Lorenz Curve

Description

Lc computes the (empirical) ordinary and generalized Lorenz curve of a vector x. Desc calculates some key figures for a Lorenz curve and produces a quick description.

Usage

Lc(x, ...)

## Default S3 method:
Lc(x, n = rep(1, length(x)), na.rm = FALSE, ...)

## S3 method for class 'formula'
Lc(formula, data, subset, na.action, ...)

## S3 method for class 'Lc'
plot(x, general = FALSE, lwd = 2, type = "l", xlab = "p", ylab = "L(p)",
     main = "Lorenz curve", las = 1, ...)

## S3 method for class 'Lclist'
plot(x, col = 1, lwd = 2, lty = 1, main = "Lorenz curve",
     xlab = "p", ylab = "L(p)", ...)

Arguments

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).

Value

A list of class "Lc" with the following components:

Note

These functions were previously published as Lc() in the ineq package and have been integrated here without logical changes.

Author(s)

Achim Zeileis <Achim.Zeileis@R-project.org>, minor changes Andri Signorell <andri@signorell.net>

References

Arnold, B. C. (1987) Majorization and the Lorenz Order: A Brief Introduction, Springer

Cowell, F. A. (2000) Measurement of Inequality in Atkinson, A. B. / Bourguignon, F. (Eds): Handbook of Income Distribution. Amsterdam.

Cowell, F. A. (1995) Measuring Inequality Harvester Wheatshef: Prentice Hall.

See Also

The original location Lc(),
inequality measures Gini(), Atkinson()

Examples

priceCarpenter <- d.pizza$price[d.pizza$driver=="Carpenter"]
priceMiller <- d.pizza$price[d.pizza$driver=="Miller"]

# compute the Lorenz curves
Lc.p <- Lc(priceCarpenter, na.rm=TRUE)
Lc.u <- Lc(priceMiller, na.rm=TRUE)
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
Atkinson(priceCarpenter, na.rm=TRUE)
Atkinson(priceMiller, na.rm=TRUE)


# 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)
plot(Lc.min)


# input of frequency tables with midpoints of classes
fl <- c(2.5,7.5,15,35,75,150)   # midpoints
n  <- c(25,13,10,5,5,2)	        # frequencies

plot(Lc(fl, n),                 # Lorenz-Curve
     panel.first=grid(10, 10),
     main="Lorenzcurve Farmers",
     xlab="Percent farmers (cumulative)",
     ylab="Percent of area (%)"
)

Gini(fl, n)

# find specific function values using appprox
x <- c(1,1,4)
lx <- Lc(x)
plot(lx)

# get interpolated function value at p = 0.55
y0 <- approx(x=lx$p, y=lx$L, xout=0.55)
abline(v=0.55, h=y0$y, lty="dotted")

# and for the inverse question
y0 <- approx(x=lx$L, y=lx$p, xout=0.6)
abline(h=0.6, v=y0$y, col="red")

text(x=0.1, y=0.65, label=expression(L^{-1}*(0.6) == 0.8), col="red")
text(x=0.65, y=0.2, label=expression(L(0.55) == 0.275))

# input of frequency tables with midpoints of classes
fl <- c(2.5,7.5,15,35,75,150)     # midpoints
n  <- c(25,13,10,5,5,2)           # frequencies

# the formula interface for Lc
lst <- Lc(count ~ cut(price, breaks=5), data=d.pizza)

plot(lst, col=1:length(lst), panel.first=grid(), lwd=2)
legend(x="topleft", legend=names(lst), fill=1:length(lst))

# Describe with Desc-function
lx <- Lc(fl, n)
Desc(lx)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(DescTools)
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/DescTools/Lc.Rd_%03d_medium.png", width=480, height=480)
> ### Name: Lc
> ### Title: Lorenz Curve
> ### Aliases: Lc Lc.default Lc.formula plot.Lc plot.Lclist lines.Lc
> ### Keywords: univar
> 
> ### ** Examples
> 
> priceCarpenter <- d.pizza$price[d.pizza$driver=="Carpenter"]
> priceMiller <- d.pizza$price[d.pizza$driver=="Miller"]
> 
> # compute the Lorenz curves
> Lc.p <- Lc(priceCarpenter, na.rm=TRUE)
> Lc.u <- Lc(priceMiller, na.rm=TRUE)
> 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
> Atkinson(priceCarpenter, na.rm=TRUE)
[1] 0.05286775
> Atkinson(priceMiller, na.rm=TRUE)
[1] 0.04893621
> 
> 
> # 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)
> plot(Lc.min)
> 
> 
> # input of frequency tables with midpoints of classes
> fl <- c(2.5,7.5,15,35,75,150)   # midpoints
> n  <- c(25,13,10,5,5,2)	        # frequencies
> 
> plot(Lc(fl, n),                 # Lorenz-Curve
+      panel.first=grid(10, 10),
+      main="Lorenzcurve Farmers",
+      xlab="Percent farmers (cumulative)",
+      ylab="Percent of area (%)"
+ )
> 
> Gini(fl, n)
[1] 0.6678112
> 
> # find specific function values using appprox
> x <- c(1,1,4)
> lx <- Lc(x)
> plot(lx)
> 
> # get interpolated function value at p = 0.55
> y0 <- approx(x=lx$p, y=lx$L, xout=0.55)
> abline(v=0.55, h=y0$y, lty="dotted")
> 
> # and for the inverse question
> y0 <- approx(x=lx$L, y=lx$p, xout=0.6)
> abline(h=0.6, v=y0$y, col="red")
> 
> text(x=0.1, y=0.65, label=expression(L^{-1}*(0.6) == 0.8), col="red")
> text(x=0.65, y=0.2, label=expression(L(0.55) == 0.275))
> 
> # input of frequency tables with midpoints of classes
> fl <- c(2.5,7.5,15,35,75,150)     # midpoints
> n  <- c(25,13,10,5,5,2)           # frequencies
> 
> # the formula interface for Lc
> lst <- Lc(count ~ cut(price, breaks=5), data=d.pizza)
> 
> plot(lst, col=1:length(lst), panel.first=grid(), lwd=2)
> legend(x="topleft", legend=names(lst), fill=1:length(lst))
> 
> # Describe with Desc-function
> lx <- Lc(fl, n)
> Desc(lx)

Sorry, don't know how to handle class(es) list (x)!

------------------------------------------------------------------------------ 
lx (list)

$xname
[1] "lx"

$label
NULL

$class
[1] "list"

$classlabel
[1] "list"

$length
[1] 4

$n
[1] 4

$NAs
[1] 0

$main
[1] "lx (list)"

> 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>