R/ss4stm.R

In psirusteam/samplesize4surveys: Sample Size Calculations for Complex Surveys

#' @import TeachingSampling
#' @export
#' 
#' @title
#' Sample Size for Estimation of Means in Stratified Sampling
#' @description 
#' This function computes the minimum sample size required for estimating a single mean, in a stratified sampling, subject to predefined errors.
#' @return 
#' The required sample size for the sample and the required sample size per stratum. 
#' @details
#' Let assume that the population U is partitioned in H strate. 
#' Under a stratified sampling, the neccesary sample size to achieve a relative margin of error \eqn{\varepsilon} is defined by: 
#' \deqn{n = \frac{(\sum_{h=1}^H w_h S_h)^2}{\frac{\varepsilon^2}{z^2_{1-\frac{\alpha}{2}}}+\frac{\sum_{h=1}^H w_h S^2_h}{N}}} 
#' Where \deqn{S^2_h = DEFF_h \sigma^2_h}
#' Then, the required sample size in each stratum is given by:
#' \deqn{n_h = n \frac{w_h S_h}{\sum_{h=1}^H w_h S_h}}
#' @author Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>
#' @param Nh Vector. The population size for each stratum.
#' @param muh Vector. The means of the variable of interest in each stratum.
#' @param sigmah Vector. The standard deviation of the variable of interest in each stratum.
#' @param DEFFh Vector. The design effect of the sample design in each stratum. By default \code{DEFFh = 1}, which corresponds to a stratified simple random sampling design. 
#' @param conf The statistical confidence. By default \code{conf = 0.95}.
#' @param rme The maximun relative margin of error that can be allowed for the estimation.
#' 
#' @references 
#' Gutierrez, H. A. (2009), \emph{Estrategias de muestreo: Diseno de encuestas y estimacion de parametros}. Editorial Universidad Santo Tomas
#' @seealso \code{\link{ss4m}}
#' 
#' @examples 
#' 
#' Nh <- c(15000, 10000, 5000)
#' muh <- c(300, 200, 100)
#' sigmah <- c(200, 100, 20)
#' DEFFh <- c(1, 1.2, 1.5)
#' 
#' ss4stm(Nh, muh, sigmah, rme=0.03)
#' ss4stm(Nh, muh, sigmah, conf = 0.99, rme=0.03)
#' ss4stm(Nh, muh, sigmah, DEFFh, conf= 0.99, rme=0.03)
#' 
#' ##########################
#' # Example with Lucy data #
#' ##########################
#' data(Lucy)
#' attach(Lucy)
#' 
#' Strata <- as.factor(paste(Zone, Level))
#' levels(Strata)
#' 
#' Nh <- summary(Strata)
#' muh <- tapply(Income, Strata, mean)
#' sigmah <- tapply(Income, Strata, sd)
#' 
#' ss4stm(Nh, muh, sigmah, DEFFh=1, conf = 0.95, rme = 0.03)
#' ss4stm(Nh, muh, sigmah, DEFFh=1.5, conf = 0.95, rme = 0.03)
#' 
#' #############################
#' # Example with BigLucy data #
#' #############################
#' data(BigLucy)
#' attach(BigLucy)
#' 
#' Nh <- summary(Zone)
#' muh <- tapply(Income, Zone, mean)
#' sigmah <- tapply(Income, Zone, sd)
#' 
#' ss4stm(Nh, muh, sigmah, DEFFh=1, conf = 0.95, rme = 0.03)
#' ss4stm(Nh, muh, sigmah, DEFFh=1.5, conf = 0.95, rme = 0.03)


ss4stm <- function(Nh, muh, sigmah, DEFFh=1, conf=0.95, rme=0.03){
  N <- sum(Nh)
  H <- length(Nh)
  Sh <- sigmah * sqrt(DEFFh)
  t <- 1 - ((1 - conf) / 2)
  qt <- qt(t, df = N - H)
  mu <- sum(Nh * muh) / N
  me <- rme * mu
  wh <- Nh / N
  n.rme <- (sum(wh * Sh))^2 / ((me / qt)^2 + (sum(wh * Sh^2) / N))
  nh <- n.rme * (wh * Sh) / sum(wh * Sh)
  
  result <- list(n.rme = ceiling(n.rme), nh = ceiling(nh))
  result 
}