Title: Estimation of Fuzzy Poverty Measures
Version: 3.0.2
Description: Estimates fuzzy measures of poverty and deprivation. It also estimates the sampling variance of these measures using bootstrap or jackknife repeated replications.
License: MIT + file LICENSE
Encoding: UTF-8
RoxygenNote: 7.3.2
Imports: dplyr, tidyr, ggplot2, sampling, ecp, stats, graphics, utils
LazyData: true
Suggests: knitr, rmarkdown, kableExtra, spelling
VignetteBuilder: knitr
Language: en-US
NeedsCompilation: no
Packaged: 2024-10-21 14:51:59 UTC; federico
Author: Federico Crescenzi ORCID iD [aut, cre], Lorenzo Mori ORCID iD [aut], Gianni Betti ORCID iD [ctb]
Maintainer: Federico Crescenzi <federico.crescenzi@unitus.it>
Depends: R (≥ 3.5.0)
Repository: CRAN
Date/Publication: 2024-10-21 15:10:02 UTC

FuzzyPovertyR: Estimation of Fuzzy Poverty Measures

Description

Estimates fuzzy measures of poverty and deprivation. It also estimates the sampling variance of these measures using bootstrap or jackknife repeated replications.

Author(s)

Maintainer: Federico Crescenzi federico.crescenzi@unitus.it (ORCID)

Authors:

Other contributors:

s3 class fuzzy poverty

Description

s3 class fuzzy poverty

Usage

FuzzyPoverty(x)

Arguments

x

an object

Value

an object of class FuzzyPoverty

s3 class fuzzy poverty

Description

s3 class fuzzy poverty

Usage

FuzzySupplementary(x)

Arguments

x

an object

Value

an object of class FuzzyPoverty

Head Count Ratio (HCR)

Description

This function calculates the head count ratio.

Usage

HCR(predicate, weight = NULL, p = 0.5, q = 0.6, poverty.line = NULL)

Arguments

predicate

A numeric vector of a predicate variable (i.e. income or expenditure)

weight

A numeric vector of sampling weights. if NULL simple random sampling weights will be used

p

The quantile to be calculated from the predicate variable. Default is the median

q

The percentage of the quantile to be used in determining the poverty line. default is 0.6

poverty.line

The poverty line. If it is NULL it is estimated from data.

Details

The head count ration is defined as the sum of the sampling weight of statistical units whose vale of the predicate variable is below the poverty line. The poverty line is usually defined as a fraction of a weighted quantile (in official statistics the median) of the predicate distribution

Value

A list containing the classification of the units as poor (TRUE) and not-poor (FALSE), the estimated Head Count Ratio, and the poverty line

Examples


N <- 100
p <- 0.5
q <- 0.6
predicate <- rchisq(N, 15) # predicate variable
HCR(predicate)

Calculation of Equivalized Poverty Predicate

Description

This function takes as input a numeric vector representing a predicate variable and turns it into its equivalised version using different equivalence scales.

Usage

eq_predicate(
  predicate,
  ncomp,
  age = NULL,
  scale.eq = "modifiedOECD",
  newscale,
  data = NULL
)

Arguments

predicate

A numeric vector (or the variable name) representing the poverty predicate (i.e. income or expenditure)

ncomp

A numerical vector (or the variable name) of the total number of components for the j-th family.

age

A numerical vector (or the variable name) of the number of components for the j-th family less than 16 years-old to be defined only for OECD scales

scale.eq

The equivalence scale. Options are: "carbonaro", "n.par" (non parametric), "OECD7050", "modifiedOECD" (Default) or "new"

newscale

a data.frame with two columns: "ncomp" defining the number of components and "s.eq" that define the corresponding

data

An optional data frame containing the variables to be used

Value

A data.frame containing the equivalised predicate variable.

References

Bernini, C., Emili, S., & Ferrante, M. R. (2024). Regional disparities in the sensitivity of wellbeing to poverty measures. In Spatial Inequalities and Wellbeing (pp. 136-157). Edward Elgar Publishing.

Betti, G. (1999). Nonparametric equivalence scales with application to Poland. Statistics Research Report.

Chanfreau, J., & Burchardt, T. (2008). Equivalence scales: rationales, uses and assumptions. Edinburgh: Scottish Government.

Examples



#Using OECD scale

eq_predicate(predicate = "HY022", ncomp = "ncomp", age = "age16",
             scale.eq = "OECD7050", data = eusilc) #OECD7050

eq_predicate(predicate = "HY022", ncomp = "ncomp", age = "age16",
             scale.eq = "modifiedOECD", data = eusilc) #modifiedOECD

#Define a new scale

newscal <- data.frame("ncomp" = c(1:7), "s.eq" = runif(7,1,10) ) # new


eq_predicate(predicate = "HY022", ncomp = "ncomp", scale.eq = "new",
             newscale = newscal, data = eusilc)

Eusilc data

Description

Eusilc data

Usage

data(eusilc)

Format

An object of class "data.frame"

HB020

Country of residence

ID

ID

HY022

Total disposable household income before social transfer

HS040

Capacity to afford paying for one week annual holiday

HS050

Capacity to afford a meal with meat

HS060

Capacity to face unexpected financial expanses

HS070

Ownership of a telephone

HS080

Ownership of a color TV

HS090

Ownership of a computer

HS100

Ownership of a washing machine

HS110

Ownership of a car

HS120

Ability to make ends meet

HS160

Problems with the dwelling: too dark, not enough light

HS170

Noise from neighbors or from the street

HS180

Pollution, crime or other environmental problems

HS190

Crime violence or vandalism in the area

HH010

Dwelling type

HH020

Tenure Status

HH040

Leaking roof, damp walls,floors,foundation

HH050

Ability to keep home adequately warm

HH081

Bath or shower in dwelling

HH091

Indoor flushing toilet for sole use of household

HX040

Household size

DB090

Household cross-sectional weight

db040

Sub-domain

stratum

Stratum

psu

Primary selection unit

ncomp

Size of the household

age16

Number of household members aged less than 16 year

eq_income

Equivalised income

Source

Created by authors following the EU-SILC structure

Fuzzy monetary poverty estimation

Description

fm_construct constructs fuzzy monetary poverty estimates.

Usage

fm_construct(
  predicate,
  weight = NULL,
  fm = "verma",
  ID = NULL,
  HCR,
  interval = c(1, 10),
  alpha = NULL,
  hh.size,
  z_min,
  z_max,
  z1,
  z2,
  b,
  z,
  breakdown = NULL,
  data = NULL,
  verbose = FALSE
)

Arguments

predicate

A numeric vector representing the poverty predicate (i.e. income or expenditure)

weight

A numeric vector of sampling weights of the same length of predicate. if NULL weights will set equal to n (n = sample size)

fm

The membership function (default is "verma". Other options are "ZBM", "belhadj2015", "belhadj2011", "chakravarty", "cerioli", "verma1999" and "TFR". See Betti et. al., 2023)

ID

A numeric or character vector of IDs. if NULL (the default) it is set as the row sequence

HCR

If fm="verma" or fm="verma1999" or fm="TFR" . The value of the head count ratio used to compute alpha so that the membership function equals the HCR

interval

If fm="verma" or fm="verma1999" or fm="TFR". A numeric vector of length two to look for the value of alpha (if not supplied)

alpha

The value of the exponent in equations of "verma", "verma1999" and "TFR". If NULL it is calculated so that it equates the expectation of the membership function to HCR.

hh.size

If fm="ZBM". A numeric vector of household size

z_min

A parameter of the membership function if fm="belhadj2011", i.e. the z_min: $mu=1 for 0 <y_i<z_min$ (see: See Betti et al., 2023)

z_max

A parameter of the membership function if fm="belhadj2011", i.e. the z_max: $mu=0 for y_i>z_max$ (see: See Betti et al., 2023)

z1

A parameter of the membership function if fm="belhadj2015" or fm="cerioli". For "belhadj2015" z1: $mu=1 for y_i<z1$ while for "cerioli" $mu=1 for 0 <y_i<z1$ (see: See Betti et al., 2023)

z2

A parameter of the membership function if fm="belhadj2015" or fm="cerioli". For "belhadj2015" z2: $mu=0 for y_i>z2$ while for "cerioli" the z1: $mu=0 for y_i>z2$ (see: See Betti et al., 2023)

b

A parameter of the membership function if fm="belhadj2015". The shape parameter (if b=1 the mf is linear between z1 and z2)

z

A parameter of the membership function if fm="chakravarty", i.e. $mu=0 for y_i>=z$ (see: See Betti et al., 2023)

breakdown

A factor of sub-domains to calculate estimates for (using the same alpha)

data

An optional data frame containing the variables to be used

verbose

Logical. whether to print the proceeding of the procedure

Details

It implements the fuzzy set approach to monetary poverty measurement where the usual dichotomy poor (1) not-poor(0) is replaced with a continuum score in $(0,1)$

Value

an object of class FuzzyMonetary containing the (fuzzy) membership function for each individual in the sample, the estimated expected value (estimate) of the function and the parameters of the membership functions (supplied or calculated). If breakdown is supplied it gives an output for each level.

References

Belhadj, B. (2011). A new fuzzy unidimensional poverty index from an information theory perspective. Empirical Economics, 40(1):687–704.

Belhadj, B. (2015). Employment measure in developing countries via minimum wage and poverty new fuzzy approach. Opsearch, 52(1):329–339.

Betti, G., Cheli, B., Lemmi, A., and Verma, V. (2006). Multidimensional and longitudinal poverty: an integrated fuzzy approach. In Betti, G. and Lemmi, A., editors, Fuzzy set approach to multidimensional poverty measurement, pages 115–137. Springer, Boston, USA.

Betti, G., D’Agostino, A., Lemmi, A., & Neri, L. (2023). The fuzzy approach to poverty measurement. In Research Handbook on Measuring Poverty and Deprivation Edited by Silber, J. (pp. 489-500). Edward Elgar Publishing.

Betti, G. and Verma, V. (1999). Measuring the degree of poverty in a dynamic and comparative context: a multi-dimensional approach using fuzzy set theory. In Proceedings, iccs-vi, volume 11, pages 289–300.

Cerioli, A. and Zani, S. (1990). A fuzzy approach to the measurement of poverty. In Income and Wealth Distribution, Inequality and Poverty: Proceedings of the Second International Conference on Income Distribution by Size: Generation, Distribution, Measurement and Applications., 272–284. Springer, Boston, USA.

Chakravarty, S. R. (2006). An Axiomatic Approach to Multidimensional Poverty Measurement via Fuzzy Sets. Fuzzy Set Approach to Multidimensional Poverty Measurement, 49-72.

Cheli, B. and Lemmi, A. (1995). A ’totally’ fuzzy and relative approach to the multidimensional analysis of poverty. 24(1):115–134.

Zedini, A. and Belhadj, B. (2015). A new approach to unidimensional poverty analysis: Application to the Tunisian case. Review of Income and Wealth, 61(3):465–476.

Examples

#The following examples are based on the dataset eusilc
#included in the package.


#fm = "verma"

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             fm = "verma", HCR = 0.154, ID = eusilc$ID)

#fm = "verma1999"
#In this example we set alpha=4.5

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             fm = "verma1999", alpha = 4.5, ID = eusilc$ID)

#fm = "TFR"
#In this example we do not use the sample weights. alpha = 4.5

fm_construct(predicate = eusilc$eq_income,
             fm = "TFR", alpha = 4.5)

#fm = "belhadj2015"

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             z1=100, z2=15000, b=2,
             fm = "belhadj2015")

#fm = "cerioli"

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             z1=100, z2=10000, fm= "cerioli")

#fm = "belhadj2011"

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             z_min=1000, z_max=8000, fm= "belhadj2011")

#fm = "chakravarty"

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             z=8000, fm= "chakravarty")

#fm = "ZBM"
#For this index have to use the household size

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             hh.size=eusilc$ncomp , fm= "ZBM")

#######################
##Including breakdown##
#######################

#fm = "verma"

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             fm = "verma", HCR = 0.154, ID = eusilc$ID,
             breakdown = eusilc$db040)

#fm = "verma1999"
#In this example we set alpha=4.5

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             fm = "verma1999", alpha = 4.5, ID = eusilc$ID,
             breakdown = eusilc$db040)

#fm = "TFR"
#In this example we do not use the sample weights. alpha = 4.5

fm_construct(predicate = eusilc$eq_income,
             fm = "TFR", alpha = 4.5,
             breakdown = eusilc$db040)

#fm = "belhadj2015"

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             z1=100, z2=15000, b=2,
             fm = "belhadj2015", breakdown = eusilc$db040)

#fm = "cerioli"

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             z1=100, z2=10000, fm= "cerioli", breakdown = eusilc$db040)

#fm = "belhadj2011"

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             z_min=1000, z_max=8000, fm= "belhadj2011",
             breakdown = eusilc$db040)

#fm = "chakravarty"

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             z=8000, fm= "chakravarty", breakdown = eusilc$db040)

#fm = "ZBM"
#For this index we have to use the household size

fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             hh.size=eusilc$ncomp , fm= "ZBM",
             breakdown = eusilc$db040)

Fuzzy monetary poverty estimation

Description

This function estimates the variance of the fuzzy monetary poverty index

Usage

fm_var(
  predicate,
  weight,
  fm,
  ID = NULL,
  type = "bootstrap_naive",
  R = 100,
  M = NULL,
  stratum,
  psu,
  f = 0.01,
  verbose = FALSE,
  HCR,
  interval = c(1, 10),
  alpha = NULL,
  hh.size,
  z_min,
  z_max,
  z1,
  z2,
  b,
  z,
  Xs,
  total,
  breakdown = NULL,
  fixed = FALSE,
  data = NULL
)

Arguments

predicate

A numeric vector representing the poverty predicate (i.e. income or expenditure)

weight

A numeric vector of sampling weights of the same length of predicate. if NULL weights will set equal to n (n = sample size)

fm

The membership function (default is "verma". Other options are "ZBM", "belhadj2015", "belhadj2011", "chakravarty", "cerioli", "verma1999" and "TFR". See Betti et. al., 2023)

ID

A numeric or character vector of IDs. if NULL (the default) it is set as the row sequence

type

The variance estimation method chosen. One between bootstrap_naive (default), bootstrap_calibrated or jackknife

R

The number of bootstrap replicates. Default is 500

M

The size of bootstrap samples. Default is nrow(data)

stratum

The vector identifying the stratum (if 'jackknife' is chosen as variance estimation technique)

psu

The vector identifying the psu (if 'jackknife' is chosen as variance estimation technique)

f

The finite population correction fraction (if 'jackknife' is chosen as variance estimation technique)

verbose

Logical. whether to print the proceeding of the variance estimation procedure

HCR

If fm="verma" or fm="verma1999" or fm="TFR" . The value of the head count ratio used to compute alpha so that the membership function equals the HCR

interval

If fm="verma" or fm="verma1999" or fm="TFR". A numeric vector of length two to look for the value of alpha (if not supplied)

alpha

The value of the exponent in equations of "verma", "verma1999" and "TFR". If NULL it is calculated so that it equates the expectation of the membership function to HCR.

hh.size

If fm="ZBM". A numeric vector of household size

z_min

A parameter of the membership function if fm="belhadj2011", i.e. the z_min: $mu=1 for 0 <y_i<z_min$ (see: See Betti et. al, 2023)

z_max

A parameter of the membership function if fm="belhadj2011", i.e. the z_max: $mu=0 for y_i>z_max$ (see: See Betti et. al, 2023)

z1

A parameter of the membership function if fm="belhadj2015" or fm="cerioli". For "belhadj2015" z1: $mu=1 for y_i<z1$ while for "cerioli" $mu=1 for 0 <y_i<z1$ (see: See Betti et. al, 2023)

z2

A parameter of the membership function if fm="belhadj2015" or fm="cerioli". For "belhadj2015" z2: $mu=0 for y_i>z2$ while for "cerioli" the z1: $mu=0 for y_i>z2$ (see: See Betti et. al, 2023)

b

A parameter of the membership function if fm="belhadj2015". The shape parameter (if b=1 the mf is linear between z1 and z2)

z

A parameter of the membership function if fm="chakravarty", i.e. $mu=0 for y_i>=z$ (see: See Betti et. al, 2023)

Xs

A matrix (i x j) of calibration variables. i number of units, j number of variables

total

A Vector of population totals of dimension 1 x j

breakdown

A factor of sub-domains to calculate estimates for (using the same alpha). If numeric will be coerced to a factor

fixed

Whether the membership function needs to be re-calculated at each bootstrap or jackknife replicate (default is FALSE)

data

An optional data frame containing the variables to be used

Value

An object of class FuzzyMonetary containing the estimate of variance with the method selected. if breakdown is not NULL, the variance is estimated for each sub-domain.

References

Belhadj, B. (2011). A new fuzzy unidimensional poverty index from an information theory perspective. Empirical Economics, 40(1):687–704.

Belhadj, B. (2015). Employment measure in developing countries via minimum wage and poverty new fuzzy approach. Opsearch, 52(1):329–339.

Betti, G., Cheli, B., Lemmi, A., and Verma, V. (2006). Multidimensional and longitudinal poverty: an integrated fuzzy approach. In Betti, G. and Lemmi, A., editors, Fuzzy set approach to multidimensional poverty measurement, pages 115–137. Springer, Boston, USA.

Betti, G., D’Agostino, A., Lemmi, A., & Neri, L. (2023). The fuzzy approach to poverty measurement. In Research Handbook on Measuring Poverty and Deprivation Edited by Silber, J. (pp. 489-500). Edward Elgar Publishing.

Betti, G. and Verma, V. (1999). Measuring the degree of poverty in a dynamic and comparative context: a multi-dimensional approach using fuzzy set theory. In Proceedings, iccs-vi, volume 11, pages 289–300.

Cerioli, A. and Zani, S. (1990). A fuzzy approach to the measurement of poverty. In Income and Wealth Distribution, Inequality and Poverty: Proceedings of the Second International Conference on Income Distribution by Size: Generation, Distribution, Measurement and Applications., 272–284. Springer, Boston, USA.

Chakravarty, S. R. (2006). An Axiomatic Approach to Multidimensional Poverty Measurement via Fuzzy Sets. Fuzzy Set Approach to Multidimensional Poverty Measurement, 49-72.

Cheli, B. and Lemmi, A. (1995). A ’totally’ fuzzy and relative approach to the multidimensional analysis of poverty. 24(1):115–134.

Zedini, A. and Belhadj, B. (2015). A new approach to unidimensional poverty analysis: Application to the Tunisian case. Review of Income and Wealth, 61(3):465–476.

Betti, G., Gagliardi, F., & Verma, V. (2018). Simplified Jackknife variance estimates for fuzzy measures of multidimensional poverty. International Statistical Review, 86(1), 68-86.

Examples

#The following examples are based on the dataset eusilc
#included in the package.

#Example 1 using bootstrap and breakdown

#fm = "verma"

fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090,
       fm = "verma", breakdown = NULL, type = "bootstrap_calibrated",
       alpha = 4, Xs = eusilc[,4:6], total = c(20, 30, 40))

#fm = "belhadj2015"

fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090,
       fm = "belhadj2015", breakdown = eusilc$db040, type = "bootstrap_naive",
       z1 = 100, z2 = 15000, b = 2)


#Example 2 using jackknife without breakdown

#fm = "verma1999"

fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090,
       fm = "verma1999",  type = "jackknife",
       stratum = eusilc$stratum , psu = eusilc$psu,
       alpha = 4)

#fm = "cerioli"

fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090,
       fm = "cerioli",  type = "jackknife",
       stratum = eusilc$stratum , psu = eusilc$psu,
       z1 = 1000, z2 = 12000)

Fuzzy supplementary poverty estimation (Step 7)

Description

Step 7. Constructs the fuzzy supplementary poverty measure based on Steps1-6.

Usage

fs_construct(steps4_5, weight, alpha, breakdown = NULL)

Arguments

steps4_5

The results from fs_equate.

weight

A numeric vector of sampling weights of length nrow(step1). if NULL weights will set equal to n (n = sample size)

alpha

The value of the exponent in the FM equation. If NULL it is calculated so that it equates the expectation of the membership function to HCR.

breakdown

A Dimension of sub-domains to calculate estimates for (using the same alpha). If numeric will be coerced to a Dimension.

Value

An object of class FuzzySupplementary containing the fuzzy membership function for each unit, the point estimate (i.e. the expected value of the function), and the alpha parameter.

References

Betti, G., Gagliardi, F., Lemmi, A., & Verma, V. (2015). Comparative measures of multidimensional deprivation in the European Union. Empirical Economics, 49(3), 1071-1100.

Betti, G., Gagliardi, F., & Verma, V. (2018). Simplified Jackknife variance estimates for fuzzy measures of multidimensional poverty. International Statistical Review, 86(1), 68-86.

Examples


#This example is based on the dataset eusilc included in the package
#The FS index is compute without and with breakdown and using an HCR = 0.12
#The step 2-5 are the following (step 1 is the eusilc dataset)
#For more on each step see the ad hoc function included in the package

#Step 2

step2 = fs_transform(eusilc[,4:23], weight = eusilc$DB090, ID = eusilc$ID)

#Step 3 is the definition of the dimension.
#For more about the step see Betti et al. (2018)

dimensions = c(1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5)

#Step 4-5 finding weights

steps4_5 = fs_weight(dimensions, step2 = step2, rho = NULL)

#Step 6 computation of alpha parameter

alpha <- fs_equate(steps4_5 = steps4_5,
                   weight = eusilc$DB090, HCR = 0.12,
                   interval = c(1,10))

#Step 7 the FS index without breakdown

fs_results = fs_construct(steps4_5 = steps4_5,
             weight = eusilc$DB090, alpha = alpha, breakdown = NULL)

#Step 7 the FS index with breakdown

fs_results = fs_construct(steps4_5 = steps4_5,
             weight = eusilc$DB090, alpha = alpha, breakdown = eusilc$db040)

Fuzzy supplementary poverty estimation (all steps)

Description

Step 1-7. Constructs the fuzzy supplementary poverty measure based without step-by-step functions.

Usage

fs_construct_all(
  data,
  weight = NULL,
  ID = NULL,
  dimensions,
  rho = NULL,
  HCR,
  interval = c(1, 10),
  alpha = NULL,
  breakdown = NULL
)

Arguments

data

A matrix or a data frame of identified items (see Step 1 of Betti et. al, 2018)

weight

A numeric vector of sampling weights. if NULL weights will set equal to n (n = sample size)

ID

A numeric or character vector of IDs. if NULL (the default) it is set as the row sequence

dimensions

A numeric vector (of length ncol(data)) of assignments of items in data to dimensions

rho

Optional critical value to be used for calculation of weights in the Kendall correlation matrix. If NULL rho is set equal to the point of largest gap between the ordered set of correlation values encountered (see Betti and Verma, 2008)

HCR

The value of the head count ratio used to compute alpha so that the expected value of the membership function equals HCR

interval

A numeric vector of length two to look for the value of alpha (if not supplied)

alpha

The value of the exponent in equations of "verma", "verma1999" and "TFR". If NULL it is calculated so that it equates the expectation of the membership function to HCR.

breakdown

A Dimension of sub-domains to calculate estimates for (using the same alpha). If numeric will be coerced to a Dimension

Value

An object of class FuzzySupplementary containing the fuzzy membership function for each unit, the point estimate (i.e. the expected value of the function), and the alpha parameter.

References

Betti, G., & Verma, V. (2008). Fuzzy measures of the incidence of relative poverty and deprivation: a multi-dimensional perspective. Statistical Methods and Applications, 17, 225-250.

Betti, G., Gagliardi, F., Lemmi, A., & Verma, V. (2015). Comparative measures of multidimensional deprivation in the European Union. Empirical Economics, 49(3), 1071-1100.

Betti, G., Gagliardi, F., & Verma, V. (2018). Simplified Jackknife variance estimates for fuzzy measures of multidimensional poverty. International Statistical Review, 86(1), 68-86.

Examples


#This example is based on the dataset eusilc included in the package
#The FS index is compute with breakdown and using an HCR = 0.12
#with summary and plot

FS <- fs_construct_all(data = eusilc[,4:23], weight = eusilc$DB090, # step 2
                       dimensions = c(1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5), # step 3
                       rho = NULL, # steps 4 and 5
                       HCR = .12, # step 6
                       breakdown = eusilc$db040) # step 7 with breakdowns
summary(FS)
plot(FS)

Fuzzy supplementary poverty estimation, finding the alpha parameter (step 6)

Description

Step 6. This function solves $E(mu)^(alpha-1) = HCR$ for alpha.

Usage

fs_equate(steps4_5, weight, HCR, interval = c(1, 10), verbose = TRUE)

Arguments

steps4_5

The results obtained from fs_weight.

weight

A numeric vector of sampling weights. if NULL weights will set equal to n (n = sample size)

HCR

The value of the head count ratio used to compute alpha so that the membership function equals the HCR

interval

The range to look for the value of alpha.

verbose

Logical. whether to print the proceeding of the procedure.

Value

The alpha parameter that solves the non-linear equation $E(mu) = HCR$

References

Betti, G., Gagliardi, F., Lemmi, A., & Verma, V. (2015). Comparative measures of multidimensional deprivation in the European Union. Empirical Economics, 49(3), 1071-1100.

Betti, G., Gagliardi, F., & Verma, V. (2018). Simplified Jackknife variance estimates for fuzzy measures of multidimensional poverty. International Statistical Review, 86(1), 68-86.

Examples

#This example is based on the dataset eusilc included in the package
#The Step 6 of the FS index is computed
#The step 2-5 are the following (step 1 is the eusilc dataset)
#For more on each step see the ad hoc function included in the package

#Step 2

step2 = fs_transform(eusilc[,4:23], weight = eusilc$DB090, ID = eusilc$ID)

#Step 3 is the definition of the dimension.
#For more about the step see Betti et al. (2018)

dimensions = c(1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5)

#Step 4-5 finding weights

steps4_5 = fs_weight(dimensions, step2 = step2, rho = NULL)

#Step 6 computation of alpha parameter

fs_equate(steps4_5 = steps4_5,
          weight = eusilc$DB090,
          HCR = 0.12, interval = c(1,10))

Fuzzy monetary poverty estimation (Step 1)

Description

Detects and inverts deprivation items for FS

Usage

fs_order(data, vec_order)

Arguments

data

a data-set of n columns with the considered items

vec_order

a vector of length n with TRUE or FALSE. True if the order of the variable is to be inverted, False otherwise

Value

A data.frame with the same item of data with inverted order for those with vec_order==TRUE

References

Betti, G., Gagliardi, F., Lemmi, A., & Verma, V. (2015). Comparative measures of multidimensional deprivation in the European Union. Empirical Economics, 49(3), 1071-1100.

Betti, G., Gagliardi, F., & Verma, V. (2018). Simplified Jackknife variance estimates for fuzzy measures of multidimensional poverty. International Statistical Review, 86(1), 68-86.

Examples


#Create data

data=data.frame("X"=rep(c(1,2,3,4),20), "Y"=rep(c(7,8,9,1),20))

#Crete vec_order

vec_order=c(TRUE,FALSE)

fs_order(data=data, vec_order)

Fuzzy supplementary poverty estimation (Step 2)

Description

Step 2. This function maps a set of answers to binary or categorical items to the (0,1) interval.

Usage

fs_transform(data, weight = NULL, ID = NULL, depr.score = "s", ...)

Arguments

data

A matrix or a data frame of identified items (see Step 1 of Betti et. al, 2018)

weight

A numeric vector of sampling weights of length nrow(step1). if NULL weights will set equal to n (n = sample size)

ID

A numeric or character vector of IDs. if NULL (the default) it is set as the row sequence

depr.score

The deprivation score to be used (see d or s in Betti et al (2018))

...

other parameters

Details

The function calculates deprivation score. To obtain consistent measures of supplementary poverty it is important that items are in the right order. Lower levels of the items have to correspond to more deprivation while higher levels of the items to a less deprivation.

Value

An object of class FuzzySupplementary containing a matrix of the same dimension of data with items mapped into the (0,1) interval

References

Betti, G., Gagliardi, F., Lemmi, A., & Verma, V. (2015). Comparative measures of multidimensional deprivation in the European Union. Empirical Economics, 49(3), 1071-1100.

Betti, G., Gagliardi, F., & Verma, V. (2018). Simplified Jackknife variance estimates for fuzzy measures of multidimensional poverty. International Statistical Review, 86(1), 68-86.

Examples

#This example is based on the dataset eusilc included in the package
#step 1 is the choice of the eusilc dataset

#Step 2

step2 = fs_transform(eusilc[,4:23], weight = eusilc$DB090, ID = eusilc$ID)

Fuzzy supplementary poverty estimation.

Description

Fuzzy supplementary poverty estimation.

Usage

fs_var(
  data,
  weight = NULL,
  ID = NULL,
  dimensions,
  HCR,
  breakdown = NULL,
  alpha,
  rho = NULL,
  type = "bootstrap_naive",
  R = 500,
  M = NULL,
  stratum,
  psu,
  f = 0.01,
  Xs,
  total,
  fixed = FALSE,
  verbose = TRUE
)

Arguments

data

A matrix or data frame of items

weight

A numeric vector of sampling weights of length nrow(step1). if NULL weights will set equal to n (n = sample size)

ID

A numeric or character vector of IDs. if NULL (the default) it is set as the row sequence

dimensions

A numeric vector (of length ncol(data)) of assignments of items in data to dimensions

HCR

The value of the head count ratio used to compute alpha so that the expected value of the membership function equals HCR

breakdown

A factor of sub-domains to calculate estimates for (using the same alpha). If numeric will be coerced to a factor

alpha

The value of the exponent in equations of "verma", "verma1999" and "TFR". If NULL it is calculated so that it equates the expectation of the membership function to HCR.

rho

Optional critical value to be used for calculation of weights in the Kendall correlation matrix. If NULL rho is set equal to the point of largest gap between the ordered set of correlation values encountered (see Betti and Verma, 2008)

type

The variance estimation method chosen. One between bootstrap_naive (default), bootstrap_calibrated or jackknife

R

The number of bootstrap replicates. Default is 500

M

The size of bootstrap samples. Default is nrow(data)

stratum

The vector identifying the stratum (if 'jackknife' is chosen as variance estimation technique)

psu

The vector identifying the psu (if 'jackknife' is chosen as variance estimation technique)

f

The finite population correction fraction (if 'jackknife' is chosen as variance estimation technique

Xs

A matrix (i x j) of calibration variables. i number of units, j number of variables

total

A Vector of population totals of dimension 1 x j

fixed

Whether the membership function needs to be re-calculated at each bootstrap or jackknife replicate (default is FALSE)

verbose

Logical. whether to print the proceeding of the variance estimation procedure

Value

An object of class FuzzySupplementary containing the estimated variance.

References

Betti, G., & Verma, V. (2008). Fuzzy measures of the incidence of relative poverty and deprivation: a multi-dimensional perspective. Statistical Methods and Applications, 17, 225-250.

Betti, G., Gagliardi, F., Lemmi, A., & Verma, V. (2015). Comparative measures of multidimensional deprivation in the European Union. Empirical Economics, 49(3), 1071-1100.

Betti, G., Gagliardi, F., & Verma, V. (2018). Simplified Jackknife variance estimates for fuzzy measures of multidimensional poverty. International Statistical Review, 86(1), 68-86.

Examples


#This example is based on the dataset eusilc included in the package
#The  variance of the FS index is compute without breakdown
#and using an alpha = 2

#############
##Bootstrap##
#############

fs_var(data = eusilc[,4:23], weight = eusilc$DB090, ID = NULL,
       dimensions = c(1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5),
       breakdown = NULL, alpha = 2,
       rho = NULL, type = 'bootstrap_naive', M = NULL, R = 2, verbose = TRUE)

Fuzzy supplementary poverty estimation (Steps 4 and 5)

Description

Step 4 and Step 5. Calculates the weights of dimensions discovered after Dimension analysis.

Usage

fs_weight(dimensions, step2, rho = NULL)

Arguments

dimensions

A numeric vector (of length ncol(data)) of assignments of items in data to dimensions

step2

The data frame resulting from step2

rho

Optional critical value to be used for calculation of weights in the Kendall correlation matrix. If NULL rho is set equal to the point of largest gap between the ordered set of correlation values encountered (see Betti and Verma, 2008)

Details

This function calculates the two set of weights w_a and w_b (see References)

Value

An object of class FuzzySupplementary with calculated weights and deprivation scores in each dimension identified.

References

Betti, G., & Verma, V. (2008). Fuzzy measures of the incidence of relative poverty and deprivation: a multi-dimensional perspective. Statistical Methods and Applications, 17, 225-250.

Betti, G., Gagliardi, F., & Verma, V. (2018). Simplified Jackknife variance estimates for fuzzy measures of multidimensional poverty. International Statistical Review, 86(1), 68-86.

Examples

#This example is based on the dataset eusilc included in the package
#The step 2-3 are the following (step 1 is the eusilc dataset)
#For more on each step see the ad hoc function included in the package

#Step 2

step2 = fs_transform(eusilc[,4:23], weight = eusilc$DB090, ID = eusilc$ID)

#Step 3 is the definition of the dimension.
#For more about the step see Betti et al. (2018)

dimensions = c(1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5)

#Step 4-5 finding weights

steps4_5 = fs_weight(dimensions, step2 = step2, rho = NULL)

The plot of a FuzzyMonetary object

Description

plot method for class "FuzzyMonetary"

Usage

## S3 method for class 'FuzzyMonetary'
plot(x, ...)

Arguments

x

An object of class "FuzzyMonetary"

...

Additional options

Value

The plot

Examples


#The following example is based on the dataset eusilc
#included in the package.


#fm = "verma"

index = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             fm = "verma", HCR = 0.154, ID = eusilc$ID)

plot(index)

The plot of a FuzzySupplementary object

Description

plot method for class "FuzzySupplementary"

Usage

## S3 method for class 'FuzzySupplementary'
plot(x, ...)

Arguments

x

An object of class "FuzzySupplementary"

...

Additional options

Value

The plot

Examples


#This example is based on the dataset eusilc included in the package
#The plot of the FS index is compute with breakdown and using an HCR = 0.12

FS <- fs_construct_all(data = eusilc[,4:23], weight = eusilc$DB090, # step 2
                       dimensions = c(1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5), # step 3
                       rho = NULL, # steps 4 and 5
                       HCR = .12, # step 6
                       breakdown = eusilc$db040) # step 7 with breakdowns
plot(FS)

The summary of a FuzzyMonetary object

Description

Summary method for class "FuzzyMonetary"

Usage

## S3 method for class 'FuzzyMonetary'
summary(object, ...)

Arguments

object

An object of class "FuzzyMonetary"

...

Additional options

Value

The summary method for class "FuzzyMonetary"

Examples



#The following example is based on the dataset eusilc
#included in the package.


#fm = "verma"

index = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090,
             fm = "verma", HCR = 0.154, ID = eusilc$ID)

summary(index)

The summary of a FuzzySupplementary object

Description

Summary method for class "FuzzySupplementary"

Usage

## S3 method for class 'FuzzySupplementary'
summary(object, ...)

Arguments

object

An object of class "FuzzySupplementary"

...

Additional options

Value

The summary method for class "FuzzySupplementary"

Examples


#This example is based on the dataset eusilc included in the package
#The summary of FS index is compute with breakdown and using an HCR = 0.12

FS <- fs_construct_all(data = eusilc[,4:23], weight = eusilc$DB090, # step 2
                       dimensions = c(1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5), # step 3
                       rho = NULL, # steps 4 and 5
                       HCR = .12, # step 6
                       breakdown = eusilc$db040) # step 7 with breakdowns
summary(FS)