# Modelling cure indicators using the curesurv R package: a tutorial using data from the French cancer registries

Judith Breaud, Olayidé Boussari, Valérie Jooste, Laura Botta, Juste Goungounga

## ► To cite this version:

Judith Breaud, Olayidé Boussari, Valérie Jooste, Laura Botta, Juste Goungounga. Modelling cure indicators using the curesurv R package: a tutorial using data from the French cancer registries. Computer Methods and Programs in Biomedicine Update, 2025, 8, pp.100222. 10.1016/j.cmpbup.2025.100222. hal-05328581

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Submitted on 23 Oct 2025

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# Modelling cure indicators using the curesurv R package: a tutorial using data from the French cancer registries

Judith Breaud<sup>a</sup>, Olayidé Boussari<sup>b,c</sup>, Valérie Jooste<sup>c,d</sup>, Laura Botta<sup>e</sup>, Juste Goungounga<sup>a,*</sup> [1]

## ARTICLE INFO

**Keywords:**

Cancer

Cure models, parametric survival models

Relative survival

Reproducibility

## ABSTRACT

Cure models are essential in survival analyses because they take into account the fraction of patients who achieve long-term survival and will not experience disease-related death. In a relative survival framework, these models estimate the excess hazard (i.e. the additional death risk attributable to the disease), while assuming that some patients will not experience such excess risk. In 2002, Phillips and colleagues introduced a class of cure models where background mortality is modelled as a scaled version of the general population's mortality. This tutorial presents the *curesurv* R package for parametric cure modelling in the relative survival setting, with and without background mortality rescaling. The package supports both the Weibull mixture cure model and the non-mixture cure model proposed by Boussari and colleagues in 2018, offering flexible tools for assessing long-term survival. The *curesurv* R package application is illustrated using testicular, breast, and prostate cancer datasets from the French FRANCIM cancer registries. This tutorial provides a practical guide to researchers in epidemiology and health economics for implementing cure fraction models and interpreting long-term survival estimates.

## 1. Introduction

Survival analysis is a statistical framework used in medical studies to model the time until the event of interest occurs, such as death or disease onset/cancer relapse. Traditional survival analysis assumes that all individuals will eventually experience the event, even if they are censored at the study end. However, this assumption is not always valid. In some cases, a subset of individuals may achieve long-term survival and will remain disease-free for the rest of their lives. For example, some patients with childhood cancer may be effectively cured and never experience cancer-related mortality [1]. Cure models extend survival analysis by taking into account this phenomenon through a cure fraction that represents the proportion of individuals who will never experience the event [2]. Statistical cure is defined as the point where the survival function reaches a plateau, indicating that the excess hazard attributable to the disease has decreased to zero.

A major challenge in cancer survival analysis is distinguishing between mortality due to the cancer of interest and mortality due to other

causes. When a patient dies, determining whether the cause was cancer or an unrelated condition is often difficult, especially when the cause-of-death data are unavailable or unreliable. This issue is particularly relevant in the competing risks framework where individuals are exposed to multiple potential causes of death, and experiencing one event prevents the occurrence of others. To address this challenge, survival analysis offers two main approaches. First, the cause-specific survival approach models the probability of survival from a cancer in the presence of competing risks [3]. However, this method requires accurate cause-of-death information to differentiate between cancer-related and non-cancer-related death, which could be subject to misclassification. To overcome this problem, the concept of "net survival" was introduced as the survival that a patient would experience if the cancer of interest was the only possible cause of death [4]. This second approach does not need information on the cause of death, which is not always available or reliable.

Cure models can be implemented in the relative survival setting to estimate the cure fraction while considering the fact that not all patients

<footnote><sup>a</sup> Université de Rennes, EHESP, CNRS, INSERM, Arènes-UMR 6051, RSMS-U1309, 35000 Rennes, France</footnote>

<footnote><sup>b</sup> Fédération Francophone de Cancérologie Digestive (FFCD), Département de Méthodologie, F-21000, Dijon, France</footnote>

<footnote><sup>c</sup> UMR 1231, EPICAD team, INSERM, Université Bourgogne-Franche-Comté, Dijon, France</footnote>

<footnote><sup>d</sup> Registre Bourguignon des Cancers Digestifs, Dijon-Bourgogne University Hospital, F-21000, Dijon, France</footnote>

<footnote><sup>e</sup> Evaluative Epidemiology Unit, Department of Epidemiology and Data Science, Fondazione IRCCS "Istituto nazionale dei Tumori", Via Venezian 1, 20133, Milan, Italy</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: juste.goungounga@ehesp.fr (J. Goungounga).</footnote>
will die from cancer. In this setting, statistical cure is reached when the net survival reaches plateau, indicating that the observed risk of surviving patients is equivalent to that of the general population [5]. Amico et al. published a comprehensive review of the existing cure models [6].

Several R packages have been developed for cure modelling in both overall and relative survival frameworks. Within the overall survival framework, the `ncpure` package [7] allows the non-parametric inference of mixture cure models (MCM). The `smcure` package [8] fits semi-parametric proportional hazards and accelerated failure time MCM. The `flexsurvcure` package [9] provides fully parametric implementations and accommodates both mixture and non-mixture cure models (NMCM). The `nltm` package extends these approaches by fitting cure or survival models within the non-linear transformation model framework [10]. More recent developments include the `penPHcure` package [11], which extends semiparametric proportional hazards cure models to time-varying covariates with penalization for variable selection, and the `geecure` package [12], which implements marginal parametric and semiparametric MCM for clustered survival data.

Within the relative survival framework, the `cuRe` package [13] fits parametric MCM and NMCM, including flexible spline-based models, while the `rstpm2` package [13] implements spline-based NMCM. Beyond these dedicated packages, the Comprehensive R Archive Network (CRAN) Survival Analysis Task View [14] provides an extensive list of R packages for survival analysis, including cure modelling approaches. A comparative table of the selected packages is presented in the Appendix.

This paper introduces the `curesurv` R package, that offers a flexible and user-friendly framework for estimating parametric cure models in both relative survival and overall survival settings. The package supports MCM and NMCM, extends them to incorporate increased other-cause mortality among cancer patients [15,16] and implements the Time to Null Excess Hazard (TNEH) model [17] for the refined estimation of cure probabilities under different mortality assumptions.

Despite their importance, cure models remain underused in cancer epidemiology. While several R packages already provide cure model implementations, `curesurv` extends these approaches by considering that cancer patients be at a different risk of dying from other causes than the general population. This feature enables more accurate net survival estimates and better reflects real-world mortality patterns. In addition, `curesurv` is the only R package that implements a cure model that allows estimating the point at which excess hazard becomes negligible, indicating when the cohort of patients studied reaches a mortality level comparable to that of the general population. By making these models more accessible, this paper aims to promote the broader use of cure models in medical research and epidemiology. The `curesurv` package, now available on CRAN, facilitates the estimation of cure indicators and supports evidence-based decision-making in cancer survival analysis.

Cancer is unique among chronic diseases because long-term survivors remain at risk of late relapse, even after many years following treatment. This has implications beyond medical research; for example, financial institutions often consider former cancer patients as high-risk when applying for loans or insurances, even after remission. In France, national estimates of cancer survival are regularly produced from the French network of cancer registries (FRANCIM) to monitor progress and evaluate cancer policies. Indicators obtained from these data play a crucial role in guiding public health strategies, with recent legislation changes to improve patient rights: in France, the 2022 Lemoine's Law granted to patients with cancer or hepatitis C the right to be "forgotten" by insurers after five years of remission [18]. In this context, accurate cure models provide essential indicators to inform policymakers and assess the policies' alignment with real-world survival patterns.

The remainder of this paper is organized as follows. Section 2 presents the statistical models. Section 3 describes the package functions and implementation. Section 4 demonstrates the use of the `curesurv` package with cancer survival data from the French network of cancer registries (FRANCIM) [19]. Finally, Section 5 discusses the results and highlights the benefits this package.

## 2. Models and computational methods

### 2.1. Net survival

In cancer survival analyses, patient survival is assessed after diagnosis by following them until they experience the event of interest (i.e., death) or are censored. Event times and censoring times are considered independent. Overall survival measures the time until death from any cause and does not differentiate between mortality due to the cancer and mortality due to other causes. In population-based studies, cancer-specific survival among cancer patients is often the indicator of interest. It is estimated using net survival, defined as the probability that patients survive their cancer in the absence of other causes of death, i.e. in "the hypothetical situation where the disease under study would be the only possible cause of death" [20,21]. The excess mortality methodology allows estimating net survival without knowledge of the cause of death by using life tables from the general population to which the patients belong and decomposing the observed mortality hazard at time $t_i \in \mathbb{R}^+$ of a patient $i$ with characteristics represented by the covariates $z_i \in \mathbb{R}^k$, into two components: i) the background mortality hazard, also known as the mortality due to other causes at time $t_i$ and estimated from the general population life tables $\lambda_p(t_i + a_i|z_{Di})$ where $a_i \in \mathbb{R}^+$ is the age of the patient at diagnosis and $z_{Di}|Cz_i$ is a subset of the covariates; and ii) the excess hazard at time $t_i$, $\lambda_e(t_i|z_i)$, that reflects the additional mortality risk specifically attributable to that cancer. The observed mortality hazard function at time $t_i$ can be expressed as:

$$ \lambda(t_i|z_i) = \lambda_p(t_i + a_i|z_{Di}) + \lambda_e(t_i|z_i) \quad (1) $$

The net survival $S_e$ of patient $i$ is defined in relation to the excess hazard:

$$ S_e(t_i|z_i) = \exp \left( - \int_0^{t_i} \lambda_e(u|z_i) du \right) \quad (2) $$

The observed survival can be expressed as a product of the population survival $S_p$ obtained from life tables and the net survival $S_c$:

$$ S(t_i|z_i) = S_p(t_i + a_i|z_{Di}) \times S_c(t_i|z_i) \quad (3) $$

In parametric models, the excess hazard and net survival depend on a set of parameters $\theta$. For a patient $i$ at time $t_i$ and with covariates $z_i$ they are written $\lambda_e(t_i|z_i; \theta)$ and $S_e(t_i|z_i; \theta)$ respectively.

To estimate cure models within the relative survival framework, `curesurv` uses maximum likelihood estimation methods. For a parametric net survival model, given a cohort of N patients and $d_i$ indicating whether patient $i$ died ($d_i = 1$) or was censored ($d_i = 0$), the log-likelihood function is:

$$ \text{LogLik}(\theta) = \sum_{i=1}^{N} \left( \log\{S_p(t_i + a_i|z_{Di})\} + \log\{S_e(t_i|z_i; \theta)\} + d_i \log\{\lambda_p(t_i + a_i|z_{Di}) + \lambda_e(t_i|z_i; \theta)\} \right) \quad (4) $$

$\log(S_p(t_i + a_i|z_i))$ does not depend on the parameters $\theta$ and thus may be omitted from this equation in the optimization process.

### 2.2. Cure models and their key indicators

In the case of cancer, it is very difficult, if not impossible, to declare a patient medically cured. There is no way to ascertain whether there are still cancer cells. After treatment, the patient goes into remission, but there is a risk of relapse/recurrence. This explains the need of a statistical method to estimate cure indicators [22]. In the relative survival framework, statistical cure is defined as the time when the net survival reaches a (non-zero) plateau. Patients still alive at that time are
considered statistically cured. There are five main cure-model specific indicators. These indicators are defined at a time $t \in \mathbb{R}^+$ for groups of patients who share the same characteristics $z \in \mathbb{R}^k$:

* The excess hazard $\lambda_e(t|z; \theta)$ of the group of patients.

* The net survival $S_e(t|z; \theta)$.

* The cure fraction, defined by $\pi(z; \theta) = \lim_{t \to +\infty} S_e(t|z; \theta)$.

* The conditional probability for a patient to be cured at time t, knowing that this patient was alive at time t, defined by $P(t|z; \theta) = \frac{\pi(z; \theta)}{S_e(t|z; \theta)}$.

* The time to cure. Different definitions of the time to cure were discussed in Boussari et al. [23]. This study will focus on $t_e$ as the time from when a patient has a probability of being cured higher that 1 - $\epsilon$, conditionally on being alive. That is $t_e(z; \theta) \in \mathbb{R}^+$ | $\forall t \ge t_e(z; \theta)$, $P(t|z; \theta) \ge 1 - \epsilon$.

## 2.3. Two classes of cure models

Two main cure model types are available to estimate these indicators: MCM and NMCM. The NMCM are also known in literature as promotion time cure models and bounded cumulative hazard models [24].

### 2.3.1. Mixture cure model

In MCM, the assumption is that the group of patients is divided into two fractions [25]:

* $\pi$ is the proportion of cured patients in the study (i.e. the cure fraction). Since we assume that they will never die from the disease under study, their net survival is 1.

* $(1 - \pi)$ is the proportion of uncured patients who will eventually die from the disease under study. Their net survival is expressed by $S_u$.

In these models, the net survival is a mixture of the net survivals of the cured and uncured patients:

$$S_e(t|z; \theta) = \pi(z_e; \theta) \times 1 + (1 - \pi(z_e; \theta))S_u(t|z_u; \theta) \quad (5)$$

And the excess hazard is:

$$\lambda_e(t|z; \theta) = \frac{(1 - \pi(z_e; \theta))f_u(t|z_u; \theta)}{\pi(z_e; \theta) + (1 - \pi(z_e; \theta))S_u(t|z_u; \theta)} \quad (6)$$

where $z_e \subset z$ is the subset of covariates acting on cure proportion $\pi$, $z_u \subset z$ is the subset of covariates acting on survival of the uncured $S_u$, and where $f_u(t|z_u; \theta)$ denotes the probability density function for $S_u(t|z_u; \theta)$.

The NMCM-specific log-likelihood is obtained by plugging Eqs. (5) and (6) into Eq. (4):

In *curesurv*, the net survival of the uncured population is parametrized as a Weibull regression. With $t \in \mathbb{R}^+$, $z_u \in \mathbb{R}^{q_u}$, $\lambda \in \mathbb{R}$, $\gamma \in \mathbb{R}$ and $\delta \in \mathbb{R}^{q_u}$:

$$S_u(t|z; \theta) = \exp(-e^{\lambda} \times (t)^{\gamma} \times e^{\delta z}) \quad (9)$$

where $\theta = (\beta_0, \beta_k, \lambda, \gamma, \delta)$. These parameters are estimated using the maximum likelihood method detailed in Eq. (7).

### 2.3.2. Non mixture cure model

In NMCM, long-term survivors are accounted for by the fact that the net survival reaches a plateau. The net survival is then modelled as:

$$S_e(t|z; \theta) = \pi(z; \theta)^{F(t|z; \theta)} \quad (10)$$

where $F(t)$ a distribution function. With $f(t|z; \theta) = F(t|z; \theta)$, the NMCM-specific log-likelihood can be written as:

$$\text{LogLik}(\theta) = \sum_{i=1}^{N} \left( \log\{S_p(t_i + a_i|z_{Di})\} + \log\{\pi(z_i; \theta)^{F(t_i|z_i; \theta)}\} \right) + d_i \log\{\lambda_p(t_i + a_i|z_{Di}) - f(t_i|z_i; \theta)\log(\pi(z_i; \theta))\} \quad (11)$$

The *curesurv* package implements the TNEH model [17]. This NMCM focuses on estimating the statistical time to cure as the time at which excess hazard becomes null. In this model, the excess hazard is based on a beta probability function. With $\alpha(z_e; \gamma) > 0$ a shape parameter depending on the subset of covariates $z_e \subset z$ through parameters $\gamma$, $\beta(\beta_0) > 1$ another shape parameter, and $\tau(z_e; \eta) > 0$ the TNEH depending on the subset of covariates $z_e$ through parameters $\eta$; and with $\theta = (\eta, \gamma, \beta_0) \in \mathbb{R}^{q_e} \times \mathbb{R}^{q_u} \times \mathbb{R}$, the excess hazard is parametrized as:

$$\lambda_e(t|z; \theta) = \left\{ \frac{t}{\tau(z_e; \eta)} \right\}^{\alpha(z_e; \gamma)-1} \left\{ 1 - \frac{t}{\tau(z_e; \eta)} \right\}^{\beta(\beta_0)-1} \quad (12)$$

The net survival is obtained by plugging Eq. (12) into (2):

$$S_e(t|z; \theta) = (\exp(-\tau(z_e; \eta)B(\alpha(z_e; \gamma); \beta(\beta_0)))) F_B\left(\frac{t}{\tau(z_e; \eta)}\alpha(z_e; \gamma); \beta(\beta_0)\right) \quad (13)$$

with $B(\alpha; \beta) = \int_0^1 u^{\alpha-1}(1-u)^{\beta-1} du$ and $F_B(x; \alpha; \beta) = \frac{\int_0^\alpha u^{\alpha-1}(1-u)^{\beta-1} du}{B(\alpha; \beta)}$.

The cure proportion corresponds to the net survival when $t \ge \tau$. We can identify that the net survival in Eq. (13) has the shape of a NMCM net survival as in Eq. (10), and that the cure fraction is:

$$\pi(z; \theta) = \exp\{-\tau(z_e; \eta)B(\alpha(z_e; \gamma); \beta(\beta_0))\} \quad (14)$$

The parameters $\theta$ are estimated using the maximum likelihood method on the log-likelihood detailed in Eq. (11).

$$\text{LogLik}(\theta) = \sum_{i=1}^{N} \left( \log\{S_p(t_i + a_i|z_{Di})\} + \log\{\pi(z_{ri}; \theta) + [1 - \pi(z_{ri}; \theta)]S_u(t_i|z_{ui}; \theta)\} \right) + d_i \log\{\lambda_p(t_i + a_i|z_{Di}) + \frac{[1 - \pi(z_{ri}; \theta)]f_u(t_i|z_{ui}; \theta)}{\pi(z_{ri}; \theta) + [1 - \pi(z_{ri}; \theta)]S_u(t_i|z_{ui}; \theta)}\} \quad (7)$$

## 2.4. Cure models with correction for mortality from other causes

It is usual, as detailed in Section 2.1, to consider that the general population's mortality hazard is a good proxy for the other-cause mortality hazard of patients with cancer. However, there is increasing evidence that cancer patients may have a different and often higher risk of

In the *curesurv* package, the cure fraction is parametrized with the logistic regression link. With $\beta_0 \in \mathbb{R}$, $z_e \in \mathbb{R}^{q_e}$ and $\beta_q \in \mathbb{R}^{q_u}$:

$$\pi(z; \theta) = \frac{1}{1 + \exp(-\beta_0 - \beta_q z_e)} \quad (8)$$
dying from other diseases compared to the general population [26-30]. If the general population's life tables are not suitable, the other-causes mortality hazard could be wrongly estimated, and the comparability assumption wouldn't be valid [31,32]. To take this into account, the "other causes" hazard $\lambda_+$ of the patients with cancer is assumed to be a multiple of the general population's mortality hazard [15]:

$$ \lambda_+(t) = \alpha\lambda_p(t) \qquad (15) $$

Therefore, the patients "other causes" survival is:

$$ S_+(t) = S_p(t)^\alpha \qquad (16) $$

and the parameter $\alpha \in \mathbb{R}^+$ is added to the list of parameters to be estimated. The log-likelihood for a cure model with correction of other causes mortality is slightly modified. Adapting Eq. (4) with (15) and (16) we get:

$$ \text{LogLik}(\theta) = \sum_{i=1}^{N} \left( \begin{array}{l} \alpha \log\{S_p(t_i + a_i|z_{Di})\} + \log\{S_e(t_i|z_i; \theta)\} \\ +d_i \log\{\alpha\lambda_p(t_i + a_i|z_{Di}) + \lambda_e(t_i|z_i; \theta)\} \end{array} \right) \qquad (17) $$

Unlike in Eq. (4) where the comparability assumption is valid and $\log(S_p(t_i+a_i|z_{Di}))$ is constant, the $\alpha$ parameter must be estimated, and the log-likelihood cannot be simplified by subtracting a constant.

# 3. Curesurv package functionalities

## 3.1. Function for model fitting

At the core of the package is the `curesurv` function to fit the models described in Section 2.

curesurv(formula, data, pophaz, cumpophaz, pophaz alpha, model, dist, link_tau, optim_func, optimizer, method_opt, ...)

The main arguments of the `curesurv` function are as follows:

* **formula**: defines the structure of the survival model using a `Surv` object, the survival time (in years), the event status, and covariates.

* **data**: specifies the data set (a `data.frame`) that contains all the variables used in the `formula` as well as other information needed for the relative survival analysis.

* **pophaz**: provides the name of the column in `data` that contains general population's mortality risk, obtained from life tables.

* **cumpophaz**: provides the name of the column in `data` that contains general population's mortality risk accumulated from study start (cumulative population hazard), obtained from life tables. This is necessary for models with corrections of other causes mortality. It is not necessary to fill this argument for models without correction, except if you intend to compare them with models with correction using AIC or likelihood ratio tests.

* **pophaz.alpha**: determines whether the model includes or not other cause mortality correction (see Section 2.4). By default, `pophaz.alpha=FALSE`.

* **model**: defines the family of model, either `model="mixture"` to fit a MCM or `model="nmixture"` to fit a NMCM

* **dist**: defines for MCM the form of uncured patients' survival function ("weib" or "eweib" for Weibull or exponentiated Weibull), and for NMCM the name of the specific model ("tneh").

* **link_tau**: defines for the TNEH model whether effect of covariates on the shape parameters is "linear" or "loglinear".

* **optim_func**: specifies which function must be used for optimization. It takes for value "optim" to use the `stats` package `optim` function and "bbmle" to use the `bbmle` package `mle2` function.

* **optimizer**: corresponds to the `bbmle` optimizer argument and is only used when `optim_func="bbmle"`

* **method_opt**: specifies the optimisation method when `optim_func="optim"`.

### 3.1.1. Mixture cure model

The following example will show how to fit a Weibull MCM with correction of other causes mortality on the simulated data set `dataweib` available in `curesurv`. For the Weibull cure model, covariates can influence two parameters: the cure fraction and the survival of uncured patients. The left side of the formula is separated in two by "|" to consider the fact that different covariates may affect the cure fraction and the survival of the uncured. The variables that affect the uncured patients' survival are on the left side of |, and the variables that affect the cure fraction's survival are on the right side of |. In this example, we suppose that the age "age_cr" affects both the cure fraction and the survival of the uncured patients, but that the cancer stage only affects the uncured patients' survival, and the gender "sexe" only affects the cure fraction.

curesurv(formula=Surv(timeobs, event) ~ age_cr + stage | age_cr+sexe, data=dataweib, pophaz="ehazard", cumpophaz="cumehazard", pophaz.alpha=TRUE, model="mixture", dist="weib", ...)

### 3.1.2. Non mixture cure model

The following example will show how to fit a TNEH NMCM with no correction for other causes mortality on the simulated dataset `dataweib` available in `curesurv`. Like in the Weibull mixture model, covariates can influence the TNEH $\tau$ or the shape parameter $\alpha$ (or both). To specify the parameter(s) influenced by a covariate, the indicators `z_tau` and `z_alpha` are used. In this example, we suppose that the age affects both the TNEH and the shape parameter, but that the stage affects only the TNEH and that the gender affects only the shape parameter. We also suppose that the covariates have a linear effect on the TNEH and shape parameter.

curesurv(formula=Surv(timeobs, event) ~ z_tau(age_cr+stage)+ z_alpha(age_cr+sexe), data=dataweib, pophaz="ehazard", pophaz.alpha=FALSE, model="nmixture", dist="tneh", link_tau="linear", ...)

## 3.2. Predictions

The package contains a build-in prediction method.

predict(object, newdata, level = 0.975, epsilon = 0.05, xmax = 10^9, ...)

The main arguments of the predict function are as follows:

* **object**: specifies the model for prediction (previously obtained with the `curesurv` function).

* **newdata**: specifies the dataset for prediction which contains patient (s) characteristics and follow-up time (e.g. 1, 2, 3 years).

* **level**: defines the $1 - \alpha/2$ order quantile of a normal distribution for the confidence intervals $1 - \alpha$. By default, `level=0.975` for a 95 % CI.

* **epsilon**: is the value of $\epsilon$ fixed to estimate the time to cure $t_e$ as defined in Section 2.2.

* **xmax**: is the maximum time stamps at which the time to cure $t_e$ is estimated.

The output of a prediction is a `data.frame` object of class `pre-dCuresurv` that contains:

* the prediction characteristics (time stamps, values of the covariates).

* the indicators described in Section 2.2: excess hazard, net survival, cure fraction, conditional probability of being cured, time to cure $t_e$.

* the confidence intervals (of confidence level $1 - \alpha$) of these indicators obtained using the delta method (see annexe A.1 for details on the delta method).
### 3.3. Plotting predictions

The custom plot method allows plotting different indicators. The method is intended for predictions made for a single individual over time. To plot predictions that don't fit this situation, such as with varying characteristics of the patient at fixed times, it is possible to use common R plotting packages (e.g. ggplot2) on the `predCuresurv` data frame.

`plot(x, fun, conf.int, conf.type, ...)`

The plot arguments are as follows:

* `x` is a `predCuresurv` object.

* `fun`: specifies the indicator to be plotted. It takes for value `"haz"` for excess hazard, `"surv"` for net survival, `"pt_cure"` for the conditional probability of being cured, `"cumhaz"` for the cumulative excess hazard, `"logcumhaz"` for logarithm of the cumulative excess hazard, and `"all"` for the excess hazard, net survival, and conditional probability of being cured.

* `conf.int`: specifies whether the confidence intervals of the plotted indicator are plotted or not.

* `conf.type`: decides the confidence interval that should be plotted. The possibilities are `"plain"`, `"log"` and `"log-log"`. All confidence intervals are calculated based on the delta method. The first option uses the standard confidence interval curve. The log option uses confidence intervals based on log(indicator), and the log-log option the intervals on log(-log(indicator)). For more details about the delta method, see supplementary file A.1.

### 3.4. Other functionalities

The AIC function is implemented in `curesurv` to compare the models fitted by the package using the Akaike Information Criterion (AIC).

The `anova` function is implemented in `curesurv` to use the Likelihood Ratio Test (LRT) on two nested models.

## 4. Application using data from the FRANCIM network of cancer registries

### 4.1. Testicular, breast, and prostate cancer data

We tested the `curesurv` package methods using data from the French FRANCIM network of cancer registries. We included patients who received a diagnosis of testicular, breast or prostate cancer between 1989 and 2015 (Table 1). We obtained the population life tables, stratified by sex, year of birth, age and department, from the French National Institute of Statistics and Economic Studies (INSEE).

In the FRANCIM database, 4521 patients had testicular cancer. Their age at diagnosis ranged between 15.18 and 95.41 years. For this study, we used the function `nessie` of the package `relsurv` to estimate the net sample size at different times, to focus on the ages at diagnosis for

<table><caption>**Table 1**
Number of patients with the indicated cancer type by French department.</caption><thead><tr><th rowspan="2">Department</th><th colspan="2">Testicular</th><th colspan="2">Breast</th><th colspan="2">Prostate</th></tr><tr><th>Count</th><th>%</th><th>Count</th><th>%</th><th>Count</th><th>%</th></tr></thead><tbody><tr><td>Calvados</td><td>483</td><td>10.68 %</td><td>12,897</td><td>11.13 %</td><td>11,965</td><td>12.92 %</td></tr><tr><td>Côtes-d'Or</td><td></td><td></td><td>9579</td><td>8.26 %</td><td></td><td></td></tr><tr><td>Doubs</td><td>483</td><td>10.68 %</td><td>8799</td><td>7.59 %</td><td>10,148</td><td>10.96 %</td></tr><tr><td>Isère</td><td>915</td><td>20.24 %</td><td>21,145</td><td>18.24 %</td><td>20,103</td><td>21.71 %</td></tr><tr><td>Loire</td><td></td><td></td><td>22,377</td><td>19.30 %</td><td></td><td></td></tr><tr><td>Atlantique</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Bas-Rhin</td><td>1201</td><td>26.56 %</td><td>19,157</td><td>16.53 %</td><td>18,049</td><td>19.94 %</td></tr><tr><td>Haut-Rhin</td><td>809</td><td>17.89 %</td><td>13,858</td><td>11.95 %</td><td>13,834</td><td>14.94 %</td></tr><tr><td>Vendée</td><td>401</td><td>8.87 %</td><td>10,850</td><td>9.36 %</td><td>9477</td><td>10.23 %</td></tr><tr><td>Tarn</td><td>229</td><td>5.06 %</td><td>7258</td><td>6.26 %</td><td>9022</td><td>9.74 %</td></tr></tbody></table>

<table><caption>**Table 2**
Estimated net sample size for patients with testicular cancer.</caption><thead><tr><th>Age class</th><th>t = 0</th><th>t = 1</th><th>t = 3</th><th>t = 5</th><th>t = 10</th><th>t = 15</th></tr></thead><tbody><tr><td>(15,20)</td><td>133</td><td>132.90</td><td>132.68</td><td>132.45</td><td>131.92</td><td>131.38</td></tr><tr><td>(20,25)</td><td>484</td><td>481.52</td><td>480.61</td><td>479.75</td><td>477.66</td><td>475.22</td></tr><tr><td>(25,30)</td><td>784</td><td>783.23</td><td>781.73</td><td>780.20</td><td>775.96</td><td>770.34</td></tr><tr><td>(30,35)</td><td>858</td><td>857.05</td><td>855.04</td><td>852.86</td><td>846.22</td><td>837.09</td></tr><tr><td>(35,40)</td><td>806</td><td>804.77</td><td>802.08</td><td>799.05</td><td>789.61</td><td>776.24</td></tr><tr><td>(40,45)</td><td>549</td><td>547.72</td><td>544.90</td><td>541.69</td><td>531.55</td><td>517.18</td></tr><tr><td>(45,50)</td><td>342</td><td>340.75</td><td>338.00</td><td>334.89</td><td>325.16</td><td>311.68</td></tr><tr><td>(50,55)</td><td>221</td><td>219.74</td><td>217</td><td>213.92</td><td>204.46</td><td>192.29</td></tr><tr><td>(55,60)</td><td>139</td><td>137.73</td><td>135.00</td><td>132.00</td><td>123.23</td><td>112.36</td></tr><tr><td>(60,65)</td><td>72</td><td>71.01</td><td>68.89</td><td>66.61</td><td>60.01</td><td>52.14</td></tr><tr><td>(65,70)</td><td>64</td><td>62.75</td><td>60.10</td><td>57.25</td><td>49.16</td><td>39.08</td></tr><tr><td>(70,75)</td><td>28</td><td>27.15</td><td>25.36</td><td>23.46</td><td>18.11</td><td>11.77</td></tr><tr><td>(75,80)</td><td>13</td><td>12.41</td><td>11.19</td><td>9.90</td><td>6.44</td><td>3.13</td></tr><tr><td>(80,85)</td><td>20</td><td>18.36</td><td>15.10</td><td>11.96</td><td>5.35</td><td>1.48</td></tr><tr><td>(85,90)</td><td>6</td><td>5.25</td><td>3.86</td><td>2.67</td><td>0.75</td><td>0.08</td></tr><tr><td>(90,95)</td><td>3</td><td>2.39</td><td>1.44</td><td>0.8</td><td>0.12</td><td>0.01</td></tr><tr><td>(95,100)</td><td>1</td><td>0.75</td><td>0.38</td><td>0.17</td><td>0.02</td><td>0.00</td></tr></tbody></table>

which there were enough patients to fit the models even 15 years after the diagnosis (Table 2). This reduced the sample to 4345 patients who were between 20 and 75 years old at the time of diagnosis.

In the FRANCIM database, 125,920 patients (all women) had breast cancer and their age at cancer diagnosis ranged from 16.2 to 111.26 years (Table 3). For this study, we selected the 95,093 patients who were between 40 and 75 years old at the time of diagnosis.

The FRANCIM database recorded 92,598 patients with prostate cancer (Table 4) whose age at diagnosis ranged between 16.84 and 102.18 years. For this study, we selected 78,531 patients who were between 40 and 80 years old at the time of diagnosis.

### 4.2. Regression modelling strategies

First, we used the breast cancer data and a simple Weibull MCM with an effect of the patient's age at diagnosis as a class in both the cure fraction and the survival of the uncured patients. This enabled a comparison between `curesurv` and the `cuRe` package, which also fits this model, and the non-parametric Pohar-Perme estimator.

Then, we implemented for each cancer type a more complex Weibull MCM, with covariates selected via the model with the best AIC. The possible covariates were age at diagnosis and year of diagnosis. For modelling purpose, continuous covariates were centred and reduced. The covariates effect on the cure fraction and on survival of the uncured fraction are supposed to be either linear or modelled as a quadratic spline. We kept the best model and compared it with the model with the same covariates, but with an added correction for other-cause mortality rates using the AIC and the LRT. Comparing a model with and without the parameter for the correction of other-cause mortality, as described by Phillips et al. [15], corresponds to case 6 in Self and Liang [33]. We considered the LRT statistic asymptotic distribution as a 50:50 mixture of a $\chi_1^2$ and a $\chi_2^2$ distribution because the null hypothesis $H_0$ corresponds to $\alpha = 1$ and $H_1$ to $\alpha$ different from 1.

Lastly, we implemented the NMCM proposed by Boussari et al. [17], with covariates selected using the model with the best AIC. The possible covariates were the same as for the Weibull MCM and affected either the TNEH $\tau$ or the shape parameter $\alpha$, or both.

### 4.3. Comparison of `curesurv` and `cuRe` packages using breast cancer data

Fig. 1 shows the net survival predicted over 15 years of follow-up by a Weibull MCM fitted with `cuRe` and with `curesurv`, and by the non-parametrical Pohar-Perme estimates. We obtained the same estimates with the two packages when `method_opt`="Nelder-Mead", but the estimates obtained with `method_opt`="L-BFGS-B" had a higher log-
<table><caption>**Table 3**
Estimated net sample size for patients with breast cancer.</caption><thead><tr><th>Age class</th><th>t = 0</th><th>t = 1</th><th>t = 3</th><th>t = 5</th><th>t = 10</th><th>t = 15</th></tr></thead><tbody><tr><td>(15,20]</td><td>6</td><td>6.00</td><td>6.00</td><td>6.00</td><td>6.00</td><td>6.00</td></tr><tr><td>(20,25]</td><td>79</td><td>78.98</td><td>78.93</td><td>78.88</td><td>78.74</td><td>78.56</td></tr><tr><td>(25,30]</td><td>575</td><td>574.78</td><td>574.32</td><td>573.83</td><td>572.32</td><td>570.16</td></tr><tr><td>(30,35]</td><td>1932</td><td>1930.99</td><td>1928.80</td><td>1926.32</td><td>1918.38</td><td>1906.60</td></tr><tr><td>(35,40]</td><td>4526</td><td>4522.45</td><td>4514.57</td><td>4505.48</td><td>4475.79</td><td>4432.12</td></tr><tr><td>(40,45]</td><td>9218</td><td>9206.76</td><td>9181.69</td><td>9152.74</td><td>9059.55</td><td>8929.47</td></tr><tr><td>(45,50]</td><td>13,692</td><td>13,666.52</td><td>13,610.14</td><td>13,575.96</td><td>13,346.63</td><td>13,078.04</td></tr><tr><td>(50,55]</td><td>14,771</td><td>14,731.16</td><td>14,644.23</td><td>14,546.90</td><td>14,249.47</td><td>13,847.52</td></tr><tr><td>(55,60]</td><td>14,636</td><td>14,579.53</td><td>14,456.96</td><td>14,319.90</td><td>13,896.06</td><td>13,305.85</td></tr><tr><td>(60,65]</td><td>15,305</td><td>15,220.95</td><td>150,370.6</td><td>14,828.61</td><td>14,165.96</td><td>13,208.43</td></tr><tr><td>(65,70]</td><td>14,815</td><td>14,960.27</td><td>14,413.74</td><td>14,094.96</td><td>13,050.94</td><td>11,424.24</td></tr><tr><td>(70,75]</td><td>12,656</td><td>12,480.50</td><td>12,086.12</td><td>11,623.02</td><td>10,049.28</td><td>7617.89</td></tr><tr><td>(75,80]</td><td>9444</td><td>9200.85</td><td>8651.49</td><td>8006.39</td><td>5919.67</td><td>3310.43</td></tr><tr><td>(80,85]</td><td>7551</td><td>7174.26</td><td>6348.04</td><td>5433.74</td><td>2961.07</td><td>975.17</td></tr><tr><td>(85,90]</td><td>4589</td><td>4155.74</td><td>3283.42</td><td>2440.96</td><td>801.54</td><td>146.64</td></tr><tr><td>(90,95]</td><td>1742</td><td>1451.47</td><td>946.82</td><td>560.24</td><td>100.11</td><td>16.5</td></tr><tr><td>(95,100]</td><td>350</td><td>256.84</td><td>127.65</td><td>58.98</td><td>8.77</td><td>1.41</td></tr><tr><td>(100,105]</td><td>32</td><td>21.53</td><td>9.86</td><td>4.59</td><td>0.71</td><td>0.12</td></tr><tr><td>(110,115]</td><td>1</td><td>0.70</td><td>0.30</td><td>0.10</td><td>0.00</td><td>0.00</td></tr></tbody></table>

<table><caption>**Table 4**
Estimated net sample size for patients with prostate cancer.</caption><thead><tr><th>Age class</th><th>t = 0</th><th>t = 1</th><th>t = 3</th><th>t = 5</th><th>t = 10</th><th>t = 15</th></tr></thead><tbody><tr><td>(15,20]</td><td>2</td><td>2.00</td><td>2.00</td><td>2.00</td><td>2.00</td><td>2.00</td></tr><tr><td>(20,25]</td><td>1</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.99</td><td>0.99</td></tr><tr><td>(30,35]</td><td>2</td><td>2.00</td><td>1.99</td><td>1.99</td><td>1.97</td><td>1.95</td></tr><tr><td>(35,40]</td><td>9</td><td>8.98</td><td>8.95</td><td>8.91</td><td>8.79</td><td>8.63</td></tr><tr><td>(40,45]</td><td>70</td><td>69.83</td><td>69.44</td><td>69.01</td><td>67.63</td><td>65.66</td></tr><tr><td>(45,50]</td><td>595</td><td>592.69</td><td>587.60</td><td>571.80</td><td>536.66</td><td>538.75</td></tr><tr><td>(50,55]</td><td>3082</td><td>3063.96</td><td>3024.62</td><td>2980.54</td><td>2846.12</td><td>2674.94</td></tr><tr><td>(55,60]</td><td>8180</td><td>8108.37</td><td>7953.87</td><td>7783.67</td><td>7286.09</td><td>6657.73</td></tr><tr><td>(60,65]</td><td>14,217</td><td>14,033.85</td><td>13,644.13</td><td>13,221.58</td><td>11,994.05</td><td>10,512.79</td></tr><tr><td>(65,70]</td><td>18,750</td><td>18,385.07</td><td>17,613.05</td><td>16,779.05</td><td>14,406.92</td><td>11,417.77</td></tr><tr><td>(70,75]</td><td>18,945</td><td>18,372.01</td><td>17,164.78</td><td>15,874.37</td><td>12,223.98</td><td>7893.16</td></tr><tr><td>(75,80]</td><td>14,699</td><td>13,972.32</td><td>12,468.14</td><td>10,896.98</td><td>6784.90</td><td>3057.85</td></tr><tr><td>(80,85]</td><td>8782</td><td>8040.68</td><td>6573.68</td><td>5162.18</td><td>2211.39</td><td>569.68</td></tr><tr><td>(85,90]</td><td>4009</td><td>3445.40</td><td>2439.80</td><td>1618.21</td><td>400.08</td><td>48.14</td></tr><tr><td>(90,95]</td><td>1108</td><td>865.86</td><td>497.55</td><td>260.86</td><td>32.59</td><td>3.55</td></tr><tr><td>(95,100]</td><td>142</td><td>98.01</td><td>43.59</td><td>18.20</td><td>2.09</td><td>0.24</td></tr><tr><td>(100,105]</td><td>5</td><td>3.17</td><td>1.24</td><td>0.46</td><td>0.04</td><td>0.00</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Comparison of the Pohar-Perme, cuRe and curesrv Weibull MCM net survival predictions in function of the categorical age groups at diagnosis, using breast cancer data.</figcaption></figure>
<caption>**Table 5**</caption>

Results of the variable selection process in the Weibull and Boussari models, for the indicated cancer types.

<table><caption>**Table 6**</caption><thead><tr><th rowspan="3"></th><th colspan="3">Weibull</th><th colspan="3">Effect on π</th><th colspan="3">Boussari</th></tr><tr><th colspan="3">Effect on S<sub>u</sub></th><th colspan="3">Effect on π</th><th colspan="3">Effect on α</th></tr><tr><th>Age</th><th>Year</th><th>Interaction</th><th>Age</th><th>Year</th><th>Interaction</th><th>Age</th><th>Year</th><th>Age</th><th>Year</th></tr></thead><tbody><tr><td>Testicular</td><td>Spline</td><td>Linear</td><td>Yes</td><td>Linear</td><td>Linear</td><td>Yes</td><td>Spline</td><td>Linear</td><td>/</td><td>/</td></tr><tr><td>Breast</td><td>Spline</td><td>Linear</td><td>Yes</td><td>Spline</td><td>Linear</td><td>Yes</td><td>Spline</td><td>Linear</td><td>/</td><td>/</td></tr><tr><td>Prostate</td><td>Spline</td><td>Linear</td><td>Yes</td><td>Spline</td><td>Linear</td><td>Yes</td><td>Spline</td><td>Linear</td><td>/</td><td>/</td></tr></tbody></table>

Comparison of different models for various cancer types.

<table><thead><tr><th rowspan="2"></th><th colspan="3">AIC (log-likelihood)</th><th rowspan="2">α [95 % CI]</th><th rowspan="2">p-value LRT</th></tr><tr><th>Weibull NC</th><th>Weibull C</th><th>Boussari</th></tr></thead><tbody><tr><td>Testicular</td><td>4030.0<br>(-2002.0)</td><td>4029.5<br>(-2000.7)</td><td>4071.8<br>(-2028.9)</td><td>1.19 [0.95,1.43]</td><td>0.193</td></tr><tr><td>Breast</td><td>216,622.8<br>(-108,294.4)</td><td>216,579.9<br>(-108,272.0)</td><td>217,631.7<br>(-108,807.9)</td><td>1.31 [1.24,1.39]</td><td>&lt;0.001</td></tr><tr><td>Prostate</td><td>238,163.8<br>(-119,068.9)</td><td>238,161.1<br>(-119,066.6)</td><td>238,569.7<br>(-119,277.9)</td><td>1.03 [1.00,1.06]</td><td>0.064</td></tr></tbody></table>

likelihood value (using Nelder-Mead for cuRe: -129 308 and for curesurv: -129 308; using L-BFGS-B for curesurv: -129 276.8).

### 4.4. Model selection results

Table 5 summarizes the results of variable selection for the Weibull and Boussari models.

Table 6 lists the criteria that explain the model selection process. For each cancer type, we fitted three models: a Boussari NMCM, a Weibull MCM without correction for other-cause mortality rates (Weibull NC), and a Weibull MCM with correction for other-causes mortality rates (Weibull C). We compared the Boussari model and the Weibull models using the AIC criterion, and the two Weibull models using the AIC criterion and a LRT to ascertain whether the model with correction for other-cause mortality performed significantly better.

For all three cancer types, the Weibull models displayed a lower AIC than the Boussari model, and the Weibull model with correction had a lower AIC than the Weibull model without correction. However, only for the breast cancer, the Weibull model with correction for other-cause mortality performed significantly better than the model without correction ($α = 1.31$ [1.24, 1.39] and LRT p-value < 0.001).

### 4.5. Cure indicators

#### 4.5.1. Effect of life table correction (on cure indicators and on cure fraction prediction)

As discussed in Section 4.4, the correction using the general population life tables significantly improved the model performance for

breast cancer. Fig. 2 presents the comparison of the cure fraction estimates in function of the age at diagnosis, obtained using the Weibull MCM with and without this correction. This figure illustrates how accounting for other-cause mortality can affect the prediction of cure indicators.

#### 4.5.2. Cure indicators obtained with the “best” model for each cancer type

The cure indicators were obtained using the Weibull model without correction for testicular and prostate cancer, and the Weibull model with correction of other-cause mortality for breast cancer.

For each cancer type, predictions were made for nine representative group of patients, defined by the combination of the 25th, 50th, and 75th percentiles of both age at diagnosis and year of diagnosis. Table 7 lists the cure fraction, time to cure $t_{0.05}$, net survival 15 years after diagnosis, and conditional probability of being cured 15 years after diagnosis, for each patient group.

For testicular cancer, the patient group with the highest cure fraction (97.57 % [96.57,98.58]) was the group that included patients with a cancer diagnosis in 2010 (75th percentile) at the age of 29 years (25th percentile). The patient group with the lowest cure fraction (93.85 % [92.11,95.61]) included patients with a cancer diagnosis in 1997 (25th percentile) at the age of 43 years (75th percentile). The cure fraction was lower than 95 % only for the patients who had a diagnosis of testicular cancer in 1997 at the age of 35 years or 43 years. This means that the time to cure $t_{0.05}$ was estimated to be 0 for all the other patient groups, and 0.19 years [0.01,0.90] and 0.84 years [0.12,2.46] for the patients who had a diagnosis of testicular cancer in 1997 at the age of 35 years or 43 years. The graphics of the predictions for patients with testicular

<figure><img src="image_3.png" /><figcaption>Fig. 2. Comparison of the cure fraction estimated using breast cancer data and the Weibull MCM with (red) and without (black) correction of the life tables.</figcaption></figure>
<caption>**Table 7**</caption>

Cancer statistics by cancer type, age at diagnosis, and year of diagnosis: $\pi$ the cure fraction, $t_{0.05}$ the time when a patient has a probability of being cured higher than 95%, $S_n(t=15)$ the net survival 15 years after diagnosis, and $P(t=15)$ the conditional probability of being cured 15 years after diagnosis.

<table><caption>**Table 7 (continued)**</caption><thead><tr><th></th><th>25th percentile</th><th>50th percentile</th><th>75th percentile</th></tr></thead><tbody><tr><td><strong>Testicular cancer</strong></td><td>29-year-old at diagnosis</td><td>35-year-old at diagnosis</td><td>43-year-old at diagnosis</td></tr><tr><td>Year of diagnosis: 1997 (25th percentile)</td><td></td><td></td><td></td></tr><tr><td>π (95% CI)</td><td>95.15<br>[93.98,96.33]</td><td>94.62<br>[93.43,95.83]</td><td>93.85<br>[92.11,95.61]</td></tr><tr><td>t<sub>v</sub> (95% CI)</td><td>0</td><td>0.19 [0.01,0.90]</td><td>0.84 [0.12,2.46]</td></tr><tr><td>S<sub>e</sub>(t=15) (95% CI)</td><td>95.18<br>[93.88,96.20]</td><td>94.76<br>[93.59,95.75]</td><td>94.20<br>[92.53,95.50]</td></tr><tr><td>P(t=15) (95% CI)</td><td>99.97<br>[99.74,100.00]</td><td>99.85<br>[98.92,99.98]</td><td>99.62<br>[97.48,99.94]</td></tr><tr><td>Year of diagnosis: 2004 (50th percentile)</td><td></td><td></td><td></td></tr><tr><td>π (95% CI)</td><td>96.65<br>[95.72,97.59]</td><td>96.12<br>[95.24,97.00]</td><td>95.28<br>[94.26,96.32]</td></tr><tr><td>t<sub>v</sub> (95% CI)</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S<sub>e</sub>(t=15) (95% CI)</td><td>96.68<br>[95.66,97.46]</td><td>96.19<br>[95.31,96.90]</td><td>95.30<br>[94.18,96.21]</td></tr><tr><td>P(t=15) (95% CI)</td><td>99.96<br>[98.94,100.00]</td><td>99.93<br>[98.62,100.00]</td><td>99.98<br>[99.81,100.00]</td></tr><tr><td>Year of diagnosis: 2010 (75th percentile)</td><td></td><td></td><td></td></tr><tr><td>π (95% CI)</td><td>97.57<br>[96.57,98.58]</td><td>97.07<br>[96.12,98.03]</td><td>96.25<br>[95.22,97.30]</td></tr><tr><td>t<sub>v</sub> (95% CI)</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S<sub>e</sub>(t=15) (95% CI)</td><td>97.60<br>[96.45,98.39]</td><td>97.11<br>[96.06,97.88]</td><td>96.25<br>[95.06,97.16]</td></tr><tr><td>P(t=15) (95% CI)</td><td>99.97<br>[82.16,100.00]</td><td>99.96<br>[93.96,100.00]</td><td>100.00<br>[99.99,100.00]</td></tr><tr><td><strong>Breast cancer</strong></td><td>50-year-old at diagnosis</td><td>58-year-old at diagnosis</td><td>66-year-old at diagnosis</td></tr><tr><td>Year of diagnosis: 1998 (25th percentile)</td><td></td><td></td><td></td></tr><tr><td>π (95% CI)</td><td>76.66 [75.09, 78.25]</td><td>78.66 [77.13, 80.22]</td><td>79.28 [77.12, 81.5]</td></tr><tr><td>t<sub>v</sub> (95% CI)</td><td>14.61 [12.68, 16.84]</td><td>11.75 [10.14, 13.62]</td><td>9.27 [7.53, 11.41]</td></tr><tr><td>S<sub>e</sub>(t=15) (95% CI)</td><td>80.48 [79.76, 81.18]</td><td>81.12 [80.16, 82.03]</td><td>80.59 [78.73, 82.31]</td></tr><tr><td>P(t=15) (95% CI)</td><td>95.25 [94.11, 96.16]</td><td>96.97 [96.13, 97.63]</td><td>98.37 [97.69, 98.86]</td></tr><tr><td>Year of diagnosis: 2004 (50th percentile)</td><td></td><td></td><td></td></tr><tr><td>π (95% CI)</td><td>80.35 [78.4, 82.35]</td><td>82.76 [81.16, 84.39]</td><td>84.58 [82.56, 86.66]</td></tr><tr><td>t<sub>v</sub> (95% CI)</td><td>14.27 [11.68, 17.43]</td><td>10.76 [8.92, 12.97]</td><td>7.61 [5.84, 9.93]</td></tr><tr><td>S<sub>e</sub>(t=15) (95% CI)</td><td>84.22 [83.55, 84.86]</td><td>85.09 [84.21, 85.92]</td><td>85.64 [83.91, 87.2]</td></tr><tr><td>P(t=15) (95% CI)</td><td>95.41 [93.41, 96.81]</td><td>97.27 [95.97, 98.15]</td><td>98.76 [97.75, 99.32]</td></tr><tr><td>Year of diagnosis: 2010 (75th percentile)</td><td></td><td></td><td></td></tr><tr><td>π (95% CI)</td><td>83.59 [81.11, 86.14]</td><td>86.21 [84.42, 88.04]</td><td>88.72 [86.7, 90.78]</td></tr><tr><td>t<sub>v</sub> (95% CI)</td><td>13.68 [10.1, 18.52]</td><td>9.52 [7.28, 12.46]</td><td>5.78 [3.88, 8.63]</td></tr><tr><td>S<sub>e</sub>(t=15) (95% CI)</td><td>87.39 [86.36, 88.35]</td><td>88.37 [87.48, 89.19]</td><td>89.56 [87.83, 91.06]</td></tr><tr><td>P(t=15) (95% CI)</td><td>95.64 [91.77, 97.72]</td><td>97.56 [95.26, 98.76]</td><td>99.06 [97.11, 99.69]</td></tr><tr><td><strong>Prostate cancer</strong></td><td>63-year-old at diagnosis</td><td>69-year-old at diagnosis</td><td>74-year-old at diagnosis</td></tr><tr><td>Year of diagnosis: 2000 (25th percentile)</td><td></td><td></td><td></td></tr><tr><td>π (95% CI)</td><td>85.12<br>[82.94,87.36]</td><td>82.76<br>[79.43,86.22]</td><td>76.17<br>[71.91,80.68]</td></tr><tr><td>t<sub>v</sub> (95% CI)</td><td>11.51 [7.88,15.87]</td><td>15.16 [9.72,21.91]</td><td>19.07<br>[12.70,26.82]</td></tr><tr><td>S<sub>e</sub>(t=15) (95% CI)</td><td>88.28<br>[87.18,89.29]</td><td>87.18<br>[85.86,88.38]</td><td>87.18<br>[85.86,88.38]</td></tr><tr><td>P(t=15) (95% CI)</td><td>96.42<br>[95.63,97.07]</td><td>94.93<br>[93.79,95.87]</td><td>94.93<br>[93.79,95.87]</td></tr><tr><td>Year of diagnosis: 2005 (50th percentile)</td><td></td><td></td><td></td></tr><tr><td>π (95% CI)</td><td>93.93<br>[92.99,94.89]</td><td>93.33<br>[92.03,94.64]</td><td>89.28<br>[87.27,91.32]</td></tr><tr><td>t<sub>v</sub> (95% CI)</td><td>1.83 [0.69,3.63]</td><td>3.19 [1.32,6.00]</td><td>8.57 [5.35,12.61]</td></tr></tbody></table>

<table><thead><tr><th></th><th>25th percentile</th><th>50th percentile</th><th>75th percentile</th></tr></thead><tbody><tr><td>S<sub>e</sub>(t=15) (95% CI)</td><td>95.22<br>[94.59,95.78]</td><td>95.04<br>[94.23,95.73]</td><td>95.04<br>[94.23,95.73]</td></tr><tr><td>P(t=15) (95% CI)</td><td>98.65<br>[97.74,99.19]</td><td>98.20<br>[96.90,98.96]</td><td>98.20<br>[96.90,98.96]</td></tr><tr><td>Year of diagnosis: 2010 (75th percentile)</td><td></td><td></td><td></td></tr><tr><td>π (95% CI)</td><td>97.67<br>[97.22,98.13]</td><td>97.61<br>[97.04,98.17]</td><td>95.60<br>[94.59,96.61]</td></tr><tr><td>t<sub>v</sub> (95% CI)</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S<sub>e</sub>(t=15) (95% CI)</td><td>98.17<br>[97.78,98.48]</td><td>98.22<br>[97.73,98.60]</td><td>98.22<br>[97.73,98.60]</td></tr><tr><td>P(t=15) (95% CI)</td><td>99.50<br>[98.08,99.87]</td><td>99.38<br>[97.27,99.86]</td><td>99.38<br>[97.28,99.86]</td></tr></tbody></table>

cancer are presented in section A.2.1 of the supplementary file.

The representative group of patients with breast cancer with the highest cure fraction was the group of patients who received the cancer diagnosis in 2010 at the age of 66 years: cure fraction = 88.72% [86.7, 90.78] and time to cure $t_{0.05}$ = 5.78 years [3.88, 8.63]. The group with the lowest cure fraction included the patients who received the cancer diagnosis in 1998 at the age of 50 years: cure fraction = 76.66% [75.09, 78.25] and time to cure $t_{0.05}$ = 14.61 years [12.68, 16.84]. The graphics of the predictions for patients with breast cancer are presented in section A.2.2 of the supplementary file.

For prostate cancer, the highest cure fraction was 97.67% [97.22,98.13] in the group of patients with a diagnosis in 2010 at the age of 63 years. The lowest cure fraction was 76.17% [71.91,80.68] in patients who received the cancer diagnosis in 2000 at the age of 74 years. For all patients with a diagnosis in 2010 (all three age groups), the cure fraction was >95% and thus the time to cure $t_{0.05}$ was 0. The longest time to cure was 19.07 years [12.70,26.82] for patients with a cancer diagnosis in 2000 at the age of 74 years. The graphics of the predictions for patients with prostate cancer are presented in section A.2.3 of the supplementary file.

Fig. 3 presents a summary map of the time to cure $t_{0.05}$ and the cure proportion, as well as the conditional probability of cure, for each cancer type and each quartile of age at diagnosis and year of diagnosis.

## 5. Discussion

The present paper introduces the new *curesurv* package developed to disseminate recent developments of cure models in cancer epidemiology. The package implements the direct estimation of TNEH without the need of the time to cure post-estimation, and it accounts for bias introduced by the use of general population life tables in cure modelling, which may fail to capture the additional mortality risks experienced by cancer patients in population-based studies. Its application using FRANCIM cancer data allowed estimating various cure indicators, including time to cure, cure fraction, net survival, and excess hazard, and demonstrating the usefulness of the package.

For breast cancer, the MCM with correction for other-cause mortality rates [15] provided a significantly better fit. This suggests that breast cancer patients are 1.31 times more likely to die from non-cancer causes compared with their counterparts in the general population. This finding is in line with the literature. Using the same correction model, Botta et al. reported a relative risk of 1.16 among US breast cancer patients [34] while European studies have estimated this excess risk between 1.2 and 1.4 depending on age [35]. In contrast, for prostate and testicular cancers, other-cause mortality rates did not differ significantly from those of the general population. This may reflect the greater heterogeneity of comorbidities among breast cancer patients prior to diagnosis [36]. Further research is needed to clarify how comorbidities influence mortality from non-cancer causes in breast cancer populations.

Dal Maso et al. [37] estimated cure indicators for various cancer types in European patients. For testicular cancer, they reported a cure
<figure><img src="image_4.png" /><figcaption>Fig. 3. Summary map of the cure indicators for the three cancers at different quartiles of age at diagnosis and year of diagnosis.</figcaption></figure>

fraction of 94 % in 2000, which is comparable to the estimated cure fractions in this study (from 93.85 % in 1997 to 96.65 % in 2004). This high cure fraction reflects the excellent prognosis of testicular cancer, as systemic therapies have rendered the disease curable in most cases [38].

For breast cancer, Dal Maso's cure fraction was of 66 % in 2000. This is lower than the cure fraction estimated in the present analysis, regardless of the age at or year of diagnosis. The α correction, according to Phillips et al. [15], applied to the breast cancer model in the present study may explain the higher cure fraction estimates. Indeed, in the study based on EUROCARE data [35], the cure fraction of female patients with a breast cancer diagnosed in 1998–2002 ranged from 72.8 % to 79 % with the model that included the correction for other-cause mortality, and from 57.4 % to 69 % with the classical model.

For prostate cancer, the cure fraction was 63 % in 2000 in the study by Dal Maso et al. and it ranged from 76.17 % to 85.12 % in the present study. This disparity could be partly explained by differences in the data sources: patients from different countries with a cancer diagnosis from 1990 to 2007 [37] and only French patients with a diagnosis between 1989 and 2015 (this study). Additionally, their cure fraction estimates did not include variations in patient age at diagnosis, unlike the present analysis.

Romain et al. [39] estimated cure indicators for various cancer types, and used the same time to cure metric ($t_{0.05}$) and FRANCIM data of patients with a cancer diagnosis between 1995 and 2009. For testicular

cancer, their estimated $t_{0.05}$ was 0 for patients with an age at diagnosis between 15 and 44 years, in agreement with the present findings (time to cure: 0 for patients with an age at diagnosis of 29 and 35 years, and ranging from 0.84 for a diagnosis in 1997 to 0 for a diagnosis in 2004 and 2010 at the age of 43 years). These findings suggest that the patients are cured “immediately after diagnosis”. However, this should be interpreted with caution, as it may be more accurate to consider cure achieved right after treatment. A value $t_{0.05}$ = 0 means that the conditional probability of being cured exceeds 95 % even at diagnosis. For cancers with such high cure probabilities, it may be more informative to consider alternative thresholds, such as $t_{0.01}$, the time at which the conditional probability of being cured exceeds 99 %.

For breast cancer, Romain et al. reported a time to cure of 10.6 years for patients aged 45–54 year at diagnosis, which is lower than the present estimations for 50-year-old patients (14.61 and 14.27 years for a diagnosis in 1998 and in 2004, respectively). This difference may be explained by modelling differences, such as the use of continuous versus categorical age and correction for other causes mortality. Similarly for patients aged 55–64 at diagnosis, the $t_{0.05}$ was 11.2 years in the study by Romain et al., while it ranged between 11.75 years in 1998 and 9.52 years in 2010 for 58-year-old patients in the present study. The estimates diverged for older patients at diagnosis. Indeed, the $t_{0.05}$ was 12.3 years for 65–75-year-old patients in the study by Romain et al., and it ranged between 5.78 and 9.27 years for 66-year-old patients in the present
study. This may be because correcting other causes mortality affects older patients the most, seeing has older people have a higher mortality rate in the general population. We found that for breast cancer, the time to statistical cure can exceed 10 years, which might be explained by its potential for slow progression and late recurrence. However, in stage specific analysis in breast cancer patients, t₀.₀₅ is achieved immediately after treatment when for the cancer is of stage I [40].

For prostate cancer, the estimates were similar to those of Romain et al., but only for patients with a diagnosis in 2000: t₀.₀₅ of 10.0 years for 55–64-year-old patients in Romain et al. and of 11.51 years for 63-year-old patients in this study. For patients aged 65–75 years at diagnosis, the t₀.₀₅ was 12.3 years in Romain et al., and in the present study it was of 15.16 years and 19.07 years for patients aged 69 and 74 years at diagnosis, respectively. These differences may be attributed to variations in the model construction. In the study by Romain et al., age at diagnosis was a categorical variable, whereas the models in the present study used age at diagnosis as a continuous variable and included year of diagnosis as an additional effect. Furthermore, their dataset included patients with a diagnosis between 1995 and 2009, while the results previously cited from this study were for a diagnosis in 2000. In 2005, the t₀.₀₅ estimated by our study ranged from 1.83 years for patients aged 63 to 8.57 years for patients aged 74 at diagnosis, and in 2010 the t₀.₀₅ was estimated at 0 years for all three patient groups. These rapid declines are likely explained by the widespread adoption of opportunistic prostate-specific antigen screening, which led to earlier detection of more curable cancers [41].

For all three cancer types, the cure fraction was higher for patients with a more recent year of diagnosis, probably linked to cancer management progress. In 2020, Culp et al. found that in France, the mortality of patients with prostate cancer was lower compared with previous years [42]. Similarly, Sancho-Garnier and Colonna observed a decrease in breast cancer mortality starting in ~1990, probably thanks to a combination of earlier cancer detection due to screening and better treatment [43].

Cure models are used more and more often for health technology assessments [44]. For example, Bullement et al. [45] considered the use of cure models to compare the ipilimumab + dacarbazine combination to placebo + dacarbazine in treatment-naive patients with advanced melanoma in order to determine the long-term survival benefits of ipilimumab (a monoclonal antibody used in immunotherapy). They tested different models, including cure models, using data of patients with 3 years of follow-up and extrapolated the 5-year survival. They found that the mixture cure model provided one of the best extrapolations of the 5-year survival.

Cure models can provide other advantages when assessing the cancer-specific mortality without need of the cause of death. When the other-cause mortality rates of patients with cancer differ significantly from the mortality of the general population (as seen in breast cancer in this paper), the new cure models that take life table corrections into account can give more reliable estimates. To the best of our knowledge, the *curesurv* package is the only R package to allow estimating cure indicators using life table corrections. By making the estimation of cure indicators accessible, *curesurv* could promote the use of cure models for health technology assessments. The practical interpretation of cure indicators in cancer epidemiology helps clinicians, researchers, and policymakers understand long-term outcomes for cancer patients and are useful for public health planning (e.g. determine when follow-up care for the cancer can be reduced or stopped, informs patients and guide resource allocation in survivorship programs.

The application using real population data showed that the Boussari NMCM had a higher AIC value in all three cancer types. This means that data fitting is better with the Weibull models. The Boussari model tends to give very high estimations of the TNEH, and therefore, the assumption that the TNEH is reached during the follow-up period is invalid. The next

step could be to apply the correction of other-cause mortality by Phillips et al. [15] to the Boussari NMCM. Another option to deal with the fact that excess hazard tends to be estimated as extremely small but not null for a long time, thus forcing τ to be very large, could be to allow an "error" parameter, such that the TNEH becomes the time to constant (and very small) excess hazard.

We developed the *curesurv* R package to fit parametric MCM and NMCM. This is the first R package to include the possibility of implementing cure models when life tables are insufficiently stratified, by using the *α* correction parameter by Phillips et al. [15]. In future developments, it may be worthwhile to explore other life table correction techniques, such as those proposed by Rubio et al. [46] and Touraine et al. [47].

This study has some limitations. Currently, the package supports the Weibull MCM, with and without the correction by Phillips et al. [15], as well as the Boussari NMCM. However, it does not accommodate other parametric models. Future work will focus on expanding the package to include more classical parametric cure models and adapting the correction proposed by Phillips and colleagues to these models. The correction parameters were assumed constant over time since diagnosis and further work is needed to understand the implication of this assumption. Moreover, the limited geographical coverage of the French cancer registries may restrict the generalization of the estimated cure indicators at the national scale. To address this limitation, future research could rely on the French National Medical Administrative Database (SNDS), which is currently evaluated to validate its use for accurately measuring cancer-related indicators [48]. The stage information was not routinely available so could not be analysed even if though type of indicators could benefit from further stratification according to prognostic factors.

## 6. Conclusion

The development of the *curesurv* package facilitates the estimation of cure indicators using established and novel models. The Weibull MCM generally provided a better fit than the Boussari NMCM for the three cancer types. Applying the correction proposed by Phillips and colleagues to adjust for other-cause mortality significantly influenced the survival estimates, especially for breast cancer. Future directions for the *curesurv* package include integrating additional parametric models and refining the life table correction methods to improve the accuracy and applicability of cure models.

## Ethics approval and consent to participate

This non-interventional population-based cohort study was approved by the French Data Protection Authority (Commission Nationale de l'Informatique et des Libertés). In accordance with French legislation, there was no requirement for written informed consent because of the use of deidentified data. The study followed the Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) reporting guideline for cohort studies.

## Additional contributions

We would like to thank the French Cancer Registries Network for providing the data.

## Data availability

Details of the code used to fit the models are available in the Github repository. The repository is accessible via https://github.com/jgoungounga/curesurv_modelling.
## CRediT authorship contribution statement

**Judith Breaud:** Writing – review & editing, Writing – original draft, Visualization, Software, Methodology, Investigation, Formal analysis, Conceptualization.
**Olayidé Boussari:** Writing – review & editing, Validation, Methodology, Investigation, Formal analysis.
**Valérie Jooste:** Writing – review & editing, Validation, Resources, Methodology, Formal analysis, Data curation, Conceptualization.
**Laura Botta:** Writing – review & editing, Validation, Methodology, Investigation.
**Juste Goungounga:** Writing – review & editing, Validation, Supervision, Project administration, Methodology, Investigation, Formal analysis, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Funding/Support

Not applicable.

Role of the Funder/Sponsor: Not applicable

## Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.cmpbup.2025.100222.

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