Lc: Lorenz Curve

In AndriSignorell/DescTools: Tools for Descriptive Statistics

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, pch = NA, ...)

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

## S3 method for class 'Lc'
lines(x, general = FALSE, lwd = 2, conf.level = NA, args.cband = NULL, ...)

## S3 method for class 'Lc'
predict(object, newdata, conf.level=NA, general=FALSE, n=1000, ...)

Arguments

x	a vector containing non-negative elements, or a Lc-object for plot and lines.
n	a vector of frequencies, must be same length as x.
na.rm	logical. Should missing values be removed? Defaults to FALSE.
general	logical. If TRUE the empirical Lorenz curve will be plotted.
col	color of the curve
lwd	the linewidth of the curve
lty	the linetype of the curve
type	type of the plot, default is line ("l").
xlab, ylab	label of the x-, resp. y-axis.
pch	the point character (default is NA, meaning no points will be drawn)
main	main title of the plot.
las	las of the axis.
formula	a formula of the form lhs ~ rhs where lhs gives the data values and rhs the corresponding groups.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").
conf.level	confidence level for the bootstrap confidence interval. Set this to NA, if no confidence band should be plotted. Default is NA.
args.cband	list of arguments for the confidence band, such as color or border (see DrawBand).
object	object of class inheriting from "Lc"
newdata	an optional vector of percentages p for which to predict. If omitted, the original values of the object are used.
...	further argument to be passed to methods.

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:

p	vector of percentages
L	vector with values of the ordinary Lorenz curve
L.general	vector with values of the generalized Lorenz curve
x	the original x values (needed for computing confidence intervals)
n	the original n values

Note

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

Author(s)

Achim Zeileis <[e-mail]>, extensions Andri Signorell <[e-mail]>

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 (%)"
)
# add confidence band
lines(Lc(fl, n), conf.level=0.95,
      args.cband=list(col=SetAlpha(DescToolsOptions("col")[2], 0.3)))

Gini(fl, n)

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

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

# and for the inverse question use approx
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)

AndriSignorell/DescTools documentation built on Dec. 1, 2024, 5:13 a.m.