R/svygini.R

In convey: Income Concentration Analysis with Complex Survey Samples

#' Gini coefficient
#'
#' Estimate the Gini coefficient, an inequality measure
#'
#' @param formula a formula specifying the income variable
#' @param design a design object of class \code{survey.design} or class \code{svyrep.design} from the \code{survey} library.
#' @param na.rm Should cases with missing values be dropped?
#' @param deff Return the design effect (see \code{survey::svymean})
#' @param linearized Should a matrix of linearized variables be returned
#' @param influence Should a matrix of (weighted) influence functions be returned? (for compatibility with \code{\link[survey]{svyby}})
#' @param return.replicates Return the replicate estimates?
#' @param ... future expansion
#'
#' @details you must run the \code{convey_prep} function on your survey design object immediately after creating it with the \code{svydesign} or \code{svrepdesign} function.
#'
#' @return Object of class "\code{cvystat}", which are vectors with a "\code{var}" attribute giving the variance and a "\code{statistic}" attribute giving the name of the statistic.
#'
#' @author Djalma Pessoa, Guilherme Jacob, and Anthony Damico
#'
#' @seealso \code{\link{svyarpr}}
#'
#' @references Guillaume Osier (2009). Variance estimation for complex indicators
#' of poverty and inequality. \emph{Journal of the European Survey Research
#' Association}, Vol.3, No.3, pp. 167-195,
#' ISSN 1864-3361, URL \url{https://ojs.ub.uni-konstanz.de/srm/article/view/369}.
#'
#' Jean-Claude Deville (1999). Variance estimation for complex statistics and estimators:
#' linearization and residual techniques. Survey Methodology, 25, 193-203,
#' URL \url{https://www150.statcan.gc.ca/n1/en/catalogue/12-001-X19990024882}.
#'
#' @keywords survey
#'
#' @examples
#' library(survey)
#' library(laeken)
#' data(eusilc) ; names( eusilc ) <- tolower( names( eusilc ) )
#'
#' # linearized design
#' des_eusilc <- svydesign( ids = ~rb030 , strata = ~db040 ,  weights = ~rb050 , data = eusilc )
#' des_eusilc <- convey_prep(des_eusilc)
#'
#' svygini( ~eqincome , design = des_eusilc )
#'
#' # replicate-weighted design
#' des_eusilc_rep <- as.svrepdesign( des_eusilc , type = "bootstrap" )
#' des_eusilc_rep <- convey_prep(des_eusilc_rep)
#'
#' svygini( ~eqincome , design = des_eusilc_rep )
#'
#' \dontrun{
#'
#' # linearized design using a variable with missings
#' svygini( ~ py010n , design = des_eusilc )
#' svygini( ~ py010n , design = des_eusilc , na.rm = TRUE )
#' # replicate-weighted design using a variable with missings
#' svygini( ~ py010n , design = des_eusilc_rep )
#' svygini( ~ py010n , design = des_eusilc_rep , na.rm = TRUE )
#'
#' # database-backed design
#' library(RSQLite)
#' library(DBI)
#' dbfile <- tempfile()
#' conn <- dbConnect( RSQLite::SQLite() , dbfile )
#' dbWriteTable( conn , 'eusilc' , eusilc )
#'
#' dbd_eusilc <-
#' 	svydesign(
#' 		ids = ~rb030 ,
#' 		strata = ~db040 ,
#' 		weights = ~rb050 ,
#' 		data="eusilc",
#' 		dbname=dbfile,
#' 		dbtype="SQLite"
#' 	)
#'
#' dbd_eusilc <- convey_prep( dbd_eusilc )
#'
#' svygini( ~ eqincome , design = dbd_eusilc )
#'
#' dbRemoveTable( conn , 'eusilc' )
#'
#' dbDisconnect( conn , shutdown = TRUE )
#'
#' }
#'
#' @export
svygini <-
  function(formula, design, ...) {
    if (length(attr(terms.formula(formula) , "term.labels")) > 1)
      stop(
        "convey package functions currently only support one variable in the `formula=` argument"
      )

    UseMethod("svygini", design)

  }


#' @rdname svygini
#' @export
svygini.survey.design <-
  function(formula ,
           design ,
           na.rm = FALSE ,
           deff = FALSE ,
           linearized = FALSE ,
           influence = FALSE ,
           ...) {
    # collect income data
    incvar <-
      model.frame(formula, design$variables, na.action = na.pass)[[1]]

    # treat missing values
    if (na.rm) {
      nas <- is.na(incvar)
      design <- design[nas == 0,]
      if (length(nas) > length(design$prob))
        incvar <- incvar[nas == 0]
      else
        incvar[nas > 0] <- 0
    }

    # collect sampling weights
    w <- 1 / design$prob

    # compute point estimate
    estimate <- CalcGini(incvar , w)

    # compute linearized function
    lin <- CalcGini_IF(incvar , w)

    # ensure length
    if (length(lin) != length(design$prob)) {
      tmplin <- rep(0 , nrow(design$variables))
      tmplin[w > 0] <- lin
      lin <- tmplin
      rm(tmplin)
      names(lin) <- rownames(design$variables)
    }

    # compute variance
    variance <-
      survey::svyrecvar(
        lin / design$prob,
        design$cluster,
        design$strata,
        design$fpc,
        postStrata = design$postStrata
      )
    variance[which(is.nan(variance))] <- NA
    colnames(variance) <-
      rownames(variance) <-
      strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]

    # compute deff
    if (is.character(deff) || deff) {
      nobs <- sum(weights(design) != 0)
      npop <- sum(weights(design))
      if (deff == "replace")
        vsrs <-
        survey::svyvar(lin , design, na.rm = na.rm) * npop ^ 2 / nobs
      else
        vsrs <-
        survey::svyvar(lin , design , na.rm = na.rm) * npop ^ 2 * (npop - nobs) /
        (npop * nobs)
      deff.estimate <- variance / vsrs
    }

    # # keep necessary linearized functions
    # lin <- lin[1 / design$prob > 0]

    # coerce to matrix
    lin <-
      matrix(lin ,
             nrow = length(lin) ,
             dimnames = list(names(lin) , strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]))

    # build result object
    rval <- estimate
    names(rval) <-
      strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]
    class(rval) <- c("cvystat" , "svystat")
    attr(rval, "var") <- variance
    attr(rval, "statistic") <- "gini"
    if (linearized)
      attr(rval, "linearized") <- lin
    if (influence)
      attr(rval , "influence")  <-
      sweep(lin , 1 , design$prob , "/")
    if (linearized |
        influence)
      attr(rval , "index") <- as.numeric(rownames(lin))
    if (is.character(deff) ||
        deff)
      attr(rval , "deff") <- deff.estimate
    rval

  }

#' @rdname svygini
#' @export
svygini.svyrep.design <-
  function(formula ,
           design ,
           na.rm = FALSE ,
           deff = FALSE ,
           linearized = FALSE ,
           return.replicates = FALSE ,
           ...) {
    # collect data
    df <- model.frame(design)
    incvar <-
      model.frame(formula, design$variables, na.action = na.pass)[[1]]

    # treat missing values
    if (na.rm) {
      nas <- is.na(incvar)
      design <- design[!nas, ]
      df <- model.frame(design)
      incvar <- incvar[!nas]
    }

    # colelct sampling weights
    ws <- weights(design, "sampling")

    # compute point estimate
    estimate <- CalcGini(incvar, ws)

    # collect analysis weights
    ww <- weights(design, "analysis")

    # compute replicates
    qq <- apply(ww, 2, function(wi)
      CalcGini(incvar, wi))

    # compute variance
    if (any(is.na(qq)))
      variance <- as.matrix(NA)
    else {
      variance <-
        survey::svrVar(qq ,
                       design$scale ,
                       design$rscales ,
                       mse = design$mse ,
                       coef = estimate)
      this.mean <- attr(variance , "means")
      variance <- as.matrix(variance)
      attr(variance , "means") <- this.mean
    }
    colnames(variance) <-
      rownames(variance) <-
      strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]

    # compute deff
    if (is.character(deff) || deff || linearized) {
      # compute linearized function
      lin <- CalcGini_IF(incvar , ws)

      # compute deff
      nobs <- length(design$pweights)
      npop <- sum(design$pweights)
      vsrs <-
        unclass(
          survey::svyvar(
            lin ,
            design,
            na.rm = na.rm,
            return.replicates = FALSE,
            estimate.only = TRUE
          )
        ) * npop ^ 2 / nobs
      if (deff != "replace")
        vsrs <- vsrs * (npop - nobs) / npop
      deff.estimate <- variance / vsrs

      # filter observation
      names(lin) <- rownames(design$variables)

      # coerce to matrix
      lin <-
        matrix(lin ,
               nrow = length(lin) ,
               dimnames = list(names(lin) , strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]))

    }

    # build result object
    rval <- estimate
    names(rval) <-
      strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]
    attr(rval, "var") <- variance
    attr(rval, "statistic") <- "gini"
    if (linearized)
      attr(rval , "linearized") <- lin
    if (linearized)
      attr(rval , "index") <- as.numeric(rownames(lin))

    # keep replicates
    if (return.replicates) {
      attr(qq , "scale") <- design$scale
      attr(qq , "rscales") <- design$rscales
      attr(qq , "mse") <- design$mse
      rval <- list(mean = rval , replicates = qq)
    }

    # add design effect estimate
    if (is.character(deff) ||
        deff)
      attr(rval , "deff") <- deff.estimate

    # return object
    class(rval) <- c("cvystat" , "svrepstat")
    rval

  }


#' @rdname svygini
#' @export
svygini.DBIsvydesign <-
  function (formula, design, ...) {
    design$variables <-
      getvars(
        formula,
        design$db$connection,
        design$db$tablename,
        updates = design$updates,
        subset = design$subset
      )

    NextMethod("svygini", design)
  }


# gini estimate function
CalcGini <-
  function(x, pw) {
    # filter observations
    x <- x[pw > 0]
    pw <- pw[pw > 0]

    # reorder
    pw <- pw[order(x)]
    x <- x[order(x)]

    # intermediate estimates
    N <- sum(pw)
    n <- length(x)
    big_t <- sum(x * pw)
    r <- cumsum(pw)
    Num <- sum((2 * r - 1) * x * pw)
    Den <- N * big_t

    # gini estimate
    (Num / Den) - 1

  }

# gini linearized function
CalcGini_IF <- function(x , pw) {
  # filter observations
  x <- x[pw > 0]
  pw <- pw[pw > 0]

  # collect indices
  ind <- names(pw)

  # reorder observations
  ordx <- order(x)
  pw <- pw[ordx]
  x <- x[ordx]

  # population size
  N <- sum(pw)

  # total income
  Y <- sum(x * pw)

  # cumulative weight
  r <- cumsum(pw)

  # partial weighted function
  G <- cumsum(x * pw)
  T1 <-
    list(value = sum(r * x * pw) , lin = (Y - G + x * pw + r * x))
  T2 <- list(value = sum(x * pw), lin = x)
  T3 <- list(value = sum(pw) , lin = rep(1 , length(x)))

  # get T1
  list_all <- list(T1 = T1, T2 = T2, T3 = T3)
  GINI <-
    contrastinf(quote((2 * T1 - T2) / (T2 * T3) - 1) , list_all)
  lingini <- as.numeric(GINI$lin)

  # flip back to original order
  lingini <- lingini[order(ordx)]

  # add indices
  names(lingini) <- ind

  # return object
  return(lingini)

}