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Juste Goungounga, Olayidé Boussari, Laura Botta, Valérie Jooste
library(curesurv)
#> Le chargement a nécessité le package : stringr
#> Le chargement a nécessité le package : survivalWhen 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:
\[ \lambda_{obs}(t_i|z_i) = \lambda_{pop}(t_i+a_i|z_{Pi}) + \lambda_{exc}(t_i|z_i) \] where \(Z_P\subset Z\).
The cumulative observed hazard can be written as: \[ \Lambda_{obs}(t) = \Lambda_{pop}(t+a) + \Lambda_{exc}(t) \] and the net survival is obtained as following: \[ S_{n}(t) = \exp(-\Lambda_{exc}(t)) \]
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.
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 \(\alpha(z;\alpha*) > 0\) .
\[ \Lambda_{exc}(t|z;\theta) = \tau(z;\tau*) B \left( \alpha(z;\alpha*), \beta \right) F_{Be} \left( \dfrac{t}{\tau(z;\tau*)} ; \alpha(z;\alpha*) , \beta \right) \]
where
B is the beta function \(\\\) \(F_{Be}\) is the cumulative distribution function (cdf) of the beta distribution
\[ S_n(t|z) = \exp(-\Lambda_{exc}(t|z)) = \exp\left(-\tau(z;\tau*) B \left( \alpha(z;\alpha*), \beta \right) F_{Be} \left( \dfrac{t}{\tau(z;\tau*)} ; \alpha(z;\alpha*),\beta \right)\right) \]
The cure fraction corresponds to the net survival at \(t = \tau\) in TNEH model. It can be expressed as:
\[ \pi(z|\theta) = \exp\left(-\Lambda_{exc}(\tau(z;\tau*)|z)\right) = \exp\left(-\tau(z;\tau*) B \left( \alpha(z;\alpha*), \beta \right)\right) \]
This quantity corresponds to the probability Pi(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:
\[ Pi(t|z) = \dfrac{\pi(z|\theta)}{S_n(t|z)} = \exp \left( \tau(z;\tau*) \left( B \left( \dfrac{t}{\tau(z;\tau*)} ; \alpha(z;\alpha*) , \beta \right) - B(\alpha, \beta) \right) \right) \] To calculates the confidence intervals of \(Pi(t|z)\), can be obtained using the delta method. The application of this method requires the partial derivatives of \(Pi(t|z)\) with respect of the parameters of the model. This can be written as:
\[ \dfrac{\partial Pi(t|z)}{\partial \theta} = \dfrac{1}{S_n(t|z)^2} \left( \dfrac{\partial \pi(z|\theta)}{\partial \theta} S_n(t|z) - \dfrac{\partial S_n(t|z)}{\partial \theta} \pi(z|\theta) \right) \]
fit_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")
#> init 5 5.5 5 lower 0 1 0 upper 100 100 100
#> next evaluation with initial values = 2fit_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)] and its 95% CI
#>
#> estimates LCI UCI
#> π0 0.8474 0.7616 0.9042
#>
#> log-likelihood: -2633.903 (for 3 degree(s) of freedom)
#> AIC: 5273.806
#>
#> n= 2000 , number of events= 949newdata1 <- with(testiscancer,
expand.grid(event = 0, time_obs = seq(0.001,10,0.001)))
p_28 <- predict(object = fit_ad_tneh_nocov, newdata = newdata1)plot(p_28)
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 \[ Var(\hat{\pi}) = \dfrac{\partial \hat{\pi}}{\partial \theta} Var(\theta) \left(\dfrac{\partial \hat{\pi}}{\partial \theta}\right)^T \]
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:
\[ Var(TTC) = Var(g(\theta;z_i)) \simeq \left(\dfrac{\partial P(t|z_i;\theta)}{\partial t}_{|t = TTC}\right)^{-2} Var(P(t|z_i;\theta))_{|t=TTC} \]
testiscancer$age_crmin <- (testiscancer$age- min(testiscancer$age)) /sd(testiscancer$age)
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")
#> init 5 2.5 5.5 5 -4 lower 0 -5 1 0 -5 upper 100 100 100 100 100
#> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
#> Warning in diag(varcov): NAs introduits lors de la conversion automatique
#> non convergence with inititial values 1
#> next evaluation with initial values = 2
#> init 7.5 -1.25 7.75 2.5 3 lower 0 -5 1 0 -5 upper 100 100 100 100 100
#> next evaluation with initial values = 3
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 2.87737 0.24078 11.950 < 2e-16
#> alpha_1_(age_crmin) -0.50128 0.07498 -6.685 2.30e-11
#> beta 5.15005 1.04335 4.936 7.97e-07
#> tau0 3.25831 0.55340 5.888 3.91e-09
#> tau_1_(age_crmin) 3.46300 1.23871 2.796 0.00518
#>
#> Estimates and their 95% CI after back-transformation
#> estimates LCI UCI
#> alpha0 2.877 2.405 3.349
#> alpha_1_(age_crmin) 2.376 2.229 2.523
#> beta 5.150 3.105 7.195
#> tau0 3.258 2.174 4.343
#> tau_1_(age_crmin) 6.721 4.293 9.149
#>
#> Cured proportion exp[-(ζ0+ζ*Z)* B((α0+α*Z)β)] and its 95% CI
#> (For each Z of (age_crmin) the others are at reference level)
#>
#> estimates LCI UCI
#> π0z_alpha0 0.9675 0.7989 0.9949
#> π0z_alpha(age_crmin) 0.9601 0.8183 0.9689
#> π(age_crmin)z_alpha0 0.9341 0.6231 0.9900
#> π(age_crmin)z_alpha(age_crmin) 0.9227 0.6613 0.9356
#>
#> log-likelihood: -2544.1 (for 5 degree(s) of freedom)
#> AIC: 5098.2
#>
#> n= 2000 , number of events= 949 #mean of age
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)#max of age
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)# predictions at time 2 years depending on 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)plot of net survival for mean and maximum age
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), xlim = c(0,30),
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$tau[1],
lty = 2,
lwd = 2,
col = "blue")
abline(v = pred_agemean$TTC[1],
lty = 2,
lwd = 2,
col = "red")
abline(v = pred_agemax$tau[1],
lty = 1,
lwd = 2,
col = "blue")
abline(v = pred_agemax$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))
val_age <- seq(min(testiscancer$age_crmin),
max(testiscancer$age_crmin),
0.1) * sd(testiscancer$age) + min(testiscancer$age)
pred_age_val <- predict(object = fit_m1_ad_tneh, newdata = newdata3)par(mfrow=c(2,2))
plot(val_age,
pred_age_val$ex_haz, type = "l",
lty=1, lwd=2,
xlab = "age",
ylab = "excess hazard")
grid()
plot(val_age,
pred_age_val$netsurv, type = "l", lty=1,
lwd=2, xlab = "age", ylab = "net survival")
grid()
plot(val_age,
pred_age_val$pt_cure, type = "l", lty=1, lwd=2,
xlab = "age",
ylab = "Pi(t)")
grid()
par(mfrow=c(1,1))
#| 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"
)
#> init 5 5.5 5 -4 lower 0 1 0 -5 upper 100 100 100 100
#> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
#> Warning in diag(varcov): NAs introduits lors de la conversion automatique
#> non convergence with inititial values 1
#> next evaluation with initial values = 2
#> init 7.5 3.25 7.5 -11 lower 0 1 0 -5 upper 100 100 100 100
#> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
#> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
#> non convergence with inititial values 2
#> next evaluation with initial values = 3
#> init 2.5 7.75 2.5 3 lower 0 1 0 -5 upper 100 100 100 100
#> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
#> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
#> non convergence with inititial values 3
#> next evaluation with initial values = 4
#> init 3.75 4.375 6.25 -14.5 lower 0 1 0 -5 upper 100 100 100 100
#> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
#> Warning in diag(varcov_star): NAs introduits lors de la conversion automatique
#> non convergence with inititial values 4
#> next evaluation with initial values = 5
#> init 8.75 8.875 1.25 -0.5 lower 0 1 0 -5 upper 100 100 100 100
#> next evaluation with initial values = 6fit_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 each Z of (age_cr) the others are at reference level)
#>
#> Estimates LCI UCI
#> π0 0.794 0.6535 0.8909
#> π(age_cr) 0.739 0.4130 0.8868
#>
#> log-likelihood: -2610.768 (for 4 degree(s) of freedom)
#> AIC: 5229.537
#>
#> n= 2000 , number of events= 949newdata2 <- with(testiscancer,
expand.grid(event = 0,
time_obs = seq(0.001, 10, 0.001),
age_cr = c(-0.9577, -0.2751, 0.2849) ))
newdata2_1stqu <- newdata2[newdata2$age_cr==-0.9577,]
newdata2_2rdqu <- newdata2[newdata2$age_cr==-0.2751,]
newdata2_3rdqu <- newdata2[newdata2$age_cr==0.2849,]
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)oldpar <- par(no.readonly = FALSE)
par(mfrow = c(2,2))
plot(p1stqu,
main = "Excess hazard for age 20",
fun = "haz")
plot(p2rdqu,
fun = "haz",
main = "Excess hazard for age 51")
plot(p3rdqu,
fun = "haz",
main = "Excess hazard for age 69")
par(mfrow = c(1,1))
par(oldpar)
#> Warning in par(oldpar): le paramètre graphique "cin" ne peut être changé
#> Warning in par(oldpar): le paramètre graphique "cra" ne peut être changé
#> Warning in par(oldpar): le paramètre graphique "csi" ne peut être changé
#> Warning in par(oldpar): le paramètre graphique "cxy" ne peut être changé
#> Warning in par(oldpar): le paramètre graphique "din" ne peut être changé
#> Warning in par(oldpar): le paramètre graphique "page" ne peut être changé
#| 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"
)
#> init 5 2.5 5.5 5 lower 0 -5 1 0 upper 100 100 100 100
#> next evaluation with initial values = 2fit_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)] and its 95% CI
#>
#> estimates LCI UCI
#> π0 0.8181 0.4760 0.9485
#> π(age_cr) 0.7709 0.4124 0.7536
#>
#> log-likelihood: -2586.138 (for 4 degree(s) of freedom)
#> AIC: 5180.275
#>
#> n= 2000 , number of events= 949p4_28 <- predict(object = fit_ad_tneh_covalpha,
newdata = newdata2_1stqu)
p4_50 <- predict(object = fit_ad_tneh_covalpha,
newdata = newdata2_2rdqu)
p4_74 <- predict(object = fit_ad_tneh_covalpha,
newdata = newdata2_2rdqu)
#| echo: true
#| message: false
#| warning: false
#| include: true
#| fig.height: 15
#| fig.width: 15
par(mfrow = c(2,2))
plot(p4_28, fun = "pt_cure")
plot(p4_50, fun = "pt_cure")
plot(p4_74, fun = "pt_cure")
par(mfrow = c(1,1))
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