R/ss4dm.R

In psirusteam/samplesize4surveys: Sample Size Calculations for Complex Surveys

#' @import TeachingSampling
#' @export
#' 
#' @title
#' The required sample size for estimating a single difference of proportions
#' @description 
#' This function returns the minimum sample size required for estimating a single proportion subjecto to predefined errors.
#' @details
#' Note that the minimun sample size to achieve a relative margin of error \eqn{\varepsilon} is defined by: 
#' \deqn{n = \frac{n_0}{1+\frac{n_0}{N}}}
#' Where \deqn{n_0=\frac{z^2_{1-\frac{alpha}{2}}S^2}{\varepsilon^2 (\mu_1 - \mu_2)^2}} and 
#' \eqn{S^2=(\sigma_1^2 + \sigma_2^2) * (1 - (T * R)) * DEFF}
#' Also note that the minimun sample size to achieve a coefficient of variation \eqn{cve} is defined by:
#' \deqn{n = \frac{S^2}{|\bar{y}_1-\bar{y}_2|^2 cve^2 + \frac{S^2}{N}}} 
#'   
#' @author Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>
#' @param N The maximun population size between the groups (strata) that we want to compare.
#' @param mu1 The value of the estimated mean of the variable of interes for the first population.
#' @param mu2 The value of the estimated mean of the variable of interes for the second population.
#' @param sigma1 The value of the estimated variance of the variable of interes for the first population.
#' @param sigma2 The value of the estimated mean of a variable of interes for the second population.
#' @param DEFF The design effect of the sample design. By default \code{DEFF = 1}, which corresponds to a simple random sampling design.
#' @param T The overlap between waves. By default \code{T = 0}.
#' @param R The correlation between waves. By default \code{R = 1}.
#' @param conf The statistical confidence. By default conf = 0.95. By default \code{conf = 0.95}.
#' @param cve The maximun coeficient of variation that can be allowed for the estimation.
#' @param rme The maximun relative margin of error that can be allowed for the estimation.
#' @param plot Optionally plot the errors (cve and margin of error) against the sample size.
#' 
#' @references 
#' Gutierrez, H. A. (2009), \emph{Estrategias de muestreo: Diseno de encuestas y estimacion de parametros}. Editorial Universidad Santo Tomas
#' @seealso \code{\link{e4p}}
#' @examples 
#' ss4dm(N=100000, mu1=50, mu2=55, sigma1 = 10, sigma2 = 12, cve=0.05, rme=0.03)
#' ss4dm(N=100000, mu1=50, mu2=55, sigma1 = 10, sigma2 = 12, cve=0.05, rme=0.03, plot=TRUE)
#' ss4dm(N=100000, mu1=50, mu2=55, sigma1 = 10, sigma2 = 12, DEFF=3.45, conf=0.99, cve=0.03, 
#'      rme=0.03, plot=TRUE)
#' 
#' #############################
#' # Example with BigLucy data #
#' #############################
#' data(BigLucy)
#' attach(BigLucy)
#' 
#' N1 <- table(SPAM)[1]
#' N2 <- table(SPAM)[2]
#' N <- max(N1,N2)
#' 
#' BigLucy.yes <- subset(BigLucy, SPAM == 'yes')
#' BigLucy.no <- subset(BigLucy, SPAM == 'no')
#' mu1 <- mean(BigLucy.yes$Income)
#' mu2 <- mean(BigLucy.no$Income)
#' sigma1 <- sd(BigLucy.yes$Income)
#' sigma2 <- sd(BigLucy.no$Income)
#' 
#' # The minimum sample size for simple random sampling
#' ss4dm(N, mu1, mu2, sigma1, sigma2, DEFF=1, conf=0.99, cve=0.03, rme=0.03, plot=TRUE)
#' # The minimum sample size for a complex sampling design
#' ss4dm(N, mu1, mu2, sigma1, sigma2, DEFF=3.45, conf=0.99, cve=0.03, rme=0.03, plot=TRUE)

ss4dm = function(N, mu1, mu2, sigma1, sigma2, DEFF = 1, conf = 0.95, 
  cve = 0.05, rme = 0.03, T = 0, R = 1, plot = FALSE) {
  
  S2 = (sigma1^2 + sigma2^2) * (1 - (T * R)) * DEFF
  Z = 1 - ((1 - conf)/2)
  n.cve <- S2/((mu1 - mu2)^2 * cve^2 + (S2/N))
  me <- rme * abs(mu1 - mu2)
  n0 <- (qnorm(Z)^2/me^2) * S2
  n.rme <- n0/(1 + (n0/N))
  
  if (plot == TRUE) {
    
    nseq = seq(100, N, 10)
    cveseq = rep(NA, length(nseq))
    meseq = rep(NA, length(nseq))
    
    for (k in 1:length(nseq)) {
      fseq = nseq[k]/N
      varseq = (1/nseq[k]) * (1 - fseq) * S2
      cveseq[k] = 100 * sqrt(varseq)/abs(mu1 - mu2)
      meseq[k] = 100 * qnorm(Z) * sqrt(varseq)
    }
    
    par(mfrow = c(1, 2))
    plot(nseq, cveseq, type = "l", lty = 2, pch = 1, col = 3, 
      ylab = "Coefficient of variation %", xlab = "Sample size")
    points(n.cve, 100 * cve, pch = 8, bg = "blue")
    abline(h = 100 * cve, lty = 3)
    abline(v = n.cve, lty = 3)
    
    plot(nseq, meseq, type = "l", lty = 2, pch = 1, col = 3, 
      ylab = "Margin of error %", xlab = "Sample size")
    points(n.rme, 100 * me, pch = 8, bg = "red")
    abline(h = 100 * me, lty = 3)
    abline(v = n.rme, lty = 3)
  }
  
  msg <- cat("With the parameters of this function: N =", N, 
    "mu1 =", mu1, "mu2 =", mu2, "sigma1 = ", sigma1, "sigma2 = ", 
    sigma2, "DEFF = ", DEFF, "conf =", conf, "T =", T, "R =", 
    R, ".\n The estimated sample size to obtain a maximun coefficient of variation of", 
    100 * cve, "% is n=", ceiling(n.cve), ".
     The estimated sample size to obtain a maximun margin of error of", 
    100 * rme, "% is n=", ceiling(n.rme), ". \n \n")
  
  result <- list(n.cve = ceiling(n.cve), n.rme = ceiling(n.rme))
  result
}