R/ss4S2.R

In samplesize4surveys: Sample Size Calculations for Complex Surveys

#' @import TeachingSampling
#' @import timeDate
#' @export
#' 
#' @title
#' The required sample size for estimating a single variance
#' @description 
#' This function returns the minimum sample size required for estimating a single variance subjecto to predefined errors.
#' @details
#' Note that the minimun sample size to achieve a particular relative margin of error \eqn{\varepsilon} is defined by: 
#' \deqn{n = \frac{n_0}{\frac{(N-1)^3}{N^2(N*K+2N+2)}+\frac{n_0}{N}}}
#' Where \deqn{n_0=\frac{z^2_{1-\frac{\alpha}{2}}*DEFF}{\varepsilon^2}}
#' Also note that the minimun sample size to achieve a particular coefficient of variation \eqn{cve} is defined by:
#' \deqn{n = \frac{N^2(N*K+2N+2)*DEFF}{cve^2*(N-1)^3+N(N*K+2N+2)*DEFF}}
#'
#' @author Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>
#' @param N The population size.
#' @param K The population excess kurtosis of the variable in the population.
#' @param DEFF The design effect of the sample design. By default \code{DEFF = 1}, which corresponds to a simple random sampling design.
#' @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 me The maximun 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 
#' ss4S2(N = 10000, K = 0, cve = 0.05, me = 0.03)
#' ss4S2(N = 10000, K = 1, cve = 0.05, me = 0.03)
#' ss4S2(N = 10000, K = 1, cve = 0.05, me = 0.05, DEFF = 2)
#' ss4S2(N = 10000, K = 1, cve = 0.05, me = 0.03, plot = TRUE)

#' 
#' #############################
#' # Example with BigLucy data #
#' #############################
#' 
#' data(BigLucy)
#' attach(BigLucy)
#' N <- nrow(BigLucy)
#' K <- kurtosis(BigLucy$Income)
#' # The minimum sample size for simple random sampling
#' ss4S2(N, K, DEFF=1, conf=0.99, cve=0.03, me=0.1, plot=TRUE)
#' # The minimum sample size for a complex sampling design
#' ss4S2(N, K, DEFF=3.45, conf=0.99, cve=0.03, me=0.1, plot=TRUE)

ss4S2 <- function(N, K=0, DEFF=1, conf=0.95, cve=0.05, me=0.03, plot=FALSE){
  Z = 1 -((1 - conf)/2)
  n0.me <- qnorm(Z)^2 * DEFF / (me^2)
  n.me <- n0.me / ((N - 1)^3/(N^2*(K*N + 2*N + 2)) + n0.me / N)
  n.cve <- N^2 * (K*N + 2*N + 2) * DEFF/(cve^2 * (N - 1)^3 + N * (K*N + 2*N + 2) * DEFF)
  
  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
      cveseq[k] = 100 * sqrt(N / nseq[k] * (1 - fseq) * DEFF * N * (K*N + 2*N + 2) / ((N - 1)^3))
      meseq[k] = qnorm(Z) * cveseq[k]
    }
    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="Relative margin of error %",xlab="Sample size")
    points(n.me,100*me, pch=8,bg = "red")
    abline(h=100*me,lty=3)
    abline(v=n.me,lty=3)
  }
  msg <- cat('With the parameters of this function: N =', N, 'DEFF = ',
             DEFF, 'conf =', conf, '.\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 relative margin of error of', 100*me, '% is n=', ceiling(n.me), '. \n \n')
  
  result <- list(n.cve = ceiling(n.cve), n.me = ceiling(n.me))
  result 
}