R/ris.R

In RTSA: 'Trial Sequential Analysis' for Error Control and Inference in Sequential Meta-Analyses

#' Minimum number of trials needed for a specific level of power
#'
#' Calculates minimum number of trials needed to achieve power in a meta-analysis with
#' heterogeneity. 
#'
#' @param outcome Metric of interest, options include "RR" (relative risk), "OR" (odds ratio), "RD" (risk difference) and "MD" (mean difference).
#' @param mc Minimal clinical relevant value provided as a numeric value. Such as 0.8 for e.g. an odds ratio of 0.8. 
#' @param tau2 Heterogeneity estimate. Can be extracted from the metaanalysis() function.
#' @param alpha The level of type I error as a percentage, the default is 0.05 corresponding to 5\%.
#' @param beta The level of type II error as a percentage, the default is 0.1 corresponding to 10\%.
#' @param side Whether a 1- or 2-sided hypothesis test is used. Options are 1 or 2.
#' @param pC Probability of event in control group. Only used for outcomes "RR", "OR" and "RD".
#' @param p1 Probability of event in treatment group. Only used for outcome "RD".
#' @param var_mc Variance of the estimated effect when outcome is "MD". Not required for outcome types "OR", "RR" or "RD".
#' @param var_random Estimated variance from the random-effects meta-analysis. Used then a meta-analysis have already been made previously.
#' @param trials Optional argument. Number of trials of interest for to provide the number of participants needed for that exact number of trials.
#'
#' @return Either a number (minimum required trials) or the minimum required
#' required trials together with a matrix of required participants per trial given
#' different number of trials.
#' @export
#'
#' @importFrom stats uniroot qnorm
#'
#' @examples
#' # Minimum number of trials for a prospective meta-analysis
#' minTrial(outcome = "RR", pC = 0.5, mc = 0.7, tau2 = 0.05, alpha = 0.05,
#' beta = 0.1, side = 2)
#' 
#' # Minimum number of trials still needed for a retrospective meta-analysis
#' # Note that retrospective sample size calculations are prone to bias
#' ma <- metaanalysis(outcome = "RR", data = perioOxy)
#' ris(outcome = "RR", mc = 0.80, ma = ma, type = "retrospective", fixed = FALSE,
#'  beta = 0.1, alpha = 0.05, side = 2)
#'
minTrial = function(outcome,
                    mc,
                    tau2,
                    alpha,
                    beta,
                    side,
                    pC = NULL,
                    p1 = NULL,
                    var_mc = NULL,
                    var_random = NULL,
                    trials = NULL) {
  if (outcome %in% c("RR", "OR")) {
    mc = log(mc)
  }

  fixed <- FALSE
  war_het <- NULL

  if(!is.null(var_random)){
    ntrial_random = function(mc, trial, alpha, beta, tau2, side,  var_random) {
      trial / ((qnorm(1 - alpha / side) + qnorm(1 - beta)) ^ 2/mc^2-1/var_random) - tau2
    }
    
    minTrial <- try(ceiling(uniroot(
      function(trial)
        ntrial_random(mc, trial, alpha, beta, tau2, side, var_random),
      interval = c(0, 1000)
    )$root), TRUE)
    if(inherits(minTrial,"try-error")){
      war_het <- c("Too low est. heterogeneity or too much information to calculate remaining RIS based on tau^2.")
    }
  } else {
    ntrial = function(mc, trial, alpha, beta, tau2, side) {
      trial / ((qnorm(1 - alpha / side) + qnorm(1 - beta)) ^ 2/mc^2) - tau2
    }

    minTrial <- ceiling(uniroot(
      function(trial)
        ntrial(mc, trial, alpha, beta, tau2, side),
      interval = c(0, 1000)
    )$root)
  }

  if(!is.null(var_random) & !is.null(war_het)){
    ntrial = function(mc, trial, alpha, beta, tau2, side) {
      trial / ((qnorm(1 - alpha / side) + qnorm(1 - beta)) ^ 2/mc^2) - tau2
    }
    
    minTrial <- ceiling(uniroot(
      function(trial)
        ntrial(mc, trial, alpha, beta, tau2, side),
      interval = c(0, 1000)
    )$root)
  }
  
  if (outcome == "RR") {
    pI <- exp(log(pC) + mc)
    var_mc <- 1 / pC + 1 / pI - 2
  } else if (outcome == "OR") {
    logit <- function(x)
      log(x / (1 - x))
    invlogit <- function(x)
      1 / (1 + exp(-x))
    pI <- invlogit(logit(pC) + mc)
    var_mc <- 1 / pI + 1 / pC + 1 / (1 - pI) + 1 / (1 - pC)
  } else if(outcome == "RD"){
    var_mc <- pC*(1-pC)+p1*(1-p1)
  }

  if(fixed == FALSE & (is.null(war_het) | minTrial != 0)){
    out.mat = matrix(NA, ncol = 4, nrow = 3)
    out.mat[1, ] <- c(minTrial, minTrial + 1, minTrial + 2, minTrial + 3)
    if(is.null(var_random) & is.null(war_het)){
      out.mat[2, ] <- ceiling(2 * var_mc / (
        mc ^ 2 * c(minTrial, minTrial + 1, minTrial + 2, minTrial + 3) / ((qnorm(1 -
                                                                                   alpha / side) + qnorm(1 - beta)) ^ 2) - tau2
      ))} else {
        out.mat[2, ] <- ceiling(2 * var_mc / (
          c(minTrial, minTrial + 1, minTrial + 2, minTrial + 3) / ((qnorm(1 -alpha / side) + qnorm(1 - beta)) ^ 2/mc ^ 2 -1/var_random) - tau2
        ))
      }
    out.mat[3, ] <- out.mat[1, ] * out.mat[2, ]
    row.names(out.mat) <- c("Trials", "Pax per trial", "Total nr of pax")

    if(!is.null(trials)){
      if(trials < minTrial){
        warning("The number of trials provided is less than the required number of trials")
      } else {
        if(is.null(var_random)){
          outTrials <- ceiling(2 * var_mc / (
            mc ^ 2 * trials / ((qnorm(1 - alpha / side) + qnorm(1 - beta)) ^ 2) - tau2
          ))} else {
            outTrials <- ceiling(2 * var_mc / ( trials / ((qnorm(1 -alpha / side) + qnorm(1 - beta)) ^ 2/mc ^ 2 -1/var_random) - tau2
            ))
          }

        out.mat <- cbind(out.mat, c(trials, outTrials, trials*outTrials
        ))
      }
    }

    return(list(minTrial = minTrial, nPax = out.mat, war_het = war_het))
  }
  else {
    return(list(war_het = war_het))
  }
}


#' Calculate required sample and trials size.
#'
#' @param outcome Choose between: "MD" (mean difference), "RR" (relative risk), "OR" (odds ratio) or "RD" (risk difference).
#' @param mc Minimum clinical relevant effect. For "OR" or "RR" set to natural scale, not log scale.
#' @param side Test type. Set to 1 or 2 depending on the test being 1- or 2-sided.
#' @param alpha The level of type I error as a percentage, the default is 0.05 corresponding to 5\%.
#' @param beta The level of type II error as a percentage, the default is 0.1 corresponding to 10\%.
#' @param fixed Should sample size be based on a fixed-effect (TRUE) or random-effects (FALSE) model. Defaults to TRUE.
#' @param sd_mc Standard deviation of estimated effect. Only needed when outcome type is "MD".
#' @param pC Probability of event in control group. Only needed when outcome type is "OR", "RR" or "RD".
#' @param p1 Probability of event in treatment group. Only needed when outcome type is "RD".
#' @param ma An optional \code{metaanalysis} object. Required for retrospective sample size calculations.
#' @param tau2 The value of the heterogeneity. Use when estimating the sample size under a random effects model. If data is provided, the estimated heterogeneity is used instead.
#' @param I2 Optional argument. Inconsistency. 
#' @param D2 Optional argument. Diversity.
#' @param type Whehter the type of calculaiton is for "prospective" meta-analysis or "retrospective" meta-analysis. If the type is retrospective, one should add a meta-analysis object to the function. See argument ma. 
#' @param trials Optional numeric argument. If one is interested in a specific number of trials.
#' @param RTSA Whether the ris function was called via the RTSA function. Purely operational argument.
#' @param ... additional arguments
#'
#' @return A list of up to 6 elements:
#' \item{settings}{A list containing the arguments provided to the \code{ris} function.}
#' \item{NF}{The total number of required participants in a fixed-effect meta-analysis if type is prospective. Contains a list if the type is retrospective, where \code{NF} is the additional required number of participants and \code{NF_full} is the total required number of participants.}
#' \item{NR_tau}{A list containing: \code{minTrial} the minimum number of trials. \code{nPax} a matrix containing four possible number of trials with the number of participants per trial and total number of participants. \code{tau2} the estimate used for the calculation. Might contain \code{NR_tau_ll} and \code{NR_tau_ul} which contain the same three elements. \code{NR_tau_ll} is based on the lower value in the confidence interval of tau2. \code{NR_tau_ul} is based on the upper value in the confidence interval for tau2. If the type is prospective the numbers are the total required. If the type is retrospective the numbers are the additional required.}
#' \item{NR_D2}{The total number of required participants in a random-effects meta-analysis adjusted by diversity (\code{D2}) if type is prospective. Contains a list if the type is retrospective, where \code{NR_D2} is the additional required number of participants and \code{NR_D2_full} is the total required number of participants.}
#' \item{NR_I2}{The total number of required participants in a random-effects meta-analysis adjusted by inconsistency (\code{I2}) if type is prospective. Contains a list if the type is retrospective, where \code{NR_I2} is the additional required number of participants and \code{NR_I2_full} is the total required number of participants.}
#'
#' @export
#' @aliases print.ris
#'
#' @examples
#' # Sample and trial size calculation for prospective meta-analysis
#' ris(outcome = "RR", mc = 0.8, pC = 0.12, fixed = TRUE, alpha = 0.05,
#' beta = 0.1, side = 2)
#'
#' # Additional sample and trial size calculation for retrospective meta-analysis 
#' # It is calculated directly from the metaanalysis() function
#' data("perioOxy")
#' ma <- metaanalysis(outcome = "RR", data = perioOxy, mc = 0.8, beta = 0.2)
#' ma$ris
#' # Or by using the two functions in sequence
#' ma <- metaanalysis(outcome = "RR", data = perioOxy)
#' ris(outcome = "RR", mc = 0.8, ma = ma, type = "retrospective", fixed = FALSE,
#'  beta = 0.2, alpha = 0.05, side = 2)
ris <-
  function(outcome,
           mc,
           side = 2,
           alpha = 0.05,
           beta = 0.1,
           fixed = TRUE,
           sd_mc = NULL,
           pC = NULL,
           p1 = NULL,
           ma = NULL,
           tau2 = NULL,
           I2 = NULL,
           D2 = NULL,
           type = "prospective",
           trials = NULL,
           RTSA = FALSE,
           ...) {
    
    # check input
    if (outcome == "MD" & (is.null(sd_mc) | is.null(mc)) & is.null(ma)) {
      stop("For continuous outcomes provide mc and sd_mc")
    }

    if (type == "prospective" & outcome %in% c("RR", "OR") & (is.null(pC) | is.null(mc)) & is.null(ma)) {
      stop("For binary outcomes OR and RR provide mc and pC.")
    }

    if (outcome == "RD" & (is.null(mc) | is.null(pC) | is.null(p1))) {
      stop("For binary outcome RD provide mc, pC and p1.")
    }

    if(type == "retrospective" & (is.null(ma))){
      stop("A metaanalysis object is required for retrospective sample size calculation")
    }

    if ((!is.null(tau2) | !is.null(I2) | !is.null(D2) ) & fixed == TRUE) {
      fixed <- FALSE
      warning("There is provided a value for tau2, I2 and/or D2. Argument fixed is changed to FALSE.")
    }
    
    if(!is.null(ma)) type = "retrospective"
    
    if (type == "retrospective" & fixed == TRUE & !is.null(ma$synthesize$peR[6])) {
      if(ma$hete_results$hete_est$tau2 != 0){
      warning("There is presence of tau2 (heterogeneity) and hence I2 and/or D2 is not 0. Consider changing fixed to FALSE.")
    }}

    # calculate theta and nu
    if (outcome %in% c("OR", "RR")) {
      # prepare helper functions
      logit <- function(x)
        log(x / (1 - x))
      invlogit <- function(x)
        1 / (1 + exp(-x))

      if(type == "retrospective" & is.null(pC)){
        pC <- sum(ma$metaPrepare$org_data$eC)/sum(ma$metaPrepare$org_data$nC)
      }
      
      if (outcome == "RR") {
        pI <- exp(log(pC) + log(mc))
      } else if (outcome == "OR") {
        pI <- invlogit(logit(pC) + log(mc))
      }
      p <- (pC + pI) / 2
      var_mc <- p * (1 - p)
      mc_nf <- pC - pI
    }
    
    if(outcome == "MD"){
      var_mc <- sd_mc^2
      mc_nf <- mc
    }

    if(outcome == "RD"){

      if(type == "retrospective" & is.null(pC)){
        pC <- sum(ma$metaPrepare$org_data$eC)/sum(ma$metaPrepare$org_data$nC)
        p1 <- sum(ma$metaPrepare$org_data$eI)/sum(ma$metaPrepare$org_data$nI)
      }

      var_mc <- pC*(1-pC)+p1*(1-p1)
      mc_nf <- mc
    }
    
    # store arguments
    args <- mget(names(formals()),sys.frame(sys.nframe()))
    
    # calculate fixed effect sample size
    NF <- 2 * (qnorm(1 - alpha / side) + qnorm(1 - beta)) ^ 2 * 2 * var_mc / mc_nf ^ 2
    NF <- ceiling(NF)

    if(!is.null(ma)){
      args[names(args) == "ma"] <- NULL
    }
    
    args$var_mc = var_mc
    
    # create output
    outlist <-
      list(
        NF = NF,
        settings = args
      )
    
    # calculate random-effects sample size
    # when used inside metaanalysis or metaanalysis object is provided
    if(!is.null(ma) & fixed == FALSE & is.null(ma$synthesize$peR)){
      fixed = TRUE
    }
    else if (!is.null(ma) & fixed == FALSE & !is.null(ma$synthesize$peR)) {

      if(outcome %in% c("RR", "OR")){
        NR_tau <- minTrial(outcome = outcome, mc = mc, alpha = alpha, side = side,
                       beta = beta, pC = pC, tau2 = ma$synthesize$U[1],
                       var_random = ma$synthesize$peR[6], trials = trials)
        war_het <- NR_tau$war_het
        if(is.null(war_het) | NR_tau$minTrial != 0){ NR_tau <- append(NR_tau, list(tau2 = ma$synthesize$ci.tau$random[1,1])) 
        } else {
          NR_tau <- NULL
        }
        if(ma$synthesize$ci.tau$random[1,2] != 0 & (is.null(war_het) | NR_tau$minTrial != 0)){
        NR_tau_ll <- minTrial(outcome = outcome, mc = mc, alpha = alpha,
                          beta = beta, pC = pC, side = side,
                          tau2 = ma$synthesize$ci.tau$random[1,2],
                          var_random = ma$synthesize$peR[6])
        NR_tau_ll <- append(NR_tau_ll, list(tau2 = ma$synthesize$ci.tau$random[1,2]))
        } else {
          NR_tau_ll <- NULL
        }
        if(!is.null(NR_tau)){
          NR_tau_ul <- minTrial(outcome = outcome, mc = mc, alpha = alpha,
                          beta = beta, pC = pC, side = side,
                          tau2 = ma$synthesize$ci.tau$random[1,3],
                          var_random = ma$synthesize$peR[6])
          NR_tau_ul <- append(NR_tau_ul, list(tau2 = ma$synthesize$ci.tau$random[1,3]))
        } else {
          NR_tau_ul <- NULL
        }
      } else if(outcome == "RD"){
        NR_tau <- minTrial(outcome = outcome, mc = mc, alpha = alpha, side = side,
                       beta = beta, pC = pC, p1 = p1, tau2 = ma$synthesize$U[1],
                       var_random = ma$synthesize$peR[6], trials = trials)
        war_het <- NR_tau$war_het
        if(is.null(war_het) | NR_tau$minTrial != 0){ NR_tau <- append(NR_tau, list(tau2 = ma$synthesize$ci.tau$random[1,1])) 
        } else {
          NR_tau <- NULL
        }
        if(ma$synthesize$ci.tau$random[1,2] != 0 & (is.null(war_het) | NR_tau$minTrial != 0)){
        NR_tau_ll <- minTrial(outcome = outcome, mc = mc, alpha = alpha,
                          beta = beta, pC = pC, p1 = p1, side = side,
                          tau2 = ma$synthesize$ci.tau$random[1,2],
                          var_random = ma$synthesize$peR[6])
        NR_tau_ll <- append(NR_tau_ll, list(tau2 = ma$synthesize$ci.tau$random[1,2]))
        } else {
          NR_tau_ll <- NULL
        }
        NR_tau_ul <- minTrial(outcome = outcome, mc = mc, alpha = alpha,
                          beta = beta, pC = pC, p1 = p1, side = side,
                          tau2 = ma$synthesize$ci.tau$random[1,3],
                          var_random = ma$synthesize$peR[6])
        NR_tau_ul <- append(NR_tau_ul, list(tau2 = ma$synthesize$ci.tau$random[1,3]))
      } else {
        NR_tau <- minTrial(outcome = outcome, mc = mc, alpha = alpha, side = side,
                       beta = beta, var_mc = var_mc, tau2 = ma$synthesize$U[1],
                       var_random = ma$synthesize$peR[6], trials = trials)
        war_het <- NR_tau$war_het
        if(is.null(war_het) | NR_tau$minTrial != 0){ NR_tau <- append(NR_tau, list(tau2 = ma$synthesize$ci.tau$random[1,1])) 
        } else {
          NR_tau <- NULL
        }
        if(ma$synthesize$ci.tau$random[1,2] != 0 & (is.null(war_het) | NR_tau$minTrial != 0)){
        NR_tau_ll <- minTrial(outcome = outcome, mc = mc, alpha = alpha,
                          beta = beta, var_mc = var_mc,
                          tau2 = ma$synthesize$ci.tau$random[1,2],
                          var_random = ma$synthesize$peR[6], side = side)
        NR_tau_ll <- append(NR_tau_ll, list(tau2 = ma$synthesize$ci.tau$random[1,2]))
        } else {
          NR_tau_ll <- NULL
        }
        NR_tau_ul <- try(minTrial(outcome = outcome, mc = mc, alpha = alpha,
                              beta = beta, var_mc = var_mc, side = side,
                              tau2 = ma$synthesize$ci.tau$random[1,3],
                              var_random = ma$synthesize$peR[6]), TRUE)
        if(inherits(NR_tau_ul,"try-error")){
          NR_tau_ul = "Too much heterogeneity to estimate an upper number of trials."
        } else {
          NR_tau_ul <- append(NR_tau_ul, list(tau2 = ma$synthesize$ci.tau$random[1,3]))
        }
      }
      
      NR_tau[names(NR_tau) == "war_het"] <- NULL
      NR_tau_ll[names(NR_tau_ll) == "war_het"] <- NULL
      NR_tau_ul[names(NR_tau_ul) == "war_het"] <- NULL
      
      # sample size for inconsistency adj. and diversity adj.
      NR_D2 <- ifelse(is.null(D2),1 / (1 - ma$synthesize$U[4]) * NF, 1 / (1 - D2) * NF)
      NR_I2 <- ifelse(is.null(I2),1 / (1 - ma$synthesize$U[3]) * NF, 1 / (1 - I2) * NF)
      NR_D2 <- ceiling(NR_D2)
      NR_I2 <- ceiling(NR_I2)
      
      # set relative to the sample size already achieved
      if(is.null(NR_tau)){
        NR_tau_full <- NULL  
      }  else {
      NR_tau_full <-
          ifelse(is.null(trials),NR_tau$nPax[3, 1],NR_tau$nPax[3, 5]) + (sum(ma$metaPrepare$org_data$nI) + sum(ma$metaPrepare$org_data$nC))
      }
        NR_I2_full <- NR_I2
        NR_D2_full <- NR_D2
        NR_I2 <- NR_I2 - (sum(ma$metaPrepare$org_data$nI) + sum(ma$metaPrepare$org_data$nC))
        NR_D2 <- NR_D2 - (sum(ma$metaPrepare$org_data$nI) + sum(ma$metaPrepare$org_data$nC))

      outlist <-
        append(outlist, list(
          NR_D2 = list("NR_D2" = NR_D2, "NR_D2_full" = NR_D2_full),
          NR_I2 = list("NR_I2" = NR_I2, "NR_I2_full" = NR_I2_full),
          NR_tau = list("NR_tau" = NR_tau, "NR_tau_ll" = NR_tau_ll,
                        "NR_tau_ul" = NR_tau_ul, "NR_tau_full" = NR_tau_full),
          war_het = war_het
        ))
    } else if(fixed == FALSE & is.null(ma)){
      # not inside metaanalysis call or metaanalysis object is not provided

      if(is.null(tau2) & is.null(I2) & is.null(D2)){
        stop("No value for heterogeneity (tau2, I2 and/or D2) is provided")
      }

      if(is.null(tau2)) tau2 <- 0
      NR_D2 = NULL
      NR_I2 = NULL

      if(outcome %in% c("RR", "OR")){
        NR_tau <- minTrial(outcome = outcome, mc = mc, alpha = alpha,
                       beta = beta, pC = pC, tau2 = tau2,
                       trials = trials, side = side)
      } else if(outcome == "RD"){
        NR_tau <- minTrial(outcome = outcome, mc = mc, alpha = alpha,
                       beta = beta, pC = pC, p1 = p1, tau2 = tau2,
                       trials = trials, side = side)
      } else{
        NR_tau <- minTrial(outcome = outcome, mc = mc, alpha = alpha,
                       beta = beta, var_mc = var_mc, tau2 = tau2,
                       trials = trials, side = side)
      }
      if(!is.null(I2)){
        NR_I2 <- 1 / (1 - I2) * NF
        NR_I2 <- ceiling(NR_I2)
      }
      if(!is.null(D2)){
        NR_D2 <- 1 / (1 - D2) * NF
        NR_D2 <- ceiling(NR_D2)
      }

      outlist <-
        append(outlist, list(
          NR_tau = NR_tau,
          NR_D2 = NR_D2,
          NR_I2 = NR_I2
        ))
    }
    
    if(type == "retrospective"){
      NF_full <- NF
      NF <- NF - (sum(ma$metaPrepare$data$nI) + sum(ma$metaPrepare$data$nC))

      outlist[names(outlist) == "NF"] <- NULL
      outlist$NF <- list("NF" = NF, "NF_full" = NF_full)
    }

    class(outlist) <- "ris"
    return(outlist)
  }


# FUNCTION | print ris ----
#' @method print ris
#' @export
print.ris <- function(x, ...) {
  if (x$settings$type == "prospective") {
    cat(
      "This is a prospective meta-analysis sample size calculation.\nThe sample size calculation assumes a",
      paste0(x$settings$side, "-sided test,"), "equal group sizes,\na type-I-error of",
      x$settings$alpha,
      "and a type-II-error of",
      paste0(x$settings$beta, "."), "\nThe minimum clinical relevant value is set to:", x$settings$mc, "for outcome metric", paste0(x$settings$outcome,".\n")
    )
    cat("Additional parameters for sample size are:\n")
    if(x$settings$outcome %in% c("RR", "OR")){
      cat("Probability of event in the control group:", paste0(round(x$settings$pC,4), "."))
    } else if(x$settings$outcome == "MD"){
      cat("Standard deviation of the mean (standard error): ", paste0(x$settings$sd_mc, "."))
    } else {
      cat("Probability of event in the control group:", round(x$settings$pC,4), "and probability of event in intervention group:", paste0(x$settings$p1, "."))
    }
    if(!is.null(x$settings$trials)){
      cat("\nNumber of planned trials:", x$settings$trials)
    }
    cat("\n\n")
    cat("Fixed-effect required information size:\n")
    cat(paste(ifelse(x$settings$RTSA,x$SMA_NF,x$NF), "participants in total. \n"))
    if (x$settings$fixed == FALSE) {
      cat("\n")
      cat("Random-effects required information size:\n")
      if (!is.null(x$settings$tau2) & is.null(x$war_het)) {
        if(!is.null(x$settings$trials)){
          cat(
            paste("Adjusted by tau^2:",
                  ifelse(x$settings$RTSA,x$SMA_tau2[3, 5],x$NR_tau$nPax[3, 5]),
                  "participants in total split over",
                  x$NR_tau$nPax[1, 5],
                  "trial(s).\n"
            )
          )  
        } else {
        cat(
          paste("Adjusted by tau^2:",
                ifelse(x$settings$RTSA,x$SMA_tau2[3, 5],x$NR_tau$nPax[3, 1]),
            "participants in total split over (at minimum)",
            x$NR_tau$nPax[1, 1],
            "trial(s).\n"
          )
        )
        }
      }
      if (!is.null(x$NR_D2)) {
        cat(paste("Adjusted by diversity (D^2):",
                  ifelse(x$settings$RTSA,x$SMA_D2,x$NR_D2),
          "participants in total.\n"
        ))
      }
      if (!is.null(x$NR_I2)) {
        cat(paste("Adjusted by inconsistency (I^2):",
                  ifelse(x$settings$RTSA,x$SMA_I2,x$NR_I2),
          "participants in total.\n"
        ))
      }
    }
    cat(
      "\nFor more information about the sample size calculation see vignette:\n'Calculating required sample size and required number of trials'."
    )
    if(!is.null(x$war_het)){cat("\n\nPlease note the following warnings:\n")
      cat("*",x$war_het)}
  } else {
    cat(
      "This is a retrospective meta-analysis sample size calculation. \nThe sample size calculation assumes a", paste0(x$settings$side, "-sided test,"), "equal group sizes,\na type-I-error of",
      x$settings$alpha,
      "and a type-II-error of",
      x$settings$beta,
      "\n\n"
    )
    cat("Fixed-effect required information size:\n")
    if(x$NF$NF < 0){
      cat("The number of required participants for a fixed-effect meta-analysis is reached.\n")
    } else {
      cat(paste(x$NF$NF, "participants in total are additionally required. \n"))}
    if (x$settings$fixed == FALSE & (!is.null(x$NR_tau) | !is.null(x$NR_D2) | !is.null(x$NR_I2))) {
      cat("\n")
      cat("Random-effects required information size:\n")
      if (!is.null(x$NR_tau$NR_tau$tau2) & is.null(x$war_het)) {
        if(!is.null(x$settings$trials)){
          cat(
            paste("Adjusted by tau^2:",
                  ifelse(x$settings$RTSA,x$SMA_tau2[3, 5],x$NR_tau$NR_tau$nPax[3, 5]),
                  "participants are additionally required in total split over ",
                  x$NR_tau$NR_tau$nPax[1, 5],
                  "trial(s).\n"
            )
          )  
        } else {
          cat(
            paste("Adjusted by tau^2:",
                  ifelse(x$settings$RTSA,x$SMA_tau2[3, 1],x$NR_tau$NR_tau$nPax[3, 1]),
                  "participants are additionally required in total split over (at minimum)",
                  x$NR_tau$NR_tau$nPax[1, 1],
                  "trial(s).\n"
            )
          )
        }
      }
      
      if (!is.null(x$NR_D2$NR_D2) & x$NR_D2$NR_D2 >= 0) {
        cat(paste("Adjusted by diversity (D^2):",
                  ifelse(x$settings$RTSA,x$SMA_D2,x$NR_D2$NR_D2),
          "participants in total are additionally required. \n"
        ))
      } else if(!is.null(x$NR_D2$NR_D2) & x$NR_D2$NR_D2 < 0){
        cat(paste("Adjusted by inconsistency (D^2):",
                  "The number of required participants for a random-effects meta-analysis is reached.\n"))
      }
      if (!is.null(x$NR_I2$NR_I2) & x$NR_I2$NR_I2 >= 0) {
        cat(paste("Adjusted by inconsistency (I^2):",
                  ifelse(x$settings$RTSA,x$SMA_I2,x$NR_I2$NR_I2),
          "participants in total are additionally required.\n"
        ))
      } else if(!is.null(x$NR_I2$NR_I2) & x$NR_I2$NR_I2 < 0){cat(paste("Adjusted by inconsistency (I^2):",
                                                             "The number of required participants for a random-effects meta-analysis is reached.\n"
      ))}
    }
    cat(
      "\nFor more information about the sample size calculation see vignette: \n'Calculating required sample size and required number of trials'."
    )
    if(!is.null(x$war_het)){cat("\n\nPlease note the following warnings:\n")
     cat("-",x$war_het)}
  }
}