Survival Endpoints: Hazard Ratio, Milestone Survival, and RMST

This vignette describes Regional Consistency Probability (RCP) calculations for three survival endpoint types: hazard ratio, milestone survival probability, and restricted mean survival time (RMST). All three endpoints share a common trial design framework, and event times are modelled by the exponential distribution.

Common trial design framework

For all survival endpoints, the following parameters define the trial design.

Parameter Symbol Description
t_a \(t_a\) Accrual period; patients enrol uniformly over \([0, t_a]\)
t_f \(t_f\) Follow-up period after accrual closes
tau \(\tau = t_a + t_f\) Total study duration (computed internally)
lambda \(\lambda\) True hazard rate under the alternative (exponential model)
lambda0 \(\lambda_0\) Historical control hazard rate
lambda_dropout \(\lambda_d\) Dropout hazard rate; NULL assumes no dropout (default)

The common parameters below are used throughout the examples in this vignette.

lambda  <- log(2) / 10   # treatment arm: median survival = 10
lambda0 <- log(2) / 5    # historical control: median survival = 5
t_a     <- 3             # accrual period
t_f     <- 10            # follow-up period
# True HR = lambda / lambda0 = 0.5

1. Hazard Ratio Endpoint

Statistical model

Under the exponential model with uniform accrual over \([0, t_a]\) and administrative censoring at \(\tau\), the expected event probability per patient is:

\[ \phi = \frac{\lambda}{\lambda + \lambda_d} \left[1 - \frac{e^{-(\lambda + \lambda_d)t_f} - e^{-(\lambda + \lambda_d)\tau}}{(\lambda + \lambda_d)\,t_a}\right] \]

The expected number of events in Region \(j\) is \(E_j = N_j\,\phi\), and the log-hazard ratio estimator has the approximate distribution:

\[ \log(\widehat{HR}_j) \sim N\!\left(\log(HR),\; \frac{1}{E_j}\right) \]

Consistency criteria

Method 1 (log-HR scale):

Letting \(\delta = \log(HR) < 0\), \(E_1 = N_1\phi\), and \(E_{-1} = (N - N_1)\phi\):

\[ \text{RCP}_{1,\log} = \Phi\!\left(\frac{-(1 - \pi)\,\delta} {\sqrt{(1 - \pi f_1)^2/E_1 + \{\pi(1 - f_1)\}^2/E_{-1}}}\right) \]

Method 1 (linear-HR scale) (Hayashi and Itoh 2018):

\[ \text{RCP}_{1,\text{lin}} = \Pr\!\left[\,(1 - \widehat{HR}_1) \geq \pi\,(1 - \widehat{HR})\,\right] \]

This is derived via the delta method. Define \(g = \log(\widehat{HR}_1) - \log(1 - \pi + \pi\,\widehat{HR})\). Under homogeneity:

\[ E[g] = \log(HR) - \log(1 - \pi + \pi\,HR) \]

\[ \mathrm{Var}(g) = \frac{1}{E_{\text{total}}} \left[ \frac{\{1 - \pi + \pi(1 - f_1)HR\}^2}{f_1\,(1 - \pi + \pi\,HR)^2} + \frac{\{\pi(1 - f_1)HR\}^2}{(1 - f_1)\,(1 - \pi + \pi\,HR)^2} \right] \]

and \(\text{RCP}_{1,\text{lin}} = \Phi(-E[g] / \sqrt{\mathrm{Var}(g)})\).

Method 2:

\[ \text{RCP}_2 = \prod_{j=1}^{J} \Pr\!\left(\widehat{HR}_j < 1\right) = \prod_{j=1}^{J} \Phi\!\left(-\delta\,\sqrt{E_j}\right) \]

Example

result_f <- rcp1armHazardRatio(
  lambda         = lambda,
  lambda0        = lambda0,
  Nj             = c(20, 80),
  t_a            = t_a,
  t_f            = t_f,
  lambda_dropout = NULL,
  PI             = 0.5,
  approach       = "formula"
)
print(result_f)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Time-to-Event (Hazard Ratio)
#> 
#>    Approach       : Closed-Form Solution
#>    True Hazard    : lambda  = 0.069315
#>    Control Hazard : lambda0 = 0.138629
#>    Sample Size    : Nj      = (20, 80)
#>    Total Size     : N       = 100
#>    Accrual Period : t_a     = 3.00
#>    Follow-up      : t_f     = 10.00
#>    Study Duration : tau     = 13.00
#>    Dropout Hazard : lambda_d = NA
#>    Threshold      : PI      = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall):
#>       Log-HR based    : 0.8935
#>       Linear-HR based : 0.9228
#>    Method 2 (All Regions Show Benefit):
#>       HR < 1          : 0.9892
result_s <- rcp1armHazardRatio(
  lambda         = lambda,
  lambda0        = lambda0,
  Nj             = c(20, 80),
  t_a            = t_a,
  t_f            = t_f,
  lambda_dropout = NULL,
  PI             = 0.5,
  approach       = "simulation",
  nsim           = 10000,
  seed           = 1
)
print(result_s)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Time-to-Event (Hazard Ratio)
#> 
#>    Approach       : Simulation-Based (nsim = 10000)
#>    True Hazard    : lambda  = 0.069315
#>    Control Hazard : lambda0 = 0.138629
#>    Sample Size    : Nj      = (20, 80)
#>    Total Size     : N       = 100
#>    Accrual Period : t_a     = 3.00
#>    Follow-up      : t_f     = 10.00
#>    Study Duration : tau     = 13.00
#>    Dropout Hazard : lambda_d = NA
#>    Threshold      : PI      = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall):
#>       Log-HR based    : 0.9019
#>       Linear-HR based : 0.9320
#>    Method 2 (All Regions Show Benefit):
#>       HR < 1          : 0.9924

Effect of dropout

Specifying lambda_dropout reduces the expected event probability \(\phi\), which in turn reduces all RCP values.

result_dropout <- rcp1armHazardRatio(
  lambda         = lambda,
  lambda0        = lambda0,
  Nj             = c(20, 80),
  t_a            = t_a,
  t_f            = t_f,
  lambda_dropout = 0.05,
  PI             = 0.5,
  approach       = "formula"
)
print(result_dropout)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Time-to-Event (Hazard Ratio)
#> 
#>    Approach       : Closed-Form Solution
#>    True Hazard    : lambda  = 0.069315
#>    Control Hazard : lambda0 = 0.138629
#>    Sample Size    : Nj      = (20, 80)
#>    Total Size     : N       = 100
#>    Accrual Period : t_a     = 3.00
#>    Follow-up      : t_f     = 10.00
#>    Study Duration : tau     = 13.00
#>    Dropout Hazard : lambda_d = 0.050000
#>    Threshold      : PI      = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall):
#>       Log-HR based    : 0.8656
#>       Linear-HR based : 0.8971
#>    Method 2 (All Regions Show Benefit):
#>       HR < 1          : 0.9793

Visualisation

plot_rcp1armHazardRatio(
  lambda    = lambda,
  lambda0   = lambda0,
  t_a       = t_a,
  t_f       = t_f,
  PI        = 0.5,
  N_vec     = c(20, 40, 100),
  J         = 3,
  nsim      = 5000,
  seed      = 1,
  base_size = 8
)

Grid plot of RCP versus f1 for a hazard ratio endpoint with HR = 0.5, showing Method 1 on log-HR and linear-HR scales and Method 2 across N = 20, 40, 100

2. Milestone Survival Endpoint

Statistical model

The treatment effect at evaluation time \(t_{\text{eval}}\) is:

\[ \delta = S(t_{\text{eval}}) - S_0 = e^{-\lambda\,t_{\text{eval}}} - S_0 \]

where \(S_0\) is the historical control survival rate at \(t_{\text{eval}}\), supplied by the user.

The asymptotic variance of the Kaplan-Meier estimator \(\hat{S}_j(t_{\text{eval}})\) is derived from Greenwood’s formula:

\[ \mathrm{Var}\!\left[\hat{S}_j(t)\right] \approx \frac{e^{-2\lambda t}}{N_j} \int_0^{t} \frac{\lambda\,e^{(\lambda + \lambda_d)u}}{G_a(u)}\,du \;=:\; \frac{v_{KM}(t)}{N_j} \]

where:

\[ G_a(u) = \begin{cases} 1 & 0 \leq u \leq t_f \\ (\tau - u)/t_a & t_f < u \leq \tau \end{cases} \]

Closed-form solution when \(t_{\text{eval}} \leq t_f\) (\(G_a(u) = 1\) throughout):

\[ v_{KM}(t) = e^{-2\lambda t} \cdot \frac{\lambda}{\lambda + \lambda_d} \left(e^{(\lambda + \lambda_d)\,t} - 1\right) \]

For \(\lambda_d = 0\) this reduces to the binomial variance \(S(t)(1 - S(t))\). When \(t_{\text{eval}} > t_f\), the integral is evaluated numerically via stats::integrate().

Consistency criteria

Method 1:

\[ \text{RCP}_1 = \Phi\!\left(\frac{(1 - \pi)\,\delta} {\sqrt{(1 - \pi f_1)^2\,v_{KM}/N_1 + \{\pi(1 - f_1)\}^2\,v_{KM}/(N - N_1)}}\right) \]

Method 2:

\[ \text{RCP}_2 = \prod_{j=1}^{J} \Phi\!\left(\frac{\delta}{\sqrt{v_{KM}/N_j}}\right) \]

Example

t_eval <- 8
S0     <- exp(-log(2) * t_eval / 5)
cat(sprintf("True S(%g) = %.4f,  S0 = %.4f,  delta = %.4f\n",
            t_eval, exp(-lambda * t_eval), S0,
            exp(-lambda * t_eval) - S0))
#> True S(8) = 0.5743,  S0 = 0.3299,  delta = 0.2445
result_f <- rcp1armMilestoneSurvival(
  lambda         = lambda,
  t_eval         = t_eval,
  S0             = S0,
  Nj             = c(20, 80),
  t_a            = t_a,
  t_f            = t_f,
  lambda_dropout = NULL,
  PI             = 0.5,
  approach       = "formula"
)
print(result_f)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Milestone Survival
#> 
#>    Approach       : Closed-Form Solution (Greenwood)
#>    True Hazard    : lambda  = 0.069315
#>    Sample Size    : Nj      = (20, 80)
#>    Total Size     : N       = 100
#>    Accrual Period : t_a     = 3.00
#>    Follow-up      : t_f     = 10.00
#>    Study Duration : tau     = 13.00
#>    Dropout Hazard : lambda_d = NA
#>    Threshold      : PI      = 0.5000
#>    Eval Time      : t_eval  = 8.00
#>    Control Surv   : S0      = 0.3299
#>    True Surv      : S_est   = 0.5743
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.8848
#>    Method 2 (All Regions > S0)     : 0.9865

Since \(t_{\text{eval}} = 8 \leq t_f = 10\), the closed-form solution is used (formula_type = "closed-form"). For \(t_{\text{eval}} > t_f\), numerical integration is applied automatically.

result_s <- rcp1armMilestoneSurvival(
  lambda         = lambda,
  t_eval         = t_eval,
  S0             = S0,
  Nj             = c(20, 80),
  t_a            = t_a,
  t_f            = t_f,
  lambda_dropout = NULL,
  PI             = 0.5,
  approach       = "simulation",
  nsim           = 10000,
  seed           = 1
)
print(result_s)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Milestone Survival
#> 
#>    Approach       : Simulation-Based (nsim = 10000)
#>    True Hazard    : lambda  = 0.069315
#>    Sample Size    : Nj      = (20, 80)
#>    Total Size     : N       = 100
#>    Accrual Period : t_a     = 3.00
#>    Follow-up      : t_f     = 10.00
#>    Study Duration : tau     = 13.00
#>    Dropout Hazard : lambda_d = NA
#>    Threshold      : PI      = 0.5000
#>    Eval Time      : t_eval  = 8.00
#>    Control Surv   : S0      = 0.3299
#>    True Surv      : S_est   = 0.5743
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.8873
#>    Method 2 (All Regions > S0)     : 0.9887

Visualisation

plot_rcp1armMilestoneSurvival(
  lambda    = lambda,
  t_eval    = t_eval,
  S0        = S0,
  t_a       = t_a,
  t_f       = t_f,
  PI        = 0.5,
  N_vec     = c(20, 40, 100),
  J         = 3,
  nsim      = 5000,
  seed      = 1,
  base_size = 8
)

Line plot of RCP versus f1 for a milestone survival endpoint at t_eval = 8, showing Method 1 and Method 2 across N = 20, 40, 100

3. RMST Endpoint

Statistical model

The restricted mean survival time (RMST) up to truncation time \(\tau^*\) is:

\[ \mu(\tau^*) = \int_0^{\tau^*} S(t)\,dt = \frac{1 - e^{-\lambda\tau^*}}{\lambda} \]

The asymptotic variance of the regional RMST estimator is:

\[ \mathrm{Var}\!\left[\hat{\mu}_j(\tau^*)\right] \approx \frac{v_{\text{RMST}}(\tau^*)}{N_j} \]

where:

\[ v_{\text{RMST}}(\tau^*) = \int_0^{\tau^*} \frac{e^{\lambda_d t}\left(1 - e^{-\lambda(\tau^* - t)}\right)^2}{\lambda\,G_a(t)}\,dt \]

Closed-form solution when \(\tau^* \leq t_f\), using \(A(r) = (e^{r\tau^*} - 1)/r\) (and \(A(0) = \tau^*\)):

\[ v_{\text{RMST}}(\tau^*) = \frac{1}{\lambda}\Bigl[ A(\lambda_d) - 2\,e^{-\lambda\tau^*}\,A(\lambda_d + \lambda) + e^{-2\lambda\tau^*}\,A(\lambda_d + 2\lambda) \Bigr] \]

For \(\lambda_d = 0\) this simplifies to:

\[ v_{\text{RMST}}(\tau^*) = \tau^* - \frac{2(1 - e^{-\lambda\tau^*})}{\lambda} + \frac{1 - e^{-2\lambda\tau^*}}{2\lambda} \]

When \(\tau^* > t_f\), the integral is split at \(t_f\) and evaluated via stats::integrate().

Consistency criteria

Method 1:

\[ \text{RCP}_1 = \Phi\!\left(\frac{(1 - \pi)\,(\mu_{\text{est}} - \mu_0)} {\sqrt{(1 - \pi f_1)^2\,v/N_1 + \{\pi(1 - f_1)\}^2\,v/(N - N_1)}}\right) \]

Method 2:

\[ \text{RCP}_2 = \prod_{j=1}^{J} \Phi\!\left(\frac{\mu_{\text{est}} - \mu_0}{\sqrt{v/N_j}}\right) \]

Example

tau_star <- 8
mu0      <- (1 - exp(-lambda0 * tau_star)) / lambda0
mu_est   <- (1 - exp(-lambda  * tau_star)) / lambda
cat(sprintf("True RMST = %.4f,  mu0 = %.4f,  delta = %.4f\n",
            mu_est, mu0, mu_est - mu0))
#> True RMST = 6.1408,  mu0 = 4.8339,  delta = 1.3069
result_f <- rcp1armRMST(
  lambda         = lambda,
  tau_star       = tau_star,
  mu0            = mu0,
  Nj             = c(20, 80),
  t_a            = t_a,
  t_f            = t_f,
  lambda_dropout = NULL,
  PI             = 0.5,
  approach       = "formula"
)
print(result_f)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Restricted Mean Survival Time (RMST)
#> 
#>    Approach       : Closed-Form Solution
#>    True Hazard    : lambda  = 0.069315
#>    Sample Size    : Nj      = (20, 80)
#>    Total Size     : N       = 100
#>    Accrual Period : t_a     = 3.00
#>    Follow-up      : t_f     = 10.00
#>    Study Duration : tau     = 13.00
#>    Dropout Hazard : lambda_d = NA
#>    Threshold      : PI      = 0.5000
#>    Trunc. Time    : tau*    = 8.00
#>    Control RMST   : mu0     = 4.8339
#>    True RMST      : mu_est  = 6.1408
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.8693
#>    Method 2 (All Regions > mu0)    : 0.9808
result_s <- rcp1armRMST(
  lambda         = lambda,
  tau_star       = tau_star,
  mu0            = mu0,
  Nj             = c(20, 80),
  t_a            = t_a,
  t_f            = t_f,
  lambda_dropout = NULL,
  PI             = 0.5,
  approach       = "simulation",
  nsim           = 10000,
  seed           = 1
)
print(result_s)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Restricted Mean Survival Time (RMST)
#> 
#>    Approach       : Simulation-Based (nsim = 10000)
#>    True Hazard    : lambda  = 0.069315
#>    Sample Size    : Nj      = (20, 80)
#>    Total Size     : N       = 100
#>    Accrual Period : t_a     = 3.00
#>    Follow-up      : t_f     = 10.00
#>    Study Duration : tau     = 13.00
#>    Dropout Hazard : lambda_d = NA
#>    Threshold      : PI      = 0.5000
#>    Trunc. Time    : tau*    = 8.00
#>    Control RMST   : mu0     = 4.8339
#>    True RMST      : mu_est  = 6.1408
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.8790
#>    Method 2 (All Regions > mu0)    : 0.9809

Visualisation

plot_rcp1armRMST(
  lambda    = lambda,
  tau_star  = tau_star,
  mu0       = mu0,
  t_a       = t_a,
  t_f       = t_f,
  PI        = 0.5,
  N_vec     = c(20, 40, 100),
  J         = 3,
  nsim      = 5000,
  seed      = 1,
  base_size = 8
)

Line plot of RCP versus f1 for an RMST endpoint with tau_star = 8, showing Method 1 and Method 2 across N = 20, 40, 100

Summary

Endpoint Effect parameter Benefit direction Variance basis Closed-form condition
Hazard Ratio \(\log(HR) = \log(\lambda/\lambda_0)\) (Method 1, log-HR scale); \(1 - HR = 1 - \lambda/\lambda_0\) (Method 1, linear-HR scale) \(\widehat{HR}_j < 1\) Expected events via \(\phi\) (Wu 2015) Always
Milestone Survival \(\delta = e^{-\lambda t_{\text{eval}}} - S_0\) \(\hat{S}_j(t) > S_0\) Greenwood’s formula \(t_{\text{eval}} \leq t_f\)
RMST \(\delta = \mu(\tau^*) - \mu_0(\tau^*)\) \(\hat{\mu}_j > \mu_0\) Squared survival difference integral \(\tau^* \leq t_f\)

References

Hayashi N, Itoh Y (2017). A re-examination of Japanese sample size calculation for multi-regional clinical trial evaluating survival endpoint. Japanese Journal of Biometrics, 38(2): 79–92. https://doi.org/10.5691/jjb.38.79

Wu J (2015). Sample size calculation for the one-sample log-rank test. Pharmaceutical Statistics, 14(1): 26–33. https://doi.org/10.1002/pst.1654