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 $X_{ik} \sim Bern(\pi_k)$. Then, the hypotheses to be tested will be: $$ H_k : \tau_k = \pi_k - \pi_0 \le 0,\ k=1,\dots,K.$$ The global null hypothesis, $H_G$, will be: $$ \pi_0 = \cdots = \pi_K. $$ The global alternative hypothesis, $H_A$, will be: $$ \pi_1 = \cdots = \pi_K = \pi_0 + \delta_1. $$ The least favourable configuration for experimental arm $k$, $LFC_k$, will be: $$ \pi_k = \pi_0 + \delta_1,\ \pi_1 = \cdots = \pi_{k-1} = \pi_{k+1} = \cdots = \pi_K = \pi_0 + \delta_0. $$ Here, $\delta_1$ and $\delta_0$ are interesting and uninteresting treatment effects respectively.

Inputs

K              <- params$K
alpha          <- params$alpha
beta           <- params$beta
pi0            <- params$pi0
delta1         <- params$delta1
delta0         <- params$delta0
ratio_type     <- params$ratio_type
ratio_init     <- params$ratio_init
ratio_scenario <- params$rario_scenario
ratio          <- params$ratio
correction     <- params$correction
power          <- params$power
integer        <- params$integer
N              <- params$large_N
n              <- params$small_n
opchar         <- params$opchar
gamma          <- params$gamma
gammaO         <- params$gammaO
plots          <- params$plots
equal_error    <- params$equal_error
equal_power    <- params$equal_power
equal_other    <- params$equal_other
shifted_power        <- params$shifted_power
delta          <- params$delta

if (correction == "bonferroni") {
  correction_text     <- "Bonferroni's correction"
} else if (correction == "dunnett") {
  correction_text     <- "Dunnett's correction"
} else if (correction == "none") {
  correction_text     <- "no multiple comparison correction"
} else if (correction == "sidak") {
  correction_text     <- "Sidak's correction"
} else if (correction == "benjamini_hochberg") {
  correction_text     <- "the Benjamini-Hochberg correction"
} else if (correction == "benjamini_yekutieli") {
  correction_text     <- "the Benjamini-Yekutieli correction"
} else if (correction == "hochberg") {
  correction_text     <- "Hochberg's correction"
} else if (correction == "holm_bonferroni") {
  correction_text     <- "the Holm-Bonferroni correction"
} else if (correction == "holm_sidak") {
  correction_text     <- "the Holm-Sidak correction"
} else if (correction == "step_down_dunnett") {
  correction_text     <- "the step-down version of Dunnett's correction"
}

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 if (ratio_type == "unequal") {
  ratio_text          <- paste0(paste(ratio_init[1:K], collapse = ", "),
                                " times that of the control arm")
} else if (ratio_type == "root_K") {
  ratio_text          <- paste0(round(1/sqrt(K), 3),
                                " times that of the control arm")
} else {
  ratio_text          <- paste0("chosen for ", ratio_type, "-optimality under",
                                " the ", ratio_scenario, "-response rate ",
                                "scenario")
}

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

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

The following choices were made:

$K = \boldsymbol{r K}$ experimental treatments will be included in the trial.
A significance level of $\alpha = \boldsymbol{r alpha}$ will be used, in combination with r correction_text.
The response rate in the control arm will be assumed to be: $\pi_0 = \boldsymbol{r pi0}$.
The r power_type_text will be controlled to level $1-\beta = \boldsymbol{r 1 - beta}$ under r power_scenario_text.
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

FWER                  <- round(opchar$FWERI1[1], 3)
if (correction %in% c("benjamini_hochberg", "benjamini_yekutieli", "hochberg",
                      "holm_bonferroni", "holm_sidak", "step_down_dunnett")) {
  punc_text_3         <- "s"
  gamma_text          <- paste0("(", paste(round(gammaO, 3),
                                           collapse = ", "), ")")
} else {
  punc_text_3         <- ""
  gamma_text          <- round(gamma, 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)
  }
}

The total required sample size is: $N = \boldsymbol{r round(N, 3)}$.
The required sample size in each arm is: $(n_0,\dots,n_K) = \boldsymbol{(r paste(round(n, 3), collapse = ", "))}$.
Therefore, the realised allocation ratios to the experimental arms are: $(r_1,\dots,r_K) = \boldsymbol{(r paste(round(ratio, 3), collapse = ", "))}$.
The maximum familywise error-rate is: $\boldsymbol{r FWER}$.
The r power_type_text is: $\boldsymbol{r power_value}$.
The following critical thresholdr punc_text_3 should be used with the chosen multiple comparison correction: $\boldsymbol{r gamma_text}$.

Operating characteristcs

library(knitr)
kable(round(params$data[, c(1:(K + 1), (2*K + 4):(4*K + 3))]), digits = 3)
kable(round(params$data[, 1:(2*K + 3)]), digits = 3)
kable(round(params$data[, -((K + 2):(4*K + 3))]), digits = 3)

Plots

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.")
}