Cumulative incidence regression for dynamic treatment regimens

Summary

Recently dynamic treatment regimens (DTRs) have drawn considerable attention, as an effective tool for personalizing medicine. Sequential Multiple Assignment Randomized Trials (SMARTs) are often used to gather data for making inference on DTRs. In this article, we focus on regression analysis of DTRs from a two-stage SMART for competing risk outcomes based on cumulative incidence functions (CIFs). Even though there are extensive works on the regression problem for DTRs, no research has been done on modeling the CIF for SMART trials. We extend existing CIF regression models to handle covariate effects for DTRs. Asymptotic properties are established for our proposed estimators. The models can be implemented using existing software by an augmented-data approximation. We show the improvement provided by our proposed methods by simulation and illustrate its practical utility through an analysis of a SMART neuroblastoma study, where disease progression cannot be observed after death.

1. Introduction

Dynamic treatment regimens (DTRs) are sets of decision rules for choosing effective treatments for individual patients, based on their characteristics and intermediate responses. Often practitioners are interested in finding the optimal DTR that leads to the most desirable outcome in the end. An efficient randomization design is the Sequential Multiple Assignment Randomized Trial (SMART), where patients are randomly assigned to the initial treatments and then randomized to available treatments in subsequent stages, as they become eligible. In this article, we focus on competing risks data from a two-stage randomization design that was motivated by a neuroblastoma study. Children in this study were first randomized to two initial treatments, and those who responded to the initial treatment were further randomized to receive one of the two maintenance options. Meanwhile, the event of interest, disease progression, cannot be observed after death.

If there were no competing-risk events, existing non-parametric methods could have been used. They either modeled a mean restricted survival time for a treatment regimen by using the inverse probability weighting (IPW) method (Lunceford and others, 2002; Wahed and Tsiatis, 2006), or generated various weighted Kaplan–Meier (KM) estimators (Guo and Tsiatis, 2005; Miyahara and Wahed, 2010), or proposed pattern-mixture estimators of the survival function of a DTR (Tang and Wahed, 2015). However, competing-risk events, such as death, commonly occur when subjects are exposed to multiple failures, and the event of interest cannot be experienced with the occurrence of competing events. In the competing-risk literature, the cumulative incidence function (CIF) from a specific event is often of interest and widely used, because it is easily interpretable and is non-parametrically identifiable. In a SMART design with competing risks endpoints, the objective generally is to find a regimen which results in a reduced probability of occurrence of the target event. Recently, Yavuz and others (2016) proposed four weighted non-parametric estimators of the CIF for a specific DTR without accounting for patient heterogeneity (covariates). Thus, the focus of this study is to model covariate effects on the CIFs of different DTRs.

The Cox regression model (Cox, 1972) and the accelerated failure model (Wei and others, 1990) are two popular methods of modeling covariate effects on survival. Fine and Gray (1999) extended Cox regression to competing risks data and proposed a proportional hazards model for CIFs. Klein and Andersen (2005) developed a parametric regression model on pseudo values of the CIF. Scheike and others (2008) proposed a direct binomial regression to model the time-varying effects of covariates on the CIF, which is more flexible than the fixed-effect Fine and Gray model. Recently, Gerds and others (2012) proposed a multinomial logistic model that handles multiple competing causes, providing another flexible alternative to the Fine and Gray model. However, these approaches are not readily applicable to a SMART study.

In SMART literature, Murphy (2003) proposed a backward searching algorithm to minimize the regret function at each step and find the best DTR at steps, considering previous history and decisions. Zhao and others (2009) used reinforcement learning and Q-learning to discover personal optimal therapies on cancer trials. Henderson and others (2010) proposed the regret-regression to predict outcomes based on the estimated regression coefficients and to use the resulting residuals for model diagnostics. Goldberg and Kosorok (2012) i introduced a novel approach on a multistage-decision problem with censored data by using Q-learning. Tang and Wahed (2015) proposed a fixed weight estimator for the cumulative hazard function in a two-stage design, under a proportional hazards assumption. However, none of the above methods can be used directly for competing risks outcomes.

Hence, we extend some existing regression models from the competing risks literature to SMARTs, particularly to two-stage randomization settings, adopting the IPW idea to account for the second-stage randomization. Our proposed methods perform an unbiased estimator for the CIF under the two-stage randomization design, while considering the covariate effects and the presence of the competing risk. In addition, no computational cost in the estimation is a benefit for the research which includes a complex treatment strategy. The rest of the article is organized as follows. We introduce two regression models in Sections 2.2 and 2.3, and extend the methods to the situation where subjects may develop the event before the second randomization in Section 2.4. To relax the assumption of Fine and Gray’s model, we further apply our idea to Scheike’s model in Section 2.5. Asymptotic properties of the methods are discussed in Section 3. Results from finite-sample simulations are given in Section 4 and the analysis of the neuroblastoma study is given in Section 5. Finally, we conclude with some remarks in Section 6.

2. Methods

2.1. Setting and data

We consider a two-stage SMART as depicted in Figure 1. Subjects are first randomly assigned to an initial treatment, either or . Subjects who respond to the initial treatment are randomly assigned to either treatment or , and non-responders are randomized to treatments or . This results in eight DTRs with where subjects will start with the initial treatment , and receive if they respond to , or , otherwise. Define as the time to the intermediate response since the initial randomization. The response to the initial treatment, , is often determined if the response time is shorter than a pre-specified time period (e.g., achieving remission within 6 months). Let and be the second treatment indicators for the responders and for non-responders to the initial treatment. The long-term outcome of interest is subject to competing-risk events. Let be the time to the first failure from competing causes since the first randomization, and let be the corresponding cause of failure, where denotes the event of interest. In practice, the first failure might happen before the subjects respond to their first treatment. If death is the primary outcome of interest, for example, patients may die before they manage to achieve remission. Thus, we use to denote randomization status, where for subjects who only have the first randomization, and for subjects whose response to the first treatment can be observed and who enter the second randomization.

Fig. 1.

A two-stage SMART setup.

Without any loss of generality, we focus on the regimens starting with the initial treatment . For a particular DTR, , we define the event time as and the corresponding cause of failure as . Let and be the event time and the cause indicator when a subject following has developed the event of interest before the second randomization. If the subject proceeds to the second randomization, and is further randomized to , we define the corresponding event time and the cause indicator as and . and are similarly defined for the treatment path . Thus, , and .

Note that , , , , , and are all counterfactuals, since a subject who is assigned to the DTR can only follow one of the three potential paths. Here, we adopt the consistency assumption (Hernan and Robins, 2010) in that if a subject follows a particular path, e.g., , the observed event time and the cause indicator for this subject are the same as the counterfactuals and . Under the random assignment of treatments, “no unmeasured confounders” and “positivity” assumptions are satisfied (Orellana and others, 2010). Here, we consider a more general setting where some subjects have developed the event of interest before the second randomization, as in our neuroblastoma example. As we are considering SMART studies, exchangeability naturally follows as the probability of subsequent assignment is independent of potential outcomes given covariates and treatment history up to this point.

Let be a time-independent covariate vector. We are interested in evaluating covariate effects on the cause-1 CIF of a DTR . That is,

(2.1)

With the definition of counterfactuals, we then further define the th patient’s event time as , and the corresponding cause of failure as . Since we adopt the consistency assumption (Rubin, 1974) to relate the uncensored survival time to the counterfactual outcomes, for the th subject who is assigned to the regimen , we have that the observed outcome is equal to the counterfactuals outcome, i.e. , for . The definition of on the second paragraph of page 5 and the above definition of ensure this consistency. Under the random assignment of treatments, the ‘no unmeasured confounders’ is naturally satisfied in the counterfactual model.

In general, there may be right censoring before we observe , and we assume that and are conditionally independent given baseline covariates. Let be the potential censoring time with . In the presence of conditionally independent censoring, one observes , and . Competing risks data from a two-stage SMART trial consist of independent and identically distributed copies of .

Similar to a simple randomized control trial (RCT), if the study ended too early, all subjects would be censored, yielding no useful information for group comparisons. This issue is more relevant to a SMART, as it has multiple stages, which requires more careful planning in the time line of a study to ensure not everyone dies or is censored during the first stage. Moreover, a SMART tends to require a larger sample than an RCT in order to ensure that enough subjects follow a specific path under consideration. In the sequel, we assume that the competing risk data come from a well-planned SMART in which all involved paths are followed by some participants.

2.2. The Fine and Gray model with fixed weights

Fine and Gray (1999) proposed a semiparametric proportional hazards model for the subdistribution of a competing risk, and assumed that

where , and is a completely unspecified, invertible, and monotone increasing function. If we define the hazard function for the CIF (or subdistribution) , the above model has the proportional hazards interpretation for the subdistribution hazards, where , with being the baseline hazard function at time . {For a particular DTR in Fine and Gray’s model, the CIF in (2.1) can be formulated as

(2.2)

where is the baseline subdistribution hazard function at time in DTR, and is the coefficient vector in DTR. To simplify the notation, we use to denote and refer to as in the rest of article, if there is no confusion. For data from two-stage randomized trials, if we apply the Fine and Gray method directly to estimate the CIF for , only the data from subjects following treatment sequences or are included in the estimation of (2.1). The estimated CIF is often biased, since this naive Fine and Gray method weighs each subject consistent with equally in the estimation. To see the potential bias, let us consider a hypothetical example. Suppose 100 subjects are randomized to follow the DTR , and 40 subjects respond to the initial treatment and the rest do not. If there were no second-stage randomization, we would expect the 40 responders to follow the treatment sequence and the 60 non-responders to follow . An unbiased estimate of the CIF for will include the information from these 40 responders and 60 non-responders. Now with the second randomization, suppose 20 responders are assigned to the second-stage treatment and 18 non-responders are assigned to . The sub-sample used by the naive Fine-Gray method consists of these 20 responders and 18 non-responders, which has a higher proportion of responders as compared to the original sample without the second-stage randomization. Therefore, if we treat responders and non-responders in this subsample equally, we tend to have a biased estimate of the CIF.

To account for the bias, we follow a similar IPW approach as in Guo and Tsiatis (2005), Miyahara and Wahed (2010), and Yavuz and others (2016). Since the proportion of responders in the sub-sample, which is used in the naive Fine and Gray method, is not the same as that in the original sample before second-stage randomization, we assign the responders and non-responders in the sub-sample weights that are inversely proportional to their probabilities of being assigned to or . In the created pseudo sub-sample, the sizes of responders and non-responders are about the same as the original sample. More specifically, let and be the true probabilities of being assigned to for responders and being assigned to for non-responders, where and . Define

as the weight for subject in the regimen. Due to randomization, the observed proportions of being assigned to sequences and are not exactly equal to the true probabilities, and consequently, they may provide more information about the randomization process. Thus, we consider using the estimated probabilities, and , from the sequences and , instead of the true probabilities, to obtain the estimated fixed weight, , for subject . As a result, the pseudo sample that this estimated fixed weight creates, has the exactly same number of subjects and the same mixture of responders and non-responders as the original sample.

Here we extend Fine and Gray’s model to the two-stage randomized trials with estimated fixed weights. For subject , similar to Fine and Gray (1999), we defined the weight as , where is the KM estimate of the survival function for censoring . To avoid the same potential bias of using the sub-sample after second randomization, we construct based on the weighted counting and at-risk processes and . Combining their weighted vital status to handle censored observations and our estimated fixed weights to let the sub-sample represent all related responders and non-responders, we define the fixed weight score function for the regimen as

(2.3)

Let be a solution to the above score (2.3). In our study, the estimator of the CIF is of greater interest. Based on the estimated , we evaluate the CIF at time with covariates by using the formula in (2.2), namely,

where is the cumulative subdistribution hazard function at time with covariates estimated as

The limiting distribution of estimators and the inference are shown in Section 3.

2.3. The Fine and Gray model with time-varying weights

The weighted Fine and Gray method described in Section 2.2 does not utilize the information on time to response. Considering that subjects are consistent with all of the regimens before they have the intermediate response, we modify the above model using time-varying weights that incorporate those subjects with weights of 1 until their response status is observed. After obtaining their response status, subjects receive the weights according to their second randomization as in Section 2.2. More specifically, the weight for subject at time is

This idea has been used in Guo and Tsiatis (2005), Miyahara and Wahed (2010), and Yavuz and others (2016) for the non-parametric setting. The corresponding time-varying weight score function for the regimen under Fine and Gray’s model is

(2.4)

where , , and , with using the time-varying weighted counting and at-risk processes. Again the solution for (2.4), , can be obtained via the Newton–Raphson algorithm. Hence, the time-varying weight estimator of the CIF, based on the estimated , at time with covariates is

where

The asymptotic properties of estimators and the inference are discussed in Section 3.

2.4. An extension to subjects without second-stage randomization

In Sections 2.2 and 2.3, our discussions have focused on the situations that all subjects enter the second randomization. In practice, it is likely that some subjects develop the event of interest before they respond to the initial treatment. These subjects are excluded from the analyses in the above models. However, if we assign all subjects to a specific regimen, e.g., , those subjects who have developed the event of interest before the second-stage randomization are following this regimen. Hence, we expand our definition of “consistency” with the regimen, and now treat those subjects who have developed the target event before they meet the response criteria as consistent with the regimen. Consequently, we extend our methods by redefining the weights. If a subject has developed an event before their response status is determined, we assign the weight as 1 and record the event time. Such a subject is denoted as in Section 2.1, and the extended time-varying weight for subject at time is written as

The score function in this case is given by replacing with in (2.4). Estimation and inferences are very similar and hence the details are left out for brevity.

2.5. Extensions of the Scheike model

Fine and Gray’s model is popular and convenient in practice with the available software. If the proportionality for sub-distribution is satisfied, the results are accurate and easy to interpret. However, this assumption may be too restrictive for a two-stage randomization study, because the covariate effects on the CIF may change when subjects switch from the initial treatment to the second-stage treatment. Though the weighted Fine and Gray model in Sections 2.3 and 2.4 can still provide reasonable estimates of the CIF as shown in our simulation studies, we now consider extending a more flexible binomial regression model proposed by Scheike and others (2008) to the two-stage randomization setting in order to capture potential time-varying covariate effects for a particular DTR. The additive Scheike model assumes that

where are the time-varying effects of , a subset of covariates, on the CIF at time , and are the fixed-effect coefficients for the rest of covariates, . The and are known link functions. If and , the Scheike model will become the proportional hazards model for subdistributions as in Fine and Gray (1999).

As before, we extend the original Scheike model to the fixed weight Scheike model for a two-stage randomization setting. Let denote the cause-1 CIF at time for subject with covariates following the regimen . The estimating equation for at time can be written as , where

with denoting for brevity, , denoting partial derivatives, and being some possibly random weights.

We use binomial regression as in Scheike and others (2008), coupled with the Newton–Raphson iteration, to obtain the estimator for the time-varying coefficients at each time point , and the estimator for the time-independent coefficients. The estimation of the CIF for given covariates can be carried out similar to the extended Fine and Gray models. Furthermore, we can establish the Scheike model with time-varying weights by using the estimated time-varying weights in Section 2.3, and also extend this model so that subjects without the second stage randomization are included by using the weights defined in Section 2.4.

However, the inferences of these estimators are much more involved. In the Appendix of supplementary material available at Biostatistics online, we give the influence functions for and under the simplified setting where all events occur only after second-randomization. We implement the fixed-weighted Scheike model and the time-varying weight Scheike model (TWSC) by treating all covariates with time-varying coefficients. The implementation is rather complicated. Therefore, we also propose an approximation based on the idea of augmenting the data for the fixed weight Scheike model. To illustrate our idea, we continue to consider the hypothetical example for in Section 2.2, where 100 subjects are assigned to the initial treatment , and 40 of them respond to . During the second-stage randomization, 20 of the 40 responders are assigned to , and 18 of the 60 non-responders are assigned to . According to Section 2.2, the fixed weight for subjects in the sequence of is , and that for subjects following is . We create an augmented data by repeating each subject in 20 times, and each subject following 33 times. This augmented data contains 400 responders and 594 non-responders, approximately the same mixture of responders and non-responders as the original sample. Thus, a well-implemented R function, “comp.risk,” in the package timereg for the Scheike and others (2008), can be directly applied to the augmented data, resulting in a consistent, though slightly less accurate, estimator of the CIF for . Since the size of the augmented data is about 10 times of the original sample, the standard deviation from the R function needs to be multiplied by the squared root of the ratio of the augmented data sample size to the original sample size. In general, the augmented data approach can be applied to other models, such as the fixed weight Fine and Gray method.

3. Asymptotic properties

In this section, we establish asymptotic properties of our proposed estimators. Because the fixed weight Fine and Gray method is a special case of the time-varying weight Fine and Gray model, we focus on time-varying weight Fine and Gray model in this section.

In the inference on the estimation of CIFs, we use the weights included the true probabilities, and . Let be 1 if and be , otherwise. Define as a counting process and as a martingale, where and . We replace by in the time-varying weighted score function, and recast this score function in terms of martingale integration, under the true . To simplify the score equation, denote , and where and . The time-varying weight score function for the regimen, under the true , is

Theorem 3.1

Under mild regularity conditions, the converges in distribution to a Gaussian process with covariance matrix . Then, the asymptotical distribution of is normally distributed with mean zero and the covariance matrix , where

The proof of Theorem 3.1 and the form of are included in the Appendix of supplementary material available at Biostatistics online. Consistent estimators of and are

where

We use in computing the weighted vital status, , where is the KM estimator with the sub-sample re-represented using . More specifically,

Also,

is the estimated Martingale for the cause-1 event, where . In , and is the estimated martingale for censoring, where

Theorem 3.2

If a consistent estimator exists and converges in distribution to a Gaussian process on an interval , where , then has the same limiting distribution as

(3.1)

where

and

The detailed proofs of Theorem 3.2 are given in the Appendix of supplementary material available at Biostatistics online. However, it is complicated to evaluate the exact limiting distribution of the CIF estimator in (3.1). Hence, we follow Fine and Gray (1999) and adopt an approximation based on random perturbation. More specifically, let be a random sample from the standard normal distribution and

To obtain the estimated variance at with covariates , we generate B samples , , and compute for . Then the standard deviation for the CIF estimator at time can be estimated by We further discuss the extension of models with . Several studies have shown that the inference results remain similar as before, even when is replaced by a consistent estimator (e.g. Yavuz and others 2016). Thus, we simply replace with in the estimation of the CIFs and the inference procedures. All the above discussions can be applied to the fixed weight Fine and Gray model.

4. Simulation

We conduct extensive simulations to study the finite-sample performance of our proposed methods. Subjects are assumed to be randomized to the initial treatments and . Thus, we focus on only subjects who are assigned to . The following three scenarios are considered to mimic a general setting: (i) subjects have developed the event of interest before they respond to ; (ii) subjects who have responded to within a specific time, say, months ( year), and have not developed either the target event or the competing event, are further randomized to treatments or ; and (iii) subjects who have not responded to and have not developed any event within months, are randomized to treatments or . Two covariates are considered, where is a standard normal variate and is from Bernoulli. To create the three scenarios, we introduce as the time in years to either the first event (a cause-1 or competing event) or the intermediate response from the initial randomization. The subjects whose exceed are treated as non-responders to the initial treatment . They are further randomized to or following a Bernoulli distribution with , and and their response times are truncated at . Assume that is independent of the covariates and follows an Exponential distribution with rate . Here is determined by and . Since then . For those with , they can either have developed an event or responded before , whichever occurring first. Given , we simulated following a Bernoulli distribution, where . When , subjects are assumed to have developed an event before the second randomization, where the event cause indicator is further simulated from a Bernoulli distribution with . When , those subjects are deemed as responders with , and further randomized to or following a Bernoulli distribution with . In our simulations, we let , or , and .

For those subjects proceeding to the second randomization, we let and denote the time to the first event and the corresponding cause indicator in a specific treatment sequence since the second randomization, where and . Assume that and follow the Fine and Gray Model:

(4.1)

where , , . It is not trivial to simulate from (4.1), as the CIFs involved are improper. Here we adopt the simulation strategy used in Fine and Gray (1999), Cheng and others (2009), and Beyersmann and others (2012), Section 5.3. A random variable is first drawn from Uniform[0,1]. If is smaller than the asymptote of the CIF, we generate by inverting the CIF. Otherwise, the CIF is not invertible, implying that the cause 2 event occurs first. We assume that the conditional distribution of , given covariates and the occurrence of type 2 event, follows and simulate from this conditional distribution and let The true regression coefficients were set as and .

Assuming that there is no delay between time to the intermediate response and time to the second randomization, the overall survival time is with the corresponding cause indicator . The CIF for cause 1 event at time for regimen , where , can be written as

(4.2)

Based on these assumptions and the Bayes rule, the first part in (4.2) has the form

For the second part in (4.2), equals to

Thus, we have simulated a cause-1 event time from subjects following the aforementioned three scenarios with the CIF given in (4.2). Finally, the censoring time, , was generated from the Exponential distribution with rate 0.1. The observed time and the observed event type indicator .

We generated samples with size 400 and repeated 2000 times. For each simulated data, we implemented six models, including the original Fine and Gray model (FG), the fixed weight Fine and Gray model (WFG), the time-varying weight Fine and Gray model (TWFG), the original Scheike Model (SC), the fixed weight Scheike Model (WSC), and the time-varying weight Scheike model (TWSC). We implemented the WFG and TWFG by solving the score functions as discussed in Sections 2.2 and 2.3, and then computed the estimated CIF as well as standard deviation based on the influence functions given in Section 3. The FG was simply a special case by setting all weights to be 1. For the Scheike models, we assumed time-varying effects for both covariates and used the cloglog link function to have a proportional hazards model. The implementation of the WSC and TWSC can be completed by solving the score functions for both models, and then using the functional delta method to estimate the variance of CIFs. The naive Scheike model can be simply run by using the R function “comp.risk.” In order to compare all models, the weights for subjects with were set equal to 1 for the FG and SC.

For each model, we computed the averages of the CIF estimates at different time points with covariates . The true CIF values were computed based on (4.2) through numerical integration. We considered two probabilities of response, or , and presented the results for and regimens in Figure 2 over a range of time points. In the Appendix Table 1 of supplementary material available at Biostatistics online, we also listed the mean of estimates (est), the empirical standard deviation (), the mean of estimated standard deviations (), and the coverage rate of 95% confidence intervals (Cov) along with the true values at time points 0.225, 0.3, 0.5, 0.75, and 1.

Fig. 2.

The estimated CIFs over time using six models. The black solid line is the true function. Gray lines for Fine–Gray-related models. Black stepwise curves for Scheike-related models. The native methods are dashed lines, the fixed weight methods are dotted lines, and the time-varying weight are long dashed lines.

The simulation results show that the naive estimators, FG and SC, tend to overestimate the CIF at early time points but not at later time points. This is as expected because the naive methods assign equal weights to those subjects who failed before the second-randomization (with quick failure) and to those subjects who were further randomized to second-stage treatments. In other words, the naive methods include more subjects who failed earlier. In contrast, the weighted estimators, WFG, WSC, TWFG, and TWSC, have much better performance in estimating the CIF, since they all properly up-weigh those subjects going through the second-stage randomization. The Scheike model has better performance than the Fine and Gray model, especially when the CIF is relatively low. However, the WFG and TWFG model still perform reasonably well across time. Moreover, though none of the models are exactly the true models, the WFG, TWFG, WSC, and TWSC still provide reliable estimation of the CIF. Consequently, the fixed and time-varying weight estimators are more reliable methods than the naive ones in finding an optimal DTR from a two-stage randomized trial.

5. Analysis of children’s neuroblastoma study

We now revisit our motivating example of the neuroblastoma study that was conducted by the Children’s Cancer Group between 1991 and 1996. Neuroblastoma is a type of cancer that starts in early nerve cells of the sympathetic nervous system and occurs most often in infancy and young children. Children with high-risk neuroblastoma have high recurrence and poor survival rates (www.cancer.org/cancer/neuroblastoma). Thus, an important clinical question is how to stop or delay disease progression and thus improve survival, by providing an optimal regimen to patients according to their states. In this section, we apply our methods to evaluate the preventive effect on disease progression of myeloablative chemotherapy and radiotherapy plus purged autologous bone marrow transplantation (ABMT) over intensive chemotherapy (Chemo) alone, followed by subsequent treatment with 13-cis-retinoic acid (cis-RA) or no further treatment (no RA) for children with high-risk neuroblastoma (Matthay and others, 2009). The study adopted a two-stage SMART design. After receiving an induction chemotherapy, 379 eligible children without progressive disease participated in the first-stage randomization, where 189 children were assigned to AMBT and 190 children were assigned to Chemo. Those children, who did not develop progressive disease after the initial treatment and were willing to be further randomized, were defined as responders and, subsequently randomized to receive either cis-RA or no RA. At the second stage, 50 of 98 ABMT responders and 52 of 105 Chemo responders received cis-RA. As reported in Matthay and others (2009), the median time of the first random assignment was about 60 days after diagnosis, and the median time of the second randomization was about 300 days after diagnosis. For simplicity, we referred to those children who did not have the second stage randomization as non-responders. Thus, four possible regimens could be constructed for this study: (i) treating with ABMT followed by cis-RA if subjects responded and no further therapy if subjects did not respond (ABMT/cis-RA), (ii) treating with ABMT followed by no RA if subjects responded and no further therapy if subjects did not respond (ABMT/no RA), and (iii) Chemo/cis-RA, and (iv) Chemo/no RA were defined similarly.

During the study a total of 269 children developed progressive disease, with 134 occurring in non-responders, a total of 23 children died before they developed the disease, with 22 in non-responders, and a total of 87 children were right censored, with 20 in non-responders. Therefore, the event of interest, the time to disease progression, could not be observed after death, which is a competing event, and the CIF is used to describe cumulative risks of disease progression in the presence of death. Furthermore, an interesting feature of the data is that the response was defined as no disease progression, and the time to response was closely related to our event of interest which is disease progression. As a result, the time-varying weight methods are not applicable; see Yavuz and others (2016) for more details. Therefore, only the fixed weight methods WFG and WSC can be applied to this dataset.

Following Matthay and others (2009), we considered five potential risk factors, age (Age), disease stage (Stage4dx), ferritin (Ferritindx), MYCN status (MYCNdx), and bone metastases (Bonesdx). Tumor pathology was not considered due to a very unbalanced sample size (6 vs.120 in ABMT and 9 vs.128 in Chemo). In the analysis, we treated Age as a continuous variable, and included the rest of covariates as dichotomous variables using Matthay and others (2009)’s definition. Following Matthay and others (2009), we excluded the missing values and used the complete data with a total of 260 children, with 120 in ABMT and 140 in Chemo. In the complete data, a total of 177 children developed progressive disease, with 95 occurring in non-responders, a total of 14 children died before they developed the disease, with 14 in non-responders, and a total of 69 children were right censored, with 17 in non-responders.

In order to compare with Matthay and others (2009), we illustrated our methods by focusing on the AMBT/cis-RA regimen, and applying the WSC model to examine if any time-varying effect exists. The estimated time-varying coefficients are given in Appendix Figure 1 of supplementary material available at Biostatistics online. The formal Kolmogorov–Smirnov test and the Cramer–von Mises test for time-varying coefficients are summarized in Table 1, which suggest that none of the covariates have a time-varying effect. Therefore, presenting the fixed-weight Fine and Gray model as the final model for the AMBT/cis-RA regimen is reliable.

Table 1.

p-Values for testing “time-varying effect” for each variable, using the Kolmogorov–Smirnov test and the Cramer–von Mises test

	p-Value of testing time-varying coefficient
Test	Age	Stage4dx	Ferritmdx	MYCNdx	Bonedx
Kolmogorov–Smirnov	0.661	0.307	0.566	0.090	0.735
Cramer–von Mises	0.174	0.777	0.276	0.334	0.287

Moreover, to compare with other regimens, we examined the time-varying effects of the five covariates using the Scehike model for the rest of three regimens. Only Age is significant with p-value = 0.02 in the Chemo/No RA regimen by the Cramer–von Mises test. However, Age is not significant with p-value = 0.07 by using the Kolmogorov–Smirnov test. Considering multiple comparisons involved in testing five covariates for four regimens, it is reasonable to assume constant covariate effects, and thus to apply the fixed-weight Fine and Gray model to each of the four DTRs. The p-values for testing the significance of covariate effects in the final models are given in Table 2. The results show that Ferritindx, MYCNdx, and Age are significant in three of the four regimens, Stage4dx is significant in only one of the four regimens, and Bonedx is not significant for all regimens. The estimated coefficients suggest that higher levels of ferritin and MYCN amplification are associated with faster disease progression. Despite that the outcomes in Matthay and others (2009) are not the same as our outcome of interest, and the subgroups included in the two analyses are different, our method has identified the same set of important covariates as those listed in Matthay and others (2009).

Table 2.

The estimated coefficients from the WFG model for the four regimens. The estimate , the estimate of standard deviation (), and the p-value for testing . Those with p values are shown in bold fonts.

		Covariate
Regimen	Estimates/p-value	Age	Stage4dx	Ferritindx	MYCNdx	Bonedx
ABMT/cis-RA	()	0.17(0.06)	0.97(0.45)	0.97(0.33)	0.51(0.32)	0.12(0.35)
	p-value	0.005	0.031	0.003	0.110	0.737
ABMT/no RA	()	0.19(0.05)	0.46(0.45)	0.07(0.31)	0.79(0.38)	0.22(0.35)
	p-value	0.001	0.312	0.828	0.039	0.532
Chemo/cis-RA	()	0.12(0.05)	0.67(0.42)	0.55(0.26)	0.73(0.27)	0.51(0.26)
	p-value	0.034	0.113	0.037	0.007	0.053
Chemo/no RA	()	0.06(0.05)	0.37(0.44)	0.68(0.26)	1.11(0.27)	0.02(0.24)
	p-value	0.287	0.407	0.009	0.001	0.942

To compare the CIFs of progressive disease over time of the four regimens with various covariate effects, we present the CIF estimates obtained by fitting the WFG model for the four regimens in Figure 3, for Ferritindx 0 or 1, and MYCNdx 0 or 1, while setting Age 3 (the median age in the data), Stage4dx 0 and Bonedx 0. From Figure 3, patients with higher level of Ferritindx or MYCN gene copy were more likely to experience progressive disease across the four regimens, which is consistent with the estimated coefficients in Table 2. Figure 3 also suggests that the Chemo/no RA regimen seems to be the worst regimen for children with high-risk neuroblastoma, whereas the AMBT/cis-RA regimen may be the optimal regimen among the four. For children with high ferritin level and no MYCN gene copy, ABMT/cis-RA and ABMT/no RA regimens seem to be comparable with negligible differences, and they both appear to perform better than the other two.

Fig. 3.

The estimated CIFs for the four regimens obtained by using the WFG method with four cases while controlling for Age = 3 years, Stage4dx = 0, and Bonedx = 0. The plots in the upper row are for Ferritindx = 0, and the plots in the lower row are for Ferritindx = 1. The plots in the left column are for MYCNdx = 0, and the plots in the right column are for MYCNdx = 1.

6. Discussion

Patient heterogeneity is of great clinical importance from a clinical trial perspective. If subjects were to follow a specific DTR, they may wonder how well they would fare from this specific treatment strategy given their own clinical characteristics. In this article, we have focused on the direct modeling of covariate effects on a specific DTR. We have demonstrated that the IPW method can be used to extend some commonly used regression models for competing-risk data to a two-stage randomization setting. The Fine and Gray and Scheike models were used as examples, though our methods can be readily applied to other models, such as the multinomial logistic model (Gerds and others, 2012). Our simulations show that the resulting weighted estimators of the CIF are still reasonably accurate, even though the underlying Fine and Gray or Scheike model may be misspecified. Therefore, we provide convenient and reliable methods to evaluate covariate effects on the CIF. The proper modeling of covariate effects on various DTRs will facilitate selection of the optimal treatment strategy for a subject with specific characteristics.

Our current method extends the existing regression models by properly weighting subjects who are consistent with the DTR of interest. Corresponding weight functions are added to the score functions as used in the original model. It would be of interest to consider double robust estimation (Tsiatis, 2007) in the future to further improve the efficiency of our proposed models.

7. Software

Software in the form of R code and a sample dataset are available at https://github.com/lingwanchen/CIRdtr.git.

Funding

The work was partially supported by an ADRC pilot award, P50AG005133, NIA to Cheng.