#' 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)
}
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.