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FuzzyPovertyR
In FuzzyPovertyR: Estimation of Fuzzy Poverty Measures
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
library(FuzzyPovertyR)
library(kableExtra)
Introduction
FuzzyPovertyR is a package for estimating fuzzy poverty indexes.
Broadly speaking, a fuzzy poverty index is an index that ranges in the
set $Q=[0,1]$ (@alkire2015multidimensional; @silber2023research). A fuzzy indexes assigns to each statistical unit a value
in this interval according to a given membership function (mf)
$\mu(x_i)\in Q$ where $x$ is a poverty predicate. The higher the value
of $\mu$, the more the individual is regarded as "poor" with respect to
the poverty predicate $x$. In socio-economic surveys $x$ may be the
equivalised disposable income, or expenditure. However, in principle $x$
may be a generic poverty predicate that the researcher needs to analyse,
for example this could be a variable that relates to access to
transports, services and other facilities.
Below we distinguish between monetary and supplementary poverty indexes
(or measures). A fuzzy monetary poverty measure is calculated over a
numeric vector of length $n$ (the available sample size). A
supplementary poverty index is calculated on a data.frame of items of
a questionnaire that relates to other dimensions of poverty other than
monetary.
The dataset coming from the package is loaded in the environment with
data(eusilc)
Fuzzy Monetary poverty measures
The package lets the user choose among different membership functions
trough the argument fm of the fm_construct function. The membership
function available are:
fm="verma" (see @cheli1995totally)
$$
\mu_i=\left(1-F_X(x_i)\right)^{\alpha-1} =
\left(\frac{\sum_{j=i+1} w_j|x_j> x_i}{\sum_{j\ge 2} w_j|x_j> x_1}\right)^{\alpha-1}
$$
where $F_X$ is the empirical cumulative distribution function of
$X$ calculated for the $i-th$ individual and $w_i$ is the sampling
weight of statistical unit $i$. The parameter $\alpha\ge 2$ is chosen so
that the average over the function equals the head count ratio (i.e. the
proportion of units whose equivalised disposable income falls below the
poverty line) of the whole population.
fm="verma1999" (see @betti1999measuring)
$$
\mu_i=(1-L_X(x_i))^{\alpha-1} =
\left(\frac{\sum_{j=i+1} w_jx_j|x_j> x_i}{\sum_{j\ge 2} w_jx_j|x_j> x_1}\right)^{\alpha-1}
$$
where $L_X$ is the Lorenz Curve of $X$ calculated for the $i-th$
individual and $w_i$ is the sampling weight of statistical unit $i$.
Again, the parameter $\alpha\ge 2$ is chosen once again so that the
average over the function equals the head count ratio of the whole
population or its estimate.
fm="verma" (see @betti2008fuzzy)
$$
\begin{split}
\mu_i&=\left(1-F_X(x_i)\right)^{\alpha-1}\left(1-L_X(x_i)\right)=\
&=\left(\frac{\sum_{j=i+1} w_j|x_j> x_i}{\sum_{j\ge 2} w_j|x_j> x_1}\right)^{\alpha-1} \left(\frac{\sum_{j=i+1} w_jx_j|x_j> x_i}{\sum_{j\ge 2} w_jx_j|x_j> x_1}\right)
\end{split}
$$
Again, the parameter $\alpha\ge 2$ is chosen once again so that the
average over the function equals the head count ratio of the whole
population or its estimate.
fm="chakravarty" (see @chakravarty2019axiomatic)
$$
\mu_i = \begin{cases}
1 & x_i = 0\
\frac{z - x_i}{z} & 0 \le x_i < z\
0 & x_i \ge z
\end{cases}
$$
where $z$ is a threshold to be chosen by the researcher.
fm="cerioli" (see @cerioli1990fuzzy) \$\$ \mu_i=
$$
\mu_i = \begin{cases}
1, \quad 0<x_i<z_1\
\frac{z_2-x_i}{z_2-z_1}, \quad z_1\le x_i<z_2\
0 \quad x_i\ge z_2
\end{cases}
$$
\$\$ the values $z_1$ and $z_2$ have to be chosen by the researcher.
fm="ZBM" (see @zedini2015new and @belhadj2010poverty)
$$
\mu_i = \begin{cases}
1 & a \le x_i < b\
\frac{-x_i}{c-b} + \frac{c}{c-b} & b \le x_i < c\
0 & x_i < a \cup x_i \ge c\
\end{cases}
$$ where $a,c,b$ are percentiles estimated with via the bootstrap
technique.
fm="belhadj2015" (see @besma2015employment)
$$
\mu_i = \begin{cases} 1 & x_i < z_1\ \mu^1 = 1-\frac{1}{2}\left(\frac{x_i-z_1}{z_1}\right)^b & z_1 \le x_i < z\ \mu^2 = 1-\frac{1}{2}\left(\frac{z_2 - x_i}{z_2}\right)^b & z \le x_i < z_2\ 0 & x_i \ge z_2 \end{cases}
$$ where $z^*$ is the flex point of $\mu_i$, $z_1$ and $z_2$ have to be
chosen by the researcher, and $b\ge 1$ is a shape parameter ruling the
degree of convexity of the function. In particular, when $b=1$ the trend
is linear.
fm="belhadj2011" (see @belhadj2011)
$$
\mu_i = \begin{cases}
1 & 0 < x_i < z_{min} \
\frac{-x_i}{z_{\max} - z_{\min}} + \frac{-z_{\max}}{z_{\max} - z_{\min}} & z_{min} \le x_i < z_{max}\
0 & x_i \ge z_{max}
\end{cases}
$$ where $z_{min}$ and $z_{MAX}$ have to be chosen by the researcher
For each of the functions below the breakdown argument can be
specified in case the user's want to obtain estimates for given
sub-domains.
Example using fm=verma, fm=verma1999 and fm=ZBM
The computation of a fuzzy poverty index that uses the
fm="verma"argument goes trough the following steps:
- Estimation of the Head Count Ratio (HCR). The package
FuzzyPovertyR provides the function HCR to estimate the Head
Count Ratio from data. It outputs a list of three elements: a
classification of units into being poor or not poor, the poverty
line, and the value itself.
hcr = HCR(predicate = eusilc$eq_income, weight = eusilc$DB090, p = 0.5, q = 0.6)$HCR # add poverty threshold
if needed, the package has a built-in function eq_predicate to
calculate the equivalised disposable income using some equivalence
scales.
- Construction of the Fuzzy Monetary measure.
verma = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL,
HCR = hcr, interval = c(1,20), alpha = NULL, fm = "verma", verbose = FALSE)
verma$fm
class(verma)
summary(verma)
plot(verma)
When alpha = NULL (the default) this function solves a non-linear
equation finding the value $\alpha$ in interval that equates the
expected value of the poverty measure to the Head Count Ratio calculated
above (see #eq-betti2006). This can be avoided by specifying a numeric
value of $\alpha$.
verma = fm_construct(predicate = eusilc$eq_income, fm = "verma", weight = eusilc$DB090, ID = NULL, interval = c(1,10), alpha = 2)
The result of fm_construct using fm="verma" is a list containing
- a
data.frame of individuals' membership functions sorted in
descending order (i.e. from most poor to least poor)
head(verma$results)
- The estimated FM measure (note that this equals the HCR if
breakdown=NULL). However, one can obtain estimates for sub-domains
using the breakdown argument as follows
verma.break = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL, HCR = hcr, interval = c(1,10), alpha = NULL, breakdown = eusilc$db040,
fm="verma")
summary(verma.break)
verma.break$estimate
- The
alpha parameter.
alpha = verma$parameters$alpha
alpha
With almost identical procedures is also possible to compute the index for fm="verma1999" and fm="TFR".
verma1999 = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL, HCR = hcr, interval = c(1,20), alpha = NULL, fm = "verma1999", verbose = FALSE)
verma1999$fm
class(verma1999)
summary(verma1999)
plot(verma1999)
TFR = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL,
HCR = hcr, interval = c(1,20), alpha = NULL, fm = "TFR", verbose = FALSE)
TFR$fm
class(TFR)
summary(TFR)
plot(TFR)
Example using fm=belhadj2015 and fm=cerioli
The construction of a fuzzy index using the membership function as
@besma2015employment or @cerioli1990fuzzy is obtained by specifying fm="belhadj2015" or fm="cerioli". Let us begin by fm="belhadj2015". For this mf the arguments z1, z2 and b need user's specification. The value z that correspond to the flex points of the mf or to the point where the two mf touch together is calculated by the function.
The parameter b $(>=1)$ rules the shape of the membership functions
(set b=1 for linearity)
z1 = 20000; z2 = 70000; b = 2
belhadj = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "belhadj2015", z1 = z1, z2 = z2, b = b)
summary(belhadj)
plot(belhadj)
Using fm="cerioli". Again we have to set the values of z1, z2 as follows:
z1 = 10000; z2 = 70000
cerioli = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "cerioli", z1 = z1, z2 = z2)
summary(cerioli)
plot(cerioli)
Example using fm=chakravarty and fm=belhadj2011
@chakravarty2019axiomatic fuzzy index is obtained setting
fm = "chakravarty". The argument z needs user's specification as follows:
z = 60000
chakravarty = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "chakravarty", z = z)
summary(chakravarty)
plot(chakravarty)
again is is possible to specify the breakdown argument to obtain
estimates at sub-domains.
chakravarty.break = fm_construct(predicate = eusilc$eq_income, eusilc$DB090, fm = "chakravarty", z = z, breakdown = eusilc$db040)
knitr::kable(data.frame(verma.break$estimate, chakravarty.break$estimate), col.names = c("Verma", "Chakravarty"), digits = 4)
Instead, for fm = "belhadj2011" the values of $z_{min}$ and $z_{MAX}$ need user's specification as follows:
zmin = 5000; zmax = 60000
belhadj2011 = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "belhadj2011", z_min = zmin, z_max = zmax)
summary(belhadj2011)
plot(belhadj2011)
Example using fm=ZBM
As previously showed to compute the index related to this mf is necessary to compute three parameters $a, b, c$ which are computed with bootstrap techniques (@zedini2015new). Moreover the definition of the index require the knowledge of the household size. The index is obtained as follows:
ZBM = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "ZBM", hh.size = eusilc$ncomp)
plot(ZBM)
Fuzzy supplementary
The package include also a multidimensional fuzzy poverty index known as Fuzzy Supplementary (FS) index (see @betti2008fuzzy and @betti2018simplified). This index is defined with multiple steps. The package has an ad-hoc function for each step (excluding the third one).
The steps are:
-
Step 1 - Identification
-
Step 2 - Transformation
-
Step 3 - Factor analysis to identify dimensions of poverty
-
Step 4 - Calculation of weights
-
Step 5 - Calculation of scores in dimensions
-
Step 6 - Calculation of the $\alpha$ parameter
-
Step 7 - Construction of the FS measure for each dimension
Step 1 - Identification
This step is pretty simple. The user has to select the columns of the
data that correspond to the items that he/she has decided to keep in the
analysis.
Step 1 is done with the following selection
# eusilc = na.omit(eusilc)
step1 = eusilc[,4:23]
If the data in the dataset are not "ordered" in the right way, i.e. the highest values represents the highest deprivation it is possible to invert them using the following function:
#Create a dataframe in which the variable X is not ordered in the right way:
data=data.frame("X"=rep(c(1,2,3,4),20), "Y"=rep(c(7,8,9,1),20))
#Crete vec_order a vector of length n with TRUE or FALSE. True if the order of the variable is to be inverted, False otherwise
vec_order=c(TRUE,FALSE)
head(fs_order(data=data, vec_order))
Step 2 - Transformation
In this step the items are mapped from their original space to the
$[0,1]$ interval using the function fs_transform (see
@betti2018simplified and @betti2008fuzzy). For each item, a positive score $s_{ij}$ is determined as follows
$$
s_{ij}= 1- d_{ij} = 1-\frac{1-F(c_{ij})}{1-F(1)},\quad i=1,\dots,n \quad \text{and}\quad j=1,\dots,k
$$
where $c_{ij}$ is the value of the category of the $j$-th item for the $i$-th household and $F(c_{ij})$ is the value of the $j$-th item cumulative function for the $i$-th household. This step is done as follows using the function fs_transform:
step2 = fs_transform(step1, weight = eusilc$DB090, ID = eusilc$ID); class(step2)
summary(step2$step2)
# step2.1 = fs_transform(step1, weight = eusilc$DB090, ID = eusilc$ID, depr.score = "d")
which outputs
knitr::kable(head(step2$step2), digits = 3, align = "c", caption = "Transformed items")
Step 3 - Factor Analysis
This fuzzy supplementary measure use factor analysis to undercover
latent dimension in the data. There are multiple approaches to get
factor analysis in R which we do not cover in this vignette, however the
user can check for example the lavaan package. Anyways, factor
analysis is not mandatory and the user may wish to undertake a different
approach to undercover a latent structure in the data. Indeed, it is
possible to skip factor analysis or to use a personal assignment of
columns in dimensions.
Regardless of the chosen method, to go trough Step 3 the user has to
specify a numeric vector of the same length of the number of items
selected in Step 1 that assigns each column to a given dimension.
dimensions = c(1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5)
Steps 4 and 5 - Calculation of weights and measures within dimensions
In this step, the weights to be assigned to each item, belonging to a given dimension $h$, are determined separately within each dimension. Such weighting procedure takes into account of two different aspects: the dispersion of the deprivation indicator and its correlation with other deprivation indicators in the given dimension. The weight of item $j$ belonging to dimension $h$ is taken as
$$
\omega_{hj}=\omega^a_{hj}\times \omega^b_{hj}, \quad h=1,\dots,H \quad \text{and}\quad j=1,\dots,k_h
$$
$\omega_{hj}^a$ is taken as proportional to the coefficient of variation of the complement to one of the positive score for the variable concerned
$$
\omega^a_{hj}\propto \frac{\sigma_{s_{hj}}}{1-\bar{s}_{hj}}
$$
where $\sigma_{s_{hj}}$ is the standard deviation of the deprivation score $s$ for item $j$ in dimension $h$ and $\bar s_{hj}$ its sample mean. The second factor is
$$
\omega^b_{hj}\propto\Biggl(\frac{1}{1+\sum_{j=1}^{k_h}\rho_{e_{hjhj^{\ast}}}|\rho_{e_{hjhj^{\ast}}}<r^\ast_{e_{hj}}}\Biggr)\times \Biggl(\frac{1}{1+\sum_{j=1}^{k_h}\rho_{e_{hjhj^{\ast}}}|\rho_{e_{hjhj^{\ast}}}\ge r^\ast_{e_{hj}}}\Biggr)
$$
where $\rho_{e_{hjhj^{\ast}}}$ is the kendall's correlation coefficient between deprivation indicators corresponding to items j and $j^{\ast}$ in the $h$-dimension and $r^\ast_{e_{hj}}$ is a critical value. Aggregation over a group of items in a particular dimension is given by a weighted mean taken over the items in that dimension $s_{hi}=\sum w_{hj} s_{hj,i}/w_{hj}$, where $w_{hj}$ is the sampling weight of the $j$-th deprivation item in the $h$-th dimension. An overall score for the $i$-th individual is calculated as the un-weighted mean:
$$
s_i=\frac{\sum_h s_{hi}}{H}
$$
Those two steps are implemented in the package with the function fs_weight. It is necessary to define a value $\rho$ which is a critical value to be used for calculation of weights in the Kendall correlation matrix. If NULL, i.e. not defined, rho is set equal to the point of largest gap between the ordered set of correlation values encountered (see Betti and Verma, 2008).
steps4_5 = fs_weight(dimensions, step2 = step2, rho = NULL); class(steps4_5)
summary(steps4_5)
plot(steps4_5)
The output is a longitudinal data frame that contains the weights
$w_a, w_b, w = w_a\times w_b$ , the deprivation score $s_{hi}$ for unit
$i$ and dimension $j$, and the overall score $s_i$ (the average over
dimensions).
knitr::kable(head(steps4_5$steps4_5), digits = 4, caption = "Results from Steps 4 and 5.")
Step 6 - Calculation of the parameter $\alpha$
This step is equivalent to that discussed in the Fuzzy Monetary section
when fm="verma", in-fact the mf is defined as:
$$
\mu_i=(1-F_S(s_i)^{\alpha-1}(1-L_S(s_i)).
$$
The function fs_equate is used to find the value of $\alpha$ such that the expected value of the mf equals the HCR.
alpha = fs_equate(steps4_5 = steps4_5, weight = eusilc$DB090, HCR = hcr, interval = c(1,10))
(alternatively a user's defined specification of the alpha argument
can be used as well.)
Step 7 - Construction of the Fuzzy Supplementary measure.
The parameter $\alpha$ estimated is used to calculate the fuzzy supplementary mf for each dimension of deprivation separately as follows. The FS indicator for the $h-th$ deprivation dimension for the $i-th$ individual is defined as combination of the $(1-F_{(S),hi})$ indicator and the $(1-L_{(S),hi})$ indicator.
$$
\begin{split}
\mu_{hi}&=\biggl(1-F_{S_{h}}(s_{hi})\Biggr)^{\alpha-1}\biggl(1-L_{S_h}(s_{hi})\Biggr)=\
&=\Biggl[\frac{\sum_{\gamma=i+1}w_{h\gamma}|s_{h\gamma}>s_{hi}}{\sum_{\gamma\ge 2}w_{h\gamma}|s_{h\gamma}>s_{h1}}\Biggr]^{\alpha-1}\Biggl[\frac{\sum_{\gamma=i+1}w_{h\gamma}s_{h\gamma}|s_{h\gamma}>s_{hi}}{\sum_{\gamma\ge 2}w_{h\gamma}s_{h\gamma}|s_{h\gamma}>s_{h1}}\Biggr]
\end{split}
$$
The function fs_construct is used to compute the FS for each dimension and overall as follows:
FS = fs_construct(steps4_5 = steps4_5, weight = eusilc$DB090, alpha = alpha, breakdown = NULL) # no breakdown
summary(FS)
FS = fs_construct(steps4_5 = steps4_5, weight = eusilc$DB090, alpha = alpha, breakdown = eusilc$db040)
FS$estimate
plot(FS)
The output of the fs_construct function is a list containing:
-
membership a list containing the FS measures for each
statistical unit in the sample. Results for each dimension can be
obtained by
-
estimate the average of the membership function for each dimension
FS$estimate
alpha the parameter $\alpha$ estimated from data.
Again, it is possible to obtain results for sub-domains by specifying
the breakdown argument
knitr::kable(fs_construct(steps4_5 = steps4_5, weight = eusilc$DB090, alpha = alpha, breakdown = eusilc$db040)$estimate, digits = 4 )
The package contains also a function named fs_construct_all which constructs the fuzzy supplementary poverty measure based without step-by-step functions.
Variance Estimation
The variance of each Fuzzy Monetary measure can be estimated either via
Bootstrap (naive or calibrated) or Jackknife Repeated Replications. We recommend the former
each time the user has no knowledge of the sampling design, while we
recommend the Jackknife when there is full information on the design and
of the PSUs (see @betti2018simplified). In the following, for the unidimensional indices, we report only the examples linked with fm=verma. For the other specification of the mf the function is identical the only changes are linked with the parameters required by the mf (see fm_construct).
Example using fm=verma
In case of fm="verma", we recommend the user to use the value of
alpha from obtained from the function fm_construct. It is possible
to specify different values of the parameter (i.e. alpha=2). We do not
recommend to leave the argument alpha=NULL for the computation of
variance.
alpha = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL, HCR = 0.12, interval = c(1,10), alpha = NULL)$alpha
boot.var = fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", type = "bootstrap_naive", HCR = .12, alpha = alpha, verbose = F, R = 10)
# plot(boot.var)
fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", type = "jackknife", HCR = .12, alpha = 9, stratum = eusilc$stratum, psu = eusilc$psu, verbose = F)
which gives the bootstrap estimate or the jackknife estimate.
If there are multiple sub-domains or sub-populations that need variance
estimation, the user can specify the breakdown to the breakdown
argument of the function fm_var. For example:
Bootstrap = fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", breakdown = eusilc$db040, type = "bootstrap_naive", HCR = hcr, alpha = alpha, verbose = FALSE) %>% summary()
Jackknife = fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", breakdown = eusilc$db040, type = "jackknife", HCR = hcr, alpha = alpha, stratum = eusilc$stratum, psu = eusilc$psu, verbose = F)%>% summary()
Bootstrap = fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", breakdown = eusilc$db040, type = "bootstrap_naive", HCR = hcr, alpha = alpha, verbose = FALSE)
plot(Bootstrap)
Bootstrap = fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", breakdown = eusilc$db040, type = "jackknife", HCR = hcr, alpha = alpha, stratum = eusilc$stratum, psu = eusilc$psu, verbose = F)
Fuzzy supplementary
variance = fs_var(data = eusilc[,4:23], weight = eusilc$DB090, ID = NULL, dimensions = dimensions, breakdown = NULL, HCR = 0.12, alpha = 2, rho = NULL, type = 'bootstrap', M = NULL, R = 2, verbose = F)
summary(variance)
Bootstrap = fs_var(data = eusilc[,4:23], weight = eusilc$DB090, ID = NULL,
dimensions = dimensions, breakdown = eusilc$db040, HCR = .12,
alpha = 2, rho = NULL, type = 'bootstrap_naive', M = NULL, R = 10, verbose = F)
plot(Bootstrap)
The following uses the Jackknife
fs_var(data = eusilc[,4:23], weight = eusilc$DB090, ID = NULL, dimensions = dimensions,
stratum = eusilc$stratum, psu = eusilc$psu, verbose = F, f = .01,
breakdown = eusilc$db040, alpha = 3, rho = NULL, type = "jackknife")%>%summary()
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knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
library(FuzzyPovertyR) library(kableExtra)
Introduction
FuzzyPovertyR is a package for estimating fuzzy poverty indexes.
Broadly speaking, a fuzzy poverty index is an index that ranges in the
set $Q=[0,1]$ (@alkire2015multidimensional; @silber2023research). A fuzzy indexes assigns to each statistical unit a value
in this interval according to a given membership function (mf)
$\mu(x_i)\in Q$ where $x$ is a poverty predicate. The higher the value
of $\mu$, the more the individual is regarded as "poor" with respect to
the poverty predicate $x$. In socio-economic surveys $x$ may be the
equivalised disposable income, or expenditure. However, in principle $x$
may be a generic poverty predicate that the researcher needs to analyse,
for example this could be a variable that relates to access to
transports, services and other facilities.
Below we distinguish between monetary and supplementary poverty indexes
(or measures). A fuzzy monetary poverty measure is calculated over a
numeric vector of length $n$ (the available sample size). A
supplementary poverty index is calculated on a data.frame of items of
a questionnaire that relates to other dimensions of poverty other than
monetary.
The dataset coming from the package is loaded in the environment with
data(eusilc)
Fuzzy Monetary poverty measures
The package lets the user choose among different membership functions
trough the argument fm of the fm_construct function. The membership
function available are:
fm="verma"(see @cheli1995totally)
$$ \mu_i=\left(1-F_X(x_i)\right)^{\alpha-1} = \left(\frac{\sum_{j=i+1} w_j|x_j> x_i}{\sum_{j\ge 2} w_j|x_j> x_1}\right)^{\alpha-1} $$
where $F_X$ is the empirical cumulative distribution function of $X$ calculated for the $i-th$ individual and $w_i$ is the sampling weight of statistical unit $i$. The parameter $\alpha\ge 2$ is chosen so that the average over the function equals the head count ratio (i.e. the proportion of units whose equivalised disposable income falls below the poverty line) of the whole population.
fm="verma1999"(see @betti1999measuring)
$$ \mu_i=(1-L_X(x_i))^{\alpha-1} = \left(\frac{\sum_{j=i+1} w_jx_j|x_j> x_i}{\sum_{j\ge 2} w_jx_j|x_j> x_1}\right)^{\alpha-1} $$
where $L_X$ is the Lorenz Curve of $X$ calculated for the $i-th$ individual and $w_i$ is the sampling weight of statistical unit $i$. Again, the parameter $\alpha\ge 2$ is chosen once again so that the average over the function equals the head count ratio of the whole population or its estimate.
fm="verma"(see @betti2008fuzzy)
$$ \begin{split} \mu_i&=\left(1-F_X(x_i)\right)^{\alpha-1}\left(1-L_X(x_i)\right)=\ &=\left(\frac{\sum_{j=i+1} w_j|x_j> x_i}{\sum_{j\ge 2} w_j|x_j> x_1}\right)^{\alpha-1} \left(\frac{\sum_{j=i+1} w_jx_j|x_j> x_i}{\sum_{j\ge 2} w_jx_j|x_j> x_1}\right) \end{split} $$
Again, the parameter $\alpha\ge 2$ is chosen once again so that the average over the function equals the head count ratio of the whole population or its estimate.
fm="chakravarty"(see @chakravarty2019axiomatic)
$$ \mu_i = \begin{cases} 1 & x_i = 0\ \frac{z - x_i}{z} & 0 \le x_i < z\ 0 & x_i \ge z \end{cases} $$ where $z$ is a threshold to be chosen by the researcher.
fm="cerioli"(see @cerioli1990fuzzy) \$\$ \mu_i=
$$ \mu_i = \begin{cases} 1, \quad 0<x_i<z_1\ \frac{z_2-x_i}{z_2-z_1}, \quad z_1\le x_i<z_2\ 0 \quad x_i\ge z_2 \end{cases} $$
\$\$ the values $z_1$ and $z_2$ have to be chosen by the researcher.
fm="ZBM"(see @zedini2015new and @belhadj2010poverty)
$$ \mu_i = \begin{cases} 1 & a \le x_i < b\ \frac{-x_i}{c-b} + \frac{c}{c-b} & b \le x_i < c\ 0 & x_i < a \cup x_i \ge c\ \end{cases} $$ where $a,c,b$ are percentiles estimated with via the bootstrap technique.
fm="belhadj2015"(see @besma2015employment)
$$ \mu_i = \begin{cases} 1 & x_i < z_1\ \mu^1 = 1-\frac{1}{2}\left(\frac{x_i-z_1}{z_1}\right)^b & z_1 \le x_i < z\ \mu^2 = 1-\frac{1}{2}\left(\frac{z_2 - x_i}{z_2}\right)^b & z \le x_i < z_2\ 0 & x_i \ge z_2 \end{cases} $$ where $z^*$ is the flex point of $\mu_i$, $z_1$ and $z_2$ have to be chosen by the researcher, and $b\ge 1$ is a shape parameter ruling the degree of convexity of the function. In particular, when $b=1$ the trend is linear.
fm="belhadj2011"(see @belhadj2011)
$$ \mu_i = \begin{cases} 1 & 0 < x_i < z_{min} \ \frac{-x_i}{z_{\max} - z_{\min}} + \frac{-z_{\max}}{z_{\max} - z_{\min}} & z_{min} \le x_i < z_{max}\ 0 & x_i \ge z_{max} \end{cases} $$ where $z_{min}$ and $z_{MAX}$ have to be chosen by the researcher
For each of the functions below the breakdown argument can be
specified in case the user's want to obtain estimates for given
sub-domains.
Example using fm=verma, fm=verma1999 and fm=ZBM
The computation of a fuzzy poverty index that uses the
fm="verma"argument goes trough the following steps:
- Estimation of the Head Count Ratio (HCR). The package
FuzzyPovertyRprovides the functionHCRto estimate the Head Count Ratio from data. It outputs a list of three elements: a classification of units into being poor or not poor, the poverty line, and the value itself.
hcr = HCR(predicate = eusilc$eq_income, weight = eusilc$DB090, p = 0.5, q = 0.6)$HCR # add poverty threshold
if needed, the package has a built-in function eq_predicate to
calculate the equivalised disposable income using some equivalence
scales.
- Construction of the Fuzzy Monetary measure.
verma = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL, HCR = hcr, interval = c(1,20), alpha = NULL, fm = "verma", verbose = FALSE) verma$fm class(verma) summary(verma) plot(verma)
When alpha = NULL (the default) this function solves a non-linear
equation finding the value $\alpha$ in interval that equates the
expected value of the poverty measure to the Head Count Ratio calculated
above (see #eq-betti2006). This can be avoided by specifying a numeric
value of $\alpha$.
verma = fm_construct(predicate = eusilc$eq_income, fm = "verma", weight = eusilc$DB090, ID = NULL, interval = c(1,10), alpha = 2)
The result of fm_construct using fm="verma" is a list containing
- a
data.frameof individuals' membership functions sorted in descending order (i.e. from most poor to least poor)
head(verma$results)
- The estimated FM measure (note that this equals the HCR if
breakdown=NULL). However, one can obtain estimates for sub-domains using thebreakdownargument as follows
verma.break = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL, HCR = hcr, interval = c(1,10), alpha = NULL, breakdown = eusilc$db040, fm="verma")
summary(verma.break) verma.break$estimate
- The
alphaparameter.
alpha = verma$parameters$alpha alpha
With almost identical procedures is also possible to compute the index for fm="verma1999" and fm="TFR".
verma1999 = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL, HCR = hcr, interval = c(1,20), alpha = NULL, fm = "verma1999", verbose = FALSE) verma1999$fm class(verma1999) summary(verma1999) plot(verma1999)
TFR = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL, HCR = hcr, interval = c(1,20), alpha = NULL, fm = "TFR", verbose = FALSE) TFR$fm class(TFR) summary(TFR) plot(TFR)
Example using fm=belhadj2015 and fm=cerioli
The construction of a fuzzy index using the membership function as
@besma2015employment or @cerioli1990fuzzy is obtained by specifying fm="belhadj2015" or fm="cerioli". Let us begin by fm="belhadj2015". For this mf the arguments z1, z2 and b need user's specification. The value z that correspond to the flex points of the mf or to the point where the two mf touch together is calculated by the function.
The parameter b $(>=1)$ rules the shape of the membership functions
(set b=1 for linearity)
z1 = 20000; z2 = 70000; b = 2 belhadj = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "belhadj2015", z1 = z1, z2 = z2, b = b)
summary(belhadj) plot(belhadj)
Using fm="cerioli". Again we have to set the values of z1, z2 as follows:
z1 = 10000; z2 = 70000 cerioli = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "cerioli", z1 = z1, z2 = z2)
summary(cerioli) plot(cerioli)
Example using fm=chakravarty and fm=belhadj2011
@chakravarty2019axiomatic fuzzy index is obtained setting
fm = "chakravarty". The argument z needs user's specification as follows:
z = 60000 chakravarty = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "chakravarty", z = z)
summary(chakravarty) plot(chakravarty)
again is is possible to specify the breakdown argument to obtain
estimates at sub-domains.
chakravarty.break = fm_construct(predicate = eusilc$eq_income, eusilc$DB090, fm = "chakravarty", z = z, breakdown = eusilc$db040)
knitr::kable(data.frame(verma.break$estimate, chakravarty.break$estimate), col.names = c("Verma", "Chakravarty"), digits = 4)
Instead, for fm = "belhadj2011" the values of $z_{min}$ and $z_{MAX}$ need user's specification as follows:
zmin = 5000; zmax = 60000 belhadj2011 = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "belhadj2011", z_min = zmin, z_max = zmax)
summary(belhadj2011) plot(belhadj2011)
Example using fm=ZBM
As previously showed to compute the index related to this mf is necessary to compute three parameters $a, b, c$ which are computed with bootstrap techniques (@zedini2015new). Moreover the definition of the index require the knowledge of the household size. The index is obtained as follows:
ZBM = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "ZBM", hh.size = eusilc$ncomp)
plot(ZBM)
Fuzzy supplementary
The package include also a multidimensional fuzzy poverty index known as Fuzzy Supplementary (FS) index (see @betti2008fuzzy and @betti2018simplified). This index is defined with multiple steps. The package has an ad-hoc function for each step (excluding the third one). The steps are:
-
Step 1 - Identification
-
Step 2 - Transformation
-
Step 3 - Factor analysis to identify dimensions of poverty
-
Step 4 - Calculation of weights
-
Step 5 - Calculation of scores in dimensions
-
Step 6 - Calculation of the $\alpha$ parameter
-
Step 7 - Construction of the FS measure for each dimension
Step 1 - Identification
This step is pretty simple. The user has to select the columns of the data that correspond to the items that he/she has decided to keep in the analysis.
Step 1 is done with the following selection
# eusilc = na.omit(eusilc) step1 = eusilc[,4:23]
If the data in the dataset are not "ordered" in the right way, i.e. the highest values represents the highest deprivation it is possible to invert them using the following function:
#Create a dataframe in which the variable X is not ordered in the right way: data=data.frame("X"=rep(c(1,2,3,4),20), "Y"=rep(c(7,8,9,1),20)) #Crete vec_order a vector of length n with TRUE or FALSE. True if the order of the variable is to be inverted, False otherwise vec_order=c(TRUE,FALSE) head(fs_order(data=data, vec_order))
Step 2 - Transformation
In this step the items are mapped from their original space to the
$[0,1]$ interval using the function fs_transform (see
@betti2018simplified and @betti2008fuzzy). For each item, a positive score $s_{ij}$ is determined as follows
$$ s_{ij}= 1- d_{ij} = 1-\frac{1-F(c_{ij})}{1-F(1)},\quad i=1,\dots,n \quad \text{and}\quad j=1,\dots,k $$
where $c_{ij}$ is the value of the category of the $j$-th item for the $i$-th household and $F(c_{ij})$ is the value of the $j$-th item cumulative function for the $i$-th household. This step is done as follows using the function fs_transform:
step2 = fs_transform(step1, weight = eusilc$DB090, ID = eusilc$ID); class(step2) summary(step2$step2) # step2.1 = fs_transform(step1, weight = eusilc$DB090, ID = eusilc$ID, depr.score = "d")
which outputs
knitr::kable(head(step2$step2), digits = 3, align = "c", caption = "Transformed items")
Step 3 - Factor Analysis
This fuzzy supplementary measure use factor analysis to undercover
latent dimension in the data. There are multiple approaches to get
factor analysis in R which we do not cover in this vignette, however the
user can check for example the lavaan package. Anyways, factor
analysis is not mandatory and the user may wish to undertake a different
approach to undercover a latent structure in the data. Indeed, it is
possible to skip factor analysis or to use a personal assignment of
columns in dimensions.
Regardless of the chosen method, to go trough Step 3 the user has to specify a numeric vector of the same length of the number of items selected in Step 1 that assigns each column to a given dimension.
dimensions = c(1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5)
Steps 4 and 5 - Calculation of weights and measures within dimensions
In this step, the weights to be assigned to each item, belonging to a given dimension $h$, are determined separately within each dimension. Such weighting procedure takes into account of two different aspects: the dispersion of the deprivation indicator and its correlation with other deprivation indicators in the given dimension. The weight of item $j$ belonging to dimension $h$ is taken as
$$ \omega_{hj}=\omega^a_{hj}\times \omega^b_{hj}, \quad h=1,\dots,H \quad \text{and}\quad j=1,\dots,k_h $$
$\omega_{hj}^a$ is taken as proportional to the coefficient of variation of the complement to one of the positive score for the variable concerned
$$ \omega^a_{hj}\propto \frac{\sigma_{s_{hj}}}{1-\bar{s}_{hj}} $$
where $\sigma_{s_{hj}}$ is the standard deviation of the deprivation score $s$ for item $j$ in dimension $h$ and $\bar s_{hj}$ its sample mean. The second factor is
$$ \omega^b_{hj}\propto\Biggl(\frac{1}{1+\sum_{j=1}^{k_h}\rho_{e_{hjhj^{\ast}}}|\rho_{e_{hjhj^{\ast}}}<r^\ast_{e_{hj}}}\Biggr)\times \Biggl(\frac{1}{1+\sum_{j=1}^{k_h}\rho_{e_{hjhj^{\ast}}}|\rho_{e_{hjhj^{\ast}}}\ge r^\ast_{e_{hj}}}\Biggr) $$
where $\rho_{e_{hjhj^{\ast}}}$ is the kendall's correlation coefficient between deprivation indicators corresponding to items j and $j^{\ast}$ in the $h$-dimension and $r^\ast_{e_{hj}}$ is a critical value. Aggregation over a group of items in a particular dimension is given by a weighted mean taken over the items in that dimension $s_{hi}=\sum w_{hj} s_{hj,i}/w_{hj}$, where $w_{hj}$ is the sampling weight of the $j$-th deprivation item in the $h$-th dimension. An overall score for the $i$-th individual is calculated as the un-weighted mean:
$$
s_i=\frac{\sum_h s_{hi}}{H}
$$
Those two steps are implemented in the package with the function fs_weight. It is necessary to define a value $\rho$ which is a critical value to be used for calculation of weights in the Kendall correlation matrix. If NULL, i.e. not defined, rho is set equal to the point of largest gap between the ordered set of correlation values encountered (see Betti and Verma, 2008).
steps4_5 = fs_weight(dimensions, step2 = step2, rho = NULL); class(steps4_5) summary(steps4_5) plot(steps4_5)
The output is a longitudinal data frame that contains the weights $w_a, w_b, w = w_a\times w_b$ , the deprivation score $s_{hi}$ for unit $i$ and dimension $j$, and the overall score $s_i$ (the average over dimensions).
knitr::kable(head(steps4_5$steps4_5), digits = 4, caption = "Results from Steps 4 and 5.")
Step 6 - Calculation of the parameter $\alpha$
This step is equivalent to that discussed in the Fuzzy Monetary section
when fm="verma", in-fact the mf is defined as:
$$
\mu_i=(1-F_S(s_i)^{\alpha-1}(1-L_S(s_i)).
$$
The function fs_equate is used to find the value of $\alpha$ such that the expected value of the mf equals the HCR.
alpha = fs_equate(steps4_5 = steps4_5, weight = eusilc$DB090, HCR = hcr, interval = c(1,10))
(alternatively a user's defined specification of the alpha argument
can be used as well.)
Step 7 - Construction of the Fuzzy Supplementary measure.
The parameter $\alpha$ estimated is used to calculate the fuzzy supplementary mf for each dimension of deprivation separately as follows. The FS indicator for the $h-th$ deprivation dimension for the $i-th$ individual is defined as combination of the $(1-F_{(S),hi})$ indicator and the $(1-L_{(S),hi})$ indicator.
$$
\begin{split}
\mu_{hi}&=\biggl(1-F_{S_{h}}(s_{hi})\Biggr)^{\alpha-1}\biggl(1-L_{S_h}(s_{hi})\Biggr)=\
&=\Biggl[\frac{\sum_{\gamma=i+1}w_{h\gamma}|s_{h\gamma}>s_{hi}}{\sum_{\gamma\ge 2}w_{h\gamma}|s_{h\gamma}>s_{h1}}\Biggr]^{\alpha-1}\Biggl[\frac{\sum_{\gamma=i+1}w_{h\gamma}s_{h\gamma}|s_{h\gamma}>s_{hi}}{\sum_{\gamma\ge 2}w_{h\gamma}s_{h\gamma}|s_{h\gamma}>s_{h1}}\Biggr]
\end{split}
$$
The function fs_construct is used to compute the FS for each dimension and overall as follows:
FS = fs_construct(steps4_5 = steps4_5, weight = eusilc$DB090, alpha = alpha, breakdown = NULL) # no breakdown summary(FS) FS = fs_construct(steps4_5 = steps4_5, weight = eusilc$DB090, alpha = alpha, breakdown = eusilc$db040) FS$estimate plot(FS)
The output of the fs_construct function is a list containing:
-
membershipalistcontaining the FS measures for each statistical unit in the sample. Results for each dimension can be obtained by -
estimatethe average of the membership function for each dimension
FS$estimate
alphathe parameter $\alpha$ estimated from data.
Again, it is possible to obtain results for sub-domains by specifying
the breakdown argument
knitr::kable(fs_construct(steps4_5 = steps4_5, weight = eusilc$DB090, alpha = alpha, breakdown = eusilc$db040)$estimate, digits = 4 )
The package contains also a function named fs_construct_all which constructs the fuzzy supplementary poverty measure based without step-by-step functions.
Variance Estimation
The variance of each Fuzzy Monetary measure can be estimated either via
Bootstrap (naive or calibrated) or Jackknife Repeated Replications. We recommend the former
each time the user has no knowledge of the sampling design, while we
recommend the Jackknife when there is full information on the design and
of the PSUs (see @betti2018simplified). In the following, for the unidimensional indices, we report only the examples linked with fm=verma. For the other specification of the mf the function is identical the only changes are linked with the parameters required by the mf (see fm_construct).
Example using fm=verma
In case of fm="verma", we recommend the user to use the value of
alpha from obtained from the function fm_construct. It is possible
to specify different values of the parameter (i.e. alpha=2). We do not
recommend to leave the argument alpha=NULL for the computation of
variance.
alpha = fm_construct(predicate = eusilc$eq_income, weight = eusilc$DB090, ID = NULL, HCR = 0.12, interval = c(1,10), alpha = NULL)$alpha
boot.var = fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", type = "bootstrap_naive", HCR = .12, alpha = alpha, verbose = F, R = 10) # plot(boot.var) fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", type = "jackknife", HCR = .12, alpha = 9, stratum = eusilc$stratum, psu = eusilc$psu, verbose = F)
which gives the bootstrap estimate or the jackknife estimate.
If there are multiple sub-domains or sub-populations that need variance
estimation, the user can specify the breakdown to the breakdown
argument of the function fm_var. For example:
Bootstrap = fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", breakdown = eusilc$db040, type = "bootstrap_naive", HCR = hcr, alpha = alpha, verbose = FALSE) %>% summary() Jackknife = fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", breakdown = eusilc$db040, type = "jackknife", HCR = hcr, alpha = alpha, stratum = eusilc$stratum, psu = eusilc$psu, verbose = F)%>% summary()
Bootstrap = fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", breakdown = eusilc$db040, type = "bootstrap_naive", HCR = hcr, alpha = alpha, verbose = FALSE) plot(Bootstrap) Bootstrap = fm_var(predicate = eusilc$eq_income, weight = eusilc$DB090, fm = "verma", breakdown = eusilc$db040, type = "jackknife", HCR = hcr, alpha = alpha, stratum = eusilc$stratum, psu = eusilc$psu, verbose = F)
Fuzzy supplementary
variance = fs_var(data = eusilc[,4:23], weight = eusilc$DB090, ID = NULL, dimensions = dimensions, breakdown = NULL, HCR = 0.12, alpha = 2, rho = NULL, type = 'bootstrap', M = NULL, R = 2, verbose = F) summary(variance)
Bootstrap = fs_var(data = eusilc[,4:23], weight = eusilc$DB090, ID = NULL, dimensions = dimensions, breakdown = eusilc$db040, HCR = .12, alpha = 2, rho = NULL, type = 'bootstrap_naive', M = NULL, R = 10, verbose = F) plot(Bootstrap)
The following uses the Jackknife
fs_var(data = eusilc[,4:23], weight = eusilc$DB090, ID = NULL, dimensions = dimensions, stratum = eusilc$stratum, psu = eusilc$psu, verbose = F, f = .01, breakdown = eusilc$db040, alpha = 3, rho = NULL, type = "jackknife")%>%summary()
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