---
title: "Overview of survival endpoint design"
output: rmarkdown::html_vignette
bibliography: gsDesign.bib
vignette: >
  %\VignetteIndexEntry{Overview of survival endpoint design}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include=FALSE}
knitr::opts_chunk$set(
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  comment = "#>",
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  dpi = 96,
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  fig.width = 7.2916667,
  fig.asp = 0.618,
  fig.align = "center",
  out.width = "80%"
)

options(width = 58)
```

## Introduction

```{r, echo=FALSE, message=FALSE, warning=FALSE}
library(gsDesign)
library(tidyr)
library(knitr)
library(tibble)
```

This article/vignette provides a summary of functions in the ***gsDesign*** package supporting design and evaluation of trial designs for time-to-event outcomes.
We do not focus on detailed output options, but what numbers summarizing the design are based on.
If you are not looking for this level of detail and just want to see how to design a fixed or group sequential design for a time-to-event endpoint, see the vignette *Basic time-to-event group sequential design using gsSurv*.

The following functions support use of the very straightforward @Schoenfeld1981 approximation for 2-arm trials:

- `nEvents()`: number of events to achieve power or power given number of events with no interim analysis.
- `zn2hr()`: approximate the observed hazard ratio (HR) required to achieve a targeted Z-value for a given number of events.
- `hrn2z()`: approximate Z-value corresponding to a specified HR and event count.
- `hrz2n()`: approximate event count corresponding to a specified HR and Z-value.

The above functions do not directly support sample size calculations.
This is done with the @LachinFoulkes method. Functions include:

- `nSurv()`: More flexible enrollment scenarios; single analysis.
- `gsSurv()`: Group sequential design extension of `nSurv()`.
- `nSurvival()`: Sample size restricted to single enrollment rate, single analysis; this has been effectively replaced and generalized by `nSurv()` and `gsSurv()`.

Output for survival design information is supported in various formats:

- `gsBoundSummary()`: Tabular summary of a design in a data frame.
- `plot.gsDesign()`: Various plot summaries of a design.
- `gsHR()`: Approximate HR required to cross a bound.

## Schoenfeld approximation support

We will assume a hazard ratio $\nu < 1$ represents a benefit of experimental treatment over control.
We let $\delta = \log\nu$ denote the so-called *natural parameter* for this case.
Asymptotically the distribution of the Cox model estimate $\hat{\delta}$ under the proportional hazards assumption is (@Schoenfeld1981)
$$\hat\delta\sim \text{Normal}(\delta=\log\nu, (1+r)^2/nr).$$
The inverse of the variance is the statistical information:
$$\mathcal I = nr/(1 + r)^2.$$
Using a Cox model to estimate $\delta$, the Wald test for $\text{H}_0: \delta=0$ can be approximated with the asymptotic variance from above as:


$$Z_W\approx \frac{\sqrt {nr}}{1+r}\hat\delta=\frac{\ln(\hat\nu)\sqrt{nr}}{1+r}.$$


Also, we know that the Wald test $Z_W$ and a standard normal version of the logrank $Z$ are both asymptotically efficient and therefore asymptotically equivalent, at least under a local hypothesis framework.
We denote the *standardized effect size* as

$$\theta = \delta\sqrt r / (1+r)= \log(\nu)\sqrt r / (1+r).$$
Letting $\hat\theta = -\sqrt r/(1+r)\hat\delta$ and $n$ representing the number of events observed, we have
$$\hat \theta \sim \text{Normal}(\theta, 1/ n).$$
Thus, the standardized Z version of the logrank is approximately distributed as

$$Z\sim\text{Normal}(\sqrt n\theta,1).$$
Treatment effect favoring experimental treatment compared to control in this notation corresponds to a hazard ratio $\nu < 1$, as well as negative values of the standardized effect $\theta$, natural parameter $\delta$ and standardized Z-test.

### Power and sample size with `nEvents()`

Based on the above, the power for the logrank test when $n$ events have been observed is approximated by

$$P[Z\le z]=\Phi(z -\sqrt n\theta)=\Phi(z- \sqrt{nr}/(1+r)\log\nu).$$
Thus, assuming $n=100$ events and $\delta = \log\nu=-\log(.7)$, and $r=1$ (equal randomization) we approximate power for the logrank test when $\alpha=0.025$ as

```{r}
n <- 100
hr <- .7
delta <- log(hr)
alpha <- .025
r <- 1
pnorm(qnorm(alpha) - sqrt(n * r) / (1 + r) * delta)
```

We can compute this with `gsDesign::nEvents()` as:

```{r}
nEvents(n = n, alpha = alpha, hr = hr, r = r)
```

We solve for the number of events $n$ to see how many events are required to obtain a desired power

$$1-\beta=P(Z\ge \Phi^{-1}(1-\alpha))$$
with

$$n = \left(\frac{\Phi^{-1} (1-\alpha)+\Phi^{-1}(1-\beta)}{\theta}\right)^2
=\frac{(1+r)^2}{r(\log\nu)^2}\left({\Phi^{-1} (1-\alpha)+\Phi^{-1}(1-\beta)}\right)^2.$$
Thus, the approximate number of events required to power for HR=0.7 with $\alpha=0.025$ one-sided and power $1-\beta=0.9$ is

```{r}
beta <- 0.1
(1 + r)^2 / r / log(hr)^2 * ((qnorm(1 - alpha) + qnorm(1 - beta)))^2
```

which, rounding up, matches (with tabular output):

```{r}
nEvents(hr = hr, alpha = alpha, beta = beta, r = 1, tbl = TRUE) %>%
  kable()
```

The notation `delta` in the above table changes the sign for the standardized treatment effect $\theta$ in the above:

```{r}
theta <- delta * sqrt(r) / (1 + r)
theta
```

The `se` in the table is the estimated standard error for the log hazard ratio $\delta=\log\hat\nu$

```{r}
(1 + r) / sqrt(331 * r)
```

### Group sequential design

We can create a group sequential design for the above problem either with $\theta$ or with the fixed design sample size. The parameter `delta` in `gsDesign()` corresponds to standardized effect size with sign changed $-\theta$ in notation used above and by @JTBook, while the natural parameter, $\log(\text{HR})$ is in the parameter `delta1` passed to `gsDesign()`.
The name of the effect size is specified in `deltaname` and the parameter `logdelta = TRUE` indicates that `delta` input needs to be exponentiated to obtain HR in the output below.
This example code can be useful in practice.
We begin by passing the number of events for a fixed design in the parameter `n.fix` (continuous, not rounded) to adapt to a group sequential design.
By rounding to integer event counts with the `toInteger()` function we increase the power slightly over the targeted 90%.

```{r}
Schoenfeld <- gsDesign(
  k = 2,
  n.fix = nEvents(hr = hr, alpha = alpha, beta = beta, r = 1),
  delta1 = log(hr)
) %>% toInteger()
Schoenfeld %>%
  gsBoundSummary(deltaname = "HR", logdelta = TRUE, Nname = "Events") %>%
  kable(row.names = FALSE)
```

### Information based design

Exactly the same result can be obtained with the following, passing the standardized effect size `theta` from above to the parameter `delta` in `gsDesign()`.

```{r, eval=FALSE}
Schoenfeld <- gsDesign(k = 2, delta = -theta, delta1 = log(hr)) %>% toInteger()
```

We noted above that the asymptotic variance for $\hat\theta$ is $1/n$ which corresponds to statistical information $\mathcal I=n$ for the parameter $\theta$.
Thus, the value

```{r}
Schoenfeld$n.I
```

corresponds both to the number of events and the statistical information for the standardized effect size $\theta$ required to power the trial at the desired level.
Note that if you plug in the natural parameter $\delta= -\log\nu > 0$, then $n.I$ returns the statistical information for the log hazard ratio.

```{r}
gsDesign(k = 2, delta = -log(hr))$n.I
```

The reader may wish to look above to derive the exact relationship between events and statistical information for $\delta$.

### Approximating boundary characteristics

Another application of the @Schoenfeld1981 method is to approximate boundary characteristics of a design. 
We will consider `zn2hr()`, `gsHR()` and `gsBoundSummary()` to approximate the treatment effect required to cross design bounds.
`zn2hr()` is complemented by the functions `hrn2z()` and `hrz2n()`.
We begin with the basic approximation used across all of these functions in this section and follow with a sub-section with example code to reproduce some of what is in the table above.

We return to the following equation from above:

$$Z\approx Z_W\approx \frac{\sqrt {nr}}{1+r}\hat\delta=\frac{\ln(\hat\nu)\sqrt{nr}}{1+r}.$$
By fixing $Z=z, n$ we can solve for $\hat\nu$ from the above:

$$\hat{\nu} = \exp(z(1+r)/\sqrt{rn}).$$
By fixing $\hat\nu$ and $z$, we can solve for the corresponding number of events required:
$$ n = (z(1+r)/\log(\hat{\nu}))^2/r.$$

### Examples

We continue with the `Schoenfeld` example event counts:

```{r}
Schoenfeld$n.I
```

We reproduce the approximate hazard ratios required to cross efficacy bounds using the Schoenfeld approximations above:

```{r}
gsHR(
  z = Schoenfeld$upper$bound, # Z-values at bound
  i = 1:2, # Analysis number
  x = Schoenfeld, # Group sequential design from above
  ratio = r # Experimental/control randomization ratio
)
```

For the following examples, we assume $r=1$.

```{r}
r <- 1
```

1) Assuming a Cox model estimate $\hat\nu$ and a corresponding event count, approximately what Z-value (p-value) does this correspond to? We use the first equation above:

```{r}
hr <- .73 # Observed hr
events <- 125 # Events in analysis

z <- log(hr) * sqrt(events * r) / (1 + r)
c(z, pnorm(z)) # Z- and p-value
```

We replicate the Z-value with

```{r}
hrn2z(hr = hr, n = events, ratio = r)
```

2) Assuming an efficacy bound Z-value and event count, approximately what hazard ratio must be observed to cross the bound? We use the second equation above:

```{r}
z <- qnorm(.025)
events <- 120
exp(z * (1 + r) / sqrt(r * events))
```

We can reproduce this with `zn2hr()` by switching the sign of `z` above; note that the default is `ratio = 1` for all of these functions and often is not specified:

```{r}
zn2hr(z = -z, n = events, ratio = r)
```

3) Finally, if we want an observed hazard ratio $\hat\nu = .8$ to represent a positive result, how many events would be need to observe to achieve a 1-sided p-value of 0.025? assuming 2:1 randomization? We use the third equation above:

```{r}
r <- 2
hr <- .8
z <- qnorm(.025)
events <- (z * (1 + r) / log(hr))^2 / r
events
```

This is replicated with

```{r}
hrz2n(hr = hr, z = z, ratio = r)
```

## Lachin and Foulkes design

For the purpose of sample size and power for group sequential design, the @LachinFoulkes is recommended based on substantial evaluation not documented further here.
We try to make clear here what some of the strengths and weaknesses of both the @LachinFoulkes method as well as its implementation in the `gsDesign::nSurv()` (fixed design) and `gsDesign::gsSurv()` (group sequential) functions.
For historical and testing purposes, we also discuss use of the less flexible `gsDesign::nSurvival()` function that was independently programmed and can be used for some limited validations of `gsDesign::nSurv()`.

### Model assumptions

Some detail in specification comes With the flexibility allowed by the @LachinFoulkes method.
The model assumes

- Piecewise constant enrollment rates with a target fixed duration of enrollment; since inter-arrival times follow a Poisson process, the actual enrollment time to achieve the targeted enrollment is random.
- A fixed minimum follow-up period.
- Piecewise exponential failure rates for the control group.
- A single, constant hazard ratio for the experimental group relative to the control group.
- Piecewise exponential loss-to-follow-up rates.
- A stratified population.
- A fixed randomization ratio of experimental to control group assignment.

Other than the proportional hazards assumption, this allows a great deal of flexibility in trial design assumptions.
While @LachinFoulkes adjusts the piecewise constant enrollment rates proportionately to derive a sample size, `gsDesign::nSurv()` also enables the approach of @KimTsiatis which fixes enrollment rates and extends the final enrollment rate duration to power the trial; the minimum follow-up period is still assumed with this approach.
We do not enable the drop-in option proposed in @LachinFoulkes.

The two practical differences the @LachinFoulkes method has from the @Schoenfeld1981 method are:

1) By assuming enrollment, failure and dropout rates the method delivers sample size $N$ as well as events required.
2) The variance for the log hazard ratio $\hat\delta$ is computed differently and both a null ($\sigma^2_0$) and alternate hypothesis ($\sigma^2_1$) variance are incorporated through the formula
$$N = \left(\frac{\Phi^{-1}(1-\alpha)\sigma_0 + \Phi^{-1}(1-\beta)\sigma_1}{\delta}\right).$$
The null hypothesis is derived by averaging the alternate hypothesis rates, weighting according to the proportion randomized in each group.

### Fixed design

We will use the same hazard ratio 0.7 as for the @Schoenfeld1981 sample size calculations above.
We assume further that the trial will enroll for a constant rate for 12 months, have a control group median of 8 months (exponential failure rate $\lambda = \log(2)/8$), a dropout rate of 0.001 per month, and 16 months of minimum follow-up. As before, we assume a randomization ratio $r=1$, one-sided Type I error $\alpha=0.025$, 90% power which is equivalent to Type II error $\beta=0.1$.

```{r}
r <- 1 # Experimental/control randomization ratio
alpha <- 0.025 # 1-sided Type I error
beta <- 0.1 # Type II error (1 - power)
hr <- 0.7 # Hazard ratio (experimental / control)
controlMedian <- 8
dropoutRate <- 0.001 # Exponential dropout rate per time unit
enrollDuration <- 12
minfup <- 16 # Minimum follow-up
Nlf <- nSurv(
  lambdaC = log(2) / controlMedian,
  hr = hr,
  eta = dropoutRate,
  T = enrollDuration + minfup, # Trial duration
  minfup = minfup,
  ratio = r,
  alpha = alpha,
  beta = beta
)
cat(paste("Sample size: ", ceiling(Nlf$n), "Events: ", ceiling(Nlf$d), "\n"))
```

Recall that the @Schoenfeld1981 method recommended
`r ceiling(nEvents(hr=hr, alpha=alpha, beta=beta, ratio=r))` events. The two methods tend to yield very similar event count recommendations, but not the same. Other methods will also differ slightly; see @LachinFoulkes. Sample size recommendations can vary more between methods.

We can get the same result with the `nSurvival()` routine since only a single enrollment, failure and dropout rate is proposed for this example.

```{r}
lambda1 <- log(2) / controlMedian
nSurvival(
  lambda1 = lambda1,
  lambda2 = lambda1 * hr,
  Ts = enrollDuration + minfup,
  Tr = enrollDuration,
  eta = dropoutRate,
  ratio = r,
  alpha = alpha,
  beta = beta
)
```

### Group sequential design

Now we produce a group sequential design with a default asymmetric design with a futility bound based on $\beta$-spending.
We round interim event counts and round up the final event count to ensure the targeted power.

```{r}
k <- 2 # Total number of analyses
lfgs <- gsSurv(
  k = 2,
  lambdaC = log(2) / controlMedian,
  hr = hr,
  eta = dropoutRate,
  T = enrollDuration + minfup, # Trial duration
  minfup = minfup,
  ratio = r,
  alpha = alpha,
  beta = beta
) %>% toInteger()
lfgs %>%
  gsBoundSummary() %>%
  kable(row.names = FALSE)
```

Although we did not use the @Schoenfeld1981 for sample size, it is still used for the approximate HR at bound calculation above:

```{r}
events <- lfgs$n.I
z <- lfgs$upper$bound
zn2hr(z = z, n = events) # Schoenfeld approximation to HR
```

### Plotting

There are various plots available. The approximate hazard ratios required to cross bounds again use the @Schoenfeld1981 approximation. For a **ggplot2** version of this plot, use the default `base = FALSE`.

```{r, fig.asp=1}
plot(lfgs, pl = "hr", dgt = 2, base = TRUE)
```

### Event accrual

The variance calculations for the Lachin and Foulkes method are mostly determined by expected event accrual under the null and alternate hypotheses.
The null hypothesis characterized above is seemingly designed so that event accrual will be similar under each hypothesis.
You can see the expected events accrued at each analysis under the alternate hypothesis with:

```{r}
tibble::tibble(
  Analysis = 1:2,
  `Control events` = lfgs$eDC,
  `Experimental events` = lfgs$eDE
) %>%
  kable()
```

It is worth noting that if events accrue at the same rate in both the null and alternate hypothesis, then the expected duration of time to achieve the targeted events would be shortened.
Keep in mind that there can be many reasons events will accrue at a different rate than in the design plan.

The expected event accrual of events over time for a design can be computed as follows:

```{r, fig.asp=1}
Month <- seq(0.025, enrollDuration + minfup, .025)
plot(
  c(0, Month),
  c(0, sapply(Month, function(x) {
    nEventsIA(tIA = x, x = lfgs)
  })),
  type = "l", xlab = "Month", ylab = "Expected events",
  main = "Expected event accrual over time"
)
```

On the other hand, if you want to know the expected time to accrue 25% of the final events and what the expected enrollment accrual is at that time, you compute using:

```{r}
b <- tEventsIA(x = lfgs, timing = 0.25)
cat(paste(
  " Time: ", round(b$T, 1),
  "\n Expected enrollment:", round(b$eNC + b$eNE, 1),
  "\n Expected control events:", round(b$eDC, 1),
  "\n Expected experimental events:", round(b$eDE, 1), "\n"
))
```

For expected accrual of events without a design returned by `gsDesign::gsSurv()`, see the help file for `gsDesign::eEvents()`.

## References