SUMMARY
The goals in clinical and cohort studies often include evaluation of the association of a time-dependent binary treatment or exposure with a survival outcome. Recently, several impactful studies targeted the association between initiation of aspirin and survival following colorectal cancer (CRC) diagnosis. The value of this exposure is zero at baseline and may change its value to one at some time point. Estimating this association is complicated by having only intermittent measurements on aspirin-taking. Commonly used methods can lead to substantial bias. We present a class of calibration models for the distribution of the time of status change of the binary covariate. Estimates obtained from these models are then incorporated into the proportional hazard partial likelihood in a natural way. We develop non-parametric, semiparametric, and parametric calibration models, and derive asymptotic theory for the methods that we implement in the aspirin and CRC study. We further develop a risk-set calibration approach that is more useful in settings in which the association between the binary covariate and survival is strong.
Keywords: Interval censoring, Last-value-carried-forward, Missing data, Proportional hazard
1. Introduction
One benefit of the Cox proportional hazards (PH) model for the analysis of time-to-event data (Cox, 1972) is the simplicity of including time-dependent covariates, while preserving desirable theoretical properties. Classical methods assume that time-dependent covariates are measured continuously. However, in practice, they are often measured intermittently, leading to bias in effect estimates if treated naïvely (Andersen and Liestøl, 2003; Cao and others, 2015). We consider a time-dependent binary covariate having zero value at baseline that may change its value to one at some point, and if a change has occurred, the covariate retains this value for the rest of the follow-up time. Real-life scenarios of this nature are widespread, including the onset of a irreversible medical condition (e.g., HIV infection, Langohr and others, 2004) or a treatment with a constant effect that is administrated in a different time for each patient (Austin, 2012). Goggins and others (1999) described data arising from a clinical trial where the goal was to study the effect of Cytomegalovirus (CMV) shedding on the risk of developing active CMV disease.
In the problems motivating this article (Chan and others, 2009; Liao and others, 2012; Hamada and others, 2017), the researchers were interested in the association between initiating aspirin use and survival after colorectal cancer (CRC) diagnosis. The data were obtained from two cohort studies: the Nurses’ Health Study (NHS) and the Health Professionals Follow-Up Study (HPFS). Following their enrollment to the studies, participants have been receiving questionnaires biennially and answering questions about life-style and other characteristics. Patients diagnosed with CRC who had been taking aspirin typically stop taking aspirin at the time of diagnosis as part of their preparations for surgery. Post-diagnosis aspirin-initiation is a time-dependent binary covariate. The time a participant started to take aspirin is known to lie within the interval between the time of the last questionnaire answered as no aspirin-taking and the time of the first questionnaire answered as aspirin-taking.
Researchers target the association between aspirin-initiation and survival and not aspirin-use and survival (Chan and others, 2009; Liao and others, 2012; Bastiaannet and others, 2012; Barron and others, 2015; Hamada and others, 2017) because aspirin-use might be stopped after initiation due to a deteriorating health status of the participants, resulting in reverse causality. A recently published study (Murphy and others, 2017) has found that among cancer patients that initiated aspirin use, the probability of continuing to take aspirin was much lower for those who were nearing death, compared to matched survivors.
The relevant existing literature is limited. Andersen and Liestøl (2003) studied the attenuation in effect estimates caused by having infrequently measured covariates. Cao and others (2015) developed kernel-based weighted score methods. Goggins and others (1999) proposed an EM algorithm to estimate the association between a binary covariate and survival time when the change-time of the binary covariate is interval-censored, and Kim (2016) proposed a modified estimating equation for the same problem, when there are no additional covariates affecting the outcome.
In this article, we propose a novel analysis framework for the problem of interval-censored change-time and suggest a two-stage approach. In the first stage, we fit a model for the change-time of the binary covariate. This model may be non-parametric, semiparametric or fully parametric, and may include baseline covariates that affect the change-time. For this first stage, we exploit existing methods, theory and efficient algorithms for interval-censored data (Sun, 2007; Huang, 1996; Anderson-Bergman, 2017; Wang and others, 2016). In the second stage, we incorporate the first-stage model into the main PH model. We construct a partial likelihood using the conditional hazard with respect to the available history, which, in the CRC and aspirin dataset, includes the timing of the last questionnaire, corresponding aspirin status and baseline covariates.
Our goal in this article is 3-fold. First, we develop a conceptual framework for the analysis of time-to-event data under the common practice of infrequent updates of a binary covariate of interest. Second, we present a rigorous analysis of data arisen from several impactful studies in the area of CRC. Third, we provide the R package ICcalib that implements our flexible methodology under a wide range of models. Unlike existing methods, our flexible approach allows to include variables related to aspirin-initiation, utilizing the available data and subject-matter knowledge in a novel way.
The rest of the article is structured as follows. In Section 2, we describe the motivating data. In Section 3, we define the main model, and in Section 4, we develop our proposed methods. Section 5 contains asymptotic properties. In Section 6, we describe our simulation study. In Section 7, we present the analysis of the CRC data and in Section 8, we reanalyze the CMV data of Goggins and others (1999). We offer concluding remarks in Section 9.
2. Data description
The data were formed from two large cohorts: the NHS, to which 121 701 female nurses enrolled in 1976, and the HPFS that began in 1986 with the enlistment of 51 529 males in various health professions. Description of the studies, and eligibility conditions for inclusion in the CRC survival studies can be found in Hamada and others (2017).
Participants have been receiving questionnaires biennially. During each questionnaire cycle, participants returned their answers in varying times. CRC researchers are interested in the first 5 or 10 years following the diagnosis. In our analyses, we considered 10 years follow-up. We limited our analyses to participants with Stages 1–3 CRC or missing stage data. In the span of 10 years, 249 CRC-related deaths were observed. Table A.1 of the supplementary material available at Biostatistics online presents descriptive statistics of the main variables.
In the three CRC studies that motivated this article (Chan and others, 2009; Liao and others, 2012; Hamada and others, 2017), evidence was presented for differential association of aspirin-initiation with CRC mortality across the following molecular subtypes, by considering the interaction terms of the aspirin-taking status and these molecular subtypes. The three molecular subtypes were PTGS2 (cyclooxygenase-2) overexpression (Chan and others, 2009), PIK3CA mutation (Liao and others, 2012), and low CD274 (PD-L1) expression (Hamada and others, 2017). These subtype definitions are not exclusive; a tumor may be included in zero, one, two, or three of these subtypes. The goal of our analyses was to evaluate the association between post-diagnosis aspirin initiations and CRC-related mortality within the subpopulation of each tumor subtype.
The post-diagnosis aspirin-initiation time varied. Figure 1 presents estimated time-to-aspirin-initiation curves, for the entire sample and by pre-diagnosis aspirin-taking status. Patients diagnosed with CRC were asked to stop taking aspirin to prepare for surgery. It is likely that patients who had been taking aspirin prior to CRC diagnosis were more inclined to initiate aspirin, once it was possible. This is consistent with the rapid drop right after the baseline in the curve for pre-diagnosis aspirin-takers.
Fig. 1.
Survival curve estimation of aspirin-initiation time for the entire sample (middle curve) and by pre-diagnosis (Predx) aspirin-taking status (takers vs non-takers; bottom vs top curves). Solid lines are the NPMLE, dashed lines are survival curves obtained from Weibull model fitting.
3. The main model
Let 
be the time to event of interest, e.g., time-to-death following CRC diagnosis. 
is possibly right censored by a censoring time 
. Let 
and 
be the observed time and censoring indicator, respectively. Of main interest is the association of the monotone non-decreasing binary covariate 
with 
. For presentational simplicity, we henceforth refer to 
as the exposure, although it can be a treatment or any other binary covariate. As explained in Section 1, when studying aspirin and survival following cancer diagnosis, researchers prefer to study the exposure aspirin-initiation and not aspirin-taking (e.g., Bastiaannet and others, 2012; Barron and others, 2015; Hamada and others, 2017 to avoid misinterpretation of the results due to reverse causality. Let 
be the change-time in the value of 
. That is, 
is the first time when 
. Let 
be a vector of covariates presumably associated with 
. For simplicity of presentation, we assume that 
is time-independent. The proposed methods can be straightforwardly applied to time-dependent covariates 
. The Cox PH model (Cox, 1972) for the hazard function of 
given 
and 
is
 |
(3.1) |
where 
is an unspecified baseline function and 
and 
are parameters to be estimated. We are mainly interested in 
. We refer to this PH model for 
as the main model. The partial likelihood for 
is
 |
(3.2) |
where 
is an at-risk indicator. If 
was known, we could simply set the time-dependent exposure to be equal to zero for all 
and to one after that time, i.e., 
. However, the data collected from questionnaires are often limited. The aspirin-initiation status 
was measured only at a series of 
discrete time points: 
, with 
. If 
was not measured at any time point before 
, then 
. When the main event is terminal (e.g., death), exposure data are available only until the event or censoring time; that is, 
. In other studies, data on the exposure can be obtained even after the main event has occurred.
Let 
denote the last time 
was measured before time 
; 
if 
was not previously measured. If 
, then 
. However, if 
, then 
can be either one or zero. Therefore, the likelihood (3.2) cannot be calculated from the observed data.
Two common methods for addressing this type of missing data are the last-value-carried-forward (LVCF) and midpoint imputation (MidI) methods. The LVCF method imputes missing values as the last observed values, 
. That is, a participant is assumed to not initiate aspirin, until the first time she reports aspirin-taking. The MidI method imputes the change-time (
) in the middle of the interval in which the change is known to have occurred. For example, if a participant reported she was not taking aspirin 1 year after diagnosis, and then reported she is taking aspirin 2.5 years after diagnosis, then MidI would assume 
for that participant; LVCF would assume 
. These are ad hoc methods that may lead to substantially biased results. This calls for a new conceptual framework.
First, we can describe the missing data problem in terms of the random variable 
. For each participant, the data available on 
include the measurement times and the corresponding exposure status. The data can be summarized in a form of an interval 
is censored into, denoted by 
, where
 |
If 
, 
is left-censored, and if 
, 
is right-censored. Note that data about measurements of 
before 
or after 
do not add information about 
.
Let 
be the history until time 
of an at-risk participant 
, where 
are covariates informative about 
(and hence on 
), and 
means that participant 
has been event-free so far. In our application, 
includes, among other covariates, the pre-diagnosis aspirin-use status, which is clearly informative about 
, as previously demonstrated in Figure 1. If a covariate affects both 
and 
, it is included in both 
and 
.
Before turning to our proposed methods, we present our assumptions. For all 
,
Assumption 1
Conditionally on
,
and
are independent,
Assumption 2
Conditionally on
and
, the probability of event at time
is independent of
,
, and
.
Assumption 3
.
Assumption 1 is the standard independent censoring assumption for the main event time 
. Assumption 2 is plausible, as it states that for a participant surviving at least until time 
, given all covariates 
, if 
is known, then the history of 
and timing of past questionnaires are not informative about the event occurrence at time 
. This assumption could be relaxed and replaced with a model for the relationship between the history of 
and 
. Assumption 3 is the standard independent censoring assumption for interval-censored data (Sun, 2007, Section 1.3.5). It states that 
and 
do not contain additional information about 
, other than that 
is within interval 
. This assumption would not exactly hold when the main event is terminal and 
and 
are associated (
). If, for example, aspirin is strongly protective against death (
and 
is large), a finite value of 
implies that the participant survived until 
, and hence it is more likely that this participants started to take aspirin early (i.e., 
close to 
). This assumption, its validity, and potential solutions to deviations from this assumptions are discussed in Sections 4.2 and 9.
4. A calibration approach
To present our methods, we first note that under model (3.1) and Assumption 2, the hazard function of 
conditional on 
is
 |
A similar mathematical derivation was presented for the case of measurement error in a time-dependent continuous covariate (Prentice, 1982). It is readily seen that 
is also a PH model (Prentice, 1982) and therefore we can consider the partial likelihood
 |
(4.1) |
The fact that 
is binary allows us to explicitly write the expectation in (4.1) as
 |
(4.2) |
where 
can be expressed using the distribution of 
as
 |
(4.3) |
In words, the probability of a positive aspirin-initiation status at time 
, conditionally on the personal history, equals to one, if the participant previously reported she is taking aspirin, and if she did not report aspirin-taking previously, it equals to the probability of a change in the aspirin-initiation status between the last questionnaire time 
and time 
, conditionally on no aspirin-initiation at time 
. Combining (4.2) and (4.3), it is evident that 
is a functional of the distribution of 
conditionally on 
and 
. If the distribution of 
was known, we could have obtained valid estimates for 
and 
by substituting (4.2) and (4.3) into 
. However, the distribution of 
is unknown for all 
.
Because the distribution of 
can be very complicated due to the conditioning on 
, we start with a simple solution. Let 
be 
without the survival information, i.e., 
. Our first proposal is to estimate 
and 
by applying two modifications to 
. First, we replace 
by 
in the expectations in (4.1) to get 
. The second modification involves replacing expectations of the form 
by estimators 
. Our ordinary calibration (OC) estimator 
is the maximizer of
 |
with respect to 
and 
. Noting the simplicity of this likelihood function, maximization can be done in a straightforward way, e.g., using the Newton–Raphson algorithm.
The expectation 
can be expressed as a functional of the distribution of 
, as in (4.2) and (4.3), with 
replaced by 
, and omitting 
in (4.3). Therefore, we first estimate the distribution of 
and then calculate 
using the estimated distribution.
4.1. Calibration models fitted from interval-censored data
Let 
be the survival function of 
. We refer to the model for the distribution of 
as the calibration model. Under Assumption 3, the likelihood of interval-censored time-to-event data is (Sun, 2007)
 |
(4.4) |
The nonparametric maximum likelihood estimator (NPMLE) estimator has been studied as an extension of the Kaplan–Meier estimator to interval-censored data. Algorithms suggested to find the NPMLE were previously developed (Turnbull, 1976; Groeneboom and Wellner, 1992; Wellner and Zhan, 1997). Consistency and asymptotic distribution (at a 
rate) were proved by Groeneboom and Wellner (1992).
Parametric or semiparametric models for the distribution of 
can also be used, and are especially appealing when the distribution of 
is likely to depend on additional covariates, previously denoted by 
. In the CRC studies, pre-diagnosis aspirin-taking status is strongly associated with post-diagnosis aspirin-initiation time (Figure 1). Additional covariates include risk factors for cardiovascular and cerebrovascular events, because aspirin is often taken to reduce the risk of these events among high-risk patients; see Section 7.
Let 
denote a vector of parameters characterizing the distribution of 
. An estimator of 
is obtained by maximizing the equivalent of (4.4) under a model for 
that possibly accommodates the covariates. See Chapter 2 of Sun (2007) for discussion of parametric models for interval-censored time-to-event data. A more flexible model is a PH regression model with an unspecified baseline hazard function. Finkelstein (1986) discussed PH models for interval-censored data and suggested discretization of the baseline hazard function. Algorithms for computation of the MLE were previously proposed and asymptotic theory was studied (Huang, 1996; Pan, 1999).
We adopt the recently developed framework of Wang and others (2016), which uses flexible I-splines (Ramsay, 1988) for the cumulative baseline hazard function. Wang and others (2016) further developed a fast EM algorithm, which we apply in our simulations and data analysis. Let 
and 
be the cumulative baseline hazard function and the survival function of 
, respectively, which under the PH model are
 |
(4.5) |
where 
is a coefficient vector relating the covariates 
to the survival function of 
, and where for all 
, 
are unknown parameters and 
are integrated spline basis functions, that are non-decreasing. The spline basis functions are calculated according to the user specification of 
interior knots and a polynomial degree for the basic functions. The resulting 
is guaranteed to be monotone increasing. See Ramsay (1988) and Wang and others (2016) for further details and discussion about I-splines in general and for the PH model, respectively.
4.2. Risk-set calibration
The OC estimator 
may suffer from asymptotic bias. It is calculated as the maximizer of 
while the partial likelihood is 
. The degree of divergence between 
and 
depends on how different 
and 
are. Recall that 
was defined by omitting 
from 
. If the probability that 
is close to one, as in the case of rare events, the bias should be attenuated. If 
has no effect on 
, 
is independent of 
and 
. If 
has a s strong effect on 
then the 
will not approximate 
very well. In that scenario, the fact that 
carries information on the distribution of 
. This implies that as the absolute value of 
, the true value of 
, increases, a larger bias may be expected.
Another source of bias stems from fitting the calibration model under the independent interval-censoring assumption (Assumption 3). However, in our studies, the time to event, 
, is informative about the censoring in the calibration model. If, for example, aspirin reduces the risk of death, then the aspirin-initiation time is more likely to be right censored (
) in non-aspirin-taking patients. This may cause bias in the estimation of 
when fitting the calibration model. As before, if the event is rare, the censoring of 
is most likely due to administrative reasons, and hence Assumption 3 approximately holds. Furthermore, under the null (i.e., when 
), the censoring interval is independent of 
. As before, larger 
typically implies more substantial bias. We investigate this point in the simulation studies in Section 6 and Appendix C of the supplementary material available at Biostatistics online. In studies with non-terminal event, Assumption 3 may be more plausible, because data on 
can be collected after the main event has occurred.
In order to reduce potential bias, we propose a risk-set calibration (RSC) procedure, an adaption of risk-set regression calibration previously developed in the context of error-prone covariates in survival analysis (Xie and others, 2001; Ye and others, 2008; Liao and others, 2011). This method uses 
, and estimate the distribution of 
by refitting the calibration model for 
at each observed event time, using only the members of the risk set at that time, so only participants with 
are used. Then, at each risk set, we plug-in the estimated distribution of 
in (4.3), leading to 
which is then substituted in (4.2) to obtain 
for 
.
The RSC is expected to lead to less bias than OC, especially when 
is large (Xie and others, 2001). However, some asymptotic bias in the RSC estimator may be expected, due to model misspecification. Even if the PH model for the distribution of 
holds at 
, it is not likely to hold for all 
. The RSC estimator is also expected to have larger variance, due to increased number of parameters, and the decreasing (in 
) sample size for the RSC models. Therefore, it is advised to use this estimator when the 
-
association is strong, and the sample size is large.
5. Asymptotic properties
We focus on the PH model under the I-splines representation for 
of Wang and others (2016). The results can be straightforwardly extended to parametric models for 
. Let 
, 
and let
 |
where 
and where for any vector 
, 
, and 
. Recall that 
is obtained by maximizing 
, or alternatively, by solving 
where
 |
with 
being the study end-time and 
the counting process associated with 
. Under certain regularity assumptions , 
. These regularity assumption and the proof are given in Appendix A of the supplementary material available at Biostatistics online. Furthermore, 
and 
can be estimated by a sandwich estimator
 |
(5.1) |
with 
being a sample version of 
given in Appendix A of the supplementary material available at Biostatistics online.
For the RSC estimator, results of similar nature are given and proved in Appendix A of the supplementary material available at Biostatistics online. Main modifications in the results are that 
, where 
is possibly different from 
, and that 
is replaced by 
, a time-dependent parameter vector.
As explained in Section 4.2, The limiting values 
and 
are not necessarily the true values of 
. Under the null (
), they are. We investigate the direction and magnitude of the asymptotic bias when 
is non-zero in Section 6 and in Appendix C of the supplementary material available at Biostatistics online.
6. Simulation study
We carried out simulation studies to assess the finite-sample performance of our methods and to compare them to the näive methods. We simulated 1000 datasets per scenario.
For the main model, the hazard function of 
was 
, with 
and 
, all independent of each other. We took the Gompertz baseline hazard function 
, and 
, 
and 
. For 
, we considered the values 
. We simulated time-to-event data with time-dependent covariate as described in Austin (2012). We took exponential censoring (mean = 5) and additional censoring at time 5. The resulting censoring rates varied between 42% and 63%, depending on the value of 
.
For the calibration model, we considered a PH calibration model in our main simulation study and used the setup of Wang and others (2016), with 
, where 
, 
, and 
and 
are the same covariates as in the main model. For each observation, 
questionnaire time points were simulated from the intervals on the equally spaced grid of 
. For example, under 
, the first questionnaire time point was simulated from 
, and the second from 
. However, to mimic the motivating studies, we considered a terminal main event, and kept only questionnaire time points before 
.
The PH calibration model fitting was done for each simulated dataset with 
equally-spaced interior knots and a quadratic order for the basic functions of the I-splines. Standard errors were estimated by (5.1) and confidence intervals were calculated using the asymptotic normal distribution.
Table 1 summarizes the results for the LVCF, the PH calibration model (PH-OC) and PH risk-set calibration models (PH-RSC), for 
. Under the null, all three methods were valid. As the association between 
and 
got stronger, a more substantial bias was observed. The OC estimator preformed generally well, with increased bias and lower coverage rates of the 95% confidence interval for the combination of 
and 
. The RSC estimator had lower bias in these scenarios. These observed biases were much smaller (in absolute value) than the bias of the LVCF estimator.
Table 1.
Simulation study results under a calibration PH model. Methods compared are LVCF, PH calibration model (PH-OC), and PH risk-set calibration models (PH-RSC). The table presents mean estimates (Mean), empirical standard deviations (EMP.SE), mean estimated standard errors (
), and empirical coverage rate of 95% confidence intervals (CP95%) for 
|
|
Method | Mean | EMP.SE |
|
CP95% |
|---|---|---|---|---|---|---|
| 0.000 | 2 | LVCF | -0.002 | 0.141 | 0.135 | 0.944 |
|
PH-OC | 0.003 | 0.183 | 0.178 | 0.950 | |
| PH-RSC | 0.004 | 0.183 | 0.177 | 0.952 | ||
| 5 | LVCF | 0.003 | 0.122 | 0.124 | 0.955 | |
| PH-OC | 0.007 | 0.138 | 0.146 | 0.956 | ||
| PH-RSC | 0.007 | 0.138 | 0.142 | 0.956 | ||
| 0.693 | 2 | LVCF | 0.462 | 0.119 | 0.118 | 0.498 |
|
PH-OC | 0.680 | 0.175 | 0.179 | 0.936 | |
| PH-RSC | 0.684 | 0.177 | 0.175 | 0.938 | ||
| 5 | LVCF | 0.572 | 0.107 | 0.110 | 0.810 | |
| PH-OC | 0.690 | 0.132 | 0.145 | 0.958 | ||
| PH-RSC | 0.689 | 0.132 | 0.137 | 0.957 | ||
| 1.609 | 2 | LVCF | 0.968 | 0.110 | 0.109 | 0.000 |
|
PH-OC | 1.472 | 0.179 | 0.210 | 0.869 | |
| PH-RSC | 1.516 | 0.190 | 0.195 | 0.897 | ||
| 5 | LVCF | 1.212 | 0.096 | 0.099 | 0.013 | |
| PH-OC | 1.577 | 0.139 | 0.166 | 0.951 | ||
| PH-RSC | 1.575 | 0.137 | 0.151 | 0.948 | ||
| 1.946 | 2 | LVCF | 1.130 | 0.112 | 0.110 | 0.000 |
|
PH-OC | 1.695 | 0.182 | 0.230 | 0.723 | |
| PH-RSC | 1.773 | 0.198 | 0.205 | 0.816 | ||
| 5 | LVCF | 1.410 | 0.097 | 0.098 | 0.000 | |
| PH-OC | 1.890 | 0.148 | 0.187 | 0.933 | ||
| PH-RSC | 1.890 | 0.147 | 0.158 | 0.929 |
Table A.2 of supplementary material available at Biostatistics online presents more results from this simulation scenario. It includes the results for 
, 
, and the MidI estimator. In terminal main event scenarios, MidI had a very large bias, even under the null. This is because a finite interval is only observed for observations with left- or interval-censored exposure times. Right-censored exposure times are dealt with differently (e.g., using LVCF, as we did) from left- or interval-censored observations, and a right-censored exposure time is the result of the main event occurring before a change in 
was observed. This creates a negative dependency between the imputed 
and 
, even under the null. In studies with non-terminal events, the MidI method is valid under the null, but not when 
. A small simulation study (not presented here) confirmed these claims.
To investigate the performance of our methods in other settings, we have considered additional simulation studies without any covariates affecting the time-to-exposure (i.e., no 
and 
in the calibration model) and compared Weibull and non-parametric calibration models when the true calibration model was Weibull, and when it was piecewise exponential. The results, presented in the supplementary materials available at Biostatistics online, generally agreed with the results we have descried in this section.
7. Results of aspirin and CRC survival analyses
Our first step was to construct a calibration model for the aspirin-initiation time. The 113 participants without available questionnaire data were not used for fitting the calibration model, and the calibration model was fitted using the remaining 1258 participants. The baseline potential covariates included gender, age-at-diagnosis, pre-diagnosis body mass index (BMI), pre-diagnosis aspirin-taking status, and the following tumor characteristics: disease stage (1–3 and missing), differentiation (poor vs well-moderate) and location (proximal colon, distal colon, or rectum). We considered three PH calibration models: (I) a model with all the aforementioned covariates, (II) a model with all non-tumor-related baseline covariates and disease stage, (III) a model with all non-tumor-related baseline covariates. Model (III) minimized the BIC (Schwarz, 1978). In addition, including strong determinants of the terminal event, such as the disease stage, is undesired. An association between disease stage and aspirin-initiation could be the result of violation of Assumption 3. That is, the association of the disease stage with aspirin-initiation could be only due to an association between disease stage and death.
Model (III) is logically sound from a subject-matter perspective. Aspirin is a preventive care for patients in high-risk for vascular diseases. Therefore, determinants of vascular diseases would also increase the probability of aspirin-initiation. The covariates in Model (III), age, gender, and BMI, are all well-established risk factors for vascular diseases. Therefore, we adopted Model (III) as our calibration model. Table 2 presents the results of fitting this PH model to the data. We used the same PH calibration model for all the analyses since there is no reason to believe that the tumor subtype, that was not even known at the time of diagnosis, informs the aspirin-initiation time.
Table 2.
The PH calibration model for aspirin-initiation time, using cubic order and 
interior knots. Covariates include pre-diagnosis aspirin status (Predx-asp, taker vs non-taker), pre-diagnosis BMI (Predx-BMI), age-at-diagnosis (Age-at-dx), and gender (Female)
Est ( ) |
|
CI |
 -value |
|
|---|---|---|---|---|
| Predx-asp | 1.135 (0.074) | 3.113 |
|
 0.001 |
| Predx-BMI | 0.027 (0.006) | 1.028 |
|
 0.001 |
| Age-at-dx | 0.010 (0.001) | 1.010 |
|
 0.001 |
| Female |
 0.181 (0.073) |
0.834 |
|
0.013 |
As suggested by Wang and others (2016), we used BIC to choose the number of equally spaced interior knots, which led to 
. We used the flexible cubic order for the spline basis functions. The number of interior knots according to the AIC criterion (Akaike, 1974) was 
. The final results did not substantially change when we took 
.
The simulation results presented in Section 6 and the Supplementary materials available at Biostatistics online have shown that when there are covariates affecting the time-to-exposure, the PH-OC estimator is preferable over the näive methods, namely LVCF and MidI. Figure 2 illustrates the difference between the methods. On its right panel, we drew the probabilities 
vs 
, for the first nine participants in our data. Once 
was observed, all methods assign 
for the rest of the relevant risk sets. The left panel of Figure 2 presents the same probabilities as the right panel of the figure, but for the first participant only. This person did not report aspirin taking during the 10 years follow-up (
). From the data, this person was a 75 years old (at diagnosis time) male, who was taking aspirin at baseline. From Table 2, we would expect the probability of this person to take aspirin to be higher than in the general population. This aligns with this person having larger estimated probabilities under the PH model comparing to the NP calibration model.
Fig. 2.
(a) First participant; (b) First 9 participants. 
using LVCF, MidI, non-parametric (NP) calibration, and the PH calibration model, for the first 9 participants. Panel (a) corresponds to the top left corner of Panel (b).
We included in the RSC models the covariates of Model (III). To avoid destabilization of the calibration model fittings, we grouped the risk sets in intervals of size 0.5. That is, we refitted the calibration model every 6 months.
Turning to the main model, we included baseline covariates that are known to be associated with CRC-related death. These included age-at-diagnosis, pre-diagnosis BMI, family history of CRC, and the following tumor characteristics: stage, differentiation, and location. Table 3 shows estimates and corresponding standard errors, confidence intervals, and 
-values for the association between aspirin-initiation and survival in the three motivating studies and in the entire data. Compared to LVCF, stronger protective associations were estimated by our methods in all three subtype-specific analyses. Compared to the OC estimates, the RSC estimates were only slightly further from null. This could be partially explained by the high censoring rate (
80%). Even though the sample sizes and case numbers were low to moderate, there was evidence for strong protective effect of aspirin for PIK3CA subtype CRC. The point estimates for aspirin in low CD274 subtype imply negative association between aspirin and death, but, possibly due to limited power, the null hypothesis was not rejected.
Table 3.
Results for the main model in the three CRC studies and in all data. Results presented for the aspirin effect. The main model also included family history of CRC, pre-diagnosis BMI, age-at-diagnosis, gender, disease stage, differentiation, and location
| Study | Method | Est (SE) |
|
CI 95%(for HR) |
 -value |
|---|---|---|---|---|---|
| All data | LVCF |
 0.34 (0.15) |
0.71 |
|
0.020 |
( ) |
PH-OC |
 0.32 (0.18) |
0.73 |
|
0.072 |
(No. of events  ) |
PH-RSC |
 0.32 (0.18) |
0.73 |
|
0.073 |
Low CD274
|
LVCF |
 0.64 (0.35) |
0.53 |
|
0.067 |
( ) |
PH-OC |
 0.77 (0.40) |
0.46 |
|
0.054 |
(No. of events  ) |
PH-RSC |
 0.79 (0.40) |
0.45 |
|
0.049 |
PIK3CA
|
LVCF |
 2.13 (0.64) |
0.12 |
|
0.001 |
( ) |
PH-OC |
 2.22 (0.65) |
0.11 |
|
0.001 |
(No. of events  ) |
PH-RSC |
 2.23 (0.66) |
0.11 |
|
0.001 |
PTGS2
|
LVCF |
 0.30 (0.20) |
0.74 |
|
0.138 |
( ) |
PH-OC |
 0.32 (0.24) |
0.73 |
|
0.185 |
(No. of events  ) |
PH-RSC |
 0.32 (0.24) |
0.72 |
|
0.181 |
8. Reanalyzing the data in Goggins and others (1999)
The motivating application of Goggins and others (1999) was an analysis of AIDS Clinical Trial Group (ACTG) 181 (Finkelstein and others, 2002). The event of interest, active CMV disease, was non-terminal. Their binary covariate was CMV shedding. Based on the joint likelihood for 
and the distribution of 
(
in their notation), they proposed an EM algorithm where the E-step is carried out with respect to the ordering of CMV shedding among the study participants. They further developed a Gibbs sampler to improve computational efficiency.
Our methodology has several improvements over the method of Goggins and others (1999). First, we exploit modern and fast procedures for estimating the distribution of 
, by the NPMLE. Second, we can (but do not have to) include parametric modeling assumptions on the distribution of 
, if appropriate. Third, since we fit the calibration and main models separately, we can include in our analysis participants without data about the covariate of interest. Finally, and most importantly, we can include measured covariates affecting the time-to-exposure 
.
We analyzed the ACTG 181 data (Finkelstein and others, 2002) using our non-parametric calibration method. The results are presented in Appendix D of the supplementary material available at Biostatistics online. We observed a divergence between the OC and RSC estimates. The RSC estimates were further away from the null, and similar to those obtained by Goggins and others (1999). The confidence intervals were quite wide, as one may obtain for hazard ratios of strong effects when the sample size is moderate.
9. Conclusion
We have presented a novel calibration approach for studying the association between a time-dependent binary exposure and survival time, when the data about the monotone time-dependent exposure is only available intermittently. Our proposed approach allows for a wide range of calibration models. In practice, an adequate model should be chosen by combination of subject-matter knowledge and the available data. The R package ICcalib implementing our methods is available from CRAN. The package is described in Appendix B of the supplementary material available at Biostatistics online. The simulations and CMV data analysis can be reproduced using the Github repository ICcalibReproduce.
When the association between exposure and survival is strong, and a terminal event is non-rare, our calibration framework may suffer from bias, that can be reduced, though not eliminated, by RSC. Unlike regression calibration methods for error-prone covariates, additional data (e.g., reliability or validation data) is not needed to fit the calibration model, as the covariate measurements are inherently part of the available data. To address deviations from Assumption 3, future research may incorporate methods for analyzing data subject to dependent interval-censoring (Sun, 2007, Section 10.5) into our approach.
The problem described in this article and the proposed conceptual framework open the way for further research. A main question of interest is whether 
can be estimated consistently from the data, and what is the nature of further assumptions and methods to ensure consistency. Our model is a type of a joint model for time-to-event and longitudinal data (Rizopoulos, 2012). Potential alternative methods may model the binary covariate directly (Faucett and others, 1998; Larsen, 2004; Rizopoulos and others, 2008). Additionally, in scenarios the binary variable may change its value again, from 1 to 0, a potential approach would model the transition times between the two stages, although the assumptions and data needed for such an approach might be different from what we described in this article. Finally, future research may consider the case of a non-terminal main event that is assessed at the same time as the covariate. Then, both the main event time and the covariate change-time are interval-censored.
In conclusion, we presented a new conceptual framework accompanied by flexible and simple methodology to preform time-to-event analysis under the problem of infrequently updated binary covariate, a common problem in medical and epidemiological research.
Supplementary Material
Acknowledgments
We thank two anonymous reviewers and the associate editor for useful comments and suggestions that improved the article. Conflict of Interest: None declared.
Funding
National Institutes of Health (NIH) (P01 CA55075, P01 CA87969, R01 CA118553, R01 CA151993, R01 CA169141, R35 CA197735, U01 CA167552, UM1 CA167552, and UM1 CA186107).
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