---
title: "Multi-stage drop-the-losers multi-arm clinical trial design for a Poisson distributed primary outcome"
date: "`r format(Sys.time(), '%d/%m/%Y')`"
params:
  alpha:         NA
  beta:          NA
  delta0:        NA
  delta1:        NA
  integer:       NA
  J:             NA
  K:             NA
  Kv:            NA
  lambda0:       NA
  maxN:          NA
  n10:           NA
  n1:            NA
  opchar:        NA
  plots:         NA
  power:         NA
  ratio_type:    NA
  ratio_init:    NA
  ratio:         NA
  swss:          NA
  equal_error:   NA
  equal_power:   NA
  equal_other:   NA
  shifted_power: NA
---

```{r, include = F}
knitr::opts_chunk$set(echo = FALSE, results = "asis")
```

# Design setting

The trial will be designed to compare $K$ experimental treatments to a shared control arm. Response $X_{ik}$, from patient $i=1,\dots,n_k$ in arm $k=0,\dots,K$, will be assumed to be distributed as $X_{ik} \sim Po(\lambda_k)$. Then, the hypotheses to be tested will be:
$$ H_k : \tau_k = \lambda_k - \lambda_0 \le 0,\ k=1,\dots,K.$$
The *global null hypothesis*, $H_G$, will be:
$$ \lambda_0 = \cdots = \lambda_K. $$
The *global alternative hypothesis*, $H_A$, will be:
$$ \lambda_1 = \cdots = \lambda_K = \lambda_0 + \delta_1. $$
The *least favourable configuration* for experimental arm $k$, $LFC_k$, will be:
$$ \lambda_k = \lambda_0 + \delta_1,\ \lambda_1 = \cdots = \lambda_{k-1} = \lambda_{k+1} = \cdots = \lambda_K = \lambda_0 + \delta_0. $$
Here, $\delta_1$ and $\delta_0$ are *interesting* and *uninteresting* treatment
effects respectively.

# Inputs

```{r, echo = F, results = "asis"}
alpha         <- params$alpha
beta          <- params$beta
delta0        <- params$delta0
delta1        <- params$delta1
integer       <- params$integer
J             <- params$J
K             <- params$K
Kv            <- params$Kv
lambda0       <- params$lambda0
maxN          <- params$maxN
n10           <- params$n10
n1            <- params$n1
opchar        <- params$opchar
plots         <- params$plots
power         <- params$power
ratio_type    <- params$ratio_type
ratio_init    <- params$ratio_init
ratio         <- params$ratio
swss          <- params$swss
equal_error   <- params$equal_error
equal_power   <- params$equal_power
equal_other   <- params$equal_other
shifted_power <- params$shifted_power
```

```{r input_text, echo = F, results = "asis"}
if (power == "marginal") {
  power_scenario_text <-
    "each of their respective least favourable configurations"
  power_type_text     <- "marginal power for each null hypothesis"
} else {
  power_scenario_text <- "the global alternative hypothesis"
  power_type_text     <- "disjunctive power"
}

if (ratio_type == "equal_all") {
  ratio_text          <- "the same as the control arm"
} else if (ratio_type == "equal_exp") {
  ratio_text          <- paste0(ratio_init[1], " times that of the control arm")
} else {
  ratio_text          <- paste0(round(1/sqrt(K), 3),
                                " times that of the control arm")
}

if (integer) {
  integer_text        <- "will"
} else {
  integer_text        <- "will not"
}

if (plots) {
  plot_text           <- "will"
} else {
  plot_text           <- "will not"
}
```

```{r output_text, echo = F, results = "asis"}
FWER                  <- round(opchar$FWERI1[1], 3)
if (power == "marginal") {
  power_type_text     <- "minimum marginal power"
  power_value         <-
    round(min(diag(as.matrix(opchar[-(1:2), (K + 4):(2*K + 3)]))), 3)
} else {
  power_scenario_text <- "the global alternative hypothesis."
  if (power == "conjunctive") {
    power_type_text   <- "conjunctive power"
    power_value       <- round(opchar$Pcon[2], 3)
  } else if (power == "disjunctive") {
    power_type_text   <- "disjunctive power"
    power_value       <- round(opchar$Pdis[2], 3)
  }
}
if (swss == "fixed") {
  swss_text  <- "is"
  swss_text2 <- round(n1, 3)
} else {
  swss_text  <- "to the control arm is"
  swss_text2 <- round(n10, 3)
}
```

#### Inputs

The following choices were made:

- $K = \boldsymbol{`r K`}$ experimental treatments will initially be included in the trial.
- $J = \boldsymbol{`r J`}$ stages will be included in the trial.
- $\boldsymbol{K} = \boldsymbol{(`r paste(Kv, collapse = ", ")`)}$ experimental treatments will be present in each stage.
- The stage-wise sample size will be __`r swss`__.
- A family-wise error-rate of $\alpha = \boldsymbol{`r alpha`}$ will be used.
- The __`r power_type_text`__ will be controlled to level $1-\beta = \boldsymbol{`r 1 - beta`}$ under __`r power_scenario_text`__.
- The event rate in the control arm will be assumed to be: $\lambda_0 = \boldsymbol{`r lambda0`}$.
- The interesting and uninteresting treatment effects will be: $\delta_1 = \boldsymbol{`r round(delta1, 3)`}$ and $\delta_0 = \boldsymbol{`r round(delta0, 3)`}$ respectively.
- The target allocation to each of the experimental arms will be: __`r ratio_text`__.
- The sample size in each arm __`r integer_text`__ be required to be an integer.
- Plots __`r plot_text`__ be produced.

#### Outputs

- The required sample size is: $\boldsymbol{`r round(maxN, 3)`}$.
- The stage-wise sample size __`r swss_text`__: __`r swss_text2`__.
- The realised allocation ratios to the experimental arms are: $(r_1,\dots,r_K) = \boldsymbol{(`r paste(rep(round(ratio, 3), K), collapse = ", ")`)}$.
- The maximum familywise error-rate is: $\boldsymbol{`r FWER`}$.
- The __`r power_type_text`__ is: $\boldsymbol{`r power_value`}$.

# Operating characteristcs

```{r, echo = F, results = "asis"}
library(knitr)
kable(round(opchar[, c(1:(K + 1), (2*K + 4):(4*K + 3))]), digits = 3)
kable(round(opchar[, 1:(2*K + 3)]), digits = 3)
kable(round(opchar[, -((K + 2):(4*K + 3))]), digits = 3)
```

# Plots

```{r, echo = F, results = "asis", dpi = 600}
if (params$plots) {
  print(params$equal_error)
  cat("\n \n \n \n \n \n")
  print(params$equal_power)
  cat("\n \n \n \n \n \n")
  print(params$equal_other)
  cat("\n \n \n \n \n \n")
  print(params$shifted_power)
} else {
  cat("Not chosen for output: select the box next to 'Plot power curves' to",
      "add plots to the report.")
}
```
