 2, 1975, pp. 269-276.
http://dx.doi.org/10.1093/biomet/62.2.269 - 8. R. Gill, “Marginal Partial Likelihood,” Scandinavian Journal of Statistics, Vol. 19, No. 2, 1992, pp. 133-137.
- 9. M. R. Kosorok, “Introduction to Empirical Processes and Semiparametric Inference,” Springer, Berlin, 2008.
http://dx.doi.org/10.1007/978-0-387-74978-5 - 10. J. P. Sy and J. M. G. Taylor, “Estimation in a Cox Proportional Hazards Cure Model,” Biometrics, Vol. 56, No. 1, 2000, pp. 227-236. http://dx.doi.org/10.1111/j.0006-341X.2000.00227.x
- 11. T. Sugimoto, “A Large Sample Study of Marginal Partial Likelihood in a Cox Cure-Mixture Regression Model,” Unpublished.
- 12. A. Y. C. Kuk and C.-H. Chen, “A Mixture Model Combining Logistic Regression with Proportional Hazards Regression,” Biometrika, Vol. 79, No. 3, 1992, pp. 531-541.
http://dx.doi.org/10.1093/biomet/79.3.531 - 13. Y. Peng and K. B. G. Dear, “A Nonparametric Mixture Model for Cure Rate Estimation,” Biometrics, Vol. 56, No. 1, 2000, pp. 237-243. http://dx.doi.org/10.1111/j.0006-341X.2000.00237.x
- 14. W. Lu and Z. Ying, “On Semiparametric Transformation Cure Models,” Biometrika, Vol. 91, No. 2, 2004, pp. 331-343.
http://dx.doi.org/10.1093/biomet/91.2.331 - 15. T. Sugimoto, T. Hamasaki and M. Goto, “Estimation from Pseudo Partial Likelihood in a Semiparametric Cure Model,” Journal of the Japanese Society of Computational Statistics, Vol. 18, No. 1, 2005, pp. 33-46.
- 16. B. W. Turnbull, “Nonparametric Estimation of a Survivorship Function with Doubly Censored Data,” Journal of the American Statistical Association, Vol. 69, No. 345, 1974, pp. 169-173. http://dx.doi.org/10.1080/01621459.1974.10480146
- 17. B. W. Turnbull, “The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data,” Journal of the Royal Statistical Society, Series B, Vol. 38, No. 3, 1976, pp. 290-295.
- 18. J. D. Kalbleisch and R. L. Prentice, “Marginal Likelihoods Based on Cox’s Regression and Life Model,” Biometrika, Vol. 60, No. 2, 1973, pp. 267-278. http://dx.doi.org/10.1093/biomet/60.2.267
- 19. P. K. Andersen and R. D. Gill, “Cox’s Regression Model for Counting Processes: A Large Sample Study,” Annals of Statistics, Vol. 10, No. 3, 1982, pp. 1100-1120.
http://dx.doi.org/10.1214/aos/1176345976 - 20. D. Collett, “Modelling Survival Data in Medical Research,” 2nd Edition, Chapman & Hall/CRC, London, 2003.
- 21. A. W. van der Vaart and J. A. Wellner, “Weak Convergence and Empirical Processes,” Springer-Verlag, New York, 1996.
http://dx.doi.org/10.1007/978-1-4757-2545-2