# Lc

0th

Percentile

##### Lorenz Curve

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.

Keywords
univar
##### Usage
Lc(x, ...)
"Lc"(x, n = rep(1, length(x)), na.rm = FALSE, ...)
"Lc"(formula, data, subset, na.action, ...)
"plot"(x, general = FALSE, lwd = 2, type = "l", xlab = "p", ylab = "L(p)", main = "Lorenz curve", las = 1, ...)
"plot"(x, col = 1, lwd = 2, lty = 1, main = "Lorenz curve", xlab = "p", ylab = "L(p)", ...)
"lines"(x, general = FALSE, lwd = 2, conf.level = NA, args.cband = NULL, ...)
##### 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.
main
main title of the plot.
las
las of the axis.
p
a numeric vector with percent points, at which the Lorenz curve will be calculated.

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).

...
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: with the following components:

##### Note

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

##### 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.

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

• Lc
• Lc.default
• Lc.formula
• plot.Lc
• plot.Lclist
• lines.Lc
##### 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 (%)"
)
lines(Lc(fl, n), conf.level=0.95,
args.cband=list(col=SetAlpha(DescToolsOptions("col"), 0.3)))

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)


Documentation reproduced from package DescTools, version 0.99.19, License: GPL (>= 2)

### Community examples

DanielPare at Oct 18, 2017 DescTools v0.99.19

###Competitive Balance between NBA and NHL for the 2015-16 season r #Nb of victories per team per league NBA1516_Win <- c(73, 67, 55, 57, 53, 56, 48, 48, 48, 40, 45, 48, 44, 44, 41, 42, 41, 42, 35, 42, 33, 32, 33, 29, 30, 33, 23, 21, 17, 10) NHL1516_Win <- c(56, 50, 49, 48, 47, 46, 47, 48, 46, 45, 46, 46, 41, 41, 41, 42, 38, 35, 38, 38, 39, 38, 35, 35, 35, 35, 34, 31, 31, 29) #Lorenz Curve NBA_lc <- Lc(NBA1516_Win) NHL_lc <- Lc(NHL1516_Win) #Ploting the Lorenz Curve for the NBA 2015-16 season plot(NHL_lc, panel.first=grid(10, 10), main="NHL (red) vs NBA (blue)", xlab="Percentage of victory", ylab="Percent of area (%)", col="red" ) #Adding the Lorenz Curve for the NHL season 2015-16 lines(NBA_lc, col="blue")