#' Zenga index
#'
#' Estimate the Zenga index, a 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.
#'
#' @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{svygini}}
#'
#' @references Lucio Barabesi, Giancarlo Diana and Pier Francesco Perri (2016). Linearization of inequality indices in the design-based framework. Statistics, 50(5), 1161-1172.
#' DOI \doi{10.1080/02331888.2015.1135924}.
#'
#' Matti Langel and Yves Tille (2012). Inference by linearization for Zenga's new inequality index: a comparison with the Gini index.
#' Metrika, 75, 1093-1110. DOI \doi{10.1007/s00184-011-0369-1}.
#'
#' Matti Langel (2012). Measuring inequality in finite population sampling.
#' PhD thesis: Universite de Neuchatel,
#' URL \url{https://doc.rero.ch/record/29204/files/00002252.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)
#'
#' svyzenga( ~eqincome , design = des_eusilc )
#'
#' # replicate-weighted design
#' des_eusilc_rep <- as.svrepdesign( des_eusilc , type = "bootstrap" )
#' des_eusilc_rep <- convey_prep(des_eusilc_rep)
#'
#' svyzenga( ~eqincome , design = des_eusilc_rep )
#'
#' \dontrun{
#'
#' # linearized design using a variable with missings
#' svyzenga( ~ py010n , design = des_eusilc )
#' svyzenga( ~ py010n , design = des_eusilc , na.rm = TRUE )
#' # replicate-weighted design using a variable with missings
#' svyzenga( ~ py010n , design = des_eusilc_rep )
#' svyzenga( ~ 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 )
#'
#' svyzenga( ~ eqincome , design = dbd_eusilc )
#'
#' dbRemoveTable( conn , 'eusilc' )
#'
#' dbDisconnect( conn , shutdown = TRUE )
#'
#' }
#'
#' @export
svyzenga <-
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("svyzenga", design)
}
#' @rdname svyzenga
#' @export
svyzenga.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
# check for negative incomes
if (any(incvar[w != 0] < 0, na.rm = TRUE))
stop("The Zenga index is defined for non-negative numeric variables only.")
# compute point estimate
estimate <- CalcZenga(incvar , w)
# compute linearized function
lin <- CalcZenga_IF(incvar , w)
# compute variance
variance <-
survey::svyrecvar(
lin / design$prob,
design$cluster,
design$strata,
design$fpc,
postStrata = design$postStrata
)
variance[which(is.nan(variance))] <- NA
colnames(variance) <-
rownames(variance) <-
strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]
# compute deff
if (is.character(deff) || deff) {
nobs <- sum(weights(design) != 0)
npop <- sum(weights(design))
if (deff == "replace")
vsrs <-
survey::svyvar(lin , design, na.rm = na.rm) * npop ^ 2 / nobs
else
vsrs <-
survey::svyvar(lin , design , na.rm = na.rm) * npop ^ 2 * (npop - nobs) /
(npop * nobs)
deff.estimate <- variance / vsrs
}
# coerce to matrix
lin <-
matrix(lin ,
nrow = length(w) ,
dimnames = list(names(w) , strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]))
# build result object
rval <- estimate
names(rval) <-
strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]
class(rval) <- c("cvystat" , "svystat")
attr(rval, "var") <- variance
attr(rval, "statistic") <- "zenga"
if (linearized)
attr(rval, "linearized") <- lin
if (influence)
attr(rval , "influence") <-
sweep(lin , 1 , design$prob , "/")
if (linearized |
influence)
attr(rval , "index") <- as.numeric(rownames(lin))
if (is.character(deff) ||
deff)
attr(rval , "deff") <- deff.estimate
rval
}
#' @rdname svyzenga
#' @export
svyzenga.svyrep.design <-
function(formula ,
design ,
na.rm = FALSE ,
deff = FALSE ,
linearized = FALSE ,
return.replicates = FALSE ,
...) {
# collect data
incvar <-
model.frame(formula, design$variables, na.action = na.pass)[[1]]
# treat missing values
if (na.rm) {
nas <- is.na(incvar)
design <- design[!nas, ]
incvar <- incvar[!nas]
}
# collect sampling weights
ws <- weights(design, "sampling")
# check for negative incomes
if (any(incvar[ws != 0] < 0, na.rm = TRUE))
stop("The Zenga index is defined for non-negative numeric variables only.")
# compute point estimate
estimate <- CalcZenga(incvar, ws)
# collect analysis weights
ww <- weights(design, "analysis")
# compute replicates
qq <- apply(ww, 2, function(wi)
CalcZenga(incvar, wi))
# compute variance
if (any(is.na(qq)))
variance <- as.matrix(NA)
else {
variance <-
survey::svrVar(qq ,
design$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 <- CalcZenga_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$variables)
# coerce to matrix
lin <-
matrix(lin ,
nrow = length(ws) ,
dimnames = list(names(ws) , strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]))
}
# build result object
rval <- estimate
names(rval) <-
strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]
attr(rval, "var") <- variance
attr(rval, "statistic") <- "zenga"
if (linearized)
attr(rval , "linearized") <- lin
if (linearized)
attr(rval , "index") <- as.numeric(rownames(lin))
# keep replicates
if (return.replicates) {
attr(qq , "scale") <- design$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 svyzenga
#' @export
svyzenga.DBIsvydesign <-
function (formula, design, ...) {
design$variables <-
getvars(
formula,
design$db$connection,
design$db$tablename,
updates = design$updates,
subset = design$subset
)
NextMethod("svyzenga", design)
}
# BDP2016 estimator
CalcZenga <- function(y.k , w.k) {
# filter observation
y.k <- y.k[w.k != 0]
w.k <- w.k[w.k != 0]
# reorder observations
ordy <- order(y.k)
y.k <- y.k[ordy]
w.k <- w.k[ordy]
# compute intermediate statistical functionals
N <- sum(w.k)
Tot <- sum(w.k * y.k)
Hy <- cumsum(w.k)
Ky <- (Tot - cumsum(w.k * y.k)) + w.k * y.k
# compute zenga
1 - sum(w.k * (N - Hy) * (Tot - Ky) / (N * Hy * Ky))
}
# BDP2016 estimator
CalcZenga_IF <- function(y.k , w.k) {
# treat null weights
y.k <- ifelse(w.k == 0 , 0 , y.k)
# reorder observations
ordy <- order(y.k)
y.k <- y.k[ordy]
w.k <- w.k[ordy]
# filter observations
wf.k <- w.k
y.k <- y.k[wf.k != 0]
w.k <- wf.k[wf.k != 0]
# compute intermediate statistical functionals
N <- sum(w.k)
Tot <- sum(w.k * y.k)
Hy <- cumsum(w.k)
Ky <- (Tot - cumsum(w.k * y.k)) + w.k * y.k
# compute linearized variable
T1 <- -(N - Hy) * (Tot - Ky) / (N * Hy * Ky)
T2 <- -sum(w.k * (Tot - Ky) / Ky) / N ^ 2
T3 <- -(y.k / N) * sum(w.k * (N - Hy) / (Hy * Ky))
T4 <-
sum(w.k * (Tot - Ky) / (Hy ^ 2 * Ky)) - cumsum(w.k * (Tot - Ky) / (Hy ^
2 * Ky)) + (w.k * (Tot - Ky) / (Hy ^ 2 * Ky))
T5 <- (Tot * y.k / N) * cumsum(w.k * (N - Hy) / (Hy * Ky ^ 2))
lin.domain <- rowSums(cbind(T1 , T2 , T3 , T4 , T5))
# return linearized variable
lin <- rep(0 , length(wf.k))
lin[wf.k != 0] <- lin.domain
lin <- lin[order(ordy)]
lin
}
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