2.3 Relative Median Income Ratio (svyrmir)

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 (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 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 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 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 0.01128911

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::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 0.0112789

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