2.4 Relative Median Poverty Gap (svyrmpg)

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.).


A 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 synthetic european union 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(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 0.5763974

The variance estimate is computed by using the approximation defined in (1.1), where the linearized variable \(z\) is defined by (1.2). 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 0.5758866

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