R/pSceptical.R

In ReplicationSuccess: Design and Analysis of Replication Studies

# FZ omputes the cumulative distribution function of z = z_S^2

FZ <- function(z, c) {
  if (z <= 0) {
    return(0)
  } else if (c == 0) {
    return(1 - 4 * (1 - stats::pnorm(sqrt(z)))^2)
  } else if (c == 1) {
    return(stats::pgamma(z, 1 / 2, 2))
  } else {
    f <- function(t, c, z) {
      t1 <- exp(-(c - 1) * z / (sqrt(1 + (c - 1) * t) - 1))
      t2 <- 1 / sqrt(t * (1 - t))
      return(t1 * t2)
    }
    myint <- stats::integrate(
      f,
      lower = 0,
      upper = 1,
      z = z,
      c = c
    )$value
    result <- 1 - myint / pi
    return(result)
  }
}



.pSceptical_ <- function(zo,
                         zr,
                         c,
                         alternative = c("one.sided", "two.sided"),
                         type = c("golden", "nominal", "controlled")) {

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

            is.numeric(zr),
            length(zr) == 1,
            is.finite(zr),

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

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

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

  z <- zSceptical(zo = zo, zr = zr, c = c)

  if (type == "nominal") {
    result <- z
    res <- z2p(z = result, alternative = "two.sided")
  }

  # if (type == "liberal") {
  #   result <- z * sqrt(2)
  #   res <- z2p(z = result, alternative = "two.sided")
  # }

  if (type == "golden") {
    phi <- (sqrt(5) + 1) / 2
    result <- z * sqrt(phi)
    res <- z2p(z = result, alternative = "two.sided")
  }

  if (type == "controlled") {
  res2 <- (1 - FZ(z = z^2, c = c))
  res <- sqrt(res2)
  }

  if (alternative == "one.sided") {
    res <- ifelse(sign(zo) == sign(zr), res / 2, 1 - res / 2)
  }

  # if (alternative == "greater") {
  #   if (zo < 0) res <- NaN
  #   if (zo > 0 && zr > 0) res <- res / 2
  #   if (zo > 0 && zr < 0) res <- 1 - res / 2
  # }
  #
  #   if(alternative == "less"){
  #     if (zo > 0) res <- NaN
  #     if (zo < 0 && zr < 0) res <- res / 2
  #     if (zo < 0 && zr > 0) res <- 1 - res / 2
  #   }
  return(res)
}

#' Computes the sceptical p-value and z-value
#'
#' Computes sceptical p-values and z-values based on the z-values of the
#' original and the replication study and the corresponding variance ratios.
#' If specified, the sceptical p-values are recalibrated.
#' @rdname pSceptical
#' @param zo Numeric vector of z-values from original studies.
#' @param zr Numeric vector of z-values from replication 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 alternative Either "one.sided" (default) or "two.sided".
#' If "one.sided", the sceptical p-value is based on a one-sided
#' assessment of replication success in the direction of the original effect estimate.
#' If "two.sided", the sceptical p-value is based on a two-sided
#' assessment of replication success regardless of the direction of the
#' original and replication effect estimate.
#' @param type Type of recalibration. Can be either "golden" (default), "nominal",
#' or "controlled". Setting \code{type} to "nominal" corresponds to no
#' recalibration as in Held et al. (2020). A recalibration is applied if
#' \code{type} is  "controlled", or "golden", and the sceptical p-value
#' can then be interpreted on the same scale as an ordinary p-value (e.g., a one-sided
#' sceptical p-value can be thresholded at the conventional 0.025 level).
#' "golden" ensures that
#' for an original study just significant at the specified \code{level},
#' replication success is only possible if the replication effect estimate is at
#' least as large as the original one.
#' "controlled" ensures exact overall Type-I error control at level \code{level}^2.
#' @return \code{pSceptical} returns the sceptical p-value.
#' @details \code{pSceptical} is the vectorized version of
#' the internal function \code{.pSceptical_}.
#' \code{\link[base]{Vectorize}} is used to vectorize the function.
#' @references
#' Held, L. (2020). A new standard for the analysis and design of replication
#' studies (with discussion). \emph{Journal of the Royal Statistical Society:
#' Series A (Statistics in Society)}, \bold{183}, 431-448.
#' \doi{10.1111/rssa.12493}
#'
#'
#' 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
#' @seealso \code{\link{sampleSizeReplicationSuccess}}, \code{\link{powerReplicationSuccess}}, \code{\link{levelSceptical}}
#' @examples
#' ## no recalibration (type = "nominal") as in Held (2020)
#' pSceptical(zo = p2z(0.01), zr = p2z(0.02), c = 2, alternative = "one.sided",
#'            type = "nominal")
#'
#' ## recalibration with golden level as in Held, Micheloud, Pawel (2020)
#' pSceptical(zo = p2z(0.01), zr = p2z(0.02), c = 2, alternative = "one.sided",
#'            type = "golden")
#'
#' ## two-sided p-values 0.01 and 0.02, relative sample size 2
#' pSceptical(zo = p2z(0.01), zr = p2z(0.02), c = 2, alternative = "one.sided")
#' ## reverse the studies
#' pSceptical(zo = p2z(0.02), zr = p2z(0.01), c = 1/2, alternative = "one.sided")
#' ## both p-values 0.01, relative sample size 2
#' pSceptical(zo = p2z(0.01), zr = p2z(0.01), c = 2, alternative = "two.sided")
#' @export
pSceptical <- Vectorize(.pSceptical_)



#' @rdname pSceptical
#' @return \code{zSceptical} returns the z-value of the sceptical p-value.
#' @examples
#'
#' zSceptical(zo = 2, zr = 3, c = 2)
#' zSceptical(zo = 3, zr = 2, c = 2)
#' @export
zSceptical <- function(zo,
                       zr,
                       c) {
    stopifnot(is.numeric(zo),
              length(zo) > 0,
              is.finite(zo),

              is.numeric(zr),
              length(zr) > 0,
              is.finite(zr),

              is.numeric(c),
              length(c) > 0,
              is.finite(c),
              0 <= c)

    ## arithmetic mean
    aritMean <- function(x, y) (x + y) / 2
    ## harmonic mean
    harmMean <- function(x, y) 2 / (1 / x + 1 / y)

    ## vectorize function in all arguments
    z2H <- harmMean(zo^2, zr^2)
    z2A <- aritMean(zo^2, zr^2)

    if (length(c) == 1) {
        z2 <- if (c == 1) {
            z2H / 2
        } else {
            (sqrt(z2A * (z2A + (c - 1) * z2H)) - z2A) / (c - 1)
        }
    } else {
        z2 <- ifelse(
            c == 1,
            z2H / 2,
            (sqrt(z2A * (z2A + (c - 1) * z2H)) - z2A) / (c - 1)
        )
    }

    return(sqrt(z2))
}