3.4 Relative Median Poverty Gap (svyrmpg)

✔️ how poor are those below the ARPT?
✔️ median poverty gap expressed as a percentage of the threshold
✔️ useful for understanding the depth of poverty
❌ not common outside of the EU
❌ not immediately interpretable in terms of income

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 (1999Deville, Jean-Claude. 1999. “Variance Estimation for Complex Statistics and Estimators: Linearization and Residual Techniques.” Survey Methodology 25 (2): 193–203. http://www.statcan.gc.ca/pub/12-001-x/1999002/article/4882-eng.pdf.) and 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.).


3.4.1 Replication Example

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 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",
    
    # 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 0.5763974

The variance estimator and the linearized variable \(z\) are both defined in Linearization-Based Variance Estimation. 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
stopifnot(all.equal(
  attr(svyrmpg( ~ eqincome , des_eusilc_ultimate_cluster) , 'var')[1] * 10000 ,
  varpoord_rmpg_calculation$all_result$var
))

# matches
stopifnot(all.equal(SE(
  svyrmpg(~ eqincome , des_eusilc_ultimate_cluster)
)[1] * 100 ,
varpoord_rmpg_calculation$all_result$se))

For additional usage examples of svyrmpg, type ?convey::svyrmpg in the R console.

3.4.2 Real World Examples

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 svyrmpg 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.

3.4.2.1 CPS-ASEC Household Income

svyrmpg( ~ htotval , cps_household_design)
##            rmpg     SE
## htotval 0.46091 0.0042
svyby( ~ htotval , ~ sex , cps_household_design , svyrmpg)
##           sex   htotval  se.htotval
## male     male 0.4382452 0.004842012
## female female 0.4824937 0.007104020

3.4.2.2 CPS-ASEC Family Income

svyrmpg( ~ ftotval , cps_family_design)
##            rmpg     SE
## ftotval 0.39583 0.0044
svyby( ~ ftotval , ~ sex , cps_family_design , svyrmpg)
##           sex   ftotval  se.ftotval
## male     male 0.3740833 0.005989913
## female female 0.4336667 0.010176008

3.4.2.3 CPS-ASEC Worker Earnings

svyrmpg( ~ pearnval , cps_ftfy_worker_design)
##             rmpg     SE
## pearnval 0.22222 0.0126
svyby( ~ pearnval , ~ sex , cps_ftfy_worker_design , svyrmpg)
##           sex  pearnval se.pearnval
## male     male 0.2222222 0.008776067
## female female 0.2222222 0.019026712

3.4.2.4 PNAD Contínua Per Capita Income

svyrmpg( ~ deflated_per_capita_income , pnadc_design , na.rm = TRUE)
##                               rmpg    SE
## deflated_per_capita_income 0.38774 0.006
svyby( ~ deflated_per_capita_income ,
       ~ sex ,
       pnadc_design ,
       svyrmpg ,
       na.rm = TRUE)
##           sex deflated_per_capita_income se.deflated_per_capita_income
## male     male                  0.3826508                   0.006056376
## female female                  0.3908435                   0.006118998

3.4.2.5 PNAD Contínua Worker Earnings

svyrmpg( ~ deflated_labor_income , pnadc_design , na.rm = TRUE)
##                          rmpg     SE
## deflated_labor_income 0.47379 0.0047
svyby( ~ deflated_labor_income , ~ sex , pnadc_design , svyrmpg , na.rm = TRUE)
##           sex deflated_labor_income se.deflated_labor_income
## male     male             0.4636005              0.005379404
## female female             0.4782603              0.001340361

3.4.2.6 SCF Family Net Worth

scf_MIcombine(with(scf_design , svyrmpg( ~ networth)))
## Multiple imputation results:
##       with(scf_design, svyrmpg(~networth))
##       scf_MIcombine(with(scf_design, svyrmpg(~networth)))
##            results          se
## networth 0.8764535 0.006946956
scf_MIcombine(with(scf_design , svyby( ~ networth, ~ hhsex , svyrmpg)))
## Multiple imputation results:
##       with(scf_design, svyby(~networth, ~hhsex, svyrmpg))
##       scf_MIcombine(with(scf_design, svyby(~networth, ~hhsex, svyrmpg)))
##          results         se
## male   0.8344603 0.01418904
## female 0.9227891 0.01180089

3.4.2.7 SCF Family Income

scf_MIcombine(with(scf_design , svyrmpg( ~ income)))
## Multiple imputation results:
##       with(scf_design, svyrmpg(~income))
##       scf_MIcombine(with(scf_design, svyrmpg(~income)))
##          results         se
## income 0.4069153 0.01961822
scf_MIcombine(with(scf_design , svyby( ~ income, ~ hhsex , svyrmpg)))
## Multiple imputation results:
##       with(scf_design, svyby(~income, ~hhsex, svyrmpg))
##       scf_MIcombine(with(scf_design, svyby(~income, ~hhsex, svyrmpg)))
##          results         se
## male   0.3609169 0.01459789
## female 0.4478632 0.02070610