A Proportional Hazards Mixture Cure Model for Subgroup Analysis: Inferential Method and an Application to Colon Cancer Data

Abstract

When determining subgroups with heterogeneous treatment effects in cancer clinical trials, the threshold of a variable that defines subgroups is often pre-determined by physicians based on their experience, and the optimality of the threshold is not well studied, particularly when the mixture cure rate model is considered. We propose a mixture cure model that allows optimal subgroups to be estimated for both the time to event for uncured subjects and the cure status. We develop a smoothed maximum likelihood method for the estimation of model parameters. An extensive simulation study shows that the proposed smoothed maximum likelihood method provides accurate estimates. Finally, the proposed mixture cure model is applied to a colon cancer study to evaluate the potential differences in the treatment effect of levamisole plus fluorouracil therapy versus levamisole alone therapy between younger and older patients. The model suggests that the difference in the treatment effect on the time to cancer recurrence for uncured patients is significant between patients younger than 67 and patients older than 67, and the younger patient group benefits more from the combined therapy than the older patient group.

1. Introduction

With the advancement of modern medical technology, precision medicine has attracted considerable attention in recent years. Subgroup analysis serves as a crucial component in precision medicine. Rothwell [1] summarized the importance of subgroup analysis in randomized controlled trials for new drug development for the following three reasons: (1) The trial results meet the primary objectives, but the drug efficacy may vary significantly across different subgroups. (2) The trial results fail to meet the primary objectives, but the drug demonstrates significant efficacy in certain subgroups. (3) There are clinical concerns regarding the treatment of specific populations, such as the elderly. In a clinical control trial subgroup analysis, the threshold of a variable that defines subgroups is often pre-fixed. For example, the MONARCH plus study, a randomized clinical trial with women with hormone receptor-positive and human epidermal growth factor receptor 2-negative advanced breast cancer [2], investigated heterogeneous treatment effects of abemaciclib plus endocrine therapy on progression-free survival in patients with two subgroups using the Cox proportional hazards (PH) model. The two subgroups are formed by age older or younger than 65 years. The age threshold 65 is a pre-fixed threshold, and its optimality remains unknown. Several studies have investigated methods for detecting the optimal threshold in the context of proportional hazards models [3,4,5,6]. Luo et al. [3] provided a theoretical proof of the asymptotic consistency of the maximum partial likelihood estimator for the unknown threshold. Pons [4] proposed a maximum partial likelihood estimator based on the profile method and established its consistency and asymptotic distribution. Chen et al. [5] developed a novel hierarchical Bayesian method to make statistical inference simultaneously on the threshold and the covariate coefficients. Deng et al. [6] proposed a change-point proportional hazards model for clustered event data, developing a maximum marginal pseudo-partial likelihood estimator for the regression coefficients and the unknown change point, as well as a supremum test based on robust score statistics to test the existence of the change point. Lee et al. [7] also proposed three statistical tests, namely, the maximal score, maximal normalized score, and maximal Wald tests and established their asymptotic properties. Further studies extend the single threshold proportional hazards model to mulitple thresholds [8,9], continous thresholds [10], random thresholds [11], single-index threshold model [12] and change-plane model [13]. More literature on subgroup analysis in hetergenous Cox model can be found in [14,15].

Many diseases and cancers are curable nowadays, and cured patients will never experience the event of interest. For example, the cure rate for patients with resected BRAF V600–mutant stage III melanoma treated by adjuvant dabrafenib plus trametinib is 54% [16], and the cure rate of acute lymphoblastic leukemia in children is 90% [17]. When the cured fraction is present in data, the classical survival models, such as the Cox PH model, are not appropriate, and cure rate models, particularly the mixture cure models, have been used extensively for analyzing survival data with a fraction of cured subjects for decade. Parametric, semiparametric, and nonparametric statistical inferential methods for the mixture cure rate model have been studied extensively in the literature [18,19,20,21].

However, the study on determining the optimal threshold or cutoff value to form subgroups with heterogeneous treatment effects in mixture cure rate models is very limited. Othus et al. [22] studied a subgroup analysis under a parametric mixture cure rate model for the heterogeneous prognostic effect of a variable. That is, they assumed different cure rates and constant hazard rates for two subgroups formed by a covariate with an unknown change-point or threshold. A smoothed likelihood approach based on the observed likelihood function was developed to estimate the parameters in the model. The model and the method were used to analyze the heterogeneous prognostic effect of age among prostate cancer patients. Shamsi et al. [23] further extended this method by considering Bayesian inference. Zhao et al. [24] proposed a semiparametric Cox PH mixture cure rate model that allows a threshold in the regression coefficient and then developed Bayesian inference for the proposed model, and then applied the model to an oropharynx cancer data to illustrate the gender effect changes under different age groups at diagnosis. Wang et al. [25] studied the Cox PH mixture cure model when a covariate effect on the failure time of uncured subjects may change when the value of the covariate exceeds a change-point or threshold, and proposed a nonparametric maximum likelihood estimation method to estimate the model. Lou et al. [26] studied a class of partly linear transformation models within the mixture cure model framework for interval-censored failure time data with change-points.

However, some key limitations remain. Although [22,23] considered heterogeneous effects in different subgroups on the cure rate and the hazard rate, they only focus on the prognostic effect of the subgroups, and they do not consider the heterogeneous treatment effect in different subgroups, which is often called the predictive effect of the subgroups. It is worth noting that both works in [24,25] only considered heterogeneous subgroup effects in the uncured population, while the heterogeneous subgroup effects in the cure probability is not considered. To fill this gap, we propose a mixture cure model that can determine thresholds for a variable to form subgroups with heterogeneous treatment effects on the time to event of interest for uncured subjects and/or on the probability of being cured. The interaction terms given in (2) and (3) model the predictive effect of the subgroups for both the uncured subjects and the cure probability.

The rest of this paper is organized as follows. Section 2 presents the mixture cure model for subgroup analysis and the proposed estimation method. In Section 3.1, an extensive simulation study is conducted to investigate the performance of the proposed estimation method. The proposed model is illustrated with a colon cancer data set in Section 3.2. Finally, discussion and conclusions are presented in Section 4 and Section 5.

2. Materials and Methods

Let $T^{*}=Z\infty +(1-Z)\tilde{T}$ be the lifetime of a patient, where Z is the cure indicator with $Z=1$ if the patient is cured and $Z=0$ otherwise, and $\tilde{T}$ is the lifetime for uncured patients. Let C be the censoring time. Then, the observed lifetime or censoring time is denoted by $T=min(T^{*},C)$, with a censoring indicator $\delta =I(T^{*}\le C)$ where $I\left(A\right)$ is an indicator function with value 1 if A is true and 0 otherwise. Let W be the treatment variable with $W=1$ for a new treatment and $W=0$ for a standard treatment, $U_1$ and $U_2$ be the two covariates or biomarkers that define subgroups with different treatment effects on $\tilde{T}$ and Z, respectively, via $I(U_1>c_1)$ and $I(U_2>c_2)$, where $c_1$ and $c_2$ are unknown thresholds or cutoff values. Let ${\mathit{X}}_1$ and ${\mathit{X}}_2$ be covariate vectors to be adjusted in the models for $\tilde{T}$ and Z, respectively, and define ${\mathit{V}}_1=({\mathit{X}}_1^{\prime },W,U_1)$ and ${\mathit{V}}_2=(1,{\mathit{X}}_2^{\prime },W,U_2)$. Then, the mixture cure rate model is given in terms of survival function as

$$\begin{array}{c}{S}_{pop}\left(t\right|{\mathit{V}}_1,{\mathit{V}}_2;{\mathit{\vartheta }}_1,{\mathit{\vartheta }}_2)=P(T^{*}>t|{\mathit{V}}_1,{\mathit{V}}_2;{\mathit{\vartheta }}_1,{\mathit{\vartheta }}_2)= \hfill \\ \hfill p({\mathit{V}}_2;{\mathit{\vartheta }}_2)+(1-p({\mathit{V}}_2;{\mathit{\vartheta }}_2))S\left(t\right|{\mathit{V}}_1;{\mathit{\vartheta }}_1),\end{array}$$

(1)

where

$$\begin{array}{c}S(t|{\mathit{V}}_1,{\mathit{\vartheta }}_1)=P(\tilde{T}>t|{\mathit{V}}_1;{\mathit{\vartheta }}_1)=S_0{(t;\mathit{\lambda })}^{exp({\mathit{\alpha }}_1^{\prime }{\mathit{X}}_1+{\gamma }_{1}W+{\theta }_{1}I(U_1>c_1)+{\eta }_{1}WI(U_1>c_1))};\end{array}$$

(2)

$$\begin{array}{c}p({\mathit{V}}_2,{\mathit{\vartheta }}_2)=P(Z=1|{\mathit{V}}_2;{\mathit{\vartheta }}_2)={\displaystyle \frac{e^{{\mathit{\beta }}_0+{\mathit{\alpha }}_2^{\prime }{\mathit{X}}_2+{\gamma }_{2}W+{\theta }_{2}I(U_2>c_2)+{\eta }_{2}WI(U_2>c_2)}}{1+e^{{\mathit{\beta }}_0+{\mathit{\alpha }}_2^{\prime }{\mathit{X}}_2+{\gamma }_{2}W+{\theta }_{2}I(U_2>c_2)+{\eta }_{2}WI(U_2>c_2)}}};\end{array}$$

(3)

${\mathit{\vartheta }}_1={({\mathit{\lambda }}^{\prime },{\mathit{\alpha }}_1^{\prime },{\gamma }_1,{\theta }_1,{\eta }_1,c_1)}^{\prime }$ and ${\mathit{\vartheta }}_2={({\beta }_0,{\mathit{\alpha }}_2^{\prime },{\gamma }_2,{\theta }_2,{\eta }_2,c_2)}^{\prime }$ are the unknown parameter vectors, and $S_0(t;\mathit{\lambda })$ is the baseline survival function with unknown parameter vector $\mathit{\lambda }$. We consider the Weibull distribution as the baseline distribution with

$$S_0(t;\mathit{\lambda })=exp(-{\lambda }_2{t}^{{\lambda }_1}),$$

where $\mathit{\lambda }=({\lambda }_1,{\lambda }_2)$. To allow a more flexible baseline distribution in the model if the Weibull distribution is not desirable in practice, we also consider a piecewise constant hazard distribution [27] with $S_0(t;\mathit{\lambda })=exp\left(-{\int }_0^t{h}_0(u;\mathit{\lambda })du\right)$, where

$$h_0(t;\mathit{\lambda })={\lambda }_k\mathrm{for}{t}_{(k-1)}\le t<t_{\left(k\right)},k=1,\dots ,J,$$

$\mathit{\lambda }=({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_J)$, and $0=t_{\left(0\right)}<t_{\left(1\right)}<t_{\left(2\right)}<\dots t_{\left(J\right)}<max\left(t_i\right)$ is a partition of J time intervals. The part of the mixture cure model in (2) is often referred to as the latency part, and the part of the model in (3) is referred to as the incidence part.

Consider a sample of n subjects. The observed values of $(T,\delta ,{\mathit{V}}_1,{\mathit{V}}_2)$ from the sample are denoted as $(t_i,{\delta }_i,{\mathit{V}}_{i1},{\mathit{V}}_{i2})$, $i=1,2,\dots ,n$. Then, the likelihood function for the unknown parameters $\mathit{\vartheta }=\left({\mathit{\vartheta }}_1,{\mathit{\vartheta }}_2\right)$ is given by

$$\begin{array}{c}L\left(\mathit{\vartheta }\right)=\prod _{i=1}^n{\left\{\left[1-p({\mathit{V}}_{i2},{\mathit{\vartheta }}_2)\right]f\left(t_i\right|{\mathit{V}}_{i1},{\mathit{\vartheta }}_1)\right\}}^{{\delta }_i} \hfill \\ \hfill {\left\{p({\mathit{V}}_{i2},{\mathit{\vartheta }}_2)+\left[1-p({\mathit{V}}_{i2},{\mathit{\vartheta }}_2)\right]S\left(t_i\right|{\mathit{V}}_{i1},{\mathit{\vartheta }}_1)\right\}}^{1-{\delta }_i},\end{array}$$

(4)

where $f(·)$ is the density function corresponding to $S(·)$. The log-likelihood function for the mixture cure rate model specified in (2) and (3) can be simplified as

$$\begin{array}{c}l\left(\mathit{\vartheta }\right)=\sum _{i=1}^n{\delta }_i\left[ln{h}_0(t_i;\mathit{\lambda })+r_1\left({\mathit{\vartheta }}_1\right)-{\Lambda }_0(t_i;\mathit{\lambda })e^{r_1\left({\mathit{\vartheta }}_1\right)}\right] \hfill \\ \hfill +\sum _{i=1}^n(1-{\delta }_i)ln[e^{r_2\left({\mathit{\vartheta }}_2\right)}+e^{-{\Lambda }_0(t_i;\mathit{\lambda })e^{r_1\left({\mathit{\vartheta }}_1\right)}}]-\sum _{i=1}^{n}ln[1+e^{r_2\left({\mathit{\vartheta }}_2\right)}],\end{array}$$

(5)

where $r_1\left({\mathit{\vartheta }}_1\right)={\mathit{\alpha }}_1^{\prime }{\mathit{X}}_{i1}+{\gamma }_1{W}_i+{\theta }_{1}I(U_{i1}>c_1)+{\eta }_1{W}_{i}I(U_{i1}>c_1)$ and $r_2\left({\mathit{\vartheta }}_2\right)={\mathit{\alpha }}_2^{\prime }{\mathit{X}}_{i2}+{\gamma }_2{W}_i+{\theta }_{2}I(U_{i2}>c_2)+{\eta }_2{W}_{i}I(U_{i2}>c_2).$

Since the indicator functions $I(U_{i1}>c_1)$ and $I(U_{i2}>c_2)$ are discontinuous at $U_{i1}$ and $U_{i2}$ when treated as a function of $c_1$ and $c_2$, respectively, the log likelihood function is discontinuous, which makes the maximization of the log likelihood function challenging. One approach is to find the profile log-likelihood function $pl(c_1,c_2)=l\left({\hat{\mathit{\vartheta }}}_{c_1,c_2}\right)$, where ${\hat{\mathit{\vartheta }}}_{c_1,c_2}=\underset{\mathit{\vartheta }\setminus \{c_1,c_2\}}{\mathrm{argmax}}l\left(\mathit{\vartheta }\right)$. Then estimates of $c_1$ and $c_2$, denoted as ${\hat{c}}_1$ and ${\hat{c}}_2$, can be obtained as $\underset{c_1,c_2}{\mathrm{argmax}}pl(c_1,c_2)$, which can also be challenging because of discontinuity in the function $pl(c_1,c_2)$, and a grid search method is often used. With ${\hat{c}}_1$ and ${\hat{c}}_2$, an estimate of $\mathit{\vartheta }\setminus \{c_1,c_2\}$ can be obtained as ${\hat{\mathit{\vartheta }}}_{{\hat{c}}_1,{\hat{c}}_2}$. This estimation method will be referred to as the PMLE method. Due to the non-smoothness of the log-likelihood function, the standard errors in this method cannot be obtained under the standard likelihood theory, and the bootstrap method may be used instead.

In this work, we propose to use kernel smoothing functions to approximate the indicator function [12,28] so that the resulting log likelihood function is a smooth function. In particular, we consider using either the normal kernel function $g(x/h)=\mathrm{\Phi }\left(x/h\right)$ or the logistic kernel function $g(x/h)=exp(x/h)/[1+exp(x/h\left)\right]$ to approximate $I\left(x\right)$, where $\mathrm{\Phi }$ is the cumulative distribution function of the standard normal distribution and h is a bandwidth. It is obvious that as $h\to 0$, if $x>0$, $g(x/h)\to 1$, otherwise $g(x/h)\to 0$. Thus, we have $I(U_{i1}>c_1)\approx g\left({\displaystyle \frac{U_{i1}>c_1}{h_1}}\right)$ and $I(U_{i2}>c_2)\approx g\left({\displaystyle \frac{U_{i2}>c_2}{h_2}}\right)$ as $h_1\to 0$ and $h_2\to 0$. Using these approximations, we can obtain a smoothed log-likelihood function as follows:

$$\begin{array}{c}sl\left(\mathit{\vartheta }\right)=\sum _{i=1}^n{\delta }_i\left[ln{h}_0(t_i;\mathit{\lambda })+{\tilde{r}}_1\left({\mathit{\vartheta }}_1\right)-{\Lambda }_0(t_i;\mathit{\lambda })e^{{\tilde{r}}_1\left({\mathit{\vartheta }}_1\right)}\right] \hfill \\ +\sum _{i=1}^n(1-{\delta }_i)ln[e^{{\tilde{r}}_2\left({\mathit{\vartheta }}_2\right)}+e^{-{\Lambda }_0(t_i;\mathit{\lambda })e^{{\tilde{r}}_1\left({\mathit{\vartheta }}_1\right)}}-\sum _{i=1}^{n}ln[1+e^{{\tilde{r}}_2\left({\mathit{\vartheta }}_2\right)}],\end{array}$$

(6)

where ${\tilde{r}}_1\left({\mathit{\vartheta }}_1\right)={\mathit{\alpha }}_1^{\prime }{\mathit{X}}_{i1}+{\gamma }_1{W}_i+{\theta }_{1}g\left({\displaystyle \frac{U_{i1}-c_1}{h_1}}\right)+{\eta }_1{W}_{i}g\left({\displaystyle \frac{U_{i1}-c_1}{h_1}}\right)$ and ${\tilde{r}}_2\left({\mathit{\vartheta }}_2\right)={\mathit{\alpha }}_2^{\prime }{\mathit{X}}_{i2}+{\gamma }_2{W}_i+{\theta }_{2}g\left({\displaystyle \frac{U_{i2}-c_2}{h_2}}\right)+{\eta }_2{W}_{i}g\left({\displaystyle \frac{U_{i2}-c_2}{h_2}}\right).$ A derivative-based method, such as the Newton-Raphson method, can be used to maximize the smoothed log-likelihood function to obtain the estimate of $\mathit{\vartheta }$. The Newton-Raphson method requires

$$U\left(\mathit{\vartheta }\right)={\displaystyle \frac{\partial sl\left(\mathit{\vartheta }\right)}{\partial \mathit{\vartheta }}},I\left(\mathit{\vartheta }\right)=-{\displaystyle \frac{{\partial }^{2}sl\left(\mathit{\vartheta }\right)}{\partial \mathit{\vartheta }\partial {\mathit{\vartheta }}^{\prime }}},$$

and the iterative process

$${\mathit{\vartheta }}^{(k+1)}={\mathit{\vartheta }}^{\left(k\right)}+I^{-1}\left({\mathit{\vartheta }}^{\left(k\right)}\right)U\left({\mathit{\vartheta }}^{\left(k\right)}\right),$$

where ${\mathit{\vartheta }}^{\left(k\right)}$ is an estimate of $\mathit{\vartheta }$ at the kth iteration. When the iterative process converges, ${\mathit{\vartheta }}^{(k+1)}$ is the maximum smoothed likelihood estimate of $\mathit{\vartheta }$ and is denoted as $\hat{\mathit{\vartheta }}$. This estimation method will be referred to as the SMLE method. The standard error of the estimates can be obtained either from $I^{-1}\left(\hat{\mathit{\vartheta }}\right)$ or using the bootstrap method. We find that the latter is numerically more stable than the former.

We can also use the profile method to estimate the parameters with the smoothed likelihood function. The profile log-likelihood function is defined as $psl(c_1,c_2)=sl\left({\hat{\mathit{\vartheta }}}_{c_1,c_2}\right)$, where ${\hat{\mathit{\vartheta }}}_{c_1,c_2}=\underset{\mathit{\vartheta }\setminus \{c_1,c_2\}}{\mathrm{argmax}}sl\left(\mathit{\vartheta }\right)$. Then estimates of $c_1$ and $c_2$, denoted as ${\hat{c}}_1$ and ${\hat{c}}_2$, can be obtained as $\underset{c_1,c_2}{\mathrm{argmax}}psl(c_1,c_2)$ using a grid search method, which can make the computation considerably more intensive than the SMLE method. With ${\hat{c}}_1$ and ${\hat{c}}_2$, an estimate of $\mathit{\vartheta }\setminus \{c_1,c_2\}$ can be obtained as ${\hat{\mathit{\vartheta }}}_{{\hat{c}}_1,{\hat{c}}_2}$. This estimation method will be referred to as the PSMLE method. The bootstrap method may be used to estimate the standard errors of the estimates.

Both SMLE and PSMLE methods rely on prespecified bandwidths $h_1$ and $h_2$. Although a smaller value of the bandwidths implies a better approximation of the likelihood function (5) by (6), too small values of the bandwidths can cause numerical errors. We suggest that $h_j$ be set to ${\hat{\sigma }}_j{n}^{-1/3}$, where ${\hat{\sigma }}_j$ denotes the sample standard deviation of $U_{ij}$’s, $j=1$, 2. Our numerical study shows that this bandwidth selection method works well.

3. Results

3.1. Empirical Study

We conduct a simulation study to investigate the performance of the proposed method. Simulated data are generated from the mixture cure rate model (1) with

$$\begin{array}{c}S\left(t\right|{\mathit{V}}_1,{\mathit{\vartheta }}_1)=e^{-t·exp({\alpha }_{11}X+{\gamma }_{1}W+{\theta }_{1}I(U>c_1)+{\eta }_{1}WI(U>c_1))},\\ p({\mathit{V}}_2,{\mathit{\vartheta }}_2)={\displaystyle \frac{exp({\beta }_0+{\alpha }_{21}X+{\gamma }_{2}W+{\theta }_{2}I(U>c_2)+{\eta }_{2}WI(U>c_2))}{1+exp({\beta }_0+{\alpha }_{21}X+{\gamma }_{2}W+{\theta }_{2}I(U>c_2)+{\eta }_{2}WI(U>c_2))}},\end{array}$$

where X is simulated from the uniform distribution over $[0,10]$, W is simulated from the Bernoulli distribution with probability 0.5, $U_1=U_2=U$ is simulated from the uniform distribution over $[-1,1]$ or standard normal distribution, ${\alpha }_{11}=-0.1$, ${\gamma }_1=-0.69$, ${\theta }_1=-0.4$, ${\eta }_1=-0.4$, ${\beta }_0=-0.5$, ${\alpha }_{21}=0.1$, ${\gamma }_2=-1.1$, ${\theta }_2=-0.5$, ${\eta }_2=-0.5$, $c_1=0$, 0.5, $c_2=-0.5$, 0, 0.5. The censoring time is simulated from the uniform distribution over $[0,20]$. The average cure rate and the censoring rates are 0.31 and 0.43, respectively. The sample size is set to $n=200$, 500, 1000, 2000.

Under each of the settings, 1000 samples were simulated and fitted with the proposed SMLE, PSMLE and PMLE methods. The baseline survival function $S_0\left(t\right)$ is assumed to be from the Weibull distribution, and the normal kernel function $g(x/h)=\mathrm{\Phi }\left(x/h\right)$ is used in the methods. The bandwidths in the SMLE and PSMLE methods are determined using the method discussed in Section 2 if not specified. All the computing work was carried out on the clusters of the Digital Research Alliance of Canada.

Table 1. Results from simulation studies with the normal kernel function in the SMLE and PSMLE method, $(c_1,c_2)=(0,0)$, and $n=200$.

	SMLE				PSMLE				PMLE
$\mathit{\vartheta }$	Bias	SD	MSE	CP	Bias	SD	MSE	CP	Bias	SD	MSE	CP
${\lambda }_1$	0.04	0.09	0.01	0.96	0.04	0.09	0.01	0.96	0.04	0.09	0.01	0.96
${\lambda }_2$	0.03	0.32	0.10	0.93	0.03	0.40	0.17	0.93	0.02	0.32	0.11	0.93
${\gamma }_1$	0.00	0.38	0.14	0.96	−0.00	0.40	0.16	0.97	−0.00	0.37	0.13	0.97
${\gamma }_2$	−0.04	0.75	0.56	0.98	−0.03	0.87	0.76	0.99	−0.03	0.75	0.56	0.99
${\theta }_1$	−0.06	0.42	0.18	0.97	−0.04	0.46	0.21	0.97	−0.05	0.43	0.19	0.97
${\theta }_2$	−0.17	0.68	0.49	0.99	−0.15	1.01	1.05	0.99	−0.28	0.84	0.78	0.99
${\eta }_1$	−0.02	0.60	0.36	0.97	−0.04	0.60	0.37	0.98	−0.03	0.63	0.40	0.98
${\eta }_2$	−0.19	1.31	1.75	1.00	−0.28	1.53	2.41	1.00	−0.31	1.49	2.32	1.00
$c_1$	0.01	0.22	0.05	0.90	0.02	0.33	0.11	0.85	0.02	0.33	0.11	0.85
$c_2$	0.03	0.32	0.10	0.86	0.02	0.48	0.23	0.78	−0.02	0.48	0.23	0.80
${\alpha }_{11}$	−0.00	0.05	0.00	0.96	−0.00	0.05	0.00	0.96	−0.00	0.05	0.00	0.96
${\alpha }_{21}$	0.00	0.08	0.01	0.97	0.00	0.08	0.01	0.97	0.00	0.08	0.01	0.98
${\beta }_0$	0.02	0.55	0.30	0.99	0.02	0.77	0.60	0.99	0.08	0.51	0.27	0.99

,

Table 2. Results from simulation studies with the normal kernel function in the SMLE and PSMLE method, $(c_1,c_2)=(0,0)$, and $n=500$.

	SMLE				PSMLE				PMLE
$\mathit{\vartheta }$	Bias	SD	MSE	CP	Bias	SD	MSE	CP	Bias	SD	MSE	CP
${\lambda }_1$	0.02	0.05	0.00	0.95	0.01	0.05	0.00	0.95	0.02	0.05	0.00	0.96
${\lambda }_2$	0.01	0.18	0.03	0.95	0.01	0.19	0.03	0.94	0.01	0.18	0.03	0.94
${\gamma }_1$	0.03	0.22	0.05	0.95	0.02	0.21	0.05	0.96	0.01	0.21	0.05	0.95
${\gamma }_2$	−0.01	0.36	0.13	0.97	−0.02	0.42	0.17	0.98	−0.02	0.35	0.12	0.98
${\theta }_1$	−0.02	0.23	0.05	0.95	−0.02	0.22	0.05	0.96	−0.01	0.21	0.04	0.96
${\theta }_2$	−0.07	0.33	0.12	0.97	−0.08	0.39	0.16	0.97	−0.09	0.29	0.09	0.97
${\eta }_1$	−0.04	0.33	0.11	0.95	−0.02	0.31	0.10	0.96	−0.01	0.29	0.09	0.96
${\eta }_2$	−0.11	0.87	0.76	0.99	−0.11	0.96	0.93	1.00	−0.11	0.91	0.84	0.99
$c_1$	−0.00	0.12	0.01	0.93	−0.00	0.15	0.02	0.94	−0.01	0.16	0.03	0.94
$c_2$	0.03	0.18	0.03	0.88	0.03	0.32	0.10	0.89	0.02	0.32	0.10	0.89
${\alpha }_{11}$	−0.00	0.03	0.00	0.97	−0.00	0.03	0.00	0.96	−0.00	0.03	0.00	0.96
${\alpha }_{21}$	0.00	0.05	0.00	0.95	0.00	0.05	0.00	0.96	0.00	0.05	0.00	0.96
${\beta }_0$	0.02	0.31	0.10	0.96	0.03	0.33	0.11	0.97	0.03	0.30	0.09	0.96

,

Table 3. Results from simulation studies with the normal kernel function in the SMLE and PSMLE method, $(c_1,c_2)=(0,0)$, and $n=1000$.

	SMLE				PSMLE				PMLE
$\mathit{\vartheta }$	Bias	SD	MSE	CP	Bias	SD	MSE	CP	Bias	SD	MSE	CP
${\lambda }_1$	0.01	0.04	0.00	0.95	0.01	0.04	0.00	0.95	0.01	0.04	0.00	0.95
${\lambda }_2$	0.01	0.13	0.02	0.95	0.01	0.13	0.02	0.96	0.01	0.13	0.02	0.95
${\gamma }_1$	0.01	0.14	0.02	0.95	0.01	0.14	0.02	0.95	0.00	0.14	0.02	0.95
${\gamma }_2$	−0.00	0.25	0.06	0.95	−0.00	0.24	0.06	0.97	−0.01	0.24	0.06	0.97
${\theta }_1$	−0.00	0.15	0.02	0.96	−0.01	0.15	0.02	0.96	0.00	0.14	0.02	0.95
${\theta }_2$	−0.03	0.21	0.05	0.96	−0.03	0.21	0.05	0.98	−0.02	0.19	0.04	0.98
${\eta }_1$	−0.01	0.21	0.04	0.96	−0.00	0.20	0.04	0.96	0.01	0.20	0.04	0.96
${\eta }_2$	−0.07	0.67	0.46	0.99	−0.04	0.59	0.35	0.98	−0.03	0.59	0.34	0.98
$c_1$	0.00	0.07	0.00	0.94	−0.00	0.09	0.01	0.95	−0.00	0.09	0.01	0.97
$c_2$	0.01	0.13	0.02	0.90	0.02	0.18	0.03	0.95	0.01	0.18	0.03	0.95
${\alpha }_{11}$	−0.00	0.02	0.00	0.95	−0.00	0.02	0.00	0.95	−0.00	0.02	0.00	0.96
${\alpha }_{21}$	−0.00	0.03	0.00	0.96	−0.00	0.03	0.00	0.96	−0.00	0.03	0.00	0.96
${\beta }_0$	0.01	0.21	0.04	0.96	0.01	0.20	0.04	0.96	0.01	0.20	0.04	0.96

and

Table 4. Results from simulation studies with the normal kernel function in the SMLE and PSMLE method, $(c_1,c_2)=(0,0)$, and $n=2000$.

	SMLE				PSMLE				PMLE
$\mathit{\vartheta }$	Bias	SD	MSE	CP	Bias	SD	MSE	CP	Bias	SD	MSE	CP
${\lambda }_1$	0.00	0.03	0.00	0.94	0.00	0.03	0.00	0.94	0.01	0.03	0.00	0.94
${\lambda }_2$	0.01	0.09	0.01	0.94	0.01	0.08	0.01	0.95	0.01	0.08	0.01	0.95
${\gamma }_1$	0.01	0.10	0.01	0.95	0.01	0.10	0.01	0.95	0.00	0.10	0.01	0.95
${\gamma }_2$	0.00	0.17	0.03	0.95	0.00	0.17	0.03	0.95	−0.00	0.17	0.03	0.95
${\theta }_1$	−0.00	0.10	0.01	0.95	−0.00	0.10	0.01	0.95	0.00	0.10	0.01	0.94
${\theta }_2$	−0.02	0.15	0.02	0.94	−0.02	0.15	0.02	0.95	−0.01	0.15	0.02	0.95
${\eta }_1$	−0.01	0.15	0.02	0.94	−0.00	0.15	0.02	0.95	0.01	0.15	0.02	0.95
${\eta }_2$	−0.04	0.47	0.22	0.97	−0.03	0.45	0.20	0.97	−0.01	0.45	0.20	0.96
$c_1$	0.00	0.04	0.00	0.95	−0.00	0.04	0.00	0.97	−0.00	0.03	0.00	0.98
$c_2$	0.02	0.08	0.01	0.92	0.02	0.10	0.01	0.97	0.01	0.09	0.01	0.98
${\alpha }_{11}$	−0.00	0.01	0.00	0.95	−0.00	0.01	0.00	0.95	−0.00	0.01	0.00	0.94
${\alpha }_{21}$	−0.00	0.02	0.00	0.96	−0.00	0.02	0.00	0.95	−0.00	0.02	0.00	0.96
${\beta }_0$	0.02	0.14	0.02	0.95	0.01	0.14	0.02	0.95	0.01	0.14	0.02	0.95

present biases, standard deviations (SD), mean squared errors (MSE), and coverage probabilities (CP) obtained from the three estimation methods for simulated data with $(c_1,c_2)=(0,0)$ and U generated from the uniform distribution. The biases, SD, MSE and CP are calculated according to (7) and (8).

$$\begin{array}{c}bias={\displaystyle \frac{{\sum }_{i=1}^n{\hat{\vartheta }}_i}{n}}-\vartheta ,SD=\sqrt{{\displaystyle \frac{1}{n-1}}{\sum }_{i=1}^n{\left({\hat{\vartheta }}_i-{\displaystyle \frac{{\sum }_{i=1}^n{\hat{\vartheta }}_i}{n}}\right)}^2},MSE=bia{s}^2+S{D}^2,\end{array}$$

(7)

$$\begin{array}{c}CP={\displaystyle \frac{{\sum }_{i=1}^{n}I(\vartheta \in ({\hat{\vartheta }}_i-1.96sd\left({\hat{\vartheta }}_i\right)),{\hat{\vartheta }}_i+1.96sd\left({\hat{\vartheta }}_i\right))}{n}}.\end{array}$$

(8)

The estimates of most parameters have small biases and MSEs under all considered sample sizes and estimation methods except for ${\theta }_2$ and ${\eta }_2$ under the small sample size of 200. When the sample size increases, the biases, SDs, and MSEs of all the estimates decrease. The SMLE method outperforms the PSMLE method as the SMLE method possesses smaller SDs and MSEs, especially in the cases with smaller sample sizes. The bias, SD and MSE of SMLE are slightly smaller in some parameters and larger in some other parameters than those of PMLE, indicating that the two estimation methods are comparable in terms of accuracy and precision. But the computational time of PMLE is much longer. The CPs of SMLE are closer to the 95% nominal level than the other three methods. The CPs of ${\eta }_2$ and $c_2$ are away from the nominal level for small sample sizes, but they are closer to the nominal level as the sample size increases.

Table 4 presents the simulation results under a large sample size ($n=2000$), demonstrating the asymptotic properties of the proposed estimation method. The estimates of SMLE for the subgroup thresholds $c_1$ and $c_2$ show minimal bias and standard deviation, confirming that the method can accurately and precisely recover the true subgroup boundaries when sufficient data are available. Even with $n=2000$, the coverage probability for $c_2$ remains slightly below the nominal 95% level. This suggests that inference on the subgroup-defining thresholds, particularly when the two subgroups are not well-separated, remains the most challenging aspect of the inference. The coverage for the interaction effect is slightly conservative. This is likely due to the combined uncertainty from estimating both the threshold and the interaction coefficient.

Table 5. Simulation results from the SMLE method with the normal and logistic kernel functions, $(c_1,c_2)=(0.5,0)$, and $n=500$.

	Normal				Logistic
$\mathit{\vartheta }$	Bias	SD	MSE	CP	Bias	SD	MSE	CP
${\lambda }_1$	0.01	0.05	0.00	0.93	0.01	0.05	0.00	0.94
${\lambda }_2$	−0.00	0.16	0.03	0.96	−0.01	0.17	0.03	0.96
${\gamma }_1$	0.01	0.16	0.03	0.97	0.02	0.18	0.03	0.97
${\gamma }_2$	0.01	0.34	0.12	0.95	0.02	0.37	0.13	0.96
${\theta }_1$	−0.03	0.25	0.07	0.93	−0.05	0.28	0.08	0.93
${\theta }_2$	−0.08	0.33	0.12	0.92	−0.11	0.35	0.14	0.92
${\eta }_1$	−0.06	0.36	0.13	0.93	−0.07	0.38	0.15	0.93
${\eta }_2$	−0.24	1.00	1.05	0.91	−0.20	0.92	0.88	0.88
$c_1$	0.01	0.13	0.02	0.95	0.03	0.17	0.03	0.96
$c_2$	0.03	0.23	0.05	0.90	0.03	0.30	0.09	0.91
${\alpha }_{11}$	−0.00	0.03	0.00	0.95	−0.00	0.03	0.00	0.96
${\alpha }_{21}$	0.00	0.05	0.00	0.96	0.00	0.05	0.00	0.96
${\beta }_0$	0.02	0.30	0.09	0.94	0.03	0.30	0.09	0.95

presents the biases, SDs, MSEs, and CPs of the estimates from the SMLE method using both the normal and logistic kernel functions for simulated data with $(c_1,c_2)=(0.5, 0)$ and U generated from the standard normal distribution. The results indicate that the choice of kernel has a meaningful impact on the estimation of ${\eta }_2$, the treatment-by-subgroup interaction effect on the cure probability. The normal kernel yields a CP of 0.91 and an MSE of 1.05, while the logistic kernel yields a CP of 0.88 and an MSE of 0.88. This indicates that the normal kernel provides better-calibrated uncertainty intervals for this key parameter. The incidence part of the model involves estimating cure status, which may be affacted by the choice of kernel function. The normal kernel’s faster tail decay provides a more localized approximation while the heavier tails of the logistic kernel may over-smooth the boundary, which appears to be crucial for accurately capturing the discontinuity in the cure probability across the subgroup boundary. For all other parameters, performance is nearly identical between the two kernels. Consequently, we recommend the normal kernel as the default specification for the proposed method based on its robust performance for the most sensitive parameter.

The results for other values $(c_1,c_2)$ and sample sizes are similar and therefore not presented here.

Table 6. Simulation results from the SMLE method for $c_1=0.5$ and different choices of $c_2$ when the kernel smoothing function is normal, the bandwith is optimal and $n=500$.

	${\mathit{c}}_2=-0.5$				${\mathit{c}}_2=0$				${\mathit{c}}_2=0.5$
$\mathit{\vartheta }$	Bias	SD	MSE	CP	Bias	SD	MSE	CP	Bias	SD	MSE	CP
${\lambda }_1$	0.01	0.05	0.00	0.93	0.01	0.05	0.00	0.93	0.01	0.05	0.00	0.93
${\lambda }_2$	0.00	0.16	0.02	0.96	−0.00	0.16	0.03	0.96	−0.00	0.17	0.03	0.96
${\gamma }_1$	0.01	0.16	0.03	0.96	0.01	0.16	0.03	0.97	0.00	0.18	0.03	0.96
${\gamma }_2$	−0.00	0.43	0.18	0.95	0.01	0.34	0.12	0.95	0.01	0.31	0.10	0.94
${\theta }_1$	−0.02	0.25	0.06	0.94	−0.03	0.25	0.07	0.93	−0.03	0.27	0.07	0.93
${\theta }_2$	−0.07	0.32	0.11	0.94	−0.08	0.33	0.12	0.92	−0.13	0.40	0.18	0.91
${\eta }_1$	−0.07	0.34	0.12	0.94	−0.06	0.36	0.13	0.93	−0.03	0.37	0.14	0.94
${\eta }_2$	−0.13	0.76	0.60	0.92	−0.24	1.00	1.05	0.91	−0.18	1.08	1.20	0.88
$c_1$	0.01	0.14	0.02	0.95	0.01	0.13	0.02	0.95	0.00	0.17	0.03	0.94
$c_2$	0.02	0.25	0.06	0.90	0.03	0.23	0.05	0.90	0.05	0.28	0.08	0.87
${\alpha }_{11}$	−0.00	0.03	0.00	0.96	−0.00	0.03	0.00	0.95	−0.00	0.03	0.00	0.96
${\alpha }_{21}$	0.00	0.04	0.00	0.96	0.00	0.05	0.00	0.96	0.00	0.05	0.00	0.95
${\beta }_0$	0.04	0.33	0.11	0.96	0.02	0.30	0.09	0.94	0.01	0.27	0.07	0.95

presents the biases, SDs, MSEs, and CPs of the estimates from the SMLE method with $c_1=0.5$ and $c_2=-0.5$, 0, 0.5, and U generated from the standard normal distribution. It investigates a fundamental practical concern that how the method’s performance depends on the location of the true subgroup threshold $c_2$. This is crucial for applications where a treatment-sensitive subgroup may be a large majority ($c_2$ low, e.g., −0.5) or a small minority ($c_2$ high, e.g., 0.5) of the population. The biases, SDs, and MSEs of the estimates of ${\theta }_2$ from the SMLE method increase and the CPs move away from the 95% nominal level as $c_2$ goes from $-0.5$ to $0.5$ because of fewer data in $U>c_2$. This demonstrates the method’s reliance on having an adequate number of subjects within the identified subgroup for reliable inference on probablity of being cured. The biases, SDs, and MSEs of ${\eta }_2$ from the SMLE method do not strictly increase as $c_2$ increases, which may be because ${\eta }_2$ also depends on W. The incidence model’s logistic link function may make the cure probability estimate more robust to moderate changes in subgroup size when other predictive information is present. However, the CP for ${\eta }_2$ remains below the nominal level across all $c_2$ values, underscoring that inference for the cure probability interaction is challenging and requires careful interpretation, regardless of subgroup size.

We also investigated the dependence of the performance of the SMLE method on the choice of the bandwidth h.

Table 7. Simulation results from the SMLE method with $h=0.5$, 0.2, and 0.1, the normal kernel function, $(c_1,c_2)=(0.5,0.5)$ and $n=500$.

	$\mathit{h}=0.5$				$\mathit{h}=0.2$				$\mathit{h}=0.1$
$\mathit{\vartheta }$	Bias	SD	MSE	CP	Bias	SD	MSE	CP	Bias	SD	MSE	CP
${\lambda }_1$	0.01	0.05	0.00	0.93	0.01	0.05	0.00	0.92	0.01	0.05	0.00	0.93
${\lambda }_2$	−0.02	0.19	0.04	0.96	−0.01	0.18	0.03	0.96	−0.00	0.17	0.03	0.96
${\gamma }_1$	0.02	0.22	0.05	0.97	0.01	0.19	0.04	0.97	0.01	0.17	0.03	0.97
${\gamma }_2$	0.02	0.36	0.13	0.95	0.01	0.31	0.10	0.94	0.00	0.30	0.09	0.94
${\theta }_1$	−0.12	0.39	0.17	0.91	−0.04	0.28	0.08	0.92	−0.02	0.26	0.07	0.93
${\theta }_2$	−0.32	0.66	0.55	0.88	−0.16	0.44	0.22	0.91	−0.12	0.37	0.15	0.92
${\eta }_1$	−0.07	0.52	0.27	0.93	−0.04	0.40	0.16	0.94	−0.03	0.37	0.14	0.93
${\eta }_2$	−0.13	1.23	1.53	0.92	−0.13	0.99	0.99	0.91	−0.15	1.02	1.06	0.91
$c_1$	0.05	0.35	0.13	0.95	0.01	0.23	0.05	0.95	0.01	0.15	0.02	0.92
$c_2$	0.19	0.60	0.39	0.88	0.07	0.36	0.13	0.89	0.04	0.24	0.06	0.86
${\alpha }_{11}$	−0.00	0.03	0.00	0.96	−0.00	0.03	0.00	0.96	−0.00	0.03	0.00	0.95
${\alpha }_{21}$	0.00	0.04	0.00	0.95	0.00	0.05	0.00	0.95	0.00	0.05	0.00	0.95
${\beta }_0$	0.04	0.30	0.09	0.95	0.02	0.28	0.08	0.96	0.01	0.27	0.07	0.95

and

Table 8. Simulation results from the SMLE method with $h=0.01$, 0.001, $\hat{\sigma }{n}^{-1/3}$, the normal kernel function, $(c_1,c_2)=(0.5,0.5)$ and $n=500$.

	$\mathit{h}=0.01$				$\mathit{h}=0.001$				$\mathit{h}=\hat{\mathit{\sigma }}{\mathit{n}}^{-1/3}$
$\mathit{\vartheta }$	Bias	SD	MSE	CP	Bias	SD	MSE	CP	Bias	SD	MSE	CP
${\lambda }_1$	0.02	0.05	0.00	0.93	0.01	0.05	0.00	0.92	0.01	0.05	0.00	0.93
${\lambda }_2$	0.00	0.15	0.02	0.94	0.01	0.13	0.02	0.93	−0.00	0.17	0.03	0.96
${\gamma }_1$	0.01	0.16	0.03	0.95	0.01	0.14	0.02	0.95	0.00	0.18	0.03	0.96
${\gamma }_2$	0.01	0.27	0.07	0.93	0.01	0.24	0.06	0.93	0.01	0.31	0.10	0.94
${\theta }_1$	−0.01	0.23	0.05	0.92	−0.01	0.20	0.04	0.92	−0.03	0.27	0.07	0.93
${\theta }_2$	−0.08	0.33	0.12	0.90	−0.07	0.27	0.08	0.90	−0.13	0.40	0.18	0.91
${\eta }_1$	−0.03	0.31	0.10	0.92	−0.03	0.26	0.07	0.91	−0.03	0.37	0.14	0.94
${\eta }_2$	−0.14	0.81	0.67	0.87	−0.03	0.49	0.24	0.91	−0.18	1.08	1.20	0.88
$c_1$	0.01	0.07	0.01	0.81	0.01	0.06	0.00	0.77	0.00	0.17	0.03	0.94
$c_2$	0.01	0.09	0.01	0.82	0.01	0.07	0.00	0.80	0.05	0.28	0.08	0.87
${\alpha }_{11}$	−0.00	0.03	0.00	0.95	−0.00	0.02	0.00	0.95	−0.00	0.03	0.00	0.96
${\alpha }_{21}$	−0.00	0.04	0.00	0.93	−0.00	0.04	0.00	0.92	0.00	0.05	0.00	0.95
${\beta }_0$	0.02	0.25	0.06	0.95	0.01	0.21	0.04	0.94	0.01	0.27	0.07	0.95

present the biases, SDs, MSEs, and CPs of the estimates from the SMLE method with the bandwidth set to 0.5, 0.2, 0.1, 0.01, 0.001, $\hat{\sigma }{n}^{-1/3}\approx 0.126$, $n=500$, the normal kernel function, and U generated from the standard normal distribution. The selection of the bandwidth h governs a critical bias-variance trade-off in the smoothed likelihood estimation. For large bandwidth ($h=0.5$), we see substantial bias in $c_2$ (0.19) and ${\theta }_2$, and the MSEs are relatively high. The coverage probabilities are below nominal for $c_2$ and ${\theta }_2$. As the bandwidth decreases, the biases for $c_2$ and ${\theta }_2$ decrease, and the MSEs also decrease. However, the coverage for $c_2$ remains below nominal. For very small bandwidths, the biases become very small and the SDs decrease, but the coverage probabilities for the threshold $c_2$ drop dramatically to around 0.8. This indicates that the confidence intervals are too narrow, likely because the bootstrap variance estimation fails when the likelihood is too sharp. The data-driven bandwidth ($h=ha{t}_{s}igma{n}^{-1/3}$) strikes a balance. It gives low bias for $c_1$ and moderate bias for $c_2$, and the coverage probabilities are much better for $c_1$ and $c_2$. For ${\theta }_2$ and ${\eta }_2$, the biases are moderate and coverage is around 0.91 and 0.88 respectively. For all the other parameters, the SDs and MSEs of the estimates from the SMLE method decrease as the value of h decreases, and the results from $h=\hat{\sigma }{n}^{-1/3}$ perform well. Therefore, the data-driven bandwidth is recommended as the default, robust specification to ensure both accuracy and valid inference.

3.2. Analysis of Colon Cancer Data

We analyzed a colon cancer data set with the proposed Weibull PH mixture cure model and the piecewise PH mixture cure model for subgroup analysis. We also fitted the data set with the regular Weibull PH mixture cure model for comparison. The data set comes from a clinical trial study designed to evaluate the benefit of the combination of fluorouracil plus levamisole as an adjuvant therapy after resection of stage III colon carcinoma compared to levamisole alone therapy [29], which is available in survival package in R. The data set contains 614 patients, among them 310 patients were treated with levamisole (Lev) alone, and 304 patients were treated with the combination of levamisole plus fluorouracil (Lev+5FU). The maximum follow-up time is 3329 days, and the outcome of interest is the days to recurrence. There are 119 and 172 recurrence events in the Lev+5FU therapy and Lev therapy, respectively, and the respective censoring rates are 60.86% and 44.52%. The Kaplan–Meier survival curves for the two treatments are presented in Figure 1. The curves level off at survival probabilities of 59.94% and 43.29%, respectively, after 2500 days of follow-up, indicating the presence of cured subjects in the sample.

We consider subgroups defined by $I(\mathrm{age}>c_1)$ in the latency part and $I(\mathrm{age}>c_2)$ in the incidence part, and fit the data set with the proposed Weibull mixture cure rate models to determine the optimal values of $c_1$ and $c_2$ and treatment differences in the subgroups. The results are presented in

Table 9. The estimates from the mixture cure models for the colon cancer data with and without thresholds in age.

	Without Thresholds in Age		With Thresholds in Age
			Weibull
Parameter	Estimate (SE)	p-Value	Estimate (SE)	p-Value
Latency part
Lev+5FU	−0.343 (0.517)	0.51	−1.453 (0.147)	<0.01
age	−0.001 (0.006)	0.88	−1.513 (0.260)	<0.01
(Lev+5FU) × age	0.008 (0.009)	0.39	1.098 (0.382)	<0.01
$c_1$	-	-	67 (0.395)	<0.01
Incidence part
Intercept	−0.520 (0.722)	0.47	−0.743 (0.225)	<0.01
Lev+5FU	−0.414 (0.892)	0.64	−0.421 (0.285)	$0.14$
age	0.004 (0.012)	0.72	−0.144 (0.440)	0.74
(Lev+5FU) × age	0.018 (0.015)	0.21	0.247 (0.508)	0.63
$c_2$	-	-	66 (1.077)	<0.01

.

Under the Weibull mixture cure model with age thresholds, the estimated optimal threshold for age is 67 for the time to recurrence among uncured patients, and 66 for the cure probability from the Weibull PH mixture cure model. However, the interaction effect between treatment and subgroup is significant only for latency. It implies that the treatment effects are significantly different in the time to recurrence for uncured patients between age younger than 67 and older than 67. For uncured patients younger than 67, the hazard ratio between the combined therapy and levamisole alone therapy is 0.23, while for patients older than 67, the hazard ratio increases to 0.7 and is statistically significant, indicating that although the combined treatment improves the survival of uncured patients, patients younger than 67 benefit more than patients older than 67. More specific, the hazard ratio of 0.23 indicates 77% reduction in risk of recurrence if the uncured patient younger than 67 takes combined treatment instead of single treatment, and the hazard ratio of 0.7 indicates 30% reduction in risk of recurrence for the uncured patient older than 67 if the patient takes the combined treatment.

The effects in the incidence part are not statistically significant, implying that none of the treatment, age, or their interaction has any significant effects on the probability of being cured.

The data set is also fitted with the regular Weibull proportional hazard mixture cure model, where age is not dichotomized using a threshold in the main and the interaction effects. The results are presented in Table 9, and none of the effects in the model are statistically significant.

4. Discussion

In this paper, we proposed a PH mixture cure model for subgroup analysis, which can determine the optimal cutoff values or thresholds for identifying treatment-sensitive subgroups in the latency and incidence parts. A smoothed maximum likelihood estimation method is developed for estimating the unknown parameters and the unknown cutoff values or thresholds by using kernel functions to approximate the indicator functions in the likelihood function. An extensive simulation study is performed to evaluate the performance of the proposed model and the estimation method, and the results demonstrate that the proposed model and estimation method can identify subgroups and estimate heterogeneous treatment effects accurately. Finally, we applied the proposed model and method to a data set from a clinical trial study of colon cancer patients treated either by a combination of levamisole plus fluorouracil or by levamisole alone and identified age subgroups and significant heterogeneous treatment effects. In comparison, the regular mixture cure model does not show evidence to support the heterogeneous treatment effects.

This work directly addresses specific methodological gaps identified in the literature on mixture cure models for subgroup analysis. As noted in the introduction, existing approaches have either focused on prognostic subgroup effects without modeling predictive subgroup effects [22,23], or have considered heterogeneous treatment effects in different subgroups only in the latency part for the uncured population, omitting the cure probability [24,25]. The methodology proposed here advances beyond these prior works by simultaneously modeling predictive treatment-by-subgroup interactions as specified in (2) and (3), and allowing these heterogeneous effects to be identified in both the incidence and latency components of the model. Therefore, our primary contribution is a flexible, likelihood-based tool that identifies subgroups with differential treatment responses across the entire cure model structure, thereby enabling more precise inference in settings where a treatment may affect whether a subject is cured, how long uncured subjects survive, or both. Consequently, our work fills a gap by offering a principled, likelihood-based tool to investigate a broader and more nuanced range of precision medicine hypotheses than previously possible with standard or partially extended cure models.

Despite the promising results, several limitations should be considered. First, the current model is designed for a single covariate or biomarker to define subgroups. In practice, it is very likely that subgroups may be defined by multiple covariates or biomarkers. Second, our reliance on kernel smoothing to approximate the indicator function introduces a dependence on two key smoothing parameters the bandwidth and the kernel smoothing function. Although smaller bandwidth value provide better estimates, too small bandwidth values may lead to numerical instability. Our simulation study was limited to comparing the logistic and normal kernels. However, the broader class of kernel functions (e.g., Epanechnikov, uniform, triangular) remains unexplored in this context, and their differing properties could potentially affect the smoothness of the likelihood and the finite-sample precision of the threshold estimates. Third, while obtaining point estimates is feasible, the computational burden becomes substantial when performing bootstrap resampling to estimate standard errors and construct confidence intervals for the threshold and regression coefficients withohut access to high-performance computing. Furthermore, the model’s performance is contingent on the specification of the baseline hazard function for the uncured population. While we considered parametric (Weibull) and semi-parametric (piecewise constant) forms, each carries limitations. The Weibull assumption may be violated if the true baseline hazard does not follow its specific monotonic form, leading to biased estimates. Conversely, the more flexible piecewise constant baseline introduces challenge of selecting the number and locations of knots, which can be subjective and influence the fitted survival structure. Finally, the model inherits the standard proportional hazards assumption for the latency part, and violations could affect the accuracy of subgroup-specific hazard ratios.

5. Conclusions

We have developed a novel proportional hazards mixture cure model tailored for subgroup identification within survival data with a cure fraction. The primary contribution of this work is a unified framework that simultaneously identifies optimal thresholds for defining subgroups and estimates differential treatment effects on both the cure probability and the survival of uncured patients, providing a valuable statistical tool for advancing precision medicine research.

This work naturally leads to several promising directions for future research. It will be of interest to generalize the work developed here to multiple covariates. We are currently working on this problem and will report the findings in a future paper. Beyond this, further extension could be explored, such as developing a nonparametric mixture cure model for subgroup analysis that relaxes the parametric or semiparametric assumptions on the baseline hazard, or considering alternative latency formulations like the accelerated failure time, proportional odds, or accelerated hazards models. Extending the methodology to non-mixture cure models for subgroup identification represents another valuable avenue. A comparison with a Bayesian estimation framework, which would facilitate the incorporation of prior information, is another important direction. Furthermore, the performance of treatment-sensitive subgroup identification, especially under varying degrees of separation between subgroups, merits systematic evaluation. In addition, the current evaluation focuses on sample sizes appropriate for the model’s complexity. Future work could systematically explore the minimal sample size required for stable performance.