convey: Income Concentration Analysis with Complex Survey Samples

Documented in svyjdiv

#' J-divergence measure
#'
#' Estimate the J-divergence measure, an entropy-based measure of inequality
#'
#' @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.
#'
#' This measure only allows for strictly positive variables.
#'
#' @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 Guilherme Jacob, Djalma Pessoa, and Anthony Damico
#'
#' @seealso \code{\link{svygei}}
#'
#' @references Nicholas Rohde (2016). J-divergence measurements of economic inequality.
#' J. R. Statist. Soc. A, v. 179, Part 3 (2016), pp. 847-870.
#' DOI \doi{10.1111/rssa.12153}.
#'
#' Martin Biewen and Stephen Jenkins (2002). Estimation of Generalized Entropy
#' and Atkinson Inequality Indices from Complex Survey Data. \emph{DIW Discussion Papers},
#' No.345,
#' URL \url{https://www.diw.de/documents/publikationen/73/diw_01.c.40394.de/dp345.pdf}.
#'
#' @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)
#'
#' svyjdiv( ~eqincome , design = subset( des_eusilc , eqincome > 0 ) )
#'
#' # replicate-weighted design
#' des_eusilc_rep <- as.svrepdesign( des_eusilc , type = "bootstrap" )
#' des_eusilc_rep <- convey_prep(des_eusilc_rep)
#'
#' svyjdiv( ~eqincome , design = subset( des_eusilc_rep , eqincome > 0 ) )
#'
#' \dontrun{
#'
#' # linearized design using a variable with missings
#' svyjdiv( ~py010n , design = subset( des_eusilc , py010n > 0 | is.na( py010n ) ) )
#' svyjdiv( ~py010n , design = subset( des_eusilc , py010n > 0 | is.na( py010n ) ), na.rm = TRUE )
#' # replicate-weighted design using a variable with missings
#' svyjdiv( ~py010n , design = subset( des_eusilc_rep , py010n > 0 | is.na( py010n ) ) )
#' svyjdiv( ~py010n , design = subset( des_eusilc_rep , py010n > 0 | is.na( py010n ) ) , 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 )
#'
#' svyjdiv( ~eqincome , design = subset( dbd_eusilc , eqincome > 0 ) )
#'
#' dbRemoveTable( conn , 'eusilc' )
#'
#' dbDisconnect( conn , shutdown = TRUE )
#'
#' }
#'
#' @export
svyjdiv <- 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("svyjdiv", design)

}

#' @rdname svyjdiv
#' @export
svyjdiv.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$prob <- ifelse(nas , Inf , design$prob)
    }

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

    # test for positive income
    if (any(incvar[w > 0] <= 0 , na.rm = TRUE))
      stop(
        "The J-divergence measure is defined for strictly positive variables only.  Negative and zero values not allowed."
      )

    # # method 1: compute components
    # U_0 <- list( value = sum( w ), lin = rep( 1, length( incvar ) ) )
    # U_1 <- list( value = sum( w * incvar ), lin = incvar )
    # T_0 <- list( value = sum( w * log( incvar ) ), lin = log( incvar ) )
    # T_1 <- list( value = sum( w * incvar * log( incvar ) ), lin = incvar * log( incvar ) )
    # list_all <- list(  U_0 = U_0, U_1 = U_1, T_0 = T_0, T_1 = T_1 )
    # estimate <- contrastinf( quote( ( T_1 / U_1 ) - ( T_0 / U_0 ) ) , list_all )
    # rval <- estimate$value
    # lin <- estimate$lin

    # method 2: compute point estimate
    estimate <- CalcJDiv(incvar , w)
    lin <- CalcJDiv_IF(incvar , w)
    lin <- ifelse(w > 0 , lin , 0)

    # treat remaining missing
    if (is.na(estimate)) {
      rval <- NA
      variance <- as.matrix(NA)
      colnames(variance) <-
        rownames(variance) <-
        names(rval) <-
        strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]
      class(rval) <- c("cvystat" , "svystat")
      attr(rval, "statistic") <- "j-divergence"
      attr(rval, "var") <- variance
      return(rval)
    }

    # 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
    }

    # coerce to matrix
    lin <-
      matrix(lin ,
             nrow = length(lin) ,
             dimnames = list(names(w) , 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") <- "j-divergence"
    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 svyjdiv
#' @export
svyjdiv.svyrep.design <-
  function (formula,
            design,
            na.rm = FALSE,
            deff = FALSE ,
            linearized = FALSE ,
            return.replicates = FALSE ,
            ...) {
    # collect income data
    incvar <-
      model.frame(formula, design$variables, na.action = na.pass)[[1]]

    if (na.rm) {
      nas <- is.na(incvar)
      design <- design[!nas, ]
      incvar <-
        model.frame(formula, design$variables, na.action = na.pass)[[1]]
    }

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

    # check for positive incomes
    if (any(incvar[ws != 0] <= 0, na.rm = TRUE))
      stop(
        "The J-divergence measure is defined for strictly positive variables only.  Negative and zero values not allowed."
      )

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

    # treat remaining missing
    if (is.na(estimate)) {
      rval <- estimate
      variance <- as.matrix(NA)
      colnames(variance) <-
        rownames(variance) <-
        names(rval) <-
        strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]
      class(rval) <- c("cvystat" , "svystat")
      attr(rval, "statistic") <- "j-divergence"
      attr(rval, "var") <- variance
      return(rval)
    }

    ### variance calculation

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

    # compute replicates
    qq <- apply(wf, 2 , function(wi)
      CalcJDiv(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 <- CalcJDiv_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

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

    }

    # build result object
    rval <- estimate
    names(rval) <-
      strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]
    class(rval) <- c("cvystat" , "svrepstat")
    attr(rval, "var") <- variance
    attr(rval, "statistic") <- "j-divergence"
    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)
      class(rval) <- c("cvystat" , "svrepstat")
    }

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

    # return object
    rval

  }

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

    NextMethod("svyjdiv", design)
  }


# point estimate function
CalcJDiv <-  function(y , w) {

  # filter observations
  w <- ifelse( y > 0 & w != 0 , w , 0 )
  y <- ifelse( w!=0 , y , 1 )

  # compute point esitmate
  N <- sum( w )
  mu <- sum( y * w ) / N
  jdiv <- ( ( y - mu ) / mu ) * log( y / mu )

  # compute point estimate
  jdiv <- ifelse( w != 0 , jdiv , 0 )
  sum( jdiv * w ) / N

}

# function to compute linearized function
CalcJDiv_IF <-  function(y , w) {

  # filter observations
  w <- ifelse( y > 0 & w != 0 , w , 0 )
  y <- ifelse( w!=0 , y , 1 )

  # compute intermediate statistics
  Ntot <- sum( w )
  Ytot <- sum( y * w )
  Ybar <- Ytot / Ntot
  jdiv <-
    sum(ifelse(w > 0 ,  w * ((y / Ybar) - 1) * log(y / Ybar) , 0)) / Ntot
  gei1 <-
    sum(w * ifelse(w > 0 , (y / Ybar) * log(y / Ybar) , 0)) / Ntot

  # linearized function under fixed mean
  u.score <- (((y / Ybar) - 1) * log(y / Ybar))
  lin.fixed <- (u.score - jdiv) / Ntot

  # derivative wrt mean
  # djdiv.dYbar <- 1/Ybar - sum( w * (y/Ybar) * ( log( y/Ybar ) + 1 ) ) / Ytot
  djdiv.dYbar <- -gei1 / Ybar
  I.Ybar <- (y - Ybar) / Ntot

  # compute final linearized function
  lin <- lin.fixed + djdiv.dYbar * I.Ybar

  # fix domains
  lin <- ifelse(w > 0 , lin , 0)

  # return final linearized function
  return(lin)

}