Lc: 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, ...)
# S3 method for default
Lc(x, n = rep(1, length(x)), na.rm = FALSE, ...)
# S3 method for formula
Lc(formula, data, subset, na.action, ...)
# S3 method for Lc
plot(x, general = FALSE, lwd = 2, type = "l", xlab = "p", ylab = "L(p)",
     main = "Lorenz curve", las = 1, pch = NA, ...)
# S3 method for Lclist
plot(x, col = 1, lwd = 2, lty = 1, main = "Lorenz curve",
     xlab = "p", ylab = "L(p)", ...)
# S3 method for Lc
lines(x, general = FALSE, lwd = 2, conf.level = NA, args.cband = NULL, ...)
# S3 method for Lc
predict(object, newdata, conf.level=NA, general=FALSE, n=1000, ...)

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

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.

Author

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

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).

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.

Examples

Run this code

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)

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