| Version: | 0.2-13 |
| Date: | 2014-07-21 |
| Title: | Measuring Inequality, Concentration, and Poverty |
| Description: | Inequality, concentration, and poverty measures. Lorenz curves (empirical and theoretical). |
| Depends: | R (≥ 2.10.0) |
| Imports: | stats, graphics, grDevices |
| License: | GPL-2 | GPL-3 |
| Packaged: | 2014-07-21 17:37:43 UTC; zeileis |
| Author: | Achim Zeileis [aut, cre], Christian Kleiber [ctb] |
| Maintainer: | Achim Zeileis <Achim.Zeileis@R-project.org> |
| NeedsCompilation: | no |
| Repository: | CRAN |
| Date/Publication: | 2014-07-21 20:10:45 |
Concentration Measures
Description
computes the concentration within a vector according to the specified concentration measure
Usage
conc(x, parameter = NULL, type = c("Herfindahl", "Rosenbluth"), na.rm = TRUE)
Herfindahl(x, parameter = 1, na.rm = TRUE)
Rosenbluth(x, na.rm = TRUE)
Arguments
x |
a vector containing non-negative elements |
parameter |
parameter of the concentration measure (if set to |
type |
character string giving the measure used to compute concentration. must be one of the strings in the default argument (the first character is sufficient). defaults to "Herfindahl". |
na.rm |
logical. Should missing values ( |
Details
conc is just a wrapper for the concentration measures of
Herfindahl and Rosenbluth
(Hall / Tiedemann / Rosenbluth). If parameter is set to NULL
the default from the respective function is used.
Value
the value of the concentration measure
References
F A Cowell: Measurement of Inequality, 2000, in A B Atkinson / F Bourguignon (Eds): Handbook of Income Distribution, Amsterdam,
F A Cowell: Measuring Inequality, 1995 Prentice Hall/Harvester Wheatshef,
M Hall / N Tidemann: Measures of Concentration, 1967, JASA 62, 162-168.
See Also
Examples
# generate vector (of sales)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
# compute Herfindahl coefficient with parameter 1
conc(x)
# compute coefficient of Hall/Tiedemann/Rosenbluth
conc(x, type="Rosenbluth")
Income Metadata from Ilocos, Philippines
Description
Income metadata from surveys conducted by the Philippines' National Statistics Office.
Usage
data(Ilocos)Format
A data frame with 632 observations of 8 variables.
- income
total income of household,
- sex
sex of household head (
"male"or"female"),- family.size
family size (sometimes averaged over two semesters),
- urbanity
factor with levels
"rural"and"urban",- province
factor indicating the particular province,
- AP.income
total household income during the APIS,
- AP.family.size
family size during APIS,
- AP.weight
APIS survey weight for each household.
Details
The data contains household income and metadata in one of the sixteen regions of the Philippines called Ilocos. The data comes from two of the NSO's surveys: the 1997 Family and Income and Expenditure Survey and the 1998 Annual Poverty Indicators Survey (APIS).
Since the APIS only has a six month reference period, the original data were rescaled using an adjustment factor from the quarterly GDP figures that can be obtained from the major sectors.
Source
National Statistics Office, Philippines: http://www.census.gov.ph/, where also the whole data set may be obtained.
Inequality Measures
Description
computes the inequality within a vector according to the specified inequality measure
Usage
ineq(x, parameter = NULL, type = c("Gini", "RS", "Atkinson", "Theil", "Kolm", "var",
"square.var", "entropy"), na.rm = TRUE)
Gini(x, corr = FALSE, na.rm = TRUE)
RS(x, na.rm = TRUE)
Atkinson(x, parameter = 0.5, na.rm = TRUE)
Theil(x, parameter = 0, na.rm = TRUE)
Kolm(x, parameter = 1, na.rm = TRUE)
var.coeff(x, square = FALSE, na.rm = TRUE)
entropy(x, parameter = 0.5, na.rm = TRUE)
Arguments
x |
a vector containing at least non-negative elements |
parameter |
parameter of the inequality measure (if set to |
type |
character string giving the measure used to compute inequality. must be one of the strings in the default argument (the first character is sufficient). defaults to "Gini". |
corr |
logical. Argument of the function |
square |
logical. Argument of the function |
na.rm |
logical. Should missing values ( |
Details
ineq is just a wrapper for the inequality measures Gini,
RS, Atkinson, Theil, Kolm,var.coeff,
entropy. If parameter is set to NULL the default from
the respective function is used.
Gini is the Gini coefficient, RS is the the Ricci-Schutz
coefficient (also called Pietra's measure), Atkinson gives
Atkinson's measure and Kolm computes Kolm's measure.
If the parameter in Theil is 0 Theil's entropy measure is
computed, for every other value Theil's second measure is
computed.
ineq(x, type="var") and var.coeff(x) respectively
compute the coefficient of variation, while
ineq(x,type="square.var") and var.coeff(x, square=TRUE)
compute the squared coefficient of variation.
entropy computes the generalized entropy, which is for
parameter 1 equal to Theil's entropy coefficient and for parameter
0 equal to the second measure of Theil.
Value
the value of the inequality measure
References
F A Cowell: Measurement of Inequality, 2000, in A B Atkinson / F Bourguignon (Eds): Handbook of Income Distribution, Amsterdam,
F A Cowell: Measuring Inequality, 1995 Prentice Hall/Harvester Wheatshef,
Marshall / Olkin: Inequalities: Theory of Majorization and Its Applications, New York 1979 (Academic Press).
See Also
Examples
# generate vector (of incomes)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
# compute Gini coefficient
ineq(x)
# compute Atkinson coefficient with parameter=0.5
ineq(x, parameter=0.5, type="Atkinson")
Lorenz Asymmetry Coefficient
Description
Coefficient of asymmetry in the Lorenz curve.
Usage
Lasym(x, n = rep(1, length(x)), interval = FALSE, na.rm = TRUE)
Arguments
x |
a vector containing non-negative elements. |
n |
a vector of frequencies, must be same length as |
interval |
logical. In the case where there are observations exactly equal to the mean, either an interval of asymmetry coefficients can be returned or their midpoint. |
na.rm |
logical. Should missing values ( |
Details
Damgaard and Weinter (2000) have suggested an additional measure for comparing inequality in in distributions (specifically for describing plant size or fecundity distributions) to accompany the Lorenz curve and Gini coefficient. It assesses the asymmetry in the Lorenz curve of the distributions.
References
C Damgaard, J Weiner: Describing Inequality in Plant Size or Fecundity, 2000. Ecology 81(4), 1139–1142.
See Also
Examples
## Examples from Damgaard & Weiner (2000)
## Figure 2
x <- rep(c(50/9, 50), c(9, 1))
y <- rep(c(2, 18), c(5, 5))
plot(table(x))
plot(table(y))
## statistics
mean(x)
mean(y)
Gini(x, corr = TRUE)
Gini(y, corr = TRUE)
Lasym(x)
Lasym(y)
## Figure 3
plot(Lc(x))
lines(Lc(y), col = "slategray")
abline(1, -1, lty = 2)
Lorenz Curve
Description
Computes the (empirical) ordinary and generalized Lorenz curve of a vector x
Usage
Lc(x, n = rep(1,length(x)), plot = FALSE)
Arguments
x |
a vector containing non-negative elements. |
n |
a vector of frequencies, must be same length as |
plot |
logical. If TRUE the empirical Lorenz curve will be plotted. |
Details
Lc(x) computes the empirical ordinary Lorenz curve of x
as well as the generalized Lorenz curve (= ordinary Lorenz curve *
mean(x)). The result can be interpreted like this: p*100 percent
have L(p)*100 percent of x.
If n is changed to anything but the default x is
interpreted as a vector of class means and n as a vector of
class frequencies: in this case Lc will compute the minimal
Lorenz curve (= no inequality within each group). A maximal curve can be
computed with Lc.mehran.
Value
A list of class "Lc" with the following components:
p |
vector of percentages |
L |
vector with values of the ordinary Lorenz curve |
L.general |
vector with values of the generalized Lorenz curve |
References
B C Arnold: Majorization and the Lorenz Order: A Brief Introduction, 1987, Springer,
F A Cowell: Measurement of Inequality, 2000, in A B Atkinson / F Bourguignon (Eds): Handbook of Income Distribution, Amsterdam,
F A Cowell: Measuring Inequality, 1995 Prentice Hall/Harvester Wheatshef.
See Also
plot.Lc, Lc.mehran,
plot.theorLc
Examples
## Load and attach income (and metadata) set from Ilocos, Philippines
data(Ilocos)
attach(Ilocos)
## extract and rescale income for the provinces "Pangasinan" und "La Union"
income.p <- income[province=="Pangasinan"]/10000
income.u <- income[province=="La Union"]/10000
## compute the Lorenz curves
Lc.p <- Lc(income.p)
Lc.u <- Lc(income.u)
## it can be seen the the inequality in La Union is higher than in
## Pangasinan because the respective Lorenz curve takes smaller values.
plot(Lc.p)
lines(Lc.u, col=2)
## the picture becomes even clearer with generalized Lorenz curves
plot(Lc.p, general=TRUE)
lines(Lc.u, general=TRUE, col=2)
## inequality measures emphasize these results, e.g. Atkinson's measure
ineq(income.p, type="Atkinson")
ineq(income.u, type="Atkinson")
## or Theil's entropy measure
ineq(income.p, type="Theil", parameter=0)
ineq(income.u, type="Theil", parameter=0)
# income distribution of the USA in 1968 (in 10 classes)
# x vector of class means, n vector of class frequencies
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
n <- c(482, 825, 722, 690, 661, 760, 745, 2140, 1911, 1024)
# compute minimal Lorenz curve (= no inequality in each group)
Lc.min <- Lc(x, n=n)
# compute maximal Lorenz curve (limits of Mehran)
Lc.max <- Lc.mehran(x,n)
# plot both Lorenz curves in one plot
plot(Lc.min)
lines(Lc.max, col=4)
# add the theoretic Lorenz curve of a Lognormal-distribution with variance 0.78
lines(Lc.lognorm, parameter=0.78)
# add the theoretic Lorenz curve of a Dagum-distribution
lines(Lc.dagum, parameter=c(3.4,2.6))
Mehran Bounds For Lorenz Curves
Description
Computes the Mehran bounds for a Lorenz curve of grouped data
Usage
Lc.mehran(x,n)
Arguments
x |
vector of class means. |
n |
vector of class frequencies. |
Value
An object of class "Lc", but containing only
p and L.
References
F Mehran: Bounds on the Gini Index Based on Observed Points of the Lorenz Curve, 1975, JASA 70, 64-66.
See Also
Examples
# income distribution of the USA in 1968 (in 10 classes)
# x vector of class means, n vector of class frequencies
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
n <- c(482, 825, 722, 690, 661, 760, 745, 2140, 1911, 1024)
# compute minimal Lorenz curve (= no inequality in each group)
Lc.min <- Lc(x, n=n)
# compute maximal Lorenz curve (limits of Mehran)
Lc.max <- Lc.mehran(x,n)
# plot both Lorenz curves in one plot
plot(Lc.min)
lines(Lc.max, col=4)
# add the theoretic Lorenz curve of a Lognormal-distribution with variance 0.78
lines(Lc.lognorm, parameter=0.78)
# add the theoretic Lorenz curve of a Dagum-distribution
lines(Lc.dagum, parameter=c(3.4,2.6))
Majorization
Description
tests whether a vector x majorizes another vector y
Usage
major(x,y)
Arguments
x, y |
vectors containing non-negative elements (with same length and same mean) |
Details
even if x and y are comparable (i.e. have same length
and same mean) it is possible that neither x majorizes y nor y majorizes x.
Value
logical. TRUE if x majorizes y (x >=(M) y), FALSE if not.
References
Marshall / Olkin: Inequalities: Theory of Majorization and Its Applications, New York 1979 (Academic Press)
See Also
Examples
# generate vectors (of incomes)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
y <- c(841, 2063, 2445, 3438, 4437, 5401, 6392, 8304, 11304, 21961)
# test whether x majorizes y (TRUE, because y is result of
# Pigou-Dalton-transfers)
major(x,y)
Pen's Parade
Description
plots Pen's Parade of a vector x
Usage
Pen(x, n = rep(1, length(x)), group = NULL,
scaled = TRUE, abline = TRUE, add = FALSE, segments = NULL,
main = "Pen's Parade", ylab = NULL, xlab = NULL,
col = NULL, lwd = NULL, las = 1, fill = NULL, ...)
Arguments
x |
a vector containing non-negative elements. |
n |
a vector of frequencies or weights, must be same length as |
group |
a factor coding different groups, must be same length as |
scaled |
logical. Should Pen's parade be divided by |
abline |
logical. Should a horizontal line for the mean be drawn? |
add |
logical. Should the plot be added to an existing plot? |
segments |
logical. Should histogram-like segments be drawn? |
col |
a (vector of) color(s) for drawing the curve. |
fill |
a (vector of) color(s) for filling the area under the curve. |
xlab, ylab |
axis labels. Suitable defaults depending on
|
main, lwd, las, ... |
further high-level |
Details
Pen's Parade is basically the inverse distribution function
(standardized by mean(x)).
Pen allows for fine control of the layout—the graphical parameters col
and fill can be vectorized if histogram-like segments are drawn
(segments = TRUE)—but implements several heuristics in choosing its
default plotting parameters. If a grouping factor group is given,
the default is to draw segments with a grey-shaded filling. If no fill color
is used, the default is to draw a thick blue curve. But as all of these are just
defaults, they can of course easily be changed. See also the examples.
References
F A Cowell: Measurement of Inequality, 2000, in A B Atkinson / F Bourguignon (Eds): Handbook of Income Distribution, Amsterdam,
F A Cowell: Measuring Inequality, 1995 Prentice Hall/Harvester Wheatshef,
J Pen: Income Distribution, 1971, Harmondsworth: Allen Lane.
See Also
Examples
# load and attach Philippine income data
data(Ilocos)
attach(Ilocos)
# plot Pen's Parade of income
Pen(income)
Pen(income, fill = hsv(0.1, 0.3, 1))
# income distribution of the USA in 1968 (in 10 classes)
# x vector of class means, n vector of class frequencies
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
n <- c(482, 825, 722, 690, 661, 760, 745, 2140, 1911, 1024)
Pen(x, n = n)
# create artificial grouping variable
myfac <- factor(c(1, 1, 1, 2, 2, 2, 3, 3, 3, 3))
Pen(x, n = n, group = myfac)
Plot Lorenz Curve
Description
plotting method for objects of class "Lc" (Lorenz curves)
Usage
## S3 method for class 'Lc'
plot(x, general=FALSE, lwd=2, xlab="p", ylab="L(p)",
main="Lorenz curve", las=1, ...) Arguments
x |
an object of class |
general |
logical. If TRUE the generalized Lorenz curve will be plotted |
lwd, xlab, ylab, main, las, ... |
high-level |
References
B C Arnold: Majorization and the Lorenz Order: A Brief Introduction, 1987, Springer,
F A Cowell: Measurement of Inequality, 2000, in A B Atkinson / F Bourguignon (Eds): Handbook of Income Distribution, Amsterdam,
F A Cowell: Measuring Inequality, 1995 Prentice Hall/Harvester Wheatshef.
See Also
Examples
## Load and attach income (and metadata) set from Ilocos, Philippines
data(Ilocos)
attach(Ilocos)
## extract and rescale income for the provinces "Pangasinan" und "La Union"
income.p <- income[province=="Pangasinan"]/10000
income.u <- income[province=="La Union"]/10000
## compute the Lorenz curves
Lc.p <- Lc(income.p)
Lc.u <- Lc(income.u)
## plot both Lorenz curves
plot(Lc.p)
lines(Lc.u, col=2)
Plot Theoretical Lorenz Curves
Description
Plotting method for objects of class "theorLc" (theoretical Lorenz
curves)
Usage
## S3 method for class 'theorLc'
plot(x, parameter=NULL, xlab="p", ylab="L(p)", lwd=2, las=1, ...)
Arguments
x |
an object of class |
parameter |
vector containing parameters of the distributions. If |
xlab, ylab, lwd, las, ... |
high-level |
References
C Dagum: Income Distribution Models, 1983, in: Johnson / Kotz (Eds): Encyclopedia of Statistical Sciences Vol.4, 27-34.
J B McDonald: Some generalized functions for the size distribution of income, 1984, Econometrica 52, 647-664.
See Also
Examples
# income distribution of the USA in 1968 (in 10 classes)
# x vector of class means, n vector of class frequencies
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
n <- c(482, 825, 722, 690, 661, 760, 745, 2140, 1911, 1024)
# compute minimal Lorenz curve (= no inequality in each group)
Lc.min <- Lc(x, n=n)
# compute maximal Lorenz curve (limits of Mehran)
Lc.max <- Lc.mehran(x,n)
# plot both Lorenz curves in one plot
plot(Lc.min)
lines(Lc.max, col=4)
# add the theoretic Lorenz curve of a Lognormal-distribution with variance 0.78
lines(Lc.lognorm, parameter=0.78)
# add the theoretic Lorenz curve of a Dagum-distribution
lines(Lc.dagum, parameter=c(3.4,2.6))
Poverty Measures
Description
computes the poverty of an (income) vector according to the specified poverty measure
Usage
pov(x, k, parameter = NULL, type = c("Watts", "Sen", "SST", "Foster"), na.rm = TRUE)
Watts(x, k, na.rm = TRUE)
Sen(x, k, na.rm = TRUE)
SST(x, k, na.rm = TRUE)
Foster(x, k, parameter = 1, na.rm = TRUE)
Arguments
x |
a vector containing at least non-negative elements |
k |
a constant giving the absolute poverty line |
parameter |
parameter of the poverty measure (if set to |
type |
character string giving the measure used to compute poverty coefficient must be one of the strings in the default argument. Defaults to "Watts". |
na.rm |
logical. Should missing values ( |
Details
pov is just a wrapper for the poverty measures of
Watts, Sen, SST, and Foster
(Foster / Greer / Thorbecke). If parameter is set to NULL the
default from the respective function is used.
Foster gives for parameter 1 the headcount ratio and for
parameter 2 the poverty gap ratio.
Value
the value of the poverty measure
References
Foster, J. E. (1984). On Economic Poverty: A Survey of Aggregate Measures. Advances in Econometrics, 3, 215–251.
Shorroks, A. F. (1995). Revisiting the Sen Poverty Index. Econometrica, 63(5), 1225–1230.
Zheng, B. (1997). Aggregate Poverty Measures. Journal of Economic Surveys, 11, 123–162.
See Also
Examples
# generate vectors (of incomes)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
y <- c(841, 2063, 2445, 3438, 4437, 5401, 6392, 8304, 11304, 21961)
# compute Watts index with poverty line 2000
pov(x, 2000)
pov(y, 2000)
# compute headcount ratio with poverty line 2000
pov(x, 2000, parameter=1, type="Foster")
pov(y, 2000, parameter=1, type="Foster")Theoretical Lorenz Curves
Description
Theoretical Lorenz curves of income distributions
Usage
theorLc(type=c("Singh-Maddala","Dagum","lognorm","Pareto","exponential"), parameter=0)
Lc.dagum(p, parameter=c(2,2))
Lc.singh(p, parameter=c(2,2))
Lc.pareto(p, parameter=2)
Lc.lognorm(p, parameter=1)
Lc.exp(p)
Arguments
type |
character string giving the income distribution. Must be one of the strings in the default argument (the first character is sufficient). Defaults to "Singh-Maddala". |
parameter |
vector containing parameter(s) of the distributions. |
p |
vector with elements from [0,1]. |
Details
Lc.dagum, Lc.singh, Lc.pareto, Lc.lognorm,
Lc.exp are theoretical Lorenz curves of income distributions.
They are functions of class "theorLc" with plot- and a lines-
method, so that they can be added into an existing Lorenz curve plot.
theorLc returns a function of class "theorLc", that is a
one of the above theoretical Lorenz curves with fixed parameters.
Lc.dagum is the Lorenz curve of the Dagum distribution (2
parameters), Lc.singh the one of the Singh-Maddala
distribution (2 parameters), Lc.pareto the one of the Pareto
distribution (1 parameter), Lc.lognorm the one of the Lognormal
distribution (1 parameter) and Lc.exp the Lorenz curve of the
exponential distribution (no parameter).
Value
A function of class "theorLc" or its value at p
respectively.
References
C Dagum: Income Distribution Models, 1983, in: Johnson / Kotz (Eds): Encyclopedia of Statistical Sciences Vol.4, 27-34.
J B McDonald: Some generalized functions for the size distribution of income, 1984, Econometrica 52, 647-664.
See Also
Examples
## Load and attach income (and metadata) set from Ilocos, Philippines
data(Ilocos)
attach(Ilocos)
## extract income for the province "Pangasinan"
income.p <- income[province=="Pangasinan"]
## plot empirical Lorenz curve and add theoretical Lorenz curve of
## a lognormal distribution with an estimate of the standard
## deviation parameter
Lc.p <- Lc(income.p)
plot(Lc.p)
lines(Lc.lognorm, parameter=sd(log(income.p)), col=4)
# vector of percentages
p <- (1:10)*0.1
# compute values of theoretic Lorenz curve of a Dagum-distribution
Lc.dagum(p, parameter=c(3.4,2.6))
# or
mydagum <- theorLc(type="Dagum", parameter=c(3.4,2.6))
mydagum(p)