A defective cure rate quantile regression model for male breast cancer data

Abstract

In this article, we particularly address the problem of assessing the impact of different prognostic factors, such as clinical stage and age, on the specific survival times of men with breast cancer when cure is a possibility. To this end, we developed a quantile regression model for survival data in the presence of long-term survivors based on the generalized Gompertz distribution in a defective version, which is conveniently reparametrized in terms of the q-th quantile and then linked to covariates via a logarithm link function. This proposal allows us to obtain how each variable affects the survival times in different quantiles. In addition, we are able to study the effects of covariates on the cure rate as well. We consider Markov Chain Monte Carlo methods to develop a Bayesian analysis in the proposed model and we evaluate its performance through Monte Carlo simulation studies. Finally, we illustrate the application of our model in a data set about male breast cancer from Brazil analyzed for the very first time.

1. Introduction

1.1. Bibliographical review

In population-based cancer survival studies, survival models that take into account a cure fraction, known as cure rate models or long-term survival models, are constantly used to simultaneously explain immune and susceptible patients to the cause of failure event of interest under study. The most common approach to the cure rate model is the standard mixture model approached by [4] and extended by [3] and later extensively studied by [2,13,14,29,39,49], among others. Although the mixture cure rate model is the most widely used in the literature, other alternative models of population modeling with cured individuals have been studied, such as (i) models based on the latent competitive risk structure [11]; (ii) models based on defective distributions, defined as distributions that are not normalized to one for some values of its parameters, a concept discussed by [1].

Based on item (i), for example, [10,44] in cancer recurrence scenarios, assume that a latent biological process of propagation of latent clonogenic tumor cells (latent competitive risks) produces the observed failure (relapse), resulting in the model known in the literature as bounded cumulative hazard model. A unified approach of the standard mixture and bounded cumulative hazard models was proposed by [46]. Rodrigues et al. [38] developed a more flexible cure rate survival model that includes a destructive process of the initial risk factors in a competitive scenario and is thus based on the biological mechanism of the occurrence of the event of interest. Subsequently, [37] proposed a new Bayesian flexible cure rate survival model which generalizes the stochastic model of [26] and has much in common with the destructive cure rate model formulated by [38]. Borges et al. [6] introduced a new cure rate survival model which extends the model of [38] by incorporating a structure of dependence between the initiated cells. Recently, [7] proposed a cure rate survival hybrid model for accommodating characteristics of unobservable stages (initiation, promotion, and progression) of carcinogenesis from survival data in the presence of latent competing risks.

Regarding item (ii), defective distributions have the advantage of allowing a cure rate without adding any extra parameters in the modeling, unlike the above-mentioned cure rate models [40]. They are obtained from standard distributions by changing the domain of the parameters in such a way that $\underset{t\to \mathrm{\infty }}{lim}S(t)=p_0\in (0,1)$. The two main distributions used for this purpose are the inverse Gaussian and Gompertz. Proposals dealing with the defective distributions have appeared in the literature, to name a few we refer to [5,8,9,16,31,36].

For all of the aforementioned survival models, inferential results are evaluated in terms of hazard ratio or on the average of the logarithm of the failure times, not directly on the failure time. However, physicians can be interested in assessing the earlier stage of the follow-up, for example, and quantile regression can directly model the lower quantile of interest to provide the covariate effects specifically for this quantile. Under this approach, it is possible to know how the regression coefficients of a given covariate change for different quantiles of survival time.

Quantile regression, introduced by [28], is particularly useful when the rate of change in the conditional quantiles, expressed by the regression coefficients, depends on the quantile. One of its main advantages concerns its flexibility for modeling data with heteroscedasticity. This feature makes quantile regression very attractive because the failure times often are quite right-skewed [47].

Even though quantile regression for survival data has been extensively studied in the literature (e.g. [24,32–34,43,47,48]), there are not many references dealing with quantile regression in a cure rate setup, known as cure rate quantile regression models; [19,42] could be mentioned as some of the examples. These proposals have included a non-parametric component either in the functional form of the regression equation or the distribution of the lifetime of the model, or both.

However, to the best of our knowledge, no one has so far examined a particular fully parametric approach to cure rate quantile regression based on defective distributions, which, as mentioned earlier, have the competitive advantage of allowing a cure rate without adding any extra parameters in the modeling process, in contrast to the standard mixture approach considered by [42]. Therefore, the main objective of this paper is to consider a quantile regression model in which the lifetime variable is a defective distribution.

Under the proposed approach, any defective distribution can be considered for the lifetime variable. Motivated by the work of [5] and by the real data features presented in the Subsection 1.2, we used the defective version of the generalized Gompertz distribution introduced by [12]. An important feature of the generalized Gompertz distribution is the different forms of the hazard function depending on the values of its parameters. This distribution is conveniently reparametrized in terms of the q-th quantile and then linked to covariates by means of a logarithm link function, which allows us to estimate the quantiles directly.

In this work, the estimation procedure is constructed under the Bayesian perspective of inference. Ibrahim et al. [25] discuss several advantages of using the Bayesian approach, such as the possibility of making exact inferences for any sample size without the need for asymptotic calculations, the ease of estimating variances and any other measures of the posteriori distribution, and the use of a priori information. There are situations where information through expert opinion and/or past experience can be expressed in the a priori distributions. However, in a scenario where no prior information about the failure time is available, non-informative priors can be considered.

An important detail of our approach is that we use a parametric quantile regression (PQR) model, similar to what [32] considered for their generalized gamma proposal, though in a classic framework. This concept was thoroughly discussed by [17], where the author argued that the idea of modeling the quantile function of a probability distribution, instead of another location parameter, for instance, adds much more flexibility to one's modeling strategy. Specifically in comparison to the usual quantile regression defined by [28], this technique presents a great advantage, as we are able to avoid a big concern about quantile regression models, which is the issue with crossing quantiles. Because we are considering the defective distribution generalized Gompertz to model these quantiles and correspondingly its cure fraction, we coin this model as a defective cure rate quantile regression model. We believe this proposal brings more adaptability to current modeling efforts of survival times.

1.2. Motivating dataset: male breast cancer diagnosed in the state of São Paulo

Breast cancer in women has been extensively studied, as well as its prognostic factors. The large number of studies is certainly comprehensible, as this type of cancer corresponds to almost 30% of new cancer cases diagnosed annually in Brazilian women, being a major cause of death in the female population (according to Brazilian National Institute of Cancer – www.inca.gov.br). In fact, this is the second most frequent cancer type that occurs in women, after non-melanoma skin neoplasms, but there is still some misunderstanding on its incidence among men as well. According to the Centers for Disease Control and Prevention (CDC – www.cdc.gov), about 1 out of every 100 breast cancers diagnosed is found in a man, however, the occurrence of this neoplasm tends to increase due to the poor quality of life and the difficulty in making an early diagnosis. In a survey of scientific articles on breast cancer in men, [22] pointed out that the incidence has increased significantly from 0.86 to 1.06 per 100,000 men over the past few decades; the highest rates occurring in North America and Europe and the lowest rates in Asia. Additionally, it was observed that men with breast cancer have the worst overall survival rates in relation to women, but this is probably due to their older age at the time of diagnosis, which corresponds to the most advanced stage of presentation of the disease, as well as higher death rates due to disease comorbidity.

In view of this problem, it is very desirable to establish which prognostic factors impact the survival of men with breast cancer, in order to understand how the disease works in this population. This information would allow physicians to prevent the overall progressive burden of this disease, with measures of control and preventive interventions in this context.

For that, we consider a male breast cancer dataset from a retrospective survey of 887 records of males diagnosed with breast in the state of São Paulo, Brazil, between 2000 and 2019, with follow-up conducted until February of 2020 and with at least two months of follow-up. Death due to breast cancer was defined as the event of interest. Those patients who did not die due to breast cancer during the follow-up period were characterized as right-censored observations. In this analysis, we want to study the impact of clinical stage, age group ( $<55$; 55–65 and $>65$ years old), and different treatments (surgery, chemotherapy, radiotherapy, and hormone therapy) in survival for men, in order to understand how the disease affects this population. The age groups were determined by mastologists from our work team.

This dataset is provided by the Fundação Oncocentro de São Paulo (FOSP), which is responsible for coordinating and monitoring the implementation of the Hospital Cancer Registry in the State of São Paulo (Brazil), in addition to systematizing and evaluating cancer care data available for the state. The FOSP is a public institution connected to the State Health Secretariat that monitors the evolution of the Oncology Care Network, assists the State Department of Health in the creation and application of prevention and health promotion programs, and monitors the evolution of cancer mortality in the state.

Of the 887 patients, $77\mathrm{\%}$ did not die during the follow-up period, that is, 685 patients have a right-censored event time. A total of 270 (30.4%) patients are under the age of 55, 280 (31.6%) are between 55 and 65 years old and 337 (38%) are older than 65 years old. Besides that, 21.1% of the patients are classified as clinical stage I, 37.2% as clinical stage II, 30.2% as clinical stage III, and 11.5% as clinical stage IV. In relation to cancer treatment, 69.1% underwent surgery, 41% had radiotherapy sessions, 62.9% did chemotherapy and 52.4% had in their treatment hormone therapy.

Figure 1 presents the Kaplan-Meier estimates grouped by each explanatory variable. Of note, there is strong evidence of long-term survivors for all groups. Among all of the variables considered in our study, those with clinical stage I had a better prognosis, and stage IV had the worst one. Intermediate-age men (between 55 and 65 years old) have the highest survival rate when compared to younger and older patients, and these last two have very similar behavior. Patients who underwent surgery, did not need or did not undergo chemotherapy, and underwent hormone therapy have the best prognosis. The behavior of those who did and did not undergo radiotherapy is similar.

Figure 1.

Kaplan-Meier estimates for the male breast cancer dataset grouped by age, clinical stage and different treatments.

Besides estimating the cure rate, physicians from our working group are interested in assessing how the regression coefficients of age, clinical stage, and the different treatments behave at different quantiles of the survival time.

Since FOSP is an anonymized dataset with no possibility of individual identification, according to Brazilian regulations of the National Research Ethics Commission (Comissão Nacional de Ética em Pesquisa – CONEP), this study does not require prior approval by the institutional ethics board.

1.3. Outline of the paper

We organize the rest of the paper as follows: the model formulation is described in Section 2. Parameter inference under a Bayesian perspective is discussed in Section 3. In Section 4, we carry out a Monte Carlo simulation study to demonstrate the performance of the proposed estimation method. An application to men with breast carcinoma is discussed in Section 5. Furthermore, some conclusions are mentioned in Section 6.

2. Background

In this section, we present the development of the proposed model. In Subsection 2.1 the generalized Gompertz distribution and its defective version are presented and the proposed reparametrization is presented in Subsection 2.2, resulting in the generalized Gompertz cure rate quantile regression model.

2.1. Generalized Gompertz distribution and its defective version

The generalized Gompertz (GG) distribution introduced by [12] considers that lifetime T conditional to parameters λ, α and θ, has density and survival functions, respectively, given by

$$f(t\mid \lambda ,\alpha ,\theta )=\mathrm{\lambda \theta exp}\{\mathrm{\alpha t}-\frac{\lambda }{\alpha }[\exp (\mathrm{\alpha t})-1]\}{(1-\exp \{-\frac{\lambda }{\alpha }[\exp (\mathrm{\alpha t})-1]\})}^{\theta -1},$$ (1)

and

$$S(t\mid \lambda ,\alpha ,\theta )=1-{(1-\exp \{-\frac{\lambda }{\alpha }[\exp (\mathrm{\alpha t})-1]\})}^{\theta },t>0$$ (2)

where $\lambda >0$ and $\alpha >0$ are scale parameters and $\theta >0$ is a shape parameter.

The GG distribution includes the following distributions as special cases: (i) generalized exponential distribution [20] when α tends to zero; (ii) Gompertz distribution when $\theta =1$; and (iii) exponential distribution when α tends to zero and $\theta =1$.

The hazard function of the GG distribution is given by

$$h(t\mid \lambda ,\alpha ,\theta )=\frac{\mathrm{\lambda \theta exp}\{\mathrm{\alpha t}-\frac{\lambda }{\alpha }[\exp (\mathrm{\alpha t})-1]\}{(1-\exp \{-\frac{\lambda }{\alpha }[\exp (\mathrm{\alpha t})-1]\})}^{\theta -1}}{1-{(1-\exp \{-\frac{\lambda }{\alpha }[\exp (\mathrm{\alpha t})-1]\})}^{\theta }}.$$

An important feature of the GG distribution is the different forms of the hazard function depending on the values of its parameters. The shape of hazard function of the GG distribution is [12]:

1. increasing if $\alpha >0$ and $\theta =1$;

2. constant if $\alpha =0$ and $\theta =1$;

3. increasing when $\theta >1$;

4. decreasing if $\alpha =0$ and $\theta <1$; and

5. bathtub if $\alpha >0$ and $\theta <1$.

The GG distribution becomes a defective distribution, characterized by having a probability density function that integrates to values less than 1 [1] if $\alpha <0$; thus being suitable for modeling survival data with long-term survivors. The corresponding cure fraction is:

$$\underset{t\to \mathrm{\infty }}{lim}S(t\mid \lambda ,\alpha ,\theta )=1-{[1-\exp \left(\frac{\lambda }{\alpha }\right)]}^{\theta }=p_0(\lambda ,\alpha ,\theta )\in [0,1].$$ (3)

Unlike standard mixture models, defective distributions have the advantage of allowing a cure rate without adding any extra parameters in the modeling [40].

In this work, we consider the GG distribution once it can be written as a defective distribution and the quantile function can be obtained in a closed form. These features make this distribution suitable for the proposed approach, besides having different shapes of the hazard function, depending on α and θ parameter values, as discussed previously. It is noteworthy that another distribution could be considered, as long as it has the defective property and it has a closed-form quantile function.

As mentioned before, one of this work's goals is to estimate the quantile function. Nevertheless, the quantile function is not well defined, as the distribution is defective. To circumvent this restriction, we propose to rewrite the defective distribution as a mixture model, i.e. we will separate the susceptible and immune populations without including an additional parameter in the modeling. The proposed approach is presented in the next subsection.

2.2. Generalized Gompertz cure rate quantile regression

To consider a quantile regression model, first, we write the model in (2) as a mixture model. In this case, we define an indicator variable Z, which takes the value 0 if the subject is immune and 1 if the subject is susceptible. Let $\mathbb{P}[Z=0]=p_0(\lambda ,\alpha ,\theta )$ and $\mathbb{P}[Z=1]=1-p_0(\lambda ,\alpha ,\theta )$, where $p_0(\lambda ,\alpha ,\theta )$ is given by (3).

Then, the survival function in (2) in the defective version (with $\alpha <0$) is given by

$$S(t\mid \lambda ,\alpha ,\theta )=p_0(\lambda ,\alpha ,\theta )+[1-p_0(\lambda ,\alpha ,\theta )]S_1(t\mid \lambda ,\alpha ,\theta ),$$ (4)

where $S_1(\cdot )$ denotes the survival function of the susceptibles, and $p_0(\lambda ,\alpha ,\theta )$ is given by equation (3). From (4), we can get an expression for $S_1$ as

$$S_1(t\mid \lambda ,\alpha ,\theta )=\frac{S(t\mid \lambda ,\alpha ,\theta )-p_0(\lambda ,\alpha ,\theta )}{1-p_0(\lambda ,\alpha ,\theta )},$$ (5)

in which $\underset{t\to \mathrm{\infty }}{lim}{S}_1(t\mid \lambda ,\alpha ,\theta )=0$, once $\underset{t\to \mathrm{\infty }}{lim}S(t\mid \lambda ,\alpha ,\theta )=p_0(\lambda ,\alpha ,\theta )$, as presented in (3).

By considering the expression in (5), $S_1$ is a proper survival function with a quantile function given by

$${\mu }_q=\mu (q\mid \lambda ,\alpha ,\theta )=\frac{\log (\lambda -\mathrm{\alpha log}\{1-q^{\frac{1}{\theta }}[1-\exp \left(\frac{\lambda }{\alpha }\right)]\})-\log (\lambda )}{\alpha },0<q<1.$$ (6)

Next, for fixed $q\in (0,1)$, we will reparameterize the distribution of the susceptible individuals in terms of q-th quantile ${\mu }_q$, such that θ can be written as

$$\theta =-\frac{\log (q)}{\log [1-\exp \left(\frac{\lambda }{\alpha }\right)]-\log \{1-\exp [-\frac{\lambda }{\alpha }(e^{\alpha {\mu }_q}-1)]\}},$$ (7)

where this result is obtained by isolating θ in equation (6).

It is important to highlight that we express the survival function for susceptible individuals ( equation (5)) as a function of the survival function of the defective generalized Gompertz distribution and the cure fraction $p_0(\lambda ,\alpha ,\theta )$. In the conventional form of the defective Gompertz distribution, the value of α determines the presence of a cure fraction: a negative α indicates a cure fraction, while a non-negative α indicates its absence. However, in the reparametrized defective generalized Gompertz distribution, the parameter space for α is restricted to $(-\mathrm{\infty },0)$. This constraint is crucial for achieving the proposed quantile parametrization, which inherently assumes the presence of a cure rate. Therefore, the proposed model is a cure rate survival model and, thus, it is advisable to use this model only after confirming the existence of a cure rate. It is important to note that this restriction does not undermine the advantage of the proposed approach compared to standard mixture models, which introduce an additional parameter specifically to account for the cure fraction. In our model, the cure fraction is implicitly embedded within the distribution itself, meaning that an extra parameter is not included for the cure fraction.

While the proposed defective distribution-based model simplifies parameterization compared to standard mixture models, it is important to recognize that identifying the presence of a cure fraction in the data is not a straightforward task. The model inherently assumes the existence of a cure fraction, without explicitly testing for its presence. Since biological reasoning alone may not be sufficient, it is crucial that practitioners thoroughly evaluate the data for evidence of a cure fraction before applying the cure rate model. For instance, a hypothesis test, such as the one proposed by [30], can be employed to assess whether immune individuals are present in the population.

If we replace (7) in (1)–(3), we have the reparameterized density and survival functions and cure fraction of the defective Generalized Gompertz distribution (dGG), which are given by, respectively,

$$\begin{array}{rl}f(t\mid \lambda ,\alpha ,{\mu }_q)& =\lambda (-\frac{\log (q)}{\log [1-\exp \left(\frac{\lambda }{\alpha }\right)]-\log \{1-\exp [-\frac{\lambda }{\alpha }(e^{\alpha {\mu }_q}-1)]\}})\\ & \times \exp \{\mathrm{\alpha t}-\frac{\lambda }{\alpha }[\exp (\mathrm{\alpha t})-1]\}\\ & \times {(1-\exp \{-\frac{\lambda }{\alpha }[\exp (\mathrm{\alpha t})-1]\})}^{-\frac{\log (q)}{\log [1-\exp \left(\frac{\lambda }{\alpha }\right)]-\log \{1-\exp [-\frac{\lambda }{\alpha }(e^{\alpha {\mu }_q}-1)]\}}-1},\end{array}$$ (8)

$$\begin{array}{rl}S(t\mid \lambda ,\alpha ,{\mu }_q)& =1-{(1-\exp \{-\frac{\lambda }{\alpha }[\exp (\mathrm{\alpha t})-1]\})}^{-\frac{\log (q)}{\log [1-\exp \left(\frac{\lambda }{\alpha }\right)]-\log \{1-\exp [-\frac{\lambda }{\alpha }(e^{\alpha {\mu }_q}-1)]\}}}\end{array}$$ (9)

and

$$p_0(\lambda ,\alpha ,{\mu }_q)=1-{[1-\exp \left(\frac{\lambda }{\alpha }\right)]}^{-\frac{\log (q)}{\log [1-\exp \left(\frac{\lambda }{\alpha }\right)]-\log \{1-\exp [-\frac{\lambda }{\alpha }(e^{\alpha {\mu }_q}-1)]\}}}.$$ (10)

Hereafter, we shall use the notation $T\sim \mathrm{dGG}(\alpha ,\lambda ,{\mu }_q,q)$ where ${\mu }_q>0$ is the quantile parameter, $\lambda >0$, $\alpha <0$, and the probability $q\in (0,1)$ is assumed to be fixed (known) according to the quantile of interest.

The proposed model is individually fitted for each quantile of interest. However, it is really important to highlight that θ, as well as α and λ remain the same regardless of the coordinate points (q, ${\mu }_q$) that are part of equation (6) of the ${\mu }_q$ and q. For example, suppose $\lambda =2$, $\alpha =-1$, and $\theta =5$. With these fixed values, the quantile function ${\mu }_q$ for different values of q is obtained by considering equation (6). For q = 0.2, ${\mu }_q=0.6786$, for q = 0.5, ${\mu }_q=1.1995$, and for q = 0.8, ${\mu }_q=2.0957$. Thus, for the same value of α and λ, all quantiles return the same value of θ by considering equation (7): for q = 0.2 and ${\mu }_q=0.6786$, $\theta =5$; for q = 0.5 and ${\mu }_q=1.1995$, $\theta =5$; and for q = 0.8, ${\mu }_q=2.0957$, θ is also 5.

Now, we build the defective generalized Gompertz quantile regression model for a fixed (known) probability $q\in (0,1)$ associated with the quantile of interest, imposing that the quantile ${\mu }_q$ of T satisfies the following functional relation:

$${\mu }_q({\mathit{\beta }}_q,\mathbf{x})={\mu }_q=\exp ({\mathbf{x}}^{\mathrm{\top }}{\mathit{\beta }}_q),$$ (11)

in which ${\mathbf{x}}^{\mathrm{\top }}=(1,x_1,\dots ,x_p)$ is the vectors of covariates and ${\mathit{\beta }}_q=({\beta }_{q0},{\beta }_{q1},\dots ,{\beta }_{\mathrm{qp}}{)}^{\mathrm{\top }}$ is the unknown vector of regression parameters to be estimated.

The parameters α and λ can also be linked to covariates, which gives more flexibility to the model. In this way, for α, we have that

$$\alpha =-\exp ({\mathbf{w}}^{\mathrm{\top }}\mathit{\gamma }),$$ (12)

in which ${\mathbf{w}}^{\mathrm{\top }}=(1,w_1,\dots ,w_r)$ are the covariates vector and $\mathit{\gamma }=({\gamma }_0,{\gamma }_1,\dots ,{\gamma }_r{)}^{\mathrm{\top }}$ is the unknown vector of parameters related to α.

Finally, for λ, we have

$$\lambda =\exp ({\mathbf{z}}^{\mathrm{\top }}\mathit{v}),$$ (13)

in which ${\mathbf{z}}^{\mathrm{\top }}=(1,z_1,\dots ,z_s)$ is the covariate vector and $\mathit{v}=(v_0,v_1,\dots ,v_s{)}^{\mathrm{\top }}$ is the relative unknown vector of parameters related to λ.

After this formulation, we have that the parameter vector for the q-quantile of interest is ${\mathit{\vartheta }}_q=({\mathit{\beta }}_q^{\mathrm{\top }},{\mathit{\gamma }}^{\mathrm{\top }},{\mathit{v}}^{\mathrm{\top }}{)}^{\mathrm{\top }}$. In the next section, we discuss how to obtain parameter estimates for this vector. Thus, the proposed model under the generalized Gompertz distribution is referred to as Generalized Gompertz Cure Rate Quantile Regression (GGCRQR).

Our formulation is related to the approach also known as distributional regression [27,41], where all parameters of the conditional distribution are explained as function of covariates, through properly chosen link functions. Although we do something similar here, we are still interested in studying the effects of each covariate in the quantile function of the Generalized Gompertz distribution, while also analyzing the conditional cure rate given by its defective version, which would put our model in a slightly different class.

3. Inference

Consider that the lifetime T is possibly not observed, that is, it is constrained by a right-censored failure time and let C denote the censoring time. In a sample of size n, we then observe the i-th lifetime $t_i=min\{T_i,C_i\}$, and i-th failure indicator ${\delta }_i=I(T_i\le C_i)$, where ${\delta }_i=1$ if $T_i$ is observed and ${\delta }_i=0$ otherwise, for $i=1,\dots ,n$.

We consider that $T_i$'s are independent random variables with density and survival functions $S(t_i\mid {\mathit{\vartheta }}_q)$ and $f(t_i\mid {\mathit{\vartheta }}_q,)$, obtained by replacing ${\mu }_{\mathrm{qi}}=\exp ({\mathbf{x}}_i^{\mathrm{\top }}{\mathit{\beta }}_q)$, ${\alpha }_i=-\exp ({\mathbf{w}}_i^{\mathrm{\top }}\mathit{\gamma })$ and ${\lambda }_i=\exp ({\mathbf{z}}_i^{\mathrm{\top }}\mathit{v})$ in equations (8) and (9), respectively, where ${\mathbf{x}}_i^{\mathrm{\top }}=(1,x_{i1},\dots ,x_{\mathrm{ip}})$, ${\mathbf{w}}_i^{\mathrm{\top }}=(1,w_{i1},\dots ,w_{\mathrm{ir}})$ and ${\mathbf{z}}_i^{\mathrm{\top }}=(1,z_{i1},\dots ,z_{\mathrm{is}})$, for $i=1,\dots ,n$. We assume that each censoring time $C_i$ is independent of lifetime $T_i$, for all $i=1,\dots ,n$, and we consider a noninformative censoring assumption, i.e. the censoring distribution does not involve the parameters of the distribution of T. Therefore, for a given $q\in (0,1)$, the likelihood function of ${\mathit{\vartheta }}_q$ can be written as

$$L({\mathit{\vartheta }}_q;\mathbf{D})\propto \prod _{i=1}^n{\left[f(t_i\mid {\mathit{\vartheta }}_q)\right]}^{{\delta }_i}{\left[S(t_i\mid {\mathit{\vartheta }}_q)\right]}^{1-{\delta }_i},$$

where $\mathbf{D}=(n,\mathbf{t},\mathit{\delta },\mathit{X},\mathit{W},\mathit{Z})$, with $\mathbf{t}=(t_1,\dots ,t_n{)}^{\mathrm{\top }},\mathit{\delta }=({\delta }_1,\dots ,{\delta }_n{)}^{\mathrm{\top }}$. Furthermore, $\mathit{X}=({\mathbf{x}}_1^{\mathrm{\top }},\dots ,{\mathbf{x}}_n^{\mathrm{\top }}),\mathit{W}=({\mathbf{w}}_1^{\mathrm{\top }},\dots ,{\mathbf{w}}_n^{\mathrm{\top }})$ and $\mathit{Z}=({\mathbf{z}}_1^{\mathrm{\top }},\dots ,{\mathbf{z}}_n^{\mathrm{\top }})$ are $n\times (p+1)$, $n\times (r+1)$ and $n\times (s+1)$ matrices containing the covariates information, respectively.

Under the Bayesian approach, we can specify the posterior distribution of ${\mathit{\vartheta }}_q$ as

$$\pi ({\mathit{\vartheta }}_q\mid \mathbf{D})\propto \pi ({\mathit{\vartheta }}_q)L({\mathit{\vartheta }}_q;\mathbf{D}),$$ (14)

where $\pi ({\mathit{\vartheta }}_q)$ is the prior distribution of ${\mathit{\vartheta }}_q$ for a fixed q.

The prior distributions for ${\mathit{\vartheta }}_q$ can be defined in the following manner, assuming that they are prior independent. We assume a normal distribution for each β, γ and v with mean 0 and variance 100. This setup will be considered in both the simulation and the application in the following sections. Nevertheless, in the supplementary material we have considered a sensitivity analysis, where we show that there is no compelling difference in the posterior distribution in the application, when we use other prior distributions than the one defined here.

The posterior distribution in (14) does not have a closed form and the parameters are estimated through simulated samples of the posterior distribution obtained by the Adaptive Metropolis algorithm with multivariate normal distribution as proposed by [21]. This approach is implemented in the statistical package LaplacesDemon [23], which provides a friendly environment for Bayesian inference within the R program [35]. The convergence of the chains can be checked through graphical diagnostics, such as trace plots and autocorrelation plots, and also through Gelman and Rubim statistics [15]. The proposed model and estimation process are available in R package ggcrqr (https://github.com/brsantos/ggcrqr), developed and made available by the authors of this article.

As a result, a sample of size M from the joint posterior distribution of ${\mathit{\vartheta }}_q$ is obtained (eliminating burn-in and jump samples). The sample from the posterior can be expressed as $({\mathit{\vartheta }}_q^{(1)},{\mathit{\vartheta }}_q^{(2)},\dots ,{\mathit{\vartheta }}_q^{(M)})$. The posterior mean of ${\mathit{\vartheta }}_q$, for instance, can be approximated by

$${\hat{\mathit{\vartheta }}}_q=\frac{1}{M}\sum _{m=1}^M{\mathit{\vartheta }}_q^{(m)},$$

and the posterior mean of the cure rate is approximated by

$${\hat{p}}_0=\frac{1}{M}\sum _{m=1}^{M}1-{[1-\exp \left(\frac{{\lambda }^{(m)}}{{\alpha }^{(m)}}\right)]}^{-\frac{\log (q)}{\log [1-\exp \left(\frac{{\lambda }^{(m)}}{{\alpha }^{(m)}}\right)]-\log \{1-\exp [-\frac{{\lambda }^{(m)}}{{\alpha }^{(m)}}(e^{{\alpha }^{(m)}{\mu }_q^{(m)}}-1)]\}}}.$$

The Conditional Predictive Ordinate (CPO) for ith observation, ${\mathrm{CPO}}_i$, is the probability to observe $t_i$ when the model is fitted without this observation. Let ${\mathbf{D}}_{-i}$, the data without ith observation. The ith CPO is given by

$${\mathrm{CPO}}_i={\int }_{{\mathrm{\Theta }}_q}f(t_i\mid {\mathit{\vartheta }}_q)\pi ({\mathit{\vartheta }}_q\mid {\mathbf{D}}_{-i})d{\mathit{\vartheta }}_q={\left\{{\int }_{{\mathrm{\Theta }}_q}\frac{\pi ({\mathit{\vartheta }}_q\mid \mathbf{D})}{f(t_i\mid {\mathit{\vartheta }}_q)}d{\mathit{\vartheta }}_q\right\}}^{-1},$$

in which ${\mathrm{\Theta }}_q$ is the parameter space of ${\vartheta }_q$.

The ${\mathrm{CPO}}_i$ can be estimated by considering M values of the posterior distribution of ${\vartheta }_q$ [18]:

$${\widehat{\mathrm{CPO}}}_i={\left\{\frac{1}{M}\sum _{m=1}^M\frac{1}{g(t_i\mid {\mathit{\vartheta }}_q^{(m)})}\right\}}^{-1},$$

and $g(t_i\mid {\mathit{\vartheta }}_q^{(m)})$ is $f(t_i\mid {\mathit{\vartheta }}_q^{(m)})$ if ${\delta }_i=1$ and $S(t_i\mid {\mathit{\vartheta }}_q^{(m)})$ if ${\delta }_i=0$, for $i=1,\dots ,n$.

High values of ${\mathrm{CPO}}_i$ indicate that the model is able to describe the ith observation adequately and, for this reason, it is natural to think of a model selection measure that is a function of the CPO. The measure LPML (Log Pseudo Marginal Likelihood) is the sum of the logarithms of the CPO's of all observations, i.e. $\mathrm{LPML}=\sum _{i=1}^n\log ({\widehat{\mathrm{CPO}}}_i)$; the higher its value, the better will be the model fit.

In order to evaluate the model's adequacy, we can evaluate the residuals that can be obtained by also considering the predictive distribution. The standardized residual for ith observation is given by:

$$r_i=\frac{t_i-E[T_i\mid {\mathbf{D}}_{-i}]}{\sqrt{Var[T_i\mid {\mathbf{D}}_{-i}]}},$$

the difference between the actual value of the observable and the expectation, divided by the square of variance, of the observable under the predictive distribution.

As presented in [18], $E[T_i\mid {\mathbf{D}}_{-i}]$ and $\mathrm{Var}[T_i\mid {\mathbf{D}}_{-i}]$ can be approximated, respectively, by:

$$\hat{E}[T_i\mid {\mathbf{D}}_{-i}]\propto \frac{1}{M}\sum _{m=1}^M{t}_{\mathrm{im}}^{\ast },$$

and

$$\widehat{\mathrm{Var}}[T_i\mid {\mathbf{D}}_{-i}]\propto \frac{1}{M}\sum _{m=1}^M(t_{\mathrm{im}}^{\ast }-\hat{E}[T_i\mid {\mathbf{D}}_{-i}]{)}^2,$$

in which $t_{\mathrm{im}}^{\ast }$ is sampled from $f(\cdot \mid {\mathit{\vartheta }}_q^{\ast (m)})$ and ${\mathit{\vartheta }}_q^{\ast (m)}$ is obtained as in the following.

For the ith observation, the importance ratio $w_i^{(m)}=g(t_i\mid {\mathit{\vartheta }}_q^{(m)})$ is calculated. Thus, we resample with replacement from the collection $\{{\mathit{\vartheta }}_q^{\ast (m)},m=1,\dots ,M\}$ with probabilities proportional to $\{w_i^{(m)},m=1,\dots ,M\}$. The resulting sample is approximately from $f(t_i\mid {\mathbf{D}}_{-i})$.

Thus, the standardized residual for ith observation is estimated by ${\hat{r}}_i=(t_i-\hat{E}[T_i\mid {\mathbf{D}}_{-i}])/\widehat{\mathrm{Var}}[T_i\mid {\mathbf{D}}_{-i}]$, for $i=1,\dots ,n$, and residual analysis can be conducted to assess whether there is any violation of the proposed regression model. For more details, see [18] in the Chapter 9.

It is worth mentioning that in a survival analysis scenario in which there is censored data, this residual analysis is only reasonable for cases of non-censorship, that is, for cases in which $\delta =1$.

4. Simulation studies

In this section, we evaluate the performance of the proposed model across different sample sizes and percentages of right censoring (pc) in simulated scenarios. The first scenario (Scenario 1) involves a single binary covariate, while the second scenario (Scenario 2) includes both a continuous and a binary covariate for three quantiles q = 0.2, 0.5, and 0.8. The third scenario (Scenario 3) presents the evaluation of the proposed model in different percentages of censored data ( $\mathrm{pc}=20\mathrm{\%},40\mathrm{\%}$ and $60\mathrm{\%}$). Finally, the last scenario (Scenario 4) presents the comparison of the proposed model with the usual standard mixture model with proper GG distribution for susceptible individuals in relation to the cure fraction estimation.

To obtain posterior quantities, the first $\mathrm{10,000}$ samples were discarded as burn-in for scenarios 1, 3, and 4, and $\mathrm{70,000}$ samples were discarded as burn-in for scenario 2. A jump size of 10 was chosen for scenarios 1, 3, and 4, and a jump size of 70 was used for scenario 2, ensuring that the correlation between the simulated values was close to zero. The convergence of the chains was evaluated through graphical diagnostic and Gelman and Rubin statistics, in which burn-in and jump values described before were considered to get good convergence results. The Gelman and Rubin values of the parameters of all scenarios are close to 1, indicating the convergence of the chains. The convergence diagnostic of an example from the first scenario is presented in the Supplementary Material and a similar diagnostic analysis was performed for all simulated scenarios.

Thus, we get M = 1000 simulated values for each parameter. We consider the posterior mean for each parameter as our point estimates.

4.1. Scenario 1: only one binary covariate

For this simulation scenario, we consider a binary covariate x (x = 0 and x = 1) and the relations: $\alpha =-\exp ({\gamma }_0+{\gamma }_{1}x)$, $\lambda =\exp (v_0)$ and $\mu =\exp ({\beta }_{q0}+{\beta }_{q1}x)$. The parameter values are ${\gamma }_0=-1.12$, ${\gamma }_1=-0.43$ and $v_0=-1$, resulting in $\alpha =-0.32$ when x = 0, $\alpha =-0.21$ when x = 1, and $\lambda =0.368$.

The proposed model is fitted for quantiles $q=0.2,0.5,0.8$, and ${\beta }_{q0}$ values are 0.83, 1.42 and 1.97, respectively, and ${\beta }_{q1}$ values are 0.77, 0.65 and 0.57, respectively. The cure rates are $p_{00}=0.78$ and $p_{01}=0.79$, and the censoring rate is 75%.

It is worth mentioning that for fixed values of α and λ, ${\beta }_{q0}=0.83$ and ${\beta }_{q1}=0.77$ for q = 0.2, ${\beta }_{q0}=1.42$ and ${\beta }_{q1}=0.65$ for q = 0.5, and ${\beta }_{q0}=1.97$ and ${\beta }_{q1}=0.57$ for q = 0.8 lead to the same θ value, that is, $\theta =3.8$ for x = 0 and $\theta =8$ for $x=1$.

The parameter values and censoring rate are based on the application section and β values are the same obtained for the hormone therapy covariate. We consider the covariate in both µ and α in order to corroborate the results obtained in the application section.

Moreover, seven sample sizes are considered: $n=50,100,200,400,800,1000,2000$. For each combination of parameter values and sample size, B = 1000 datasets are generated by considering the Algorithm 1.

In Figure 2, we can check that the bias decreases as the sample sizes increase since point estimates do approximate their respective true values. Moreover, in general, we have that the mean standard error, represented by the bars in the plot, is relatively large for small sample sizes, such as 50, but this number rapidly decreases as the sample sizes get larger as well. An important observation for this simulation study is that the cure fractions estimates present low bias values for n greater than 50 and for all q considered.

Figure 2.

Average of the posterior mean of the parameters (dots) and mean standard error (bars) for the different combinations of $q=\{0.2,0.5,0.8\}$ and sample size ($n= 50,100,200,400,800,1000,2000$) in the scenario with one binary covariate (Scenario 1). The red line represents the true value for each parameter. Values g0, g1 b0, b1 represent ${\gamma }_0$, ${\gamma }_1$, ${\beta }_0$, ${\beta }_1$, respectively.

In Figure 3 we compare the coverage probability of 95% credible intervals for each parameter, considering Highest Probability Density (HPD) and equal-tailed credible intervals. We can observe that the coverage probabilities when n = 50 are far from the nominal value for q = 0.2 and q = 0.5. For the other sample sizes, the coverage probabilities are close to 0.95, mainly for sample sizes greater than 400.

Figure 3.

Coverage probability of each parameter for the different combinations for $q=\{0.2,0.5,0.8\}$ and n = 50, 100, 200, 400, 800, 1000, 2000 in the scenario with one binary covariate (Scenario 1). The gray line represents the nominal value of 0.95. Values g0, g1, b0, b1 represent ${\gamma }_0$, ${\gamma }_1$, ${\beta }_0$, ${\beta }_1$, respectively.

Additionally, for some parameters and for larger sample sizes, the coverage probability stays under the nominal value of 95% when q = 0.8. However, this behavior was not observed in the second scenario that follows and, for $p_{01}$ and $p_{00}$, the coverage probabilities are close to the nominal value for sample sizes equal to or greater than n = 800.

4.2. Scenario 2: one binary covariate and one continuous covariate

We consider a binary covariate x (x = 0 and x = 1) and a continuous covariate y. The relations are $\alpha =-\exp ({\gamma }_0+{\gamma }_{1}x+{\gamma }_{2}y)$, $\lambda =\exp (v_0)$ and $\mu =\exp ({\beta }_{q0}+{\beta }_{q1}x+{\beta }_{q2}y)$. The parameter values are ${\gamma }_0=-1.12$, ${\gamma }_1=-0.43$, ${\gamma }_2=0.02$ and $v_0=-1$.

The proposed model is fitted for quantiles $q=0.2,0.5,0.8$, and ${\beta }_{q0}$ values are 0.83, 1.42 and 1.97, respectively, ${\beta }_{q1}$ values are 0.77, 0.65 and 0.57, respectively, and ${\beta }_{q2}$ values are $-0.11$, $-0.07$ and $-0.05$, respectively. In this scenario, the cure rates also depend on the values of the continuous variable y, and the censoring rate is 30%.

Moreover, seven sample sizes are considered: $n=50,100,200,400,800,1000,2000$. For each combination of parameter values and sample size, B = 1000 datasets are generated by considering the Algorithm 2.

In Figure 4, we can check that the bias decreases as the sample sizes increase since the point estimates do approximate their respective true values. The bias is closer and closer to zero as n increases, starting from n = 200. The mean standard error, represented by the bars in the plot, is relatively large for small sample sizes, such as n = 50 and n = 100, but this number rapidly decreases as the sample sizes get larger as well.

Figure 4.

Average of the posterior mean of the parameters (dots) and mean standard error (bars) for the different combinations of $q=\{0.2,0.5,0.8\}$ and sample size ($n= 50,100,200,400,800,1000,2000$) in the scenario with one binary covariate and one continuous covariate (Scenario 2). The red line represents the true value for each parameter. Values g0, g1, g2, b0, b1 and b2 represent ${\gamma }_0$, ${\gamma }_1$, ${\gamma }_2$, ${\beta }_0$, ${\beta }_1$, ${\beta }_2$, respectively.

In Figure 5 we compare the coverage probability of 95% credible intervals for each parameter, considering HPD and equal-tailed credible intervals. We can observe that, in general, the coverage probabilities are close to the nominal as the sample size increases. For q = 0.5 and ${\gamma }_0$, ${\gamma }_2$, ${\beta }_0$, and ${\beta }_2$ parameters the coverage probabilities are underestimated for the largest sample size. However, this distance is not so discrepant, once these coverage probabilities are $90\mathrm{\%}$ at least.

Figure 5.

Coverage probability of each parameter for the different combinations for $q=\{0.2,0.5,0.8\}$ and n = 50, 100, 200, 400, 800, 1000, 2000 in the scenario with one binary covariate and one continuous covariate (Scenario 2). The gray line represents the nominal value of 0.95. Values g0, g1, g2, b0, b1, b2 represent ${\gamma }_0$, ${\gamma }_1$, ${\gamma }_2$, ${\beta }_0$, ${\beta }_1$, ${\beta }_2$, respectively.

4.3. Scenario 3: effect of censoring rate

In this scenario, the idea is to evaluate the effect of censoring rate in the parameter estimation. For that, we consider $\alpha =-\exp ({\gamma }_0+{\gamma }_{1}x)$, $\lambda =\exp (v_0)$ and $\mu =\exp ({\beta }_{q0}+{\beta }_{q1}x)$, when q = 0.5 (median). The parameter values are ${\gamma }_0=-3.8$, ${\gamma }_1=-0.01$, ${\beta }_{q0}=3.45$, ${\beta }_{q1}=-0.5$ and $v_0=-3$, resulting in $\alpha =-0.022$, $\mu =31.5$ and $\theta =2.5$ when x = 0, $\alpha =-0.020$, $\mu =19.1$ and $\theta =1.34$ when $x=1$, and $\lambda =0.05$. The cure rates are $p_{00}=0.25$ and $p_{01}=0.11$ and seven sample sizes are considered: n = 50, 100, 200, 400, 800, 1000, 2000, besides three percentages of right censoring data: 20%, 40% and 60%. For each combination of sample size and censoring rate, B = 1000 datasets are generated by considering the Algorithm 1.

We can observe that the bias, as well as the mean standard error, decreases as the sample sizes increase since the point estimates do approximate their respective true values (Figure 6). In general, for a fixed sample size, the mean standard error is larger for a larger censoring rate; however, the results are reasonable for a sample size bigger than 800 regardless of the censoring rate.

Figure 6.

Average of the posterior mean of the parameters (dots) and mean standard error (bars) for the different combinations of censoring rate ( $\mathrm{pc}=20\mathrm{\%},40\mathrm{\%},60\mathrm{\%}$) and sample size ( $n=50,100,200,400,800,1000,2000$) for Scenario 3. The red line represents the true value for each parameter. Values g0, g1 b0, b1 represent ${\gamma }_0$, ${\gamma }_1$, ${\beta }_0$, ${\beta }_1$, respectively.

It is important to note that for a censoring rate equal to 20%, in almost every parameter the average bias is nearly zero. For larger censoring rates such as 40% or 60%, there is bias for smaller sample sizes, up to 400 observations. So it is clear that there is a correspondence between censoring rates and sample sizes. For larger censoring rates, we need larger sample sizes to obtain better results with regards to bias.

4.4. Scenario 4: comparison of cure fraction estimators

In this subsection, the proposed GGCRQR model is compared to the standard mixture model with proper GG distribution for susceptible individuals, which is called here as SMM, in relation to the cure fraction estimation. It is worth mentioning that SMM is not a quantile regression model, but the idea here is to evaluate the effectiveness of the proposed model in estimating the cure fraction, which is comparable between the models.

We consider a binary covariate x (x = 0 and x = 1) and the relations: $\alpha =-\exp ({\gamma }_0)=-\exp (-3,77)=-0.02$ and $\lambda =\exp (v_0)=\exp (-2.99)=0.05$. The models are fitted for quantile q = 0.5 (median), and ${\beta }_{q0}$ is 3.45 and ${\beta }_{q1}$ values is $-0.23$. The cure rates are $p_{00}=0.26$ and $p_{01}=0.20$, and the censoring rate is 25%. For each sample size $n=50,100,300,500,1000$ and 2000, B = 1000 datasets are generated by considering the Algorithm 1.

When observing Figure 7, we see that the proposed GGCRQR model exhibits excellent behavior in estimating cure fractions, since the average of the posterior averages is close to the true fraction, presenting a bias very close to 0, especially for larger sample sizes. Furthermore, the mean squared error decreases with increasing sample size, indicating good performance of the proposed model. The SMM approach exhibits greater bias and mean squared error than the proposed approach for all sample sizes.

Figure 7.

Average of the posterior mean of cure fraction (symbols) and mean standard error (bars) for each sample size (n = 50, 100, 300, 500, 1000, 2000) of GGCRQR and SMM models (Scenario 4). The red line represents the true value for each parameter.

While it is true that largest difference between the estimated cure fraction with the proposed GGCRQR and the SMM approaches is 0.0043, which is not relatively a huge number in relation to the true values, it is important to remark that the former approach has additional benefits. The proposed GGCRQR model is able to effectively estimate different quantiles of interest, as shown in the previous simulations studies. Hence, this scenario adds that our proposal also does not suffer from any loss in efficiency when estimating the cure of fraction, when we contrast it to the SMM approach, which is a comparable model in this setting.

It is worth noting that the data for this simulation scenario were generated using the Algorithm 1, since in this article we are interested in evaluating the behavior of the proposed model in different scenarios (different quantiles of interest, sample sizes, and percentage of censored data).

5. Application

About 1 out of every 100 breast cancers diagnosed is found in a man (www.cdc.gov). Given this unequal distribution, it is not surprising that there is still some misunderstanding of its incidence and prognostic factors among men. In this analysis, we want to study the impact of age, clinical stage, and treatments (surgery, chemotherapy, radiotherapy, and hormone therapy) in survival for men, to understand how the disease affects this population.

As already said in the Subsection 1.2, we consider a male breast cancer dataset from a retrospective survey of n = 887 records of males diagnosed with breast in the state of São Paulo, Brazil, between 2000 and 2019, with follow-up conducted until February of 2020 and with at least two months of follow-up. It is important to revisit the discussion from Scenario 3 of the simulation studies (Section 4), where there is a relationship between censoring rates and sample sizes. Larger sample sizes are necessary for higher censoring rates to achieve reasonable bias. In this application, the censoring percentage is 77% (as described in Subsection 1.2). However, when considering the results from Scenario 1 of the simulation studies, which has a censoring percentage of 75%, a sample size above 800 already yields satisfactory results, which aligns with our case.

The dataset considered here and the routines used to estimate the models are available in the following link: https://github.com/brsantos/ggcrqr. The descriptive analyses can be seen in Subsection 1.2.

In this section, the GGCRQR model is fitted in two versions. In the first one (version 1), the covariates are included only in the quantile function ${\mu }_q$ by considering the relation in (11):

$$\begin{array}{rl}{\mu }_q& =\exp ({\beta }_{q0}+{\beta }_{q1}\times {\mathrm{stage}}_{\mathrm{II}}+{\beta }_{q2}\times {\mathrm{stage}}_{\mathrm{III}}+{\beta }_{q3}\times {\mathrm{stage}}_{\mathrm{IV}}+{\beta }_{q4}\times {\mathrm{age}}_{\ge 55\mathrm{\&}\le 65}\\ & +{\beta }_{q5}\times {\mathrm{age}}_{>65}+{\beta }_{q6}\times \mathrm{surgery}+{\beta }_{q7}\times \mathrm{chemo}\\ & +{\beta }_{q8}\times \mathrm{hormone}+{\beta }_{q9}\times \mathrm{radio}),\end{array}$$ (15)

where the model is fitted for the q-quantile of interest. If q = 0.5, the influence of the covariates is studied in relation to the median lifetime. Here we fit the proposed model for each $q=\{0.10,\dots ,0.90\}$ in order to illustrate how the covariates' effects change depending on the quantile of interest. Thus, the coefficients ${\beta }_{\mathrm{qi}}$, $i=0,1,\dots ,9$, also vary with q.

In the second version (version 2), in addition to considering covariates in the quantile function as in (15), we also consider covariates in α when considering the relation presented in (12). The idea to include covariates in α is to make the model more flexible. The inclusion of covariates in the α parameter can improve the cure fractions estimations since these quantities also depend on α.

For each version, the adaptive Metropolis–Hastings algorithm was run, discarding the first $\mathrm{50,000}$ iterations as burn-in samples and using a jump of size 100 to avoid correlation problems, with a sample size of M = 2000. The convergence of the chain was evaluated by multiple runs of the algorithm from different starting values and was monitored through graphical analysis, where we were able to obtain good convergence results.

Figure 8 presents the coefficients values by considering q varying from 0.1 to 0.9 by 0.1 for the ${\beta }_{q9}$ parameter in (15). The solid line represents the posterior mean value and the dotted lines represent the 95% highest probability density (HPD) interval. There is no evidence for quantiles of the survival times of susceptible individuals who underwent radiotherapy to be different from those who did not have radiotherapy, the reference category, for both versions of the GGCRQR model. Because of this, the two versions models are fitted again but without the radiotherapy variable. Thus, the quantile function ${\mu }_q$ is given by:

$$\begin{array}{rl}{\mu }_q& =\exp ({\beta }_{q0}+{\beta }_{q1}\times {\mathrm{stage}}_{\mathrm{II}}+{\beta }_{q2}\times {\mathrm{stage}}_{\mathrm{III}}+{\beta }_{q3}\times {\mathrm{stage}}_{\mathrm{IV}}\\ & +{\beta }_{q4}\times {\mathrm{age}}_{\ge 55\mathrm{\&}\le 65}+{\beta }_{q5}\times {\mathrm{age}}_{>65}+{\beta }_{q6}\times \mathrm{surgery}\\ & +{\beta }_{q7}\times \mathrm{chemo}+{\beta }_{q8}\times \mathrm{hormone}).\end{array}$$ (16)

Figure 8.

Posterior estimates for the regression parameter ${\beta }_{q9}$ given quantiles, q, from equation (15) for version 1 (in the left) and for version 2 (in the right). The solid lines denote the posterior mean, while the dashed are the respective HPD credible interval.

For the version with covariates in α, the selected model that presents the highest LPML is that with only surgery and hormone therapy, this is

$$\alpha =-\exp ({\gamma }_0+{\gamma }_1\times \mathrm{surgery}+{\gamma }_2\times \mathrm{hormone}).$$ (17)

Figures 9 and 10 present the coefficient values by considering q varying from 0.1 to 0.9 by 0.1 for the parameters in equation (16) for the model with covariates only in the quantile function and for the model with covariates in the quantile function and in the α parameter, as in equation (17), respectively. The solid line represents the posterior mean value and the dotted lines represent the 95% HPD interval. It is important to note how some coefficients vary along the different quantiles. For both GGCRQR model versions, all the coefficients HPD intervals do not include the zero value for all values of q considered.

Figure 9.

Posterior estimates for the regression parameter $\beta$ given quantiles, q, presented in equation (16), with covariates only in the quantile function – version 1. The solid lines denote the posterior mean, while the dashed are the respective HPD credible interval.

Figure 10.

Posterior estimates for the regression parameter $\beta$ given quantiles, q, presented in equation (16), with covariates in the quantile function and in α as in equation (17) – version 2. The solid lines denote the posterior mean, while the dashed are the respective HPD credible interval.

Table 1 presents the posterior mean and HPD 95% credible intervals for parameters related to α and λ, by considering equations (12) and (13), respectively, for models versions 1 and 2. It is worth highlighting that, as mentioned in Subsection 2.2, the estimates of α and λ are the same regardless of q and therefore they are presented only once in the table.

Table 1.

Posterior estimates for the parameters involved in α and λ parameters.

	Version 1 – with covariates only in ${\mu }_q$		Version 2 – with covariates in ${\mu }_q$ and α
Parameter	Posterior mean	95% HPD CI	Posterior mean	95% HPD CI
${\gamma }_0$	−1.604	−1.923   −1.322	−1.117	−1.487   −0.751
${\gamma }_1$	–	–	−0.194	−0.462   −0.079
${\gamma }_2$	–	–	−0.435	−0.701   −0.165
$v_0$	−1.147	−1.516   −0.812	−1.017	−1.378   −0.614

Since we have the posterior draws for each parameter we can apply the function in (10) for each draw so we have a posterior sample of the cure fraction. This allows us to check the posterior density for the combination of clinical stage and other covariates regarding its cure fraction as well. These densities are shown in Figures 11 and 12 for model version 1 and for model version 2, respectively.

Figure 11.

Posterior densities for the cure fraction parameter for each combination of clinical stage and other covariates by considering model version 1.

Figure 12.

Posterior densities for the cure fraction parameter for each combination of clinical stage and other covariates by considering model version 2.

The LPML values for each model version by considering different q values are presented in Table 2. As one can see, version 2 (model with covariates in ${\mu }_q$ and in α) presents the highest LPML value for all q values. Because of this, this is the chosen GGCRQR model and its results are interpreted in the following. The residuals analysis was performed and it is presented in the Supplementary Material.

Table 2.

LPML for model versions 1 and 2 for all q values considered.

q	Version 1 – with covariates only in ${\mu }_q$	Version 2 – with covariates in ${\mu }_q$ and α
0.1	−708.82	−705.44
0.2	−708.72	−705.21
0.3	−708.88	−704.45
0.4	−708.53	−704.22
0.5	−709.18	−704.07
0.6	−708.38	−703.88
0.7	−709.06	−703.73
0.8	−709.08	−703.98
0.9	−708.72	−704.10

By analyzing Figure 10, it is possible to note that the quantile-based model defined in (16) is able to estimate the effects for different quantiles and display contrasting differences along these quantiles. For the clinical stage variable, the effect is increasingly negative, that is, the lifetime is shorter as the stage of the disease increases, with stage I as the reference category. Still, the effect of this variable for the lifetime quantiles is clearly different, mainly for stages III and IV. Men aged between 55 and 65 years old have the highest survival time, and men aged over 65 years old have the lowest survival time. In relation to cancer treatments, those men who did surgery present the highest lifetime as well as those who underwent hormone therapy. On the other hand, those men undergoing chemotherapy have a shorter survival time.

In relation to the cure fraction (Figure 12), one can observe the ordering of the probability of cure for the different clinical stages is consistent with the common prognosis of this carcinoma, as people in Stage I have a higher probability of being cured, and men at Stage IV of the cancer diagnosis have a lower probability of being cured in comparison to other stages. Fixed the clinical stage, men older than 55 and younger than 65 years old have a higher probability of cure, in comparison with the other groups, and men aged over 65 years old have the lowest cure fraction, as well as those undergoing chemotherapy. It seems that whether or not to have surgery and whether or not to perform hormone therapy does not make much difference in the cure fraction, which confirms what can be observed in Kaplan Meier's estimates in Figure 1.

We observed that those men who underwent chemotherapy have the lowest survival. However, this result needs to be analyzed with caution. Individuals who undergo chemotherapy generally have the disease in a more severe phase.

6. Concluding remarks

While the number of studies for female breast cancer is high, data about this carcinoma in the male population is still very limited [45]. Here we provide more information on this disease in the Brazilian population of men, where we discuss the association of clinical stage, age, and treatments on the survival times of men enduring this type of cancer. Motivated by a demand from oncologists on the project team, the interest was to assess the impact of prognostic factors on different quantiles of survival time when the cure rate is present. Our approach considered the use of a parametric quantile regression model, where we can observe how these variables affect directly the interest quantiles of the survival times. By selecting a defective distribution in the form of the Generalized Gompertz distribution we are able to explain simultaneously both quantiles of the survival times and the cure fraction, given the covariates, without adding new parameters to the probability distribution. The proposed model and estimation process are available in R package ggcrqr (https://github.com/brsantos/ggcrqr).

We have shown that our estimation approach based on MCMC draws effectively reach the correct values in four simulation scenarios, while also presenting its main advantages in the application. Based on the extensive simulation study and by considering the overall results (regardless the quantile and the percentage of censoring data), we have observed that the proposed model presents a great result for sample sizes greater than n = 800. Additionally, in one of our simulation studies, we compared our approach with a standard mixture model concerning the cure fraction estimation, as per the suggestion of one of the referees, and our proposed model showed a satisfactory performance.

Here we discussed how the quantile parameterization specifies a more complete picture of the conditional distribution of survival times, as we can study how each variable influences the different quantiles of this distribution. The parametric quantile regression approach also avoids the issue of quantile crossing, which is a common problem discussed in the literature for quantile regression models.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplemental Material

Supplemental data for this article can be accessed online at http://dx.doi.org/10.1080/02664763.2024.2428272.