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%%\VignetteIndexEntry{Standard Methods for Point Estimation of Indicators on Social Exclusion and Poverty using the R Package laeken}
%%\VignetteDepends{laeken}
%%\VignetteKeywords{social exclusion, poverty, indicators, point estimation}
%%\VignettePackage{laeken}


\begin{document}


\title{Standard Methods for Point Estimation of Indicators on Social Exclusion and Poverty using the \proglang{R} Package \pkg{laeken}}
\author{Matthias Templ$^{1}$, Andreas Alfons$^{2}$}
\date{}

\maketitle

\setlength{\footnotesep}{11pt}
\footnotetext[1]{
  \begin{tabular}[t]{l}
  Zurich University of Applied Sciences\\
  E-mail: \href{mailto:matthias.templ@zhaw.ch}{matthias.templ@zhaw.ch}
  \end{tabular}
}
\footnotetext[2]{
  \begin{tabular}[t]{l}
  Erasmus School of Economics, Erasmus University Rotterdam\\
  E-mail: \href{mailto:alfons@ese.eur.nl}{alfons@ese.eur.nl}
  \end{tabular}
}


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@


\paragraph{Abstract}
This vignette demonstrates the use of the \proglang{R} package \pkg{laeken} for
standard point estimation of indicators on social exclusion and poverty
according to the definitions by Eurostat. The package contains synthetically
generated data for the European Union Statistics on Income and Living
Conditions (EU-SILC), which is used in the code examples throughout the paper.
Furthermore, the basic object-oriented design of the package is discussed. Even
though the paper is focused on showing the functionality of package
\pkg{laeken}, it also provides a brief mathematical description of the
implemented indicators.


% ------------
% introduction
% ------------

\section{Introduction}

The \emph{European Union Statistics on Income and Living Conditions} (EU-SILC)
is a panel survey conducted in EU member states and other European countries,
and serves as basis for measuring risk-of-poverty and social cohesion in Europe.
%and for evaluating the Lisbon~2010 strategy and for monitoring the
%Europe~2020 goals of the European Union.
A short overview of the $11$ most important indicators on social exclusion and
poverty according to \cite{EU-SILC04} %and \cite{EU-SILC09}
is given in the following.

\paragraph{Primary indicators}
\begin{enumerate}
\item At-risk-of-poverty rate (after social transfers)
\begin{enumerate}[a.]
\item At-risk-of-poverty rate by age and gender
\item At-risk-of-poverty rate by most frequent activity status and gender
\item At-risk-of-poverty rate by household type
\item At-risk-of-poverty rate by accommodation tenure status
\item At-risk-of-poverty rate by work intensity of the household
\item At-risk-of-poverty threshold (illustrative values)
\end{enumerate}
\item Inequality of income distribution: S80/S20 income quintile share ratio
\item At-persistent-risk-of-poverty rate by age and gender ($60\%$ median)
\item Relative median at-risk-of-poverty gap, by age and gender
\newcounter{enumi_last}
\setcounter{enumi_last}{\value{enumi}}
\end{enumerate}

\paragraph{Secondary indicators}
\begin{enumerate}
\setcounter{enumi}{\value{enumi_last}}
\item Dispersion around the at-risk-of-poverty threshold
\item At-risk-of-poverty rate anchored at a moment in time
\item At-risk-of-poverty rate before social transfers by age and gender
\item Inequality of income distribution: Gini coefficient
\item At-persistent-risk-of-poverty rate, by age and gender ($50\%$ median)
\setcounter{enumi_last}{\value{enumi}}
\end{enumerate}

\paragraph{Other indicators}
\begin{enumerate}
\setcounter{enumi}{\value{enumi_last}}
\item Mean equivalized disposable income
\item The gender pay gap
\end{enumerate}

\paragraph{}

Note that especially the Gini coefficient is very well studied due to its
importance in many fields of research.

The add-on package \pkg{laeken} \citep{laeken} aims is to bring
functionality for the estimation of indicators on social exclusion and poverty
to the statistical environment \proglang{R} \citep{RDev}. In the examples in
this vignette, standard estimates for the most important indicators are computed
according to the Eurostat definitions \citep{EU-SILC04, EU-SILC09}. More
sophisticated methods that are less influenced by outliers are described in
vignette \code{laeken-pareto} \citep{alfons11a}, while the basic framework for
variance estimation is discussed in vignette \code{laeken-variance}
\citep{templ11b}. Those documents can be viewed from within \proglang{R} with
the following commands:
<<eval=FALSE>>=
vignette("laeken-pareto")
vignette("laeken-variance")
@
Morover, a general introduction to package \pkg{laeken} is published as
\citet{alfons13b}.

The example data set of package \pkg{laeken}, which is called \code{eusilc} and
consists of $14\,827$ observations from $6\,000$ households, is used throughout
the paper. It was synthetically generated from Austrian EU-SILC survey data
from 2006 using the data simulation methodology proposed by \citet{alfons11c}
and implemented in the \proglang{R} package \pkg{simPopulation} \citep{simPopulation}.
The first three observations of the synthetic data set \code{eusilc} are
printed below.

<<>>=
library("laeken")
data("eusilc")
head(eusilc, 3)
@

Only a few of the large number of variables in the original survey are included
in the example data set. The variable names are rather cryptic codes, but these
are the standardized names used by the statistical agencies. Furthermore, the
variables \code{hsize} (household size), \code{age}, \code{eqSS} (equivalized
household size) and \code{eqIncome} (equivalized disposable income) are not
included in the standardized format of EU-SILC data, but have been derived from
other variables for convenience. Moreover, some very sparse income components
were not included in the the generation of this synthetic data set. Thus the
equivalized household income is computed from the available income components.

For the remainder of the paper, the variable \code{eqIncome} (equivalized
disposable income) is of main interest. Other variables are in some cases used
to break down the data in order to evaluate the indicators on the resulting
subsets.

It is important to note that EU-SILC data are in practice conducted through
complex sampling designs with different inclusion probabilities for the
observations in the population, which results in different weights for the
observations in the sample. Furthermore, calibration is typically performed for
non-response adjustment of these initial design weights. Therefore, the
sample weights have to be considered for all estimates, otherwise biased
results are obtained.

The rest of the paper is organized as follows. Section \ref{sec:design} briefly
illustrates the basic object-oriented design of the package. The calculation of
the equivalized household size and the equivalized disposable income is then
described in Section \ref{sec:income}. Afterwards, Section~\ref{sec:w}
introduces the Eurostat definitions of the weighted median and weighted
quantiles, which are required for the estimation of some of the indicators. In
Section~\ref{sec:ind}, a mathematical description of the most important
indicators on social exclusion and poverty is given and their estimation with
package \pkg{laeken} is demonstrated. Section~\ref{sec:sub} discusses a useful
subsetting method, and Section~\ref{sec:concl} concludes.


% ------------
% basic design
% ------------

\section{Basic design of the package}
\label{sec:design}

The implementation of the package follows an object-oriented design using
\proglang{S3} classes \citep{chambers92}. Its aim is to provide functionality
for point and variance estimation of Laeken indicators with a single command,
even for different years and domains. Currently, the following indicators are
available in the \proglang{R} package \pkg{laeken}:
\begin{itemize}
    \item \emph{At-risk-of-poverty rate}: function \code{arpr()}
    \item \emph{Quintile share ratio}: function \code{qsr()}
    \item \emph{Relative median at-risk-of-poverty gap}: function \code{rmpg()}
    \item \emph{Dispersion around the at-risk-of-poverty threshold}: also function \code{arpr()}
    \item \emph{Gini coefficient}: function \code{gini()}
\end{itemize}
Note that the implementation strictly follows the Eurostat definitions
\citep{EU-SILC04,EU-SILC09}.
%In addition, robust estimators are also implemented. Here, the focus is on
%Pareto tail modeling.


\subsection{Class structure}
In this section, the class structure of package \pkg{laeken} is briefly
discussed. Section~\ref{sec:indicator} describes the basic class
\code{"indicator"}, while the different subclasses for the specific indicators
are listed in Section~\ref{sec:classes}.


\subsubsection{Class \code{"indicator"}} \label{sec:indicator}
The basic class \code{"indicator"} acts as the superclass for all classes in
the package corresponding to specific indicators. It consists of the following
components:
%
\begin{description}
  \item[\code{value}:] A numeric vector containing the point estimate(s).
  \item[\code{valueByStratum}:] A \code{data.frame} containing the point estimates by domain.
  \item[\code{varMethod}:] A character string specifying the type of variance estimation used.
  \item[\code{var}:] A numeric vector containing the variance estimate(s).
  \item[\code{varByStratum}:] A \code{data.frame} containing the variance estimates by domain.
  \item[\code{ci}:] A numeric vector or matrix containing the confidence interval(s).
  \item[\code{ciByStratum}:] A \code{data.frame} containing the confidence intervals by domain.
  \item[\code{alpha}:] The confidence level is given by $1 - $\code{alpha}.
  \item[\code{years}:] A numeric vector containing the different years of the survey.
  \item[\code{strata}:] A character vector containing the different strata of the breakdown.
%  \item[\code{seed}:] The seed of the random number generator before the computations.
\end{description}

These list components are inherited by each indicator in the package.
One of the most important features of \pkg{laeken} is that indicators can be
evaluated for different years and domains. The latter of which can be regions
(e.g., NUTS2), but also any other breakdown given by a categorical variable
(see the examples in Section~\ref{sec:ind}).

In any case, the advantage of the object-oriented implementation is the
possibility of sharing code among the indicators. To give an example, the
following methods for the basic class \code{"indicator"} are implemented in the
package:
<<>>=
methods(class="indicator")
@
The \code{print()} and \code{subset()} methods are called by their respective
generic functions if an object inheriting from class \code{"indicator"} is
supplied. While the \code{print()} method defines the output of objects
inheriting from class \code{"indicator"} shown on the \proglang{R} console, the
\code{subset()} method allows to extract subsets of an object inheriting from
class \code{"indicator"} and is discussed in detail in Section~\ref{sec:sub}.
Furthermore, the function \code{is.indicator()} is available to test whether an
object is of class \code{"indicator"}.

\subsubsection{Additional classes} \label{sec:classes}
For the specific indicators on social exclusion and poverty, the following
classes are implemented in package \pkg{laeken}:
%
\begin{itemize}
  \item Class \code{"arpr"} with the following additional components:
  \begin{description}
    \item[\code{p}:] The percentage of the weighted median used for the
    at-risk-of-poverty threshold.
    \item[\code{threshold}:] The at-risk-of-poverty threshold(s).
  \end{description}
  \item Class \code{"qsr"} with no additional components.
  \item Class \code{"rmpg"} with the following additional components:
  \begin{description}
    \item[\code{threshold}:] The at-risk-of-poverty threshold(s).
  \end{description}
  \item Class \code{"gini"} with no additional components.
\end{itemize}
%
All these classes are subclasses of the basic class \code{"indicator"} and
therefore inherit all its components and methods. In addition, functions to
test whether an object is a member of one of these subclasses are implemented.
Similarly to \code{is.indicator()}, these are called \code{is.foo()}, where
\code{foo} is the name of the respective class (e.g., \code{is.arpr()}).


% -----------------------------
% equivalized disposable income
% -----------------------------

\section{Calculation of the equivalized disposable income}
\label{sec:income}

For each person, the equivalized disposable income is defined as the total
household disposable income divided by the equivalized household size. It
follows that each person in the same household receives the same equivalized
disposable income.

The total disposable income of a household is calculated by adding together the
personal income received by all of the household members plus the income
received at the household level. The equivalized household size is defined
according to the modified OECD scale, which gives a weight of 1.0 to the first
adult, 0.5 to other household members aged 14 or over, and 0.3 to household
members aged less than 14 \citep{EU-SILC04, EU-SILC09}.

In practice, the equivalized disposable income needs to be computed from the
income components included in EU-SILC for the estimation of the indicators on
social exclusion and poverty. Therefore, this section outlines how to perform
this step with package \pkg{laeken}, even though the variable \code{eqIncome}
containing the equivalized disposable income is already available in the
example data set \code{eusilc}. Note that not all variables that are required
for an exact computation of the equivalized income are included in the
synthetic example data. However, the functions of the package can be applied in
exactly the same manner to real EU-SILC data.

First, the equivalized household size according to the modified OECD scale
needs to be computed. This can be done with the function \code{eqSS()}, which
requires the household ID and the age of the individuals as arguments. In the
example data, household~ID and age are stored in the variables \code{db030}
and \code{age}, respectively. It should be noted that the variable
\code{age} is not in the standardized format of EU-SILC data and needs to be
calculated from the data beforehand. Nevertheless, these computations are very
simple and are therefore not shown here \citep[for details, see][]{EU-SILC09}.
The following two lines of code calculate the equivalized household size, add
it to the data set, and print the first eight observations of the variables
involved.

<<>>=
eusilc$eqSS <- eqSS("db030", "age", data=eusilc)
head(eusilc[,c("db030", "age", "eqSS")], 8)
@

Then the equivalized disposable income can be computed with the function
\code{eqInc()}. It requires the following information to be supplied: the
household~ID, the household income components to be added and subtracted,
respectively, the personal income components to be  added and subtracted,
respectively, as well as the equivalized household size. With the following
commands, the equivalized disposable income is calculated and added to the data
set, after which the first eight observations of the important variables in
this context are printed.

<<>>=
hplus <- c("hy040n", "hy050n", "hy070n", "hy080n", "hy090n", "hy110n")
hminus <- c("hy130n", "hy145n")
pplus <- c("py010n", "py050n", "py090n", "py100n",
    "py110n", "py120n", "py130n", "py140n")
eusilc$eqIncome <- eqInc("db030", hplus, hminus,
    pplus, character(), "eqSS", data=eusilc)
head(eusilc[,c("db030", "eqSS", "eqIncome")], 8)
@
%
Note that the net income is considered in this example, therefore no personal
income component needs to be subtracted \citep[see][]{EU-SILC04, EU-SILC09}.
This is reflected in the call to \code{eqInc()} by the use of an empty
character vector \code{character()} for the corresponding argument.


% ------------------
% weighted quantiles
% ------------------

\section{Weighted median and quantile estimation}
\label{sec:w}

Some of the indicators on social exclusion and poverty require the estimation
of the median income or other quantiles of the income distribution. Hence
functions that strictly follow the definitions according to \citet{EU-SILC04,
EU-SILC09} are implemented in package \pkg{laeken}. They are used internally
for the estimation of the respective indicators, but can also be called by the
user directly.

In the analysis of income distributions, the median income is typically of
higher interest than the arithmetic mean. This is because income distributions
commonly are strongly right-skewed with a heavy tail of \emph{representative
outliers} (correctly measured units that are not unique to the population) and
\emph{nonrepresentative outliers} (either measurement errors or correct
observations that can be considered unique in the population). Therefore, the
center of the distribution is more reliably estimated by a weighted median than
by a weighted mean, as the latter is highly influenced by extreme values.

In mathematical terms, quantiles are defined as $q_{p} := F^{-1}(p)$,
where $F$ is the distribution function on the population level and $0 \leq p
\leq 1$. The median as an important special case is given by $p = 0.5$. For the
following definitions, let $n$ be the number of observations in the sample, let
$\boldsymbol{x} := (x_{1}, \ldots, x_{n})'$ denote the equivalized disposable
income with \mbox{$x_{1} \leq \ldots \leq x_{n}$}, and let $\boldsymbol{w} :=
(w_{i}, \ldots, w_{n})'$ be the corresponding personal sample weights. Weighted
quantiles for the estimation of the population values according to
\citet{EU-SILC04, EU-SILC09} are then given by
\begin{equation} \label{eq:wq}
\hat{q}_{p} = \hat{q}_{p} (\boldsymbol{x}, \boldsymbol{w}) :=
\begin{cases}
  \frac{1}{2} (x_{j} + x_{j+1}), & \quad \text{if } \sum_{i=1}^{j} w_{i} = p
                               \sum_{i=1}^{n} w_{i}, \\
  x_{j+1}, & \quad \text{if } \sum_{i=1}^{j} w_{i} < p \sum_{i=1}^{n} w_{i} <
             \sum_{i=1}^{j+1} w_{i}.
\end{cases}
\end{equation}

This definition of weighted quantiles is available in \pkg{laeken} through the
function \code{weightedQuantile()}. The following command computes the weighed
20\% quantile, the weighted median, and the weighted 80\% quantile. In the
context of social exclusion indicators, these are of most importance.
% -----
<<keep.source=TRUE>>=
weightedQuantile(eusilc$eqIncome, eusilc$rb050,
    probs = c(0.2, 0.5, 0.8))
@
% -----
For the important special case of the weighted median, the function
\code{weightedMedian()} is available for convenience.
% -----
<<>>=
weightedMedian(eusilc$eqIncome, eusilc$rb050)
@

In addition, the functions \code{incMedian()} and \code{incQuintile()} are more
tailored towards application in the case of indicators on social exclusion and
poverty and provide a similar interface as the functions for the indicators
(see Section~\ref{sec:ind}). In particular, they allow to supply an additional
variable to be used as tie-breakers for sorting, and to compute the weighted
median and income quintiles, respectively, for several years of the survey.
With the following lines of code, the median income as well as the
\engordnumber{1} and \engordnumber{4} income quintile (i.e., the weighted 20\%
and 80\% quantiles) are estimated.
<<>>=
incMedian("eqIncome", weights = "rb050", data = eusilc)
incQuintile("eqIncome", weights = "rb050", k = c(1, 4), data = eusilc)
@


% -------------------
% selected indicators
% -------------------

\section{Indicators on social exclusion and poverty}
\label{sec:ind}

In this section, the most important indicators on social exclusion and poverty
are described in detail. Furthermore, the functionality of package \pkg{laeken}
to estimate these indicators is demonstrated.

It should be noted that all functions for the implemented indicators provide a
very similar interface. Most importantly, it is possible to compute estimates
for several years of the survey and different subdomains with a single command.
Furthermore, the functions allow to supply an additional variable to be used as
tie-breakers for sorting. However, not all of the implemented functionality is
shown in this vignette. For a complete description of the functions and their
arguments, the reader is referred to the corresponding \proglang{R} help pages.

In addition, only point estimation of the indicators on social exclusion and
poverty is illustrated here, statistical significance of these estimates is not
discussed. The functionality for variance estimation of the indicators is
described in the package vignette \code{laeken-variance} \citep{templ11b}.

For the following definitions of the estimators according to \citet{EU-SILC04,
EU-SILC09}, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})'$ be the equivalized
disposable income with $x_{1} \leq \ldots \leq x_{n}$ and let $\boldsymbol{w}
:= (w_{i}, \ldots, w_{n})'$ be the corresponding personal sample weights,
where $n$ denotes the number of observations. Furthermore, define the following
index sets for a certain threshold $t$:
\begin{align}
I_{< t} &:= \{ i \in \{ 1, \ldots, n \} : x_{i} < t \},\label{eq:01-Ilt}\\
I_{\leq t} &:= \{ i \in \{ 1, \ldots, n \} : x_{i} \leq t \},\label{eq:01-Ileqt}\\
I_{> t} &:= \{ i \in \{ 1, \ldots, n \} : x_{i} > t\}\label{eq:01-Igt}.
\end{align}


\subsection{At-risk-at-poverty rate} \label{sec:ARPR}
In order to define the \emph{at-risk-of-poverty rate} (ARPR), the
\emph{at-risk-of-poverty threshold} (ARPT) needs to be introduced first,
which is set at $60\%$ of the national median equivalized disposable income.
Then the at-risk-at-poverty rate is defined as the proportion of persons with
an equivalized disposable income below the at-risk-at-poverty threshold
\citep{EU-SILC04, EU-SILC09}. In a more mathematical notation, the
at-risk-at-poverty rate is defined as
\begin{equation} \label{eq:ARPR}
ARPR := P(x < 0.6 \cdot q_{0.5}) \cdot 100,% = F(0.6 \cdot q_{0.5}) \cdot 100,
\end{equation}
where $q_{0.5} := F^{-1}(0.5)$ denotes the population median (50\% quantile)
and $F$ is the distribution function of the equivalized income on the
population level.

For the estimation of the at-risk-at-poverty rate from a sample, the sample
weights need to be taken into account.
%Let $n$ be the number of observations in the sample, let $\boldsymbol{x} :=
%(x_{1}, \ldots, x_{n})'$ denote the equivalized disposable income with
%\mbox{$x_{1} \leq \ldots \leq x_{n}$}, and let $\boldsymbol{w} := (w_{i},
%\ldots, w_{n})'$ be the corresponding personal sample weights. Then the
%at-risk-at-poverty threshold is estimated by
First, the at-risk-at-poverty threshold is estimated by
\begin{equation} \label{eq:ARPT}
\widehat{ARPT} = 0.6 \cdot \hat{q}_{0.5},
\end{equation}
where $\hat{q}_{0.5}$ is the weighted median as defined in
Equation~(\ref{eq:wq}).
%Furthermore, define an index set of observations with an equivalized disposable
%income below the estimated at-risk-at-poverty threshold as
%\begin{equation}
%I_{< \widehat{ARPT}} := \{ i \in \{ 1, \ldots, n \} : x_{i} < \widehat{ARPT} \}.
%\end{equation}
%With these definitions, the at-risk-at-poverty rate can be estimated by
Then the at-risk-at-poverty rate can be estimated by
\begin{equation}
\widehat{ARPR} := \frac{\sum_{i \in I_{< \widehat{ARPT}}} w_{i}}{\sum_{i=1}^{n}
w_{i}} \cdot 100,
\end{equation}
where $I_{< \widehat{ARPT}}$ is an index set of persons with an equivalized
disposable income below the estimated at-risk-of-poverty threshold as defined
in Equation~(\ref{eq:01-Ilt}).

In package \pkg{laeken}, the functions \code{arpt()} and \code{arpr()} are
implemented for the estimation of the at-risk-of-poverty threshold and the
at-risk-of-poverty rate. Whenever sample weights are available in the data,
they should be supplied as the \code{weights} argument. Even though
\code{arpt()} is called internally by \code{arpr()}, it can also be called by
the user directly.
<<>>=
arpt("eqIncome", weights = "rb050", data = eusilc)
arpr("eqIncome", weights = "rb050", data = eusilc)
@

It is also possible to use these functions for the estimation of the indicator
\emph{dispersion around the at-risk-of-poverty threshold}, which is defined as
the proportion of persons with an equivalized disposable income
below $40\%$, $50\%$ and $70\%$ of the national weighted median equivalized
disposable income. The proportion of the median equivalized income to be used
can thereby be adjusted via the argument \code{p}.
<<>>=
arpr("eqIncome", weights = "rb050", p = 0.4, data = eusilc)
arpr("eqIncome", weights = "rb050", p = 0.5, data = eusilc)
arpr("eqIncome", weights = "rb050", p = 0.7, data = eusilc)
@

In order to compute estimates for different subdomains, a breakdown variable
simply needs to be supplied as the \code{breakdown} argument. Note that in this
case the same overall at-risk-of-poverty threshold is used for all subdomains
\citep[see][]{EU-SILC04, EU-SILC09}. The following command computes the overall
estimate, as well as estimates for all NUTS2 regions.
<<>>=
arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc)
@

However, any kind of breakdown can be supplied, e.g., the breakdowns defined by
\citet{EU-SILC04, EU-SILC09}. With the following lines of code, a breakdown
variable with all possible combinations of age categories and gender is defined
and added to the data set, before it is used to compute estimates for the
corresponding domains.
<<>>=
ageCat <- cut(eusilc$age, c(-1, 16, 25, 50, 65, Inf), right=FALSE)
eusilc$breakdown <- paste(ageCat, eusilc$rb090, sep=":")
arpr("eqIncome", weights = "rb050", breakdown = "breakdown", data = eusilc)
@
Clearly, the results are even more heterogeneous than for the breakdown into
NUTS2 regions.

%The results are even more different when considering household size
%(\code{hsize}) and citizenship (\code{pb220a}) as the domain level for
%estimation.
%<<>>=
%eusilc$breakdown <- paste(eusilc$hsize, eusilc$pb220a, sep=":")
%arpr("eqIncome", weights = "rb050", breakdown = "breakdown", data = eusilc)
%@


\subsection{Quintile share ratio}

The income \emph{quintile share ratio} (QSR) is defined as the ratio of the sum
of the equivalized disposable income received by the 20\% of the population
with the highest equivalized disposable income to that received by the 20\% of
the population with the lowest equivalized disposable income \citep{EU-SILC04,
EU-SILC09}.

For the estimation of the quintile share ratio from a sample, let
$\hat{q}_{0.2}$ and $\hat{q}_{0.8}$ denote the weighted 20\% and 80\%
quantiles, respectively, as defined in Equation~(\ref{eq:wq}). Using index sets
$I_{\leq \hat{q}_{0.2}}$ and $I_{> \hat{q}_{0.8}}$ as defined in
Equations~(\ref{eq:01-Ileqt}) and~(\ref{eq:01-Igt}), respectively, the quintile
share ratio is estimated by
\begin{equation}
\widehat{QSR} := \frac{\sum_{i \in I_{> \hat{q}_{0.8}}} w_{i} x_{i}}{\sum_{i
\in I_{\leq \hat{q}_{0.2}}} w_{i} x_{i}}.
\end{equation}

With package \pkg{laeken}, the quintile share ratio can be estimated using the
function \code{qsr()}. As for the at-risk-of-poverty rate, sample weights can
be supplied via the \code{weights} argument.
<<>>=
qsr("eqIncome", weights = "rb050", data = eusilc)
@

Computing estimates for different subdomains is again possible by specifying
the \code{breakdown} argument. In the following example, estimates for each
NUTS2 region are computed in addition to the overall estimate.
<<>>=
qsr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc)
@

Nevertheless, it should be noted that the quintile share ratio is highly
influenced by outliers \citep[see][]{hulliger09a, alfons10b}. Since the upper
tail of income distributions virtually always contains nonrepresentative
outliers, robust estimators of the quintile share ratio should preferably be
used. Thus robust semi-parametric methods based on Pareto tail modeling are
implemented in package \pkg{laeken} as well. Their application is discussed in
vignette \code{laeken-pareto} \citep{alfons11a}.


\subsection{Relative median at-risk-of-poverty gap (by age and gender)}

The \emph{relative median at-risk-of-poverty gap} (RMPG) is defined as the
difference between the median equivalized disposable income of persons below
the at-risk-of-poverty threshold and the at-risk of poverty threshold itself,
expressed as a percentage of the at-risk-of-poverty threshold \citep{EU-SILC04,
EU-SILC09}.

%Let $wmed_{(poor)}$ the weighted median of the people who having an income
%below $ARPR$ defined in Equation~\ref{eq:ARPR}. Then the relative median
%at-risk-of-poverty gap is estimated by
%\begin{displaymath}
%RMPG = \frac{ARPR - wmed_{(poor)}}{ARPR} \cdot 100
%\end{displaymath}

For the estimation of the relative median at-risk-of-poverty gap from a sample,
let $\widehat{ARPT}$ be the estimated at-risk-of-poverty threshold according to
Equation~(\ref{eq:ARPT}), and let $I_{< \widehat{ARPT}}$ be an index set of
persons with an equivalized disposable income below the
estimated at-risk-of-poverty threshold as defined in
Equation~(\ref{eq:01-Ilt}). Using this index set, define $\boldsymbol{x}_{<
\widehat{ARPT}} := (x_{i})_{i \in I_{< \widehat{ARPT}}}$ and
$\boldsymbol{w}_{< \widehat{ARPT}} := (w_{i})_{i \in I_{< \widehat{ARPT}}}$.
Furthermore, let $\hat{q}_{0.5} (\boldsymbol{x}_{< \widehat{ARPT}},
\boldsymbol{w}_{< \widehat{ARPT}})$ be the corresponding weighted median
according to the definition in Equation~(\ref{eq:wq}). Then the relative median
at-risk-of-poverty gap is estimated by
\begin{equation}
\widehat{RMPG} = \frac{\widehat{ARPT} - \hat{q}_{0.5} (\boldsymbol{x}_{<
\widehat{ARPT}}, \boldsymbol{w}_{< \widehat{ARPT}})}{\widehat{ARPT}} \cdot 100.
\end{equation}

In package \pkg{laeken}, the function \code{rmpg()} is implemented for the
estimation of the relative median at-risk-of-poverty gap. If available in the
data, sample weights should be supplied as the \code{weights} argument. Note
that the function \code{arpt()} for the estimation of the at-risk-of-poverty
threshold is called internally (cf. function \code{arpr()} for the
at-risk-of-poverty rate in Section~\ref{sec:ARPR}).

<<>>=
rmpg("eqIncome", weights = "rb050", data = eusilc)
@

Estimates for different subdomains can be computed by making use of the
\code{breakdown} argument. With the following command, the overall estimate and
estimates for all NUTS2 regions are computed.
<<>>=
rmpg("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc)
@

For the relative median at-risk-of-poverty gap, the breakdown by age and gender
is of particular interest. In the following example, a breakdown variable with
all possible combinations of age categories and gender is defined and added to
the data set. Afterwards, estimates for the corresponding domains are computed.
<<>>=
ageCat <- cut(eusilc$age, c(-1, 16, 25, 50, 65, Inf), right=FALSE)
eusilc$breakdown <- paste(ageCat, eusilc$rb090, sep=":")
rmpg("eqIncome", weights = "rb050", breakdown = "breakdown", data = eusilc)
@


\subsection{Gini coefficient}

The \emph{Gini coefficient} is defined as the relationship of cumulative shares
of the population arranged according to the level of equivalized disposable
income, to the cumulative share of the equivalized total disposable income
received by them \citep{EU-SILC04, EU-SILC09}.

For the estimation of the Gini coefficient from a sample, the sample weights
need to be taken into account. In mathematical terms, the Gini coefficient is
estimated by
\begin{equation}
\widehat{Gini} :=  100 \left[ \frac{2 \sum_{i=1}^{n} \left( w_{i} x_{i}
\sum_{j=1}^{i} w_{j} \right) - \sum_{i=1}^{n} w_{i}^{\phantom{i}2}
x_{i}}{\left( \sum_{i=1}^{n} w_{i} \right) \sum_{i=1}^{n} \left(w_{i} x_{i}
\right)} - 1 \right].
\end{equation}

The function \code{gini()} is available in \pkg{laeken} to estimate the Gini
coefficient. As for the other indicators, sample weights can be specified with
the \code{weights} argument.
<<>>=
gini("eqIncome", weights = "rb050", data = eusilc)
@

Using the \code{breakdown} argument in the following command, estimates for the
NUTS2 regions are computed in addition to the overall estimate.
<<>>=
gini("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc)
@

Since outliers have a strong influence on the Gini coefficient, robust
estimators are preferred to the standard estimation described above
\citep[see][]{alfons10b}. Vignette \code{laeken-pareto} \citep{alfons11a}
describes how to apply the robust semi-parametric methods implemented in
package \pkg{laeken}.


% ------------------
% extracting subsets
% ------------------

\section{Extracting information using  the \code{subset()} method}
\label{sec:sub}

If estimates of an indicator have been computed for several subdomains, it may
sometimes be desired to extract the results for some domains of particular
interest. In package \pkg{laeken}, this is implemented by taking advantage of
the object-oriented design of the package. Each of the functions for the
indicators described in Section~\ref{sec:ind} returns an object belonging to a
class of the same name as the respective function, e.g., function \code{arpr()}
returns an object of class \code{"arpr"}. All these classes thereby inherit
from the basic class \code{"indicator"} (see Section~\ref{sec:design}).
<<>>=
a <- arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc)
print(a)
is.arpr(a)
is.indicator(a)
class(a)
@

To extract a subset of results from such an object, a \code{subset()} method
for the class \code{"indicator"} is implemented in \pkg{laeken}. The method
\code{subset.indicator()} is hidden from the user and is called internally by
the generic function \code{subset()} whenever an object of class
\code{"indicator"} is supplied. In the following example, the estimates of the
at-risk-of-poverty rate for the regions Lower Austria and Vienna are extracted
from the object computed above.
<<>>=
subset(a, strata = c("Lower Austria", "Vienna"))
@


% -----------
% conclusions
% -----------

\section{Conclusions}
\label{sec:concl}

This vignette demonstrates the use of package \pkg{laeken} for point estimation
of the European Union indicators on social exclusion and poverty. Since the
description of the indicators in \citet{EU-SILC04, EU-SILC09} is weak from a
mathematical point of view, a more precise notation is given in this paper.
Currently, the most important indicators are implemented in \pkg{laeken}. Their
estimation is made easy with the package, as it is even possible to compute
estimates for several years and different subdomains with a single command.

Concerning the inequality indicators quintile share ratio and Gini coefficient,
it is clearly visible from their definitions that the standard estimators are
highly influenced by outliers \citep[see also][]{hulliger09a, alfons10b}.
Therefore, robust semi-parametric methods are implemented in \pkg{laeken} as
well. These are described in vignette \code{laeken-pareto}
\citep{alfons11a}, while variance and confidence interval estimation for the
indicators on social exclusion and poverty with package \pkg{laeken} is treated
in vignette \code{laeken-variance} \citep{templ11b}.


% ---------------
% acknowledgments
% ---------------

\section*{Acknowledgments}
This work was partly funded by the European Union (represented by the European
Commission) within the 7$^{\mathrm{th}}$ framework programme for research
(Theme~8, Socio-Economic Sciences and Humanities, Project AMELI (Advanced
Methodology for European Laeken Indicators), Grant Agreement No. 217322). Visit
\url{http://ameli.surveystatistics.net} for more information on the project.


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