vignettes/02-poverty.Rmd
02-poverty.RmdPoverty is an issue discussed since long time ago. As Ravallion (2016) points out, Aristotle and Confucius discussed ideas about poverty. In fact, Aristotle’s ideas influenced Thomas Aquinas, one of the pillars of Western philosophy. Since then, societies changed, modifying the theories of justice underlying the idea of poverty.
As the concept and the ethics towards poverty change, so does its measurement. From basic measures like the headcount rate to more complex metrics, such as the FGT index, poverty measurement evolved. Nowadays, poverty measures estimates are calculated using household surveys and censuses (Deaton 1997). Yet, only recently the aspects of statistical inference combining such measures and survey designs were explored1. These advances become even more important given the recent efforts in poverty mapping, an analytical method that combined poverty analysis and small area estimation, like Elbers, Lanjouw, and Lanjouw (2003) and Bedi, Coudouel, and Simler (2007).
The following subsections shows how poverty estimates and their
sampling errors can be estimated using simple commands from the
convey package.
The at-risk-of-poverty threshold (ARPT) is a measure used to define the people whose incomes imply a low standard of living in comparison to the general living standards. I.e., even though some people are not below the effective poverty line, those below the ARPT can be considered “almost deprived”.
This measure is defined as \(0.6\) times the median income for the entire population:
\[
arpt = 0.6 \times median(y),
\] where, \(y\) is the income
variable and median is estimated for the whole population.
The details of the linearization of the arpt are discussed
by Deville (1999) and Osier (2009).
A replication example
The R vardpoor package (Breidaks,
Liberts, and Ivanova 2016), created by researchers at the Central
Statistical Bureau of Latvia, includes a arpt 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)
#> Loading required package: grid
#> Loading required package: Matrix
#> Loading required package: survival
#>
#> Attaching package: 'survey'
#> The following object is masked from 'package:graphics':
#>
#> dotchart
# 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 arpt coefficient
# using the R vardpoor library
varpoord_arpt_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,
# arpt coefficient function
type = "linarpt",
# poverty threshold range
order_quant = 50L ,
# 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_arpt_calculation$all_result$value
#> [1] 10859.24
coef( svyarpt( ~ eqincome , des_eusilc ) )
#> eqincome
#> 10859.24
# linearized variables do match
# vardpoor
lin_arpt_varpoord<- varpoord_arpt_calculation$lin_out$lin_arpt
# convey
lin_arpt_convey <- attr(svyarpt( ~ eqincome , des_eusilc ),"lin")
# check equality
all.equal(lin_arpt_varpoord, lin_arpt_convey )
#> [1] TRUE
# variances do not match exactly
attr( svyarpt( ~ eqincome , des_eusilc ) , 'var' )
#> eqincome
#> eqincome 2564.027
varpoord_arpt_calculation$all_result$var
#> [1] 2559.442
# standard errors do not match exactly
varpoord_arpt_calculation$all_result$se
#> [1] 50.59093
SE( svyarpt( ~ eqincome , des_eusilc ) )
#> eqincome
#> eqincome 50.63622The variance estimate is computed by using the approximation defined
in @ref(eq:var), where the linearized variable \(z\) is defined by @ref(eq:lin). The
functions convey::svyarpt and
vardpoor::linarpt 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( svyarpt( ~ eqincome , des_eusilc_ultimate_cluster ) , 'var' )
#> eqincome
#> eqincome 2559.442
varpoord_arpt_calculation$all_result$var
#> [1] 2559.442
# matches
varpoord_arpt_calculation$all_result$se
#> [1] 50.59093
SE( svyarpt( ~ eqincome , des_eusilc_ultimate_cluster ) )
#> eqincome
#> eqincome 50.59093For additional usage examples of svyarpt, type
?convey::svyarpt in the R console.
The at-risk-of-poverty rate (ARPR) is the share of persons with an
income below the at-risk-of-poverty threshold (arpt). The
logic behind this measure is that although most people below the ARPT
cannot be considered “poor”, they are the ones most vulnerable to
becoming poor in the event of a negative economic phenomenon.
The ARPR is a composite estimate, taking into account both the
sampling error in the proportion itself and that in the ARPT estimate.
The details of the linearization of the arpr and are
discussed by Deville (1999) and Osier (2009).
A replication example
The R vardpoor package (Breidaks,
Liberts, and Ivanova 2016), created by researchers at the Central
Statistical Bureau of Latvia, includes a ARPR 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 vardpoor 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 arpr coefficient
# using the R vardpoor library
varpoord_arpr_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,
# arpr coefficient function
type = "linarpr",
# poverty threshold range
order_quant = 50L ,
# 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_arpr_calculation$all_result$value
#> [1] 14.44422
coef( svyarpr( ~ eqincome , des_eusilc ) ) * 100
#> eqincome
#> 14.44422
# linearized variables do match
# vardpoor
lin_arpr_varpoord<- varpoord_arpr_calculation$lin_out$lin_arpr
# convey
lin_arpr_convey <- attr(svyarpr( ~ eqincome , des_eusilc ),"lin")
# check equality
all.equal(lin_arpr_varpoord,100*lin_arpr_convey )
#> [1] "Mean relative difference: 0.2264738"
# variances do not match exactly
attr( svyarpr( ~ eqincome , des_eusilc ) , 'var' ) * 10000
#> eqincome
#> eqincome 0.07599778
varpoord_arpr_calculation$all_result$var
#> [1] 0.08718569
# standard errors do not match exactly
varpoord_arpr_calculation$all_result$se
#> [1] 0.2952722
SE( svyarpr( ~ eqincome , des_eusilc ) ) * 100
#> eqincome
#> eqincome 0.2756769The variance estimate is computed by using the approximation defined
in @ref(eq:var), where the linearized variable \(z\) is defined by @ref(eq:lin). The
functions convey::svyarpr and
vardpoor::linarpr 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( svyarpr( ~ eqincome , des_eusilc_ultimate_cluster ) , 'var' ) * 10000
#> eqincome
#> eqincome 0.07586194
varpoord_arpr_calculation$all_result$var
#> [1] 0.08718569
# matches
varpoord_arpr_calculation$all_result$se
#> [1] 0.2952722
SE( svyarpr( ~ eqincome , des_eusilc_ultimate_cluster ) ) * 100
#> eqincome
#> eqincome 0.2754305For additional usage examples of svyarpr, type
?convey::svyarpr in the R console.
The relative median income ratio (rmir) is the ratio of the median income of people aged above a value (65) to the median of people aged below the same value. In mathematical terms,
\[ rmir = \frac{median\{y_i; age_i >65 \}}{median\{y_i; age_i \leq 65 \}}. \]
The details of the linearization of the rmir and are
discussed by Deville (1999) and Osier (2009).
A replication example
The R vardpoor package (Breidaks,
Liberts, and Ivanova 2016), created by researchers at the Central
Statistical Bureau of Latvia, includes a rmir 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 vardpoor 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 rmir coefficient
# using the R vardpoor library
varpoord_rmir_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,
# age variable
age = "age",
# rmir coefficient function
type = "linrmir",
# poverty threshold range
order_quant = 50L ,
# 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_rmir_calculation$all_result$value
#> [1] 0.9330361
coef( svyrmir( ~ eqincome , des_eusilc, age = ~age ) )
#> eqincome
#> 0.9330361
# linearized variables do match
# vardpoor
lin_rmir_varpoord<- varpoord_rmir_calculation$lin_out$lin_rmir
# convey
lin_rmir_convey <- attr(svyrmir( ~ eqincome , des_eusilc, age = ~age ),"lin")
# check equality
all.equal(lin_rmir_varpoord, lin_rmir_convey[,1] )
#> [1] TRUE
# variances do not match exactly
attr( svyrmir( ~ eqincome , des_eusilc, age = ~age ) , 'var' )
#> eqincome
#> eqincome 0.000127444
varpoord_rmir_calculation$all_result$var
#> [1] 0.0001272137
# standard errors do not match exactly
varpoord_rmir_calculation$all_result$se
#> [1] 0.0112789
SE( svyrmir( ~ eqincome , des_eusilc , age = ~age) )
#> eqincome
#> eqincome 0.01128911The variance estimate is computed by using the approximation defined
in @ref(eq:var), where the linearized variable \(z\) is defined by @ref(eq:lin). The
functions convey::svyrmir and
vardpoor::linrmir 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( svyrmir( ~ eqincome , des_eusilc_ultimate_cluster , age = ~age ) , 'var' )
#> eqincome
#> eqincome 0.0001272137
varpoord_rmir_calculation$all_result$var
#> [1] 0.0001272137
# matches
varpoord_rmir_calculation$all_result$se
#> [1] 0.0112789
SE( svyrmir( ~ eqincome , des_eusilc_ultimate_cluster, age = ~age ) )
#> eqincome
#> eqincome 0.0112789For additional usage examples of svyrmir, type
?convey::svyrmir in the R console.
The relative median poverty gap (rmpg) is the relative
difference between the median income of people having income below the
arpt and the arpt itself:
\[
rmpg = \frac{median\{y_i, y_i<arpt\}-arpt}{arpt}
\] The details of the linearization of the rmpg are
discussed by Deville (1999) and Osier (2009).
A replication example
The R vardpoor package (Breidaks,
Liberts, and Ivanova 2016), created by researchers at the Central
Statistical Bureau of Latvia, includes a rmpg 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 vardpoor 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 rmpg coefficient
# using the R vardpoor library
varpoord_rmpg_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,
# rmpg coefficient function
type = "linrmpg",
# poverty threshold range
order_quant = 50L ,
# 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_rmpg_calculation$all_result$value
#> [1] 18.9286
coef( svyrmpg( ~ eqincome , des_eusilc ) ) * 100
#> eqincome
#> 18.9286
# linearized variables do match
# vardpoor
lin_rmpg_varpoord<- varpoord_rmpg_calculation$lin_out$lin_rmpg
# convey
lin_rmpg_convey <- attr(svyrmpg( ~ eqincome , des_eusilc ),"lin")
# check equality
all.equal(lin_rmpg_varpoord, 100*lin_rmpg_convey[,1] )
#> [1] TRUE
# variances do not match exactly
attr( svyrmpg( ~ eqincome , des_eusilc ) , 'var' ) * 10000
#> eqincome
#> eqincome 0.332234
varpoord_rmpg_calculation$all_result$var
#> [1] 0.3316454
# standard errors do not match exactly
varpoord_rmpg_calculation$all_result$se
#> [1] 0.5758866
SE( svyrmpg( ~ eqincome , des_eusilc ) ) * 100
#> eqincome
#> eqincome 0.5763974The variance estimate is computed by using the approximation defined
in @ref(eq:var), where the linearized variable \(z\) is defined by @ref(eq:lin). The
functions convey::svyrmpg and
vardpoor::linrmpg 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( svyrmpg( ~ eqincome , des_eusilc_ultimate_cluster ) , 'var' ) * 10000
#> eqincome
#> eqincome 0.3316454
varpoord_rmpg_calculation$all_result$var
#> [1] 0.3316454
# matches
varpoord_rmpg_calculation$all_result$se
#> [1] 0.5758866
SE( svyrmpg( ~ eqincome , des_eusilc_ultimate_cluster ) ) * 100
#> eqincome
#> eqincome 0.5758866For additional usage examples of svyrmpg, type
?convey::svyrmpg in the R console.
Median income below the at-risk-of-poverty- threshold (poormed) is
median of incomes of people having the income below the
arpt:
\[
poormed = median\{y_i; y_i< arpt\}
\] The details of the linearization of the poormed
are discussed by Deville (1999) and Osier (2009).
A replication example
The R vardpoor package (Breidaks,
Liberts, and Ivanova 2016), created by researchers at the Central
Statistical Bureau of Latvia, includes a poormed 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 vardpoor 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 poormed coefficient
# using the R vardpoor library
varpoord_poormed_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,
# poormed coefficient function
type = "linpoormed",
# poverty threshold range
order_quant = 50L ,
# 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_poormed_calculation$all_result$value
#> [1] 8803.735
coef( svypoormed( ~ eqincome , des_eusilc ) )
#> eqincome
#> 8803.735
# linearized variables do match
# vardpoor
lin_poormed_varpoord<- varpoord_poormed_calculation$lin_out$lin_poormed
# convey
lin_poormed_convey <- attr(svypoormed( ~ eqincome , des_eusilc ),"lin")
# check equality
all.equal(lin_poormed_varpoord, lin_poormed_convey )
#> [1] TRUE
# variances do not match exactly
attr( svypoormed( ~ eqincome , des_eusilc ) , 'var' )
#> eqincome
#> eqincome 5311.47
varpoord_poormed_calculation$all_result$var
#> [1] 5302.086
# standard errors do not match exactly
varpoord_poormed_calculation$all_result$se
#> [1] 72.81542
SE( svypoormed( ~ eqincome , des_eusilc ) )
#> eqincome
#> eqincome 72.87983The variance estimate is computed by using the approximation defined
in @ref(eq:var), where the linearized variable \(z\) is defined by @ref(eq:lin). The
functions convey::svypoormed and
vardpoor::linpoormed 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( svypoormed( ~ eqincome , des_eusilc_ultimate_cluster ) , 'var' )
#> eqincome
#> eqincome 5302.086
varpoord_poormed_calculation$all_result$var
#> [1] 5302.086
# matches
varpoord_poormed_calculation$all_result$se
#> [1] 72.81542
SE( svypoormed( ~ eqincome , des_eusilc_ultimate_cluster ) )
#> eqincome
#> eqincome 72.81542For additional usage examples of svypoormed, type
?convey::svypoormed in the R console.
Foster, Greer, and Thorbecke (1984) proposed a family of indicators to measure poverty. This class of \(FGT\) measures, can be defined as
\[ p=\frac{1}{N}\sum_{k\in U}h(y_{k},\theta ), \]
where
\[ h(y_{k},\theta )=\left[ \frac{(\theta -y_{k})}{\theta }\right] ^{\gamma }\delta \left\{ y_{k}\leq \theta \right\} , \]
where: \(\theta\) is the poverty threshold; \(\delta\) the indicator function that assigns value \(1\) if the condition \(\{y_{k}\leq \theta \}\) is satisfied and \(0\) otherwise, and \(\gamma\) is a non-negative constant.
If \(\gamma =0\), the FGT(0) equals the poverty headcount ratio, which accounts for the spread of poverty. If \(\gamma =1\), FGT(1) is the mean of the normalized income shortfall of the poor. By doing so, the measure takes into account both the spread and the intensity of poverty. When \(\gamma =2\), the relative weight of larger shortfalls increases even more, which yields a measure that accounts for poverty severity, i.e., the inequality among the poor. This way, a transfer from a poor person to an even poorer person would reduce the FGT(2).
Although Foster, Greer, and Thorbecke (1984) already presented a decomposition for the FGT(2) index, Aristondo, De La Vega, and Urrutia (2010) provided a general formula that decomposes the FGT(\(\gamma\)) for any \(\gamma \geqslant 2\). Put simply, any such FGT(\(\gamma\)) index can be seen as function of the headcount ratio, the average normalized income gap among the poor and a generalized entropy index of the normalized income gaps among poor. In mathematical terms,
\[ FGT_\gamma = FGT_0 \cdot I^\gamma \cdot \big[ 1 + \big( \gamma^2 -\gamma \big) GEI_\gamma^* \big] , \text{ } \gamma \geq 2 \]
where \(I\) is the average normalized income gap among the poor and \(GEI_\gamma^*\) is a generalized entropy index of such income gaps among the poor.
This result is particularly useful, as one can attribute cross-sectional differences of a FGT index to differences in the spread, depth and inequality of poverty.
The FGT poverty class and its decomposition is implemented in the
library convey by the function svyfgt and
svyfgtdec, respectively. The argument
thresh_type of this function defines the type of poverty
threshold adopted. There are three possible choices:
abs – fixed and given by the argument thresh_valuerelq – a proportion of a quantile fixed by the argument
proportion and the quantile is defined by the argument
order.relm – a proportion of the mean fixed the argument
proportion
The quantile and the mean involved in the definition of the threshold
are estimated for the whole population. When \(\gamma=0\) and \(\theta= .6*MED\) the measure is equal to
the indicator arpr computed by the function
svyarpr. The linearization of the FGT(0) is presented in
Berger and Skinner (2003).
Next, we give some examples of the function svyfgt to
estimate the values of the FGT poverty index.
Consider first the poverty threshold fixed (\(\gamma=0\)) in the value \(10000\). The headcount ratio (FGT0) is
svyfgt(~eqincome, des_eusilc, g=0, abs_thresh=10000)
fgt0 SE
eqincome 0.11444 0.0027The poverty gap ratio (FGT(1)) (\(\gamma=1\)) index for the poverty threshold fixed at the same value is
svyfgt(~eqincome, des_eusilc, g=1, abs_thresh=10000)
fgt1 SE
eqincome 0.032085 0.0011To estimate the FGT(0) with the poverty threshold fixed at \(0.6* MED\) we fix the argument
type_thresh="relq" and use the default values for
percent and order:
svyfgt(~eqincome, des_eusilc, g=0, type_thresh= "relq")
fgt0 SE
eqincome 0.14444 0.0028that matches the estimate obtained by
svyarpr(~eqincome, design=des_eusilc, .5, .6)
arpr SE
eqincome 0.14444 0.0028To estimate the poverty gap ratio with the poverty threshold equal to \(0.6*MEAN\), we use:
svyfgt(~eqincome, des_eusilc, g=1, type_thresh= "relm")
fgt1 SE
eqincome 0.051187 0.0011A replication example
In July 2006, Jenkins (2008) presented at the North American Stata Users’ Group Meetings on the stata Atkinson Index command. The example below reproduces those statistics.
In order to match the presentation’s results using the
svyfgt function from the convey library, the poverty
threshold was considered absolute despite being directly estimated from
the survey sample. This effectively treats the variance of the estimated
poverty threshold as zero; svyfgt does not account for the
uncertainty of the poverty threshold when the level has been stated as
absolute with the abs_thresh= parameter. In general, we
would instead recommend using either relq or
relm in the type_thresh= parameter in order to
account for the added uncertainty of the poverty threshold calculation.
This example serves only to show that svyfgt behaves
properly as compared to other software.
Load and prepare the same data set:
# load the convey package
library(convey)
# load the survey library
library(survey)
# load the foreign library
library(foreign)
# create a temporary file on the local disk
tf <- tempfile()
# store the location of the presentation file
presentation_zip <- "http://repec.org/nasug2006/nasug2006_jenkins.zip"
# download jenkins' presentation to the temporary file
download.file( presentation_zip , tf , mode = 'wb' )
# unzip the contents of the archive
presentation_files <- unzip( tf , exdir = tempdir() )
# load the institute for fiscal studies' 1981, 1985, and 1991 data.frame objects
x81 <- read.dta( grep( "ifs81" , presentation_files , value = TRUE ) )
x85 <- read.dta( grep( "ifs85" , presentation_files , value = TRUE ) )
x91 <- read.dta( grep( "ifs91" , presentation_files , value = TRUE ) )
# NOTE: we recommend using ?convey::svyarpt rather than this unweighted calculation #
# calculate 60% of the unweighted median income in 1981
unwtd_arpt81 <- quantile( x81$eybhc0 , 0.5 ) * .6
# calculate 60% of the unweighted median income in 1985
unwtd_arpt85 <- quantile( x85$eybhc0 , 0.5 ) * .6
# calculate 60% of the unweighted median income in 1991
unwtd_arpt91 <- quantile( x91$eybhc0 , 0.5 ) * .6
# stack each of these three years of data into a single data.frame
x <- rbind( x81 , x85 , x91 )Replicate the author’s survey design statement from stata code..
. ge poor = (year==1981)*(x < $z_81) + (year==1985)*(x < $z_85) + (year==1991)*(x < $z_91)
. * account for clustering within HHs
. svyset hrn [pweight = wgt].. into R code:
# initiate a linearized survey design object
y <- svydesign( ~ hrn , data = x , weights = ~ wgt )
# immediately run the `convey_prep` function on the survey design
z <- convey_prep( y )Replicate the author’s headcount ratio results with stata..
. svy: mean poor if year == 1981
(running mean on estimation sample)
Survey: Mean estimation
Number of strata = 1 Number of obs = 9772
Number of PSUs = 7476 Population size = 5.5e+07
Design df = 7475
--------------------------------------------------------------
| Linearized
| Mean Std. Err. [95% Conf. Interval]
-------------+------------------------------------------------
poor | .1410125 .0044859 .132219 .149806
--------------------------------------------------------------
. svy: mean poor if year == 1985
(running mean on estimation sample)
Survey: Mean estimation
Number of strata = 1 Number of obs = 8991
Number of PSUs = 6972 Population size = 5.5e+07
Design df = 6971
--------------------------------------------------------------
| Linearized
| Mean Std. Err. [95% Conf. Interval]
-------------+------------------------------------------------
poor | .137645 .0046531 .1285235 .1467665
--------------------------------------------------------------
. svy: mean poor if year == 1991
(running mean on estimation sample)
Survey: Mean estimation
Number of strata = 1 Number of obs = 6468
Number of PSUs = 5254 Population size = 5.6e+07
Design df = 5253
--------------------------------------------------------------
| Linearized
| Mean Std. Err. [95% Conf. Interval]
-------------+------------------------------------------------
poor | .2021312 .0062077 .1899615 .2143009
--------------------------------------------------------------..using R code:
headcount_81 <-
svyfgt(
~ eybhc0 ,
subset( z , year == 1981 ) ,
g = 0 ,
abs_thresh = unwtd_arpt81
)
headcount_81
#> fgt0 SE
#> eybhc0 0.14101 0.0045
confint( headcount_81 , df = degf( subset( z , year == 1981 ) ) )
#> 2.5 % 97.5 %
#> eybhc0 0.1322193 0.1498057
headcount_85 <-
svyfgt(
~ eybhc0 ,
subset( z , year == 1985 ) ,
g = 0 ,
abs_thresh = unwtd_arpt85
)
headcount_85
#> fgt0 SE
#> eybhc0 0.13764 0.0047
confint( headcount_85 , df = degf( subset( z , year == 1985 ) ) )
#> 2.5 % 97.5 %
#> eybhc0 0.1285239 0.1467661
headcount_91 <-
svyfgt(
~ eybhc0 ,
subset( z , year == 1991 ) ,
g = 0 ,
abs_thresh = unwtd_arpt91
)
headcount_91
#> fgt0 SE
#> eybhc0 0.20213 0.0062
confint( headcount_91 , df = degf( subset( z , year == 1991 ) ) )
#> 2.5 % 97.5 %
#> eybhc0 0.1899624 0.2143Confirm this replication applies for the normalized poverty gap as well, comparing stata code..
. ge ngap = poor*($z_81- x)/$z_81 if year == 1981
. svy: mean ngap if year == 1981
(running mean on estimation sample)
Survey: Mean estimation
Number of strata = 1 Number of obs = 9772
Number of PSUs = 7476 Population size = 5.5e+07
Design df = 7475
--------------------------------------------------------------
| Linearized
| Mean Std. Err. [95% Conf. Interval]
-------------+------------------------------------------------
ngap | .0271577 .0013502 .0245109 .0298044
--------------------------------------------------------------..to R code:
norm_pov_81 <-
svyfgt(
~ eybhc0 ,
subset( z , year == 1981 ) ,
g = 1 ,
abs_thresh = unwtd_arpt81
)
norm_pov_81
#> fgt1 SE
#> eybhc0 0.027158 0.0014
confint( norm_pov_81 , df = degf( subset( z , year == 1981 ) ) )
#> 2.5 % 97.5 %
#> eybhc0 0.02451106 0.02980428For additional usage examples of svyfgt, type
?convey::svyfgt in the R console.
The measure proposed in Watts (1968) satisfies a number of desirable poverty measurement axioms and is known to be one of the first distribution-sensitive poverty measures, as noted by Haughton and Khandker (2009). It is defined as
\[ Watts = \frac{1}{N} \sum_{i \in U} \log{ \bigg( \frac{y_i}{\theta} \bigg) \delta ( y_i \leqslant \theta) }. \]
Morduch (1998) points out that the Watts poverty index can provide an estimate of the expected time to exit poverty. Given the expected growth rate of income per capita among the poor, \(g\), the expected time taken to exit poverty \(T_\theta\) would be
\[ T_\theta = \frac{Watts}{g}. \]
The Watts poverty index also has interesting decomposition properties. Blackburn (1989) proposed a decomposition for the Watts poverty index, rewriting it in terms of the headcount ratio, the Watts poverty gap ratio and the mean log deviaton of poor incomes2. Mathematically,
\[ Watts = FGT_0 \big( I_w + L_* \big) \]
where \(I_w = \log(\theta/\mu_*)\)
is the Watts poverty gap ratio3 and \(L_*\) is the mean log deviation of incomes
among the poor. This can be estimated using the svywattsdec
function.
This result can also be interpreted as a decomposition of the time taken to exit poverty, since
\[ \begin{aligned} T_\theta &= \frac{Watts}{g} \\ &= \frac{FGT_0}{g} \big( I_w + L_* \big) \end{aligned} \]
As Morduch (1998) points out, if the income among the poor is equally distributed (i.e., \(L_*=0\)), the time taken to exit poverty is simply \(FGT_0 I_w / g\). Therefore, \(FGT_0 L_* / g\) can be seen as the additional time needed to exit poverty as a result of the inequality among the poor.
For instance, see Deville (1999), Berger and Skinner (2003), Bhattacharya (2007), and Osier (2009).↩︎
The mean log deviation (also known as Theil-L or Bourguignon-Theil index) is an inequality measure of the generalized entropy class. The family of generalized entropy indices is discussed in the next chapter.↩︎
\(\mu_*\) stands for the average income among the poor.↩︎