---
title: "Survival Endpoints: Hazard Ratio, Milestone Survival, and RMST"
output:
  rmarkdown::html_vignette:
    toc: true
    toc_depth: 3
    number_sections: false
vignette: >
  %\VignetteIndexEntry{Survival Endpoints: Hazard Ratio, Milestone Survival, and RMST}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse   = TRUE,
  comment    = "#>",
  fig.width  = 7,
  fig.height = 4.5,
  out.width  = "100%",
  dpi        = 96
)
library(SingleArmMRCT)
```

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.

```{r}
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

```{r}
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)
```

```{r}
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)
```

### Effect of dropout

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

```{r}
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)
```

### Visualisation

```{r fig.height=6, fig.alt="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"}
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
)
```

---

## 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

```{r}
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))
```

```{r}
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)
```

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.

```{r}
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)
```

### Visualisation

```{r fig.alt="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"}
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
)
```

---

## 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

```{r}
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))
```

```{r}
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)
```

```{r}
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)
```

### Visualisation

```{r fig.alt="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"}
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
)
```

---

## Summary

```{r echo=FALSE}
tbl <- data.frame(
  Endpoint = c("Hazard Ratio", "Milestone Survival", "RMST"),
  `Effect parameter` = c(
    "$\\log(HR) = \\log(\\lambda/\\lambda_0)$ (Method 1, log-HR scale); $1 - HR = 1 - \\lambda/\\lambda_0$ (Method 1, linear-HR scale)",
    "$\\delta = e^{-\\lambda t_{\\text{eval}}} - S_0$",
    "$\\delta = \\mu(\\tau^*) - \\mu_0(\\tau^*)$"
  ),
  `Benefit direction` = c(
    "$\\widehat{HR}_j < 1$",
    "$\\hat{S}_j(t) > S_0$",
    "$\\hat{\\mu}_j > \\mu_0$"
  ),
  `Variance basis` = c(
    "Expected events via $\\phi$ (Wu 2015)",
    "Greenwood's formula",
    "Squared survival difference integral"
  ),
  `Closed-form condition` = c(
    "Always",
    "$t_{\\text{eval}} \\leq t_f$",
    "$\\tau^* \\leq t_f$"
  ),
  check.names = FALSE
)
knitr::kable(tbl, align = "lllll")
```

---

## 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
