R/effectSizeReplicationSuccess.R

In ReplicationSuccess: Design and Analysis of Replication Studies

.effectSizeReplicationSuccess_ <- function(zo,
                                           c = 1,
                                           level = 0.025,
                                           alternative = c("one.sided", "two.sided"),
                                           type = c("golden", "nominal", "controlled")){

    stopifnot(is.numeric(zo),
              length(zo) == 1,
              is.finite(zo),

              is.numeric(c),
              length(c) == 1,
              !is.na(c), !is.nan(c),
              0 <= c,

              is.numeric(level),
              length(level) == 1,
              is.finite(level),
              0 < level, level < 1,

              !is.null(alternative))
    alternative <- match.arg(alternative)

    stopifnot(!is.null(type))
    type <- match.arg(type)

    alphas <- levelSceptical(level = level,
                             alternative = alternative,
                             type = type,
                             c = c)
    zalphas <- p2z(p = alphas, alternative = alternative)
    K <- zo^2 / zalphas^2

    if (zalphas > abs(zo)) {
      warning(paste("Replication success is not achievable at this level as |zo| =",
                    abs(round(zo, 2)), " < ", round(p2z(levelSceptical(level = level,
                                                        alternative = alternative,
                                                        type = type,
                                                        c = c), alternative = alternative), 3)))
      d <- NA
    } else {
        d <- if(c < Inf) sqrt(1 + c/(K - 1))/(sqrt(K * c)) else 1/sqrt(K * (K - 1))
    }

    return(d)
}

#' Computes the minimum relative effect size to achieve replication success
#' with the sceptical p-value
#'
#' The minimum relative effect size (replication to original) to achieve replication success
#' with the sceptical p-value
#' is computed based on the result of the original study and the corresponding variance ratio.
#'
#' @param zo Numeric vector of z-values from original studies.
#' @param c Numeric vector of variance ratios of the original and replication effect estimates.
#' This is usually the ratio of the sample size of the replication study to the sample
#' size of the original study.
#' @param level Threshold for the calibrated sceptical p-value.
#'  Default is 0.025.
#' @param alternative Specifies if \code{level} is "one.sided" (default) or "two.sided".
#' If "one.sided", then effect size calculations are based on a one-sided assessment of
#' replication success in the direction of the original effect estimate.
#' @param type Type of recalibration. Can be either "golden" (default), "nominal" (no recalibration),
#'  or "controlled". "golden" ensures that for an original study just significant at
#' the specified \code{level}, replication success is only possible for
#' replication effect estimates larger than the original one.
#' "controlled" ensures exact overall Type-I error control at level \code{level}^2.
#' @return The minimum relative effect size to achieve replication success
#' with the sceptical p-value.
#' @details \code{effectSizeReplicationSuccess} is the vectorized version of
#' the internal function \code{.effectSizeReplicationSuccess_}.
#' \code{\link[base]{Vectorize}} is used to vectorize the function.
#' @references Held, L., Micheloud, C., Pawel, S. (2022). The assessment of
#'     replication success based on relative effect size. The Annals of Applied
#'     Statistics. 16:706-720. \doi{10.1214/21-AOAS1502}
#'
#' Micheloud, C., Balabdaoui, F., Held, L. (2023).
#' Beyond the two-trials rule: Type-I error control and sample size planning
#' with the sceptical p-value. \url{https://arxiv.org/abs/2207.00464}
#' @author Leonhard Held, Charlotte Micheloud, Samuel Pawel, Florian Gerber
#' @seealso \code{\link{sampleSizeReplicationSuccess}}, \code{\link{levelSceptical}}
#' @examples
#' po <- c(0.001, 0.002, 0.01, 0.02, 0.025)
#' zo <- p2z(po, alternative = "one.sided")
#'
#' effectSizeReplicationSuccess(zo = zo, c = 1, level = 0.025,
#'                              alternative = "one.sided", type = "golden")
#'
#' effectSizeReplicationSuccess(zo = zo, c = 10, level = 0.025,
#'                              alternative = "one.sided", type = "golden")
#' effectSizeReplicationSuccess(zo = zo, c = 10, level = 0.025,
#'                              alternative = "one.sided", type = "controlled")
#' effectSizeReplicationSuccess(zo = zo, c= 2, level = 0.025,
#'                              alternative = "one.sided", type = "nominal")
#'
#' effectSizeReplicationSuccess(zo = zo, c = 2, level = 0.05,
#'                              alternative = "two.sided", type = "nominal")
#' @export
effectSizeReplicationSuccess <- Vectorize(.effectSizeReplicationSuccess_)