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#' Atkinson index
#'
#' Estimate the Atkinson index, 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 epsilon a parameter that determines the sensivity towards inequality in the bottom of the distribution. Defaults to \code{epsilon = 1}.
#' @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 Guilherme Jacob, Djalma Pessoa and Anthony Damico
#'
#' @seealso \code{\link{svygei}}
#'
#' @references 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}.
#'
#' 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)
#'
#' # replicate-weighted design
#' des_eusilc_rep <- as.svrepdesign( des_eusilc , type = "bootstrap" )
#' des_eusilc_rep <- convey_prep(des_eusilc_rep)
#'
#'
#' # subset all designs to positive income and non-missing records only
#' des_eusilc_pos_inc <- subset( des_eusilc , eqincome > 0 )
#' des_eusilc_rep_pos_inc <- subset( des_eusilc_rep , eqincome > 0 )
#'
#'
#' # linearized design
#' svyatk( ~eqincome , des_eusilc_pos_inc, epsilon = .5 )
#' svyatk( ~eqincome , des_eusilc_pos_inc )
#' svyatk( ~eqincome , des_eusilc_pos_inc, epsilon = 2 )
#'
#' # replicate-weighted design
#' svyatk( ~eqincome , des_eusilc_rep_pos_inc, epsilon = .5 )
#' svyatk( ~eqincome , des_eusilc_rep_pos_inc )
#' svyatk( ~eqincome , des_eusilc_rep_pos_inc, epsilon = 2 )
#'
#'
#' # subsetting
#' svyatk( ~eqincome , subset(des_eusilc_pos_inc, db040 == "Styria"), epsilon = .5 )
#' svyatk( ~eqincome , subset(des_eusilc_pos_inc, db040 == "Styria") )
#' svyatk( ~eqincome , subset(des_eusilc_pos_inc, db040 == "Styria"), epsilon = 2 )
#'
#' svyatk( ~eqincome , subset(des_eusilc_rep_pos_inc, db040 == "Styria"), epsilon = .5 )
#' svyatk( ~eqincome , subset(des_eusilc_rep_pos_inc, db040 == "Styria") )
#' svyatk( ~eqincome , subset(des_eusilc_rep_pos_inc, db040 == "Styria"), epsilon = 2 )
#'
#' \dontrun{
#'
#' # linearized design using a variable with missings (but subsetted to remove negatives)
#' svyatk( ~py010n , subset(des_eusilc, py010n > 0 | is.na(py010n)), epsilon = .5 )
#' svyatk( ~py010n , subset(des_eusilc, py010n > 0 | is.na(py010n)), epsilon = .5 , na.rm=TRUE )
#'
#' # replicate-weighted design using a variable with missings (but subsetted to remove negatives)
#' svyatk( ~py010n , subset(des_eusilc_rep, py010n > 0 | is.na(py010n)), epsilon = .5 )
#' svyatk( ~py010n , subset(des_eusilc_rep, py010n > 0 | is.na(py010n)), epsilon = .5 , 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 )
#'
#'
#' # subset all designs to positive income and non-missing records only
#' dbd_eusilc_pos_inc <- subset( dbd_eusilc , eqincome > 0 )
#'
#'
#' # database-backed linearized design
#' svyatk( ~eqincome , dbd_eusilc_pos_inc, epsilon = .5 )
#' svyatk( ~eqincome , dbd_eusilc_pos_inc )
#' svyatk( ~eqincome , dbd_eusilc_pos_inc, epsilon = 2 )
#'
#' svyatk( ~eqincome , subset(dbd_eusilc_pos_inc, db040 == "Styria"), epsilon = .5 )
#' svyatk( ~eqincome , subset(dbd_eusilc_pos_inc, db040 == "Styria") )
#' svyatk( ~eqincome , subset(dbd_eusilc_pos_inc, db040 == "Styria"), epsilon = 2 )
#'
#' # database-backed linearized design using a variable with missings
#' # (but subsetted to remove negatives)
#' svyatk( ~py010n , subset(dbd_eusilc, py010n > 0 | is.na(py010n)), epsilon = .5 )
#' svyatk( ~py010n , subset(dbd_eusilc, py010n > 0 | is.na(py010n)), epsilon = .5 , na.rm=TRUE )
#'
#'
#' dbRemoveTable( conn , 'eusilc' )
#'
#' dbDisconnect( conn , shutdown = TRUE )
#'
#' }
#'
#' @export
svyatk <-
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"
)
if ('epsilon' %in% names(list(...)) &&
list(...)[["epsilon"]] <= 0)
stop("epsilon= must be positive.")
UseMethod("svyatk", design)
}
#' @rdname svyatk
#' @export
svyatk.survey.design <-
function(formula ,
design ,
epsilon = 1 ,
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
# treat non-positive incomes
if ( any( incvar[w != 0] <= 0 , na.rm = TRUE ) )
stop(
"The Atkinson Index is defined for strictly positive variables only. Negative and zero values not allowed."
)
# compute point estimate
estimate <- CalcAtkinson(incvar , w , epsilon)
# compute linearized function
lin <- CalcAtkinson_IF(incvar , w , epsilon)
# compute variance
variance <-
survey::svyrecvar(
lin / design$prob,
design$cluster,
design$strata,
design$fpc,
postStrata = design$postStrata
)
variance[which(is.nan(variance))] <- NA
colnames(variance) <-
rownames(variance) <-
strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]
# compute deff
if (is.character(deff) || deff) {
nobs <- sum(weights(design) != 0)
npop <- sum(weights(design))
if (deff == "replace")
vsrs <-
survey::svyvar(lin , design, na.rm = na.rm) * npop ^ 2 / nobs
else
vsrs <-
survey::svyvar(lin , design , na.rm = na.rm) * npop ^ 2 * (npop - nobs) /
(npop * nobs)
deff.estimate <- variance / vsrs
}
# coerce to matrix
lin <-
matrix(lin ,
nrow = length(lin) ,
dimnames = list( rownames( design$variables ) , 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") <- "atkinson"
attr(rval, "epsilon") <- epsilon
if (influence)
attr(rval , "influence") <-
sweep(lin , 1 , design$prob , "/")
if (linearized)
attr(rval, "linearized") <- lin
if (linearized |
influence)
attr(rval , "index") <- as.numeric(rownames(lin))
if (is.character(deff) ||
deff)
attr(rval , "deff") <- deff.estimate
rval
}
#' @rdname svyatk
#' @export
svyatk.svyrep.design <-
function(formula ,
design ,
epsilon = 1 ,
na.rm = FALSE ,
deff = FALSE ,
linearized = FALSE ,
return.replicates = FALSE ,
...) {
# collect data
incvar <-
model.frame(formula, design$variables, na.action = na.pass)[[1]]
# treat missing values
if (na.rm) {
nas <- is.na(incvar)
design <- design[!nas, ]
incvar <- incvar[!nas]
}
# colelct sampling weights
ws <- weights(design, "sampling")
# treat non-positive incomes
if ( any( incvar[ws != 0] <= 0 , na.rm = TRUE ) )
stop(
"The Atkinson Index is defined for strictly positive variables only. Negative and zero values not allowed."
)
# compute point estimate
estimate <- CalcAtkinson(incvar , ws , epsilon)
# collect analysis weights
ww <- weights(design, "analysis")
# compute replicates
qq <- apply(ww, 2, function(wi)
CalcAtkinson(incvar , wi , epsilon))
# compute variance
if (any(is.na(qq)))
variance <- as.matrix(NA)
else {
variance <-
survey::svrVar(qq ,
design$scale ,
design$rscales ,
mse = design$mse ,
coef = estimate)
this.mean <- attr(variance , "means")
variance <- as.matrix(variance)
attr(variance , "means") <- this.mean
}
colnames(variance) <-
rownames(variance) <-
strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]
# compute deff
if (is.character(deff) || deff || linearized) {
# compute linearized function
lin <- CalcAtkinson_IF(incvar , ws , epsilon)
# compute deff
nobs <- length(design$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( rownames( design$variables ) , strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]))
}
# build result object
rval <- estimate
names(rval) <-
strsplit(as.character(formula)[[2]] , ' \\+ ')[[1]]
attr(rval, "var") <- variance
attr(rval, "statistic") <- "atkinson"
if (linearized)
attr(rval, "linearized") <- lin
if (linearized )
attr(rval , "index") <- as.numeric(rownames(lin))
# keep replicates
if (return.replicates) {
attr(qq , "scale") <- design$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 svyatk
#' @export
svyatk.DBIsvydesign <-
function (formula, design, ...) {
design$variables <-
getvars(
formula,
design$db$connection,
design$db$tablename,
updates = design$updates,
subset = design$subset
)
NextMethod("svyatk", design)
}
CalcAtkinson <-
function(yk , wk , epsilon) {
# filter observations
wk <- ifelse( yk > 0 & wk != 0 , wk , 0 )
yk <- ifelse( wk !=0 , yk , 1 )
# compute intermediate statistics
U0 <- sum( wk )
U1 <- sum( wk * yk )
T0 <- sum( wk * log( yk ) )
T1 <- sum( wk * yk * log( yk ) )
# compute values
if (epsilon == 1) {
res <- 1 - ( U0 / U1 ) * exp( T0 / U0 )
} else {
U.neps <- sum( wk * yk^( 1 - epsilon ) )
afac <- epsilon / ( 1 - epsilon )
bfac <- 1 / ( 1 - epsilon )
res <- 1 - ( U.neps^bfac ) / ( U0^afac * U1)
}
res
}
CalcAtkinson_IF <-
function(yk , wk , epsilon) {
# filter observations
wk <- ifelse( yk > 0 & wk != 0 , wk , 0 )
yk <- ifelse( wk !=0 , yk , 1 )
U0 <- sum( wk )
U1 <- sum( wk * yk )
T0 <- sum( wk * log( yk ) )
T1 <- sum( wk * yk * log( yk ) )
if (epsilon == 1) {
A1 <- CalcAtkinson( yk , wk , 1 )
L1 <- ( A1 - 1 ) * (1 - T0 / U0 ) / U0
L2 <- yk * ( 1 - A1 ) / U1
L3 <- log( yk ) * ( A1 - 1 ) / U0
lin <- rowSums( cbind( L1 , L2 , L3 ) )
} else {
U.neps <- sum( wk * yk^( 1 - epsilon ) )
afac <- epsilon / ( 1 - epsilon )
bfac <- 1 / ( 1 - epsilon )
L1 <- afac * ( U.neps^bfac ) / ( U1 * U0^bfac )
L2 <- yk * ( U.neps^bfac ) / ( U0^afac * U1^2 )
L3 <- - bfac * ( yk^( 1 - epsilon ) ) * ( U.neps^afac ) / ( U0^afac * U1 )
lin <- rowSums( cbind( L1 , L2 , L3 ) )
}
lin <- ifelse( wk != 0 , lin , 0 )
lin
}
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