In mjg211/multiarm: Design of single- and multi-stage multi-arm clinical trials

knitr::opts_chunk$set(echo = F, 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

alpha       <- params$alpha
beta        <- params$beta
delta0      <- params$delta0
delta1      <- params$delta1
e           <- params$e
efix        <- params$efix
f           <- params$f
ffix        <- params$ffix
integer     <- params$integer
J           <- params$J
K           <- params$K
lambda0     <- params$lambda0
lower       <- params$lower
maxN        <- params$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
stopping    <- params$stopping
swss        <- params$swss
upper       <- params$upper

if (power == "marginal") {
  power_type_text     <- "marginal power for each null hypothesis"
  power_scenario_text <-
    "each of their respective least favourable configurations"
} else {
  power_scenario_text <- "the global alternative hypothesis"
  if (power == "conjunctive") {
    power_type_text   <- "conjunctive power"
  } else if (power == "disjunctive") {
    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 (lower == "fixed") {
  lower_prelim <- "fixed value"
} else if (lower == "obf") {
  lower_prelim <- "O'Brien-Fleming"
} else if (lower == "pocock") {
  lower_prelim <- "Pocock"
} else if (lower == "triangular") {
  lower_prelim <- "triangular"
}

if (upper == "fixed") {
  upper_prelim <- "fixed value"
} else if (upper == "obf") {
  upper_prelim <- "O'Brien-Fleming"
} else if (upper == "pocock") {
  upper_prelim <- "Pocock"
} else if (upper == "triangular") {
  upper_prelim <- "triangular"
}

if (lower == "fixed") {
  lower_text          <- paste0(lower_prelim, ", with a fixed value of ", ffix)
} else {
  lower_text          <- lower_prelim
}

if (upper == "fixed") {
  upper_text          <- paste0(upper_prelim, ", with a fixed value of ", efix)
} else {
  upper_text          <- upper_prelim
}

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

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

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.
A maximum of $J = \boldsymbol{r J}$ stages will be allowed in the trial.
A r stopping stopping rule will be used.
The stage-wise sample size will be r swss.
The lower stopping boundaries will be r lower_text.
The upper stopping boundaries will be r upper_text.
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 maximum 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: $\boldsymbol{r power_value}$.
The lower stopping boundaries are: $\boldsymbol{r paste0("(", paste(round(f, 3), collapse = ", "), ")")}$.
The upper stopping boundaries are: $\boldsymbol{r paste0("(", paste(round(e, 3), collapse = ", "), ")")}$.

Operating characteristcs

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

if (params$plots) {
  print(params$boundaries)
  cat("\n \n \n \n \n \n")
  print(params$equal_error)
  cat("\n \n \n \n \n \n")
  print(params$equal_power)
  cat("\n \n \n \n \n \n")
  print(params$equal_other)
  cat("\n \n \n \n \n \n")
  print(params$equal_sample_size)
  cat("\n \n \n \n \n \n")
  print(params$shifted_power)
  cat("\n \n \n \n \n \n")
  print(params$shifted_sample_size)
  cat("\n \n \n \n \n \n")
  print(params$pmf_N)
} else {
  cat("Not chosen for output: select the box next to 'Plot power curves' to",
      "add plots to the report.")
}