Author: Juste Goungounga, Judith Breaud, Olayidé Boussari, Valérie Jooste

# Vignette_tneh

#### Juste Goungounga, Judith Breaud, Olayidé Boussari, Valérie Jooste

    library(curesurv)
    #> Le chargement a nécessité le package : stringr
    #> Le chargement a nécessité le package : survival

# How to analyse survival data using Time-To-Null excess hazard model (TNEH model)

## Additive hazard assumption in net survival setting

When the cause of death is unknown, the most common method to estimate the cancer-related survival is net survival. Its estimation assumes that the observed hazard \(\lambda_{obs}\) is equal to the sum of the known background mortality hazard in the general population \(\lambda_{pop}\) (obtained from national Statistic Institutes such as INSEE in France) and the excess hazard (due to cancer) \(\lambda_{exc}\). For one individual \(i\), this relation can be expressed as:

\[ \lambda_{obs}(t_i|z_i) = \lambda_{pop}(t_i+a_i|z_{pop,i}) + \lambda_{exc}(t_i|z_i) \] where \(Z_{pop}\subset Z\) are the variables of the mortality table and of the model respectively.

The cumulative observed hazard can be written as: \[ \Lambda_{obs}(t|z) = \Lambda_{pop}(t+a|z) + \Lambda_{exc}(t|z) \] and the net survival is obtained as following: \[ S_{n}(t|z) = \exp(-\Lambda_{exc}(t|z)) \]

# TNEH model

The TNEH model is a relatively recent excess hazard model developed by Boussari et al. \(\\\)

The particularity of this model is that it enables the estimation, at the same time as the classical parameters of a model of the excess rate, of a quantity which is obtained by post-estimation by the classical models: it concerns the time after which the excess rate becomes null i.e. the cure point.

## Instantaneous excess hazard

The excess hazard proposed can be expressed as following:

\[ \lambda_{exc}(t|z;\theta) = \left(\dfrac{t}{\tau(z;\tau*)}\right)^{\alpha(z;\alpha*)-1} \left(1 - \dfrac{t}{\tau(z;\tau*)}\right)^{\beta-1} 1_{\left\{0 \le t \le \tau(z;\tau*)\right\}} \]

where : \(\\\)

\(\tau(z;\tau*) > 0\) is the time to cure, depends on covariates z and vector of parameters \(\tau*\). It corresponds to the vector of parameters fitting the time-to-null excess hazard. \(\\\)

\(\alpha(z;\alpha*) > 0\) and \(\beta > 1\) are shape parameters. With \(\beta>1\), the excess hazard is forced to be null and continuous in \(\tau(z;\tau*)\). \(\\\)

The vector of parameters to be estimated is \(\theta = (\alpha*, \beta, \tau*)\) with \(\alpha(z;\alpha*) > 0\) .

## Cumulative excess hazard

\[ \Lambda_{exc}(t|z;\theta) = \tau(z;\tau*) B \left( \alpha(z;\alpha*), \beta \right) F_{Be} \left( \dfrac{t}{\tau(z;\tau*)} ; \alpha(z;\alpha*) , \beta \right) \]

where

B is the beta function \(\\\) \(F_{Be}\) is the cumulative distribution function (cdf) of the beta distribution

## Net survival

\[ S_n(t|z) = \exp(-\Lambda_{exc}(t|z)) = \exp\left(-\tau(z;\tau*) B \left( \alpha(z;\alpha*), \beta \right) F_{Be} \left( \dfrac{t}{\tau(z;\tau*)} ; \alpha(z;\alpha*),\beta \right)\right) \]

## The cure fraction \(\pi\)

The cure fraction corresponds to the net survival at \(t = \tau\) in TNEH model. It can be expressed as:

\[ \pi(z|\theta) = \exp\left(-\Lambda_{exc}(\tau(z;\tau*)|z)\right) = \exp\left(-\tau(z;\tau*) B \left( \alpha(z;\alpha*), \beta \right)\right) \]

## The probability P(t)

This quantity corresponds to the probability P(t) of being cured at a given time t after diagnosis knowing that he/she was alive up to time t. It can be expressed as following:

\[ P(t|z) = \dfrac{\pi(z|\theta)}{S_n(t|z)} = \exp \left( \tau(z;\tau*) \left( B \left( \dfrac{t}{\tau(z;\tau*)} ; \alpha(z;\alpha*) , \beta \right) - B(\alpha, \beta) \right) \right) \] To calculates the confidence intervals of \(P(t|z)\), can be obtained using the delta method. The application of this method requires the partial derivatives of \(P(t|z)\) with respect of the parameters of the model. This can be written as:

\[ \dfrac{\partial P(t|z)}{\partial \theta} = \dfrac{1}{S_n(t|z)^2} \left( \dfrac{\partial \pi(z|\theta)}{\partial \theta} S_n(t|z) - \dfrac{\partial S_n(t|z)}{\partial \theta} \pi(z|\theta) \right) \]

# Fit of tneh model using R

## Without covariates

There are no covariables acting on parameters \(\tau\) (\(\tau=\tau_0\)) and \(\alpha\) (\(\alpha=\alpha_0\))

    fit_ad_tneh_nocov <- curesurv(Surv(time_obs, event) ~ 1,
                                 pophaz = "ehazard",
                                 cumpophaz = "cumehazard",
                                 model = "nmixture", dist = "tneh",
                                 link_tau = "linear",
                                 data = testiscancer,
                                 method_opt = "L-BFGS-B")

    summary(fit_ad_tneh_nocov)
    #> Call:
    #> curesurv(formula = Surv(time_obs, event) ~ 1, data = testiscancer,
    #>     pophaz = "ehazard", cumpophaz = "cumehazard", model = "nmixture",
    #>     dist = "tneh", link_tau = "linear", method_opt = "L-BFGS-B")
    #>
    #>          coef se(coef)      z      p
    #> alpha0 2.1841   0.1032 21.166 <2e-16
    #> beta   4.4413   0.5178  8.577 <2e-16
    #> tau0   5.1018   0.5397  9.452 <2e-16
    #>
    #> Estimates and their 95% CI after back-transformation
    #>        estimates   LCI   UCI
    #> alpha0     2.184 1.982 2.386
    #> beta       4.441 3.426 5.456
    #> tau0       5.102 4.044 6.160
    #>
    #> Cured proportion exp[-ζ0* B((α0+α*Z)β)]
    #> For the reference individual (each Z at 0)
    #> [1] 0.8474
    #>
    #> log-likelihood: -2633.903 (for 3 degree(s) of freedom)
    #>  AIC: 5273.806
    #>
    #>   n= 2000 , number of events= 949

We can see that the time-to-cure \(\tau=\tau_0\) is estimated at 5.102 years, and the cure proportion is estimated at 84.74%.

### Prediction

We predict for varying time after diagnosis

    newdata1 <- with(testiscancer,
      expand.grid(event = 0, time_obs  = seq(0.1,10,0.1)))
    p_28 <- predict(object = fit_ad_tneh_nocov, newdata = newdata1)

### Plot of different estimators (hazard, survival, probability of being cured conditionally on being alive)

    plot(p_28)

### Cure fraction estimation precision

The confidence intervals at \(1-\alpha\) level for the cure fraction \(\pi\) can be written as:

\[ \left[\hat{\pi} \pm z_{1 - \alpha / 2} \sqrt{Var(\hat{\hat{\pi}})}\right] \] where \[ Var(\hat{\pi}) = \dfrac{\partial \hat{\pi}}{\partial \theta} Var(\theta) \left(\dfrac{\partial \hat{\pi}}{\partial \theta}\right)^T \]

    p_28[1,c("lower_bound_pi_tneh","cured","upper_bound_pi_tneh")]
    #>   lower_bound_pi_tneh     cured upper_bound_pi_tneh
    #> 1           0.8253941 0.8473745           0.8693549

### Time-to-cure estimation

We search the time \(\text{t}=\text{TTC}_i\) from which \(\text{P}_i(t) = 1-\epsilon\). \(\epsilon\) can be fixed to 0.95.

The variance formula can be expressed as:

\[ Var(TTC) = Var(g(\theta;z_i)) \simeq \left(\dfrac{\partial P(t|z_i;\theta)}{\partial t}_{|t = TTC}\right)^{-2} Var(P(t|z_i;\theta))_{|t=TTC} \]

    p_28[1,c("lower_bound_TTC_tneh","time_to_cure_ttc","upper_bound_TTC_tneh")]
    #>   lower_bound_TTC_tneh time_to_cure_ttc upper_bound_TTC_tneh
    #> 1             1.825476         2.083375             2.341273

## With covariates acting both on parameters tau and alpha

We create a new age variable : age_crmin the reduced age and “centered” around the age of the youngest patient

     testiscancer$age_crmin <- (testiscancer$age- min(testiscancer$age)) /
                  sd(testiscancer$age)

This model has the variable age_crmin acting on both \(\alpha\) and \(\tau\) : (\(\alpha=\alpha_0+Z_{age\_crmin}\times\alpha_1\) et \(\tau=\tau_0+Z_{age\_crmin}\times\tau_1\))

    fit_m1_ad_tneh <- curesurv(Surv(time_obs, event) ~ z_tau(age_crmin) +
                              z_alpha(age_crmin),
                              pophaz = "ehazard",
                              cumpophaz = "cumehazard",
                              model = "nmixture", dist = "tneh",
                              link_tau = "linear",
                              data = testiscancer,
                              method_opt = "L-BFGS-B")

     summary(fit_m1_ad_tneh)
    #> Call:
    #> curesurv(formula = Surv(time_obs, event) ~ z_tau(age_crmin) +
    #>     z_alpha(age_crmin), data = testiscancer, pophaz = "ehazard",
    #>     cumpophaz = "cumehazard", model = "nmixture", dist = "tneh",
    #>     link_tau = "linear", method_opt = "L-BFGS-B")
    #>
    #>                         coef se(coef)      z        p
    #> alpha0               2.87676  0.24110 11.932  < 2e-16
    #> alpha_1_(age_crmin) -0.50110  0.07506 -6.676 2.46e-11
    #> beta                 5.15288  1.04648  4.924 8.48e-07
    #> tau0                 3.25984  0.55516  5.872 4.31e-09
    #> tau_1_(age_crmin)    3.46628  1.24242  2.790  0.00527
    #>
    #> Estimates and their 95% CI after back-transformation
    #>                     estimates   LCI   UCI
    #> alpha0                  2.877 2.404 3.349
    #> alpha_1_(age_crmin)     2.376 2.229 2.523
    #> beta                    5.153 3.102 7.204
    #> tau0                    3.260 2.172 4.348
    #> tau_1_(age_crmin)       6.726 4.291 9.161
    #>
    #> Cured proportion exp[-(ζ0+ζ*Z)* B((α0+α*Z)β)] and its 95% CI
    #> For the reference individual (each Z at 0)
    #> [1] 0.9675
    #>
    #> log-likelihood: -2544.1 (for 5 degree(s) of freedom)
    #>  AIC: 5098.2
    #>
    #>   n= 2000 , number of events= 949

For the reference individual (that is the patient with age_crmin=0, so the youngest patient, approximately 20y old at diagnosis), the cure proportion is estimated at 96,75% and the time to null excess hazard at 3.258 year. For an individual whose age_ is_crmin is 1 (that is they are aged the standard deviation more than the youngest, that is approximately 39y old at diagnosis), the time to null excess hazard is 6.721 year.

### Predictions :

With varying time for patient of mean age

      #time varying prediction for patient with age mean
    newdata1 <- with(testiscancer,
                     expand.grid(event = 0,
                                 age_crmin = mean(age_crmin),
                                 time_obs  = seq(0.001,10,0.1)))

     pred_agemean <- predict(object = fit_m1_ad_tneh, newdata = newdata1)

With varying time for oldest patient

    #time varying prediction for patient with age max
    newdata2 <- with(testiscancer,
                      expand.grid(event = 0,
                                  age_crmin = max(age_crmin),
                                  time_obs  = seq(0.001,10,0.1)))

    pred_agemax <- predict(object = fit_m1_ad_tneh, newdata = newdata2)

At 2 years after diagnostic for patients of different ages

    # predictions at time 2 years with varying age

       newdata3 <- with(testiscancer,
          expand.grid(event = 0,
                      age_crmin = seq(min(testiscancer$age_crmin),
                                      max(testiscancer$age_crmin), 0.1),
                      time_obs  = 2))

    pred_age_val <- predict(object = fit_m1_ad_tneh, newdata = newdata3)
    val_age <- seq(min(testiscancer$age_crmin),
                   max(testiscancer$age_crmin),
                   0.1) * sd(testiscancer$age) +  min(testiscancer$age)

### Plot of main indicators as a function of time for two patients of age mean and age max

    par(mfrow = c(2, 2),cex = 1.0)

    plot(pred_agemax$time,pred_agemax$ex_haz,type = "l",lty = 1,lwd = 2,
         xlab = "Time since diagnosis", ylab = "excess hazard")

    lines(pred_agemean$time,pred_agemean$ex_haz,type = "l",lty = 2,lwd = 2)

    legend("topright",horiz = FALSE,
           legend = c("hE(t) age.max = 79.9", "hE(t) age.mean = 50.8"),
           col = c("black", "black"),lty = c(1, 2, 1, 1, 2, 2))
    grid()

    plot(pred_agemax$time,pred_agemax$netsurv,type = "l",lty = 1,lwd = 2,
        ylim = c(0, 1),xlab = "Time since diagnosis",ylab = "net survival")

    lines(pred_agemean$time,pred_agemean$netsurv,type = "l",lty = 2,lwd = 2)

    legend("bottomleft",horiz = FALSE,
           legend = c("Sn(t) age.max = 79.9", "Sn(t) age.mean = 50.8"),
           col = c("black", "black"),lty = c(1, 2, 1, 1, 2, 2))
    grid()

    plot(pred_agemax$time,pred_agemax$pt_cure,type = "l",lty = 1,lwd = 2,ylim = c(0, 1),
         xlab = "Time since diagnosis",ylab = "probability of being cured P(t)")

    lines(pred_agemean$time,pred_agemean$pt_cure,type = "l",lty = 2,lwd = 2)

    abline(v = pred_agemean$tau[1],lty = 2,lwd = 2,col = "blue")
    abline(v = pred_agemean$time_to_cure_ttc[1],lty = 2,lwd = 2,col = "red")
    abline(v = pred_agemax$tau[1],lty = 1,lwd = 2,col = "blue")
    abline(v = pred_agemax$time_to_cure_ttc[1],lty = 1,lwd = 2,col = "red")
    grid()

    legend("bottomright",horiz = FALSE,
           legend = c("P(t) age.max = 79.9","P(t) age.mean = 50.8",
                      "TNEH age.max = 79.9","TTC age.max = 79.9",
                     "TNEH age.mean = 50.8","TTC age.mean = 50.8"),
          col = c("black", "black", "blue", "red", "blue", "red"),
          lty = c(1, 2, 1, 1, 2, 2))
    par(oldpar)

### Plot of the main indicators 2 years after diagnosis, for patients of varying ages

    par(mfrow=c(2,2))
    plot(val_age,pred_age_val$ex_haz,
         type = "l",lty=1, lwd=2,
         xlab = "age",ylab = "excess hazard 2y after diagnosis")
    grid()

     plot(val_age,pred_age_val$netsurv,
          type = "l", lty=1,lwd=2,ylim=c(0,1),
          xlab = "age", ylab = "net survival 2y after diagnosis")
         grid()

     plot(val_age,pred_age_val$pt_cure, type = "l", lty=1, lwd=2,ylim=c(0,1),
         xlab = "age",ylab = "P(t) 2y after diagnosis")
         grid()

    plot(val_age,pred_age_val$cured, type = "l", lty=1, lwd=2,ylim=c(0,1),
         xlab = "age", ylab = "cure proportion")
         grid()

    par(oldpar)

## With covariates acting only on parameters adjusting the time-to-null excess hazard tau

Effects of centered and reduced age on \(\tau\) (\(tau=\tau_0+Z_{age\_cr}\times\tau_1\), \(\alpha=\alpha_0\))

    #| echo: true
    #| label: withtauonly
    #| warning: false
    #| message: false

    fit_ad_tneh_covtau <- curesurv(
      Surv(time_obs, event) ~ z_tau(age_cr),
      pophaz = "ehazard",
      cumpophaz = "cumehazard",
      model = "nmixture",
      dist = "tneh",
      link_tau = "linear",
      data = testiscancer,
      method_opt = "L-BFGS-B"
    )
    #> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
    #> Warning in diag(varcov): NAs introduits lors de la conversion automatique
    #> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
    #> Warning in diag(varcov): NAs introduits lors de la conversion automatique
    #> Warning in par(oldpar): appel de par(new=TRUE) sans graphe
    #> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
    #> Warning in diag(varcov): NAs introduits lors de la conversion automatique
    #> Warning in par(oldpar): appel de par(new=TRUE) sans graphe
    #> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
    #> Warning in diag(varcov): NAs introduits lors de la conversion automatique
    #> Warning in par(oldpar): appel de par(new=TRUE) sans graphe
    #> Warning in par(oldpar): appel de par(new=TRUE) sans graphe

    summary(fit_ad_tneh_covtau)
    #> Call:
    #> curesurv(formula = Surv(time_obs, event) ~ z_tau(age_cr), data = testiscancer,
    #>     pophaz = "ehazard", cumpophaz = "cumehazard", model = "nmixture",
    #>     dist = "tneh", link_tau = "linear", method_opt = "L-BFGS-B")
    #>
    #>                  coef se(coef)      z        p
    #> alpha0         1.9753   0.1299 15.206  < 2e-16
    #> beta           5.3066   1.1286  4.702 2.58e-06
    #> tau0           7.4380   1.8109  4.107 4.00e-05
    #> tau_1_(age_cr) 2.3159   0.8529  2.715  0.00662
    #>
    #> Estimates and their 95% CI after back-transformation
    #>                estimates   LCI    UCI
    #> alpha0             1.975 1.721  2.230
    #> beta               5.307 3.095  7.519
    #> tau0               7.438 3.889 10.987
    #> tau_1_(age_cr)     9.754 8.082 11.426
    #>
    #> Cured proportion exp[-(ζ0+ζ*Z)* B((α0+α*Z)β)] and its 95% CI
    #> For the reference individual (each Z at 0)
    #> [1] 0.794
    #>
    #> log-likelihood: -2610.768 (for 4 degree(s) of freedom)
    #>  AIC: 5229.537
    #>
    #>   n= 2000 , number of events= 949

This model estimates a cure proportion of 79.4% for individuals of mean age (51.05 year old at diagnosis) with a time to null excess hazard 7.44 year. For an individual of age at diagnosis 70.01 (age_cr=1), the time to null excess hazard is estimated at 9.754. This model estimates that the time to null excess hazard increases by 2.3159 when age_cr increases by (that is, when age at diagnosis increases by 18.96)

### Prediction

    summary(testiscancer$age_cr)
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #> -1.6362 -0.9358  0.0276  0.0000  0.9525  1.5240
    summary(testiscancer$age)
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>   20.02   33.30   51.57   51.05   69.11   79.95
    newdata2 <- with(testiscancer,
                     expand.grid(event = 0,
                                 time_obs  = seq(0.1, 10, 0.1),
                                 age_cr = c(-0.9358, 0.0276, 0.9525) ))
    newdata2_1stqu <- newdata2[newdata2$age_cr==-0.9358,]
    newdata2_2rdqu <- newdata2[newdata2$age_cr==0.0276,]
    newdata2_3rdqu <- newdata2[newdata2$age_cr==0.9525,]

    p1stqu <- predict(object = fit_ad_tneh_covtau, newdata = newdata2_1stqu)
    p2rdqu <- predict(object = fit_ad_tneh_covtau, newdata = newdata2_2rdqu)
    p3rdqu <- predict(object = fit_ad_tneh_covtau, newdata = newdata2_3rdqu)

### Plot of excess hazard at varying times for 3 different ages at diagnosis (33.3, 51.6, 69,1)

    par(mfrow=c(2,2))

      plot(p1stqu,
         main = "Excess hazard for age 33.3",
         fun = "haz")

      plot(p2rdqu,
         fun = "haz",
         main = "Excess hazard for age 51.57")

      plot(p3rdqu,
         fun = "haz",
         main = "Excess hazard for age 69.11")

    par(oldpar)

## With covariates acting only on scale parameter alpha

Effect of age_cr on only \(\alpha\) (\(\alpha=\alpha_0+Z_{age\_cr}*\alpha_1\), \(\tau=\tau_0\))

    #| echo: true
    #| label: only_covariate_on_alpha
    #| message: false
    #| warning: false

    fit_ad_tneh_covalpha <-
      curesurv(
        Surv(time_obs, event) ~ z_alpha(age_cr),
        pophaz = "ehazard",
        cumpophaz = "cumehazard",
        model = "nmixture",
        dist = "tneh",
        link_tau = "linear",
        data = testiscancer,
        method_opt = "L-BFGS-B"
      )

    summary(fit_ad_tneh_covalpha)
    #> Call:
    #> curesurv(formula = Surv(time_obs, event) ~ z_alpha(age_cr), data = testiscancer,
    #>     pophaz = "ehazard", cumpophaz = "cumehazard", model = "nmixture",
    #>     dist = "tneh", link_tau = "linear", method_opt = "L-BFGS-B")
    #>
    #>                      coef se(coef)      z        p
    #> alpha0            2.06862  0.11152 18.550  < 2e-16
    #> alpha_1_(age_cr) -0.46785  0.06331 -7.389 1.48e-13
    #> beta              4.77703  0.81573  5.856 4.74e-09
    #> tau0              6.09881  1.11797  5.455 4.89e-08
    #>
    #> Estimates and their 95% CI after back-transformation
    #>                  estimates   LCI   UCI
    #> alpha0               2.069 1.850 2.287
    #> alpha_1_(age_cr)     1.601 1.477 1.725
    #> beta                 4.777 3.178 6.376
    #> tau0                 6.099 3.908 8.290
    #>
    #> Cured proportion exp[-ζ0* B((α0+α*Z)β)]
    #> For the reference individual (each Z at 0)
    #> [1] 0.8181
    #>
    #> log-likelihood: -2586.138 (for 4 degree(s) of freedom)
    #>  AIC: 5180.275
    #>
    #>   n= 2000 , number of events= 949

The time to null excess hazard is estimated at 6.099 year for all individuals, and the cure proportion is estimated at 81.81% for patient of mean age (51.05 yo at diagnosis)

### Predictions

    p4_33.3 <- predict(object = fit_ad_tneh_covalpha,
                     newdata = newdata2_1stqu)
    p4_51.6 <- predict(object = fit_ad_tneh_covalpha,
                     newdata = newdata2_2rdqu)
    p4_69.1 <- predict(object = fit_ad_tneh_covalpha,
                     newdata = newdata2_3rdqu)

### Plot of estimation of probability P(t) of being cured at a given time t after diagnosis knowing that he/she was alive up to time t

    par(mfrow=c(2,2))

      plot(p4_33.3,
         main = "Pt_cure for age 33.3",
         fun = "pt_cure")

      plot(p4_51.6,
         fun = "pt_cure",
         main = "Pt_cure for age 51.6")

      plot(p4_69.1,
         fun = "pt_cure",
         main = "Pt_cure for age 69.1")
    par(oldpar)

## Comparing models

We fitted 4 models and to compare them we can use the AIC criteria

    AIC(fit_ad_tneh_nocov,fit_ad_tneh_covalpha,fit_ad_tneh_covtau,fit_m1_ad_tneh)
    #>   fit_ad_tneh_nocov fit_ad_tneh_covalpha fit_ad_tneh_covtau fit_m1_ad_tneh
    #> 1          5273.806             5180.275           5229.537         5098.2

The best model is the one with the lowest AIC : it’s the one with an effect of age on both $ $ and \(\alpha\).

# Session info

    date()
    #> [1] "Fri Mar  7 13:39:05 2025"
    sessionInfo()
    #> R version 4.4.2 RC (2024-10-23 r87274 ucrt)
    #> Platform: x86_64-w64-mingw32/x64
    #> Running under: Windows 10 x64 (build 19045)
    #>
    #> Matrix products: default
    #>
    #>
    #> locale:
    #> [1] LC_COLLATE=C                   LC_CTYPE=French_France.utf8
    #> [3] LC_MONETARY=French_France.utf8 LC_NUMERIC=C
    #> [5] LC_TIME=French_France.utf8
    #>
    #> time zone: Europe/Paris
    #> tzcode source: internal
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] curesurv_0.1.2 survival_3.7-0 stringr_1.5.1
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] vctrs_0.6.5         cli_3.6.3           knitr_1.48
    #>  [4] rlang_1.1.4         xfun_0.49           Formula_1.2-5
    #>  [7] highr_0.11          stringi_1.8.4       jsonlite_1.8.9
    #> [10] glue_1.8.0          htmltools_0.5.8.1   sass_0.4.9
    #> [13] rmarkdown_2.29      grid_4.4.2          evaluate_1.0.1
    #> [16] jquerylib_0.1.4     fastmap_1.2.0       numDeriv_2016.8-1.1
    #> [19] yaml_2.3.10         lifecycle_1.0.4     compiler_4.4.2
    #> [22] rngWELL_0.10-10     randtoolbox_2.0.5   rstudioapi_0.17.1
    #> [25] lattice_0.22-6      digest_0.6.37       R6_2.5.1
    #> [28] splines_4.4.2       magrittr_2.0.3      bslib_0.8.0
    #> [31] Matrix_1.7-1        tools_4.4.2         cachem_1.1.0