✔️ direct relationship with the Lorenz curve
✔️ most used inequality measure
✔️ [0,1]-bounded, easy to compare
✔️ allows for zero incomes
❌ hard to interpret without a graphical device
❌ more sensitive to outliers (when compared to `svyzenga`)
❌ https://pubmed.ncbi.nlm.nih.gov/26292521/ and https://www.jstor.org/stable/2142862The Gini index (or Gini coefficient) is one approach to turn the inequality presented by the Lorenz curve into a single number. In essence, it is twice the area between the equality curve and the real Lorenz curve — that is:
\[ \begin{aligned} G &= 2 \bigg( \int_{0}^{1} pdp - \int_{0}^{1} L(p)dp \bigg) \\ \therefore G &= 1 - 2 \int_{0}^{1} L(p)dp \end{aligned} \]
where \(G=0\) in case of perfect equality and \(G = 1\) in the case of perfect inequality.
The estimator proposed by Osier (2009Osier, Guillaume. 2009. “Variance Estimation for Complex Indicators of Poverty and Inequality.” Journal of the European Survey Research Association 3 (3): 167–95. http://ojs.ub.uni-konstanz.de/srm/article/view/369.) is defined as:
\[ \widehat{G} = \frac{ 2 \sum_{i \in S} w_i r_i y_i - \sum_{i \in S} w_i y_i }{ \hat{N} \hat{Y} } - 1 \]
The linearized formula of \(\widehat{G}\) is used to calculate the SE.
The R vardpoor package (Breidaks, Liberts, and Ivanova 2016Breidaks, Juris, Martins Liberts, and Santa Ivanova. 2016. “Vardpoor: Estimation of Indicators on Social Exclusion and Poverty and Its Linearization, Variance Estimation.” Riga, Latvia: CSB.), created by researchers at the Central Statistical Bureau of Latvia, includes a Gini coefficient calculation using the ultimate cluster method. The example below reproduces those statistics.
Load and prepare the same data set:
# load the convey package
library(convey)
# load the survey library
library(survey)
# load the vardpoor library
library(vardpoor)
# load the laeken library
library(laeken)
# load the synthetic EU statistics on income & living conditions
data(eusilc)
# make all column names lowercase
names(eusilc) <- tolower(names(eusilc))
# add a column with the row number
dati <- data.table::data.table(IDd = 1:nrow(eusilc), eusilc)
# calculate the gini coefficient
# using the R vardpoor library
varpoord_gini_calculation <-
varpoord(
# analysis variable
Y = "eqincome",
# weights variable
w_final = "rb050",
# row number variable
ID_level1 = "IDd",
# row number variable
ID_level2 = "IDd",
# strata variable
H = "db040",
N_h = NULL ,
# clustering variable
PSU = "rb030",
# data.table
dataset = dati,
# gini coefficient function
type = "lingini",
# get linearized variable
outp_lin = TRUE
)
# construct a survey.design
# using our recommended setup
des_eusilc <-
svydesign(
ids = ~ rb030 ,
strata = ~ db040 ,
weights = ~ rb050 ,
data = eusilc
)
# immediately run the convey_prep function on it
des_eusilc <- convey_prep(des_eusilc)
# coefficients do match
varpoord_gini_calculation$all_result$value## [1] 26.49652## eqincome
## 26.49652# linearized variables do match
# varpoord
lin_gini_varpoord <- varpoord_gini_calculation$lin_out$lin_gini
# convey
lin_gini_convey <-
attr(svygini( ~ eqincome , des_eusilc , linearized = TRUE) , "linearized")
# check equality
all.equal(lin_gini_varpoord , (100 * as.numeric(lin_gini_convey)))## [1] TRUE## eqincome
## eqincome 0.03790739## [1] 0.03783931## [1] 0.1945233## eqincome
## eqincome 0.1946982the variance estimator and the linearized variable \(z\) are both defined in Linearization-Based Variance Estimation. The functions convey::svygini and vardpoor::lingini produce the same linearized variable \(z\).
However, the measures of uncertainty do not line up, because library(vardpoor) defaults to an ultimate cluster method that can be replicated with an alternative setup of the survey.design object.
# within each strata, sum up the weights
cluster_sums <-
aggregate(eusilc$rb050 , list(eusilc$db040) , sum)
# name the within-strata sums of weights the `cluster_sum`
names(cluster_sums) <- c("db040" , "cluster_sum")
# merge this column back onto the data.frame
eusilc <- merge(eusilc , cluster_sums)
# construct a survey.design
# with the fpc using the cluster sum
des_eusilc_ultimate_cluster <-
svydesign(
ids = ~ rb030 ,
strata = ~ db040 ,
weights = ~ rb050 ,
data = eusilc ,
fpc = ~ cluster_sum
)
# again, immediately run the convey_prep function on the `survey.design`
des_eusilc_ultimate_cluster <-
convey_prep(des_eusilc_ultimate_cluster)
# matches
attr(svygini( ~ eqincome , des_eusilc_ultimate_cluster) , 'var') * 10000## eqincome
## eqincome 0.03783931## [1] 0.03783931## [1] 0.1945233## eqincome
## eqincome 0.1945233For additional usage examples of svygini, type ?convey::svygini in the R console.
This section displays example results using nationally-representative surveys from both the United States and Brazil. We present a variety of surveys, levels of analysis, and subpopulation breakouts to provide users with points of reference for the range of plausible values of the svygini function.
To understand the construction of each survey design object and respective variables of interest, please refer to section 1.4 for CPS-ASEC, section 1.5 for PNAD Contínua, and section 1.6 for SCF.
## gini SE
## htotval 0.48846 0.002## sex htotval se.htotval
## male male 0.4723461 0.003034223
## female female 0.5008580 0.002499741## gini SE
## ftotval 0.45816 0.0023## sex ftotval se.ftotval
## male male 0.4400040 0.003677892
## female female 0.4745831 0.002801257## gini SE
## pearnval 0.412 0.0026## sex pearnval se.pearnval
## male male 0.4185334 0.003179673
## female female 0.3914812 0.004673079## gini SE
## deflated_per_capita_income 0.51845 0.0032## sex deflated_per_capita_income se.deflated_per_capita_income
## male male 0.5202218 0.003394589
## female female 0.5165412 0.003210982## gini SE
## deflated_labor_income 0.48606 0.0036## sex deflated_labor_income se.deflated_labor_income
## male male 0.4906702 0.004082281
## female female 0.4711193 0.003826630## Multiple imputation results:
## with(scf_design, svygini(~networth))
## scf_MIcombine(with(scf_design, svygini(~networth)))
## results se
## networth 0.8299712 0.003921153## Multiple imputation results:
## with(scf_design, svyby(~networth, ~hhsex, svygini))
## scf_MIcombine(with(scf_design, svyby(~networth, ~hhsex, svygini)))
## results se
## male 0.8160695 0.005037685
## female 0.8258288 0.012458833## Multiple imputation results:
## with(scf_design, svygini(~income))
## scf_MIcombine(with(scf_design, svygini(~income)))
## results se
## income 0.6070385 0.01059348## Multiple imputation results:
## with(scf_design, svyby(~income, ~hhsex, svygini))
## scf_MIcombine(with(scf_design, svyby(~income, ~hhsex, svygini)))
## results se
## male 0.5987009 0.01197205
## female 0.4633805 0.01269733