deffectsize: Point effect size estimates and confidence intervals

Documented in cohen_CI

#' Function to compute CI around Cohen's effect size estimators
#'
#' @param m1 the average score of the first group
#' @param m2 the average score of the second group
#' @param sd1 the standard deviation the first group
#' @param sd2 the standard deviation the second group
#' @param n1 the first sample size
#' @param n2 the second sample size
#' @param conf.level confidence level of the interval
#' @param var.equal a logical variable indicating whether to assume equality of population variances.
#' If TRUE the pooled variance is used to estimate the standard error (= Cohen's d or Hedges' g). Otherwise, the square root of the non pooled
#' average of both variance estimates is used to estimate the standard error (Cohen's d' or Hedges' g').
#' @param unbiased a logical variable indicating whether to compute the biased or unbiased estimator.
#' If TRUE, unbiased estimator is computed (Hedges' g or Hedges' g'). Otherwise, bias estimator is computed (Cohen's d or Cohen's d').
#' @param alternative a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less".
#'
#' @export cohen_CI
#'
#' @exportS3Method cohen_CI default
#' @exportS3Method print cohen_CI
#'
#' @keywords Cohen's effect sizes, confidence interval
#' @return Returns Cohen's estimators of effect size and (1-alpha)% confidence interval around it, standard error
#' @importFrom stats na.omit sd pt uniroot

cohen_CI <- function(m1,m2,sd1,sd2,n1,n2,conf.level,var.equal,unbiased, alternative) UseMethod("cohen_CI")

cohen_CIEst <- function(m1,m2,sd1,sd2,n1,n2,
                        conf.level=.95,
                        var.equal=FALSE,
                        unbiased=TRUE,
                        alternative="two.sided"){

  param <- data.frame(m1,m2,sd1,sd2,n1,n2)
  vect <- NULL
  for (i in seq_len(length(param))){
    if(inherits(param[,i],c("numeric","integer"))==FALSE){
      vect <-   c(vect,names(param[i]))
    } else {vect=vect}
  }

  if(inherits(c(m1,m2,sd1,sd2,n1,n2),c("numeric","integer"))==FALSE){
    if(length(vect)==1){
      obj <- vect
      alert="is neither numeric nor integer"
    } else if (length(vect)>1){
      obj <-paste(paste(vect[-length(vect)],collapse=", "),"and",vect[length(vect)])
      alert="are neither numeric nor integer"
    }

    stop(paste(obj,alert))

  }

  if(var.equal==TRUE){

    pooled_sd <- sqrt(((n1-1)*sd1^2+(n2-1)*sd2^2)/(n1+n2-2))
    t_obs <- (m1-m2)/sqrt(pooled_sd^2*(1/n1+1/n2))
    df <- n1+n2-2
    cohen.d <- (m1-m2)/pooled_sd

    if(unbiased==TRUE){
      corr <- gamma(df/2)/(sqrt(df/2)*gamma((df-1)/2))
    } else {corr <- 1}

    if(corr=="NaN"){
      alert2="Correction for bias is only for small sample sizes. Use 'unbiased=FALSE'"
      stop(alert2)
    } else {ES <- cohen.d*corr}

    if(alternative=="two.sided"){

      # lower limit = limit of lambda such as 1-pt(q=t_obs, df=df, ncp = lambda) = (1-conf.level)/2 = alpha/2
      f=function(lambda,rep) 1-pt(q=t_obs, df=df, ncp = lambda)-rep
      out=uniroot(f,c(0,2),rep=(1-conf.level)/2,extendInt = "yes")
      lambda.1 <- out$root
      delta.1 <- lambda.1*sqrt(1/n1+1/n2) # lambda = delta * sqrt[n1n2/(n1+n2)]
                                          # <--> delta = lambda*sqrt(1/n1+1/n2)

      # upper limit = limit of lambda such as pt(q=t_obs, df=df, ncp = lambda) = (1-conf.level)/2 = alpha/2
      f=function(lambda,rep) pt(q=t_obs, df=df, ncp = lambda)-rep
      out=uniroot(f,c(0,2),rep=(1-conf.level)/2,extendInt = "yes")
      lambda.2 <- out$root
      delta.2 <- lambda.2*sqrt(1/n1+1/n2)# lambda = delta * sqrt[n1n2/(n1+n2)]
                                         # <--> delta = lambda*sqrt(1/n1+1/n2)

      result <- c(delta.1*corr, delta.2*corr)

    } else if (alternative == "greater"){

      # lower limit = limit of lambda such as 1-pt(q=t_obs, df=df, ncp = lambda) = (1-conf.level) = alpha
      f=function(lambda,rep) 1-pt(q=t_obs, df=df, ncp = lambda)-rep
      out=uniroot(f,c(0,2),rep=1-conf.level,extendInt = "yes")
      lambda.1 <- out$root
      delta.1 <- lambda.1*sqrt(1/n1+1/n2)   # See explanation in two.sided CI

      # upper limit = +Inf
      delta.2 <- +Inf
      result <- c(delta.1*corr, delta.2) # if our expectation is mu1 > mu2, then we expect that (mu1-mu2)> 0 and therefore
                                         # we want to check only the lower limit of the CI

    } else if (alternative == "less"){

      # lower limit = -Inf
      delta.1 <- -Inf # if our expectation is mu1 < mu2, then we expect that (mu1-mu2)< 0 and therefore
                      # we want to check only  the upper limit of the CI

      # upper limit = limit of lambda such as pt(q=t_obs, df=df, ncp = lambda) = (1-conf.level) = alpha
      f=function(lambda,rep) pt(q=t_obs, df=df, ncp = lambda)-rep
      out=uniroot(f,c(0,2),rep=1-conf.level,extendInt = "yes")
      lambda.2 <- out$root
      delta.2 <- lambda.2*sqrt(1/n1+1/n2) # See explanation in two.sided CI

      result <- c(delta.1, delta.2*corr)

    }

  } else if (var.equal==FALSE){

    cohen.d <- (m1-m2)/sqrt((sd1^2+sd2^2)/2)
    df <- ((n1-1)*(n2-1)*(sd1^2+sd2^2)^2)/((n2-1)*sd1^4+(n1-1)*sd2^4)
    t_obs <- (sqrt(n1*n2)*(m1-m2))/sqrt(n2*sd1^2+n1*sd2^2)

    if(unbiased==TRUE){
      corr <- gamma(df/2)/(sqrt(df/2)*gamma((df-1)/2))
    } else {corr <- 1}

    if(corr=="NaN"){
      alert2="Correction for bias is only for small sample sizes. Use 'unbiased=FALSE'"
      stop(alert2)
    } else {ES <- cohen.d*corr}

    if(alternative=="two.sided"){

      # lower limit = limit of lambda such as 1-pt(q=t_obs, df=df, ncp = lambda) = (1-conf.level)/2 = alpha/2
      f=function(lambda,rep) 1-pt(q=t_obs, df=df, ncp = lambda)-rep
      out=uniroot(f,c(0,2),rep=(1-conf.level)/2,extendInt = "yes")
      lambda.1 <- out$root
      delta.1 <- lambda.1*sqrt((2*(n2*sd1^2+n1*sd2^2))/(n1*n2*(sd1^2+sd2^2)))

      # upper limit = limit of lambda such as pt(q=t_obs, df=df, ncp = lambda) = (1-conf.level)/2 = alpha/2
      f=function(lambda,rep) pt(q=t_obs, df=df, ncp = lambda)-rep
      out=uniroot(f,c(0,2),rep=(1-conf.level)/2,extendInt = "yes")
      lambda.2 <- out$root
      delta.2 <- lambda.2*sqrt((2*(n2*sd1^2+n1*sd2^2))/(n1*n2*(sd1^2+sd2^2))) # lambda = delta * sqrt([n1n2(sd1^2+sd2^2)]/[2*(n2sd1^2+n1sd2^2)])
                                                                              # <--> delta = lambda * sqrt([2*(n2sd1^2+n1sd2^2)]/[n1n2(sd1^2+sd2^2)])

      result <- c(delta.1*corr, delta.2*corr)

    } else if (alternative == "greater"){

      # lower limit = limit of lambda such as 1-pt(q=t_obs, df=df, ncp = lambda) = (1-conf.level) = alpha
      f=function(lambda,rep) 1-pt(q=t_obs, df=df, ncp = lambda)-rep
      out=uniroot(f,c(0,2),rep=1-conf.level,extendInt = "yes")
      lambda.1 <- out$root
      delta.1 <- lambda.1*sqrt((2*(n2*sd1^2+n1*sd2^2))/(n1*n2*(sd1^2+sd2^2))) # See explanation in two.sided CI

      # upper limit = +Inf
      delta.2 <- +Inf # if our expectation is mu1 > mu2, then we expect that (mu1-mu2)> 0 and therefore
                      # we want to check only the lower limit of the CI

      result <- c(delta.1*corr, delta.2)

    } else if (alternative == "less"){

      # lower limit = -Inf
      delta.1 <- -Inf # if our expectation is mu1 < mu2, then we expect that (mu1-mu2)< 0 and therefore
                      # we want to check only the upper limit of the CI

      # upper limit = limit of lambda such as pt(q=t_obs, df=df, ncp = lambda) = (1-conf.level) = alpha
      f=function(lambda,rep) pt(q=t_obs, df=df, ncp = lambda)-rep
      out=uniroot(f,c(0,2),rep=1-conf.level,extendInt = "yes")
      lambda.2 <- out$root
      delta.2 <- lambda.2*sqrt((2*(n2*sd1^2+n1*sd2^2))/(n1*n2*(sd1^2+sd2^2))) # See explanation in two.sided CI

      result <- c(delta.1, delta.2*corr)


    }

  }

  # print results
  meth <- "Confidence interval around the cohen's estimate"

  # Return results in list()
  invisible(
    list(ES = ES,
         conf.level = conf.level,
         CI = result)
  )

}

# Adding a default method in defining a function called cohen_CI.default
cohen_CI.default <- function(m1,m2,sd1,sd2,
                             n1,n2,conf.level=.95,
                             var.equal=FALSE,
                             unbiased=TRUE,
                             alternative="two.sided"){

  out <- cohen_CIEst(m1,m2,sd1,sd2,n1,n2,conf.level,var.equal,unbiased,alternative)
  out$ES <- out$ES
  out$call <- match.call()
  out$CI <- out$CI
  out$conf.level <- out$conf.level

  class(out) <- "cohen_CI"
  out
}

print.cohen_CI <- function(x,...){
  cat("Call:\n")
  print(x$call)

  cat("\nEffect size estimate :\n")
  print(round(x$ES,3))

  cat(paste0("\n",x$conf.level*100," % confidence interval around effect size estimate:\n"))
  print(round(x$CI,3))

}