---
title: "Vaccine efficacy trial design"
output:
  rmarkdown::html_document:
    code_folding: show
bibliography: gsDesign.bib
vignette: >
  %\VignetteIndexEntry{Vaccine efficacy trial design}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include=FALSE}
knitr::opts_chunk$set(
  collapse = FALSE,
  comment = "#>",
  dev = "ragg_png",
  dpi = 96,
  fig.retina = 1,
  fig.width = 7.2916667,
  fig.asp = 0.618,
  fig.align = "center",
  out.width = "80%"
)

options(width = 58)
```

```{r, message=FALSE, warning=FALSE, class.source="fold-hide"}
library(gsDesign)
library(gt)
library(dplyr)
```

# Introduction

This article explores a method of approximating a design using the exact binomial method of @ChanBohidar by a time-to-event design using the method of @LachinFoulkes.
This allows use of spending functions to derive boundaries for the exact method.
The time-to-event design can not only be used to set boundaries for the @ChanBohidar method, but to allow specification of enrollment duration and study duration to determine enrollment rates and sample size required.
This vignette also illustrates the concept of super-superiority often used in prevention studies.

# Parameterization

We begin with the assumption that we will a require a large sample size due to an endpoint with a small incidence rate.
This could apply to a vaccine study or other prevention study with a relatively small number of events expected.

## Exact binomial approach

Paralleling the notation of @ChanBohidar, we assume $N_C, P_C$ to be binomial sample size and probability of an event for each participant assigned to control; for the experimental treatment group, these are labelled $N_E, P_E$.
Vaccine efficacy is defined as

$$\pi = 1 - P_E/P_C.$$

The parameter $\pi$ is often labelled as $VE$ for vaccine efficacy.
Taking into account the randomization ratio $r$ (experimental / control) the approximate probability that any given event is in the experimental group is

$$
\begin{aligned}
p &= rP_E/(rP_E+ P_C)\\
  &= r/(r + P_C/P_E)\\
  &= r/(r + (1-\pi)^{-1}).
\end{aligned}
$$

This can be inverted to obtain

$$\pi = 1 - \frac{1}{r(1/p-1)}. $$

For our example of interest, we begin with an alternate hypothesis vaccine efficacy $\pi_1 = 0.7$ and experimental:control randomization ratio $r=3$.
This converts to an alternate hypothesis (approximate) probability that any event is in the experimental group of

```{r}
pi1 <- .7
ratio <- 3
p1 <- ratio / (ratio + 1 / (1 - pi1))
p1
```

We use the inversion formula to revert this to $\pi_1 = 0.7$

```{r}
1 - 1 / (ratio * (1 / p1 - 1))
```

Letting the null hypothesis vaccine efficacy be $\pi_0 = 0.3$, our exact binomial null hypothesis probability that an event is in the experimental group is

```{r}
pi0 <- .3
p0 <- ratio / (ratio + 1 / (1 - pi0))
p0
```

We also translate several vaccine efficacy values to proportion of events in the experimental group:

```{r}
ve <- c(.5, .6, .65, .7, .75, .8)
prob_experimental <- ratio / (ratio + 1 / (1 - ve))
tibble(VE = ve, "P(Experimental)" = prob_experimental) %>%
  gt() %>%
  tab_options(data_row.padding = px(1)) %>%
  fmt_number(columns = 2, decimals = 3)
```



Chapter 12 of @JTBook walks through how to design and analyze such a study using a fixed or group sequential design. This may have advantages and disadvantages compared to what is proposed here.
However, the time-to-event approximation gives not only proposed bounds, but also sample size and study duration approximations.

# The time-to-event approach

For a time-to-event formulation with exponential failure rates $\lambda_C$ for control and $\lambda_E$ for experimental group assigned participants, we would define

$$\pi = 1 - \lambda_E / \lambda_C$$

which is 1 minus the hazard ratio often used in time-to-event studies.
In the following we examine how closely the time-to-event method using asymptotic distributional assumptions can approximate an appropriate exact binomial design.
We will also define a planned number of events at each of $K$ planned analyses by $D_k, 1\le k\le K$.

# Generating a design

We begin by specifying parameters.
The `alpha` and `beta` parameters will not be met exactly due to the discrete group sequential probability calculations performed.
Thus, you may need to adjust these parameters slightly to ensure your final design operating characteristics are within a targeted range.
The current version includes only designs that use non-binding futility bounds or no futility bounds.
The design is generated by first using asymptotic theory for a time-to-event design with specified spending functions.
This design is then adapted to a design using the exact binomial method of @ChanBohidar.
The randomization ratio (experimental/control) was assumed to be 3:1 as in the @SPUTNIK2021 trial.

```{r}
alpha <- 0.023 # Type I error; this was adjusted from .025 to ensure Type I error control
beta <- 0.09 # Type II error (1 - power); this was reduced from .1 to .09 to ensure power
k <- 3 # number of analyses in group sequential design
timing <- c(.45, .7) # Relative timing of interim analyses compared to final
sfu <- sfHSD # Efficacy bound spending function (Hwang-Shih-DeCani)
sfupar <- -4 # Parameter for efficacy spending function
sfl <- sfLDOF # Futility bound spending function (O'Brien-Fleming-like here)
sflpar <- 1 # Futility bound spending function parameter; this is the standard O'Brien-Fleming
timename <- "Month" # Time unit
failRate <- .002 # Exponential failure rate
dropoutRate <- .0001 # Exponential dropout rate
enrollDuration <- 8 # Enrollment duration
trialDuration <- 24 # Planned trial duration
VE1 <- .7 # Alternate hypothesis vaccine efficacy
VE0 <- .3 # Null hypothesis vaccine efficacy
ratio <- 3 # Experimental/Control enrollment ratio
test.type <- 4 # 1 for one-sided, 4 for non-binding futility
```

## The time-to-event design

Now we generate the design. If resulting alpha and beta do not satisfy requirements, adjust parameters above until a satisfactory result is obtained.

```{r}
# Derive Group Sequential Design
# This determines final sample size
x <- gsSurv(
  k = k, test.type = test.type, alpha = alpha, beta = beta, timing = timing,
  sfu = sfu, sfupar = sfupar, sfl = sfl, sflpar = sflpar,
  lambdaC = failRate, eta = dropoutRate,
  # Translate vaccine efficacy to HR
  hr = 1 - VE1, hr0 = 1 - VE0,
  R = enrollDuration, T = trialDuration,
  minfup = trialDuration - enrollDuration, ratio = ratio
)
```

Now we convert this to a design with integer event counts at analyses.
This is achieved by rounding interim analysis event counts from the above design and rounding up the final analysis event count. This will result in a slight change in event fractions at interim analyses as well as a slight change from the targeted 90% power.
We now explain the rationale behind the spending function choices.
Recall that the hazard ratio (HR) is 1 minus the VE.
The `~HR at bound` represents the approximate hazard ratio required to cross a bound.
Thus, small HR's at the interim analyses along with small cumulative efficacy spending suggest crossing an interim efficacy bound would provide a result strong enough to potentially justify the new treatment.
The hazard ratio of ~0.7 (VE ~ 0.3) for the interim 1 futility bound mean that the efficacy trend would be no better than the null hypothesis if the futility bound were crossed.
The second analysis futility bound with approximate VE of 0.5 would be worth discussion with a data monitoring committee as well as other planners for the trial; a custom spending function \citep{AndClark} could be used to set both the first and second interim bounds to desired levels.

```{r}
xx <- toInteger(x)
gsBoundSummary(xx,
  tdigits = 1, logdelta = TRUE, deltaname = "HR", Nname = "Events",
  exclude = c("B-value", "CP", "CP H1", "PP")
) %>%
  gt() %>%
  tab_header(
    title = "Initial group sequential approximation",
    subtitle = "Integer event counts at analyses"
  ) %>%
  tab_options(data_row.padding = px(1))
```

A textual summary for the design is:

```{r, results='asis'}
cat(summary(xx, timeunit = "months"))
```

# Converting to an exact binomial design

We now convert this to an exact binomial design.
The bound counts are described in this initial table displayed.
`N` is the total event count, `a` the maximum number of events in the experimental group to cross the efficacy bound.
For example, if 13 or fewer of 32 events at interim 1 are in the experimental group the efficacy bound has been crossed.
The futility bound is in `b`; at the first interim, if 22 or more of 32 total events are in the experimental group then the futility bound would be crossed and the alternate hypothesis could be rejected.
The second and third tables below give probabilities of crossing the upper (futility) and lower (efficacy) bounds under the null (`Theta` = 0.6774`) and alternate (`Theta` = 0.4737) hypotheses, respectively; these calculations are done under the exact binomial distribution assumptions.
Total Type I error of 0.0187 (third table, next to last row) is below the targeted 0.025.
Total power of 0.9145 (third table, last row) is above the targeted 90%.

```{r}
xb <- toBinomialExact(x)
```

## Combined summary table

```{r}
# Function to print summary table for VE design
veTable <- function(xbDesign, tteDesign, ve) {
  # Analysis, N, Time, Total Cases, Success(Cases, VE, VE lower CI, Spend), Futility (Cases, VE, Spend), Type I Error, Power table
  ratio <- tteDesign$ratio
  prob_experimental <- ratio / (ratio + 1 / (1 - ve))
  power_table <- gsBinomialExact(
    k = xbDesign$k, theta = prob_experimental,
    n.I = xbDesign$n.I, a = xbDesign$lower$bound,
    b = xbDesign$upper$bound
  )$lower$prob
  # Cumulative sum within rows
  power_table <- apply(power_table, 2, cumsum)
  colnames(power_table) <- paste(ve * 100, "%", sep = "")
  out_tab <- tibble(
    Analysis = 1:tteDesign$k,
    Time = tteDesign$T,
    N = as.vector(round(tteDesign$eNC + tteDesign$eNE)),
    Cases = xbDesign$n.I,
    Success = xb$lower$bound,
    Futility = xb$upper$bound,
    ve_efficacy = 1 - 1 / (ratio * (xbDesign$n.I / xbDesign$lower$bound - 1)), # Efficacy bound
    ve_futility = 1 - 1 / (ratio * (xbDesign$n.I / xbDesign$upper$bound - 1)), # Futility bound
    alpha = as.vector(cumsum(
      gsBinomialExact(
        k = k, theta = xbDesign$theta[1], n.I = xbDesign$n.I,
        a = xbDesign$lower$bound, b = xbDesign$n.I + 1
      )$lower$prob
    )),
    beta = as.vector(cumsum(xbDesign$upper$prob[, 2]))
  )
  out_tab <- cbind(out_tab, power_table)
}
```

```{r}
veTable(xb, x, ve) %>%
  gt() %>%
  fmt_number(columns = 2, decimals = 1) %>%
  fmt_number(columns = c(7:8, 11:16), decimals = 2) %>%
  fmt_number(columns = 9:10, decimals = 4) %>%
  tab_spanner(label = "Experimental Cases at Bound", columns = 5:6, id = "cases") %>%
  tab_spanner(label = "Power by Vaccine Efficacy", columns = 11:16, id = "power") %>%
  tab_spanner(label = "Error Spending", columns = 9:10, id = "spend") %>%
  tab_spanner(label = "Vaccine Efficacy at Bound", columns = 7:8, id = "vebound") %>%
  cols_label(
    ve_efficacy = "Efficacy",
    ve_futility = "Futility"
  ) %>%
  tab_footnote(
    footnote = "Cumulative spending at each analysis",
    locations = cells_column_spanners(spanners = "spend")
  ) %>%
  tab_footnote(
    footnote = "Experimental case counts to cross between success and futility counts do not stop trial",
    locations = cells_column_spanners(spanners = "cases")
  ) %>%
  tab_footnote(
    footnote = "Exact vaccine efficacy required to cross bound",
    locations = cells_column_spanners(spanners = "vebound")
  ) %>%
  tab_footnote(
    footnote = "Cumulative power at each analysis by underlying vaccine efficacy",
    locations = cells_column_spanners(spanners = "power")
  ) %>%
  tab_footnote(
    footnote = "Efficacy spending ignores non-binding futility bound",
    location = cells_column_labels(columns = alpha)
  ) %>%
  tab_header("Design Bounds and Operating Characteristics")
```

Next, we look at $\alpha$- and $\beta$-spending for the time-to-event design to compare to the exact bound restricted by the discrete possible counts at bounds.
For both $\alpha$- and $\beta$-spending, the exact binomial design has no more spending at each analysis than allowed by each spending function.

```{r}
# Cumulative beta-spending at lower bounds
cumsum(xx$lower$spend) # Asymptotic design
# Cumulative alpha-spending at efficacy bounds
cumsum(gsProbability(k = xx$k, n.I = xx$n.I, a = rep(-20, xx$k), b = xx$upper$bound, theta = 0)$upper$prob)
```

# Summary

We have provided an extended example to show that a @ChanBohidar exact binomial using spending function bounds can be derived in a two-step process that delivers sample size and bounds by 1) deriving a related time-to-event design using asymptotic methods and then 2) converting to an exact binomial design.
Adjustments were made to target Type I and Type II error probabilities in the asymptotic approximation to ensure the exact binomial Type I and Type II error rates were achieved. The method seems a reasonable and straightforward approach to develop a complete design.

# References
