Hmisc (version 5.1-3)

GiniMd: Gini's Mean Difference

Description

GiniMD computes Gini's mean difference on a numeric vector. This index is defined as the mean absolute difference between any two distinct elements of a vector. For a Bernoulli (binary) variable with proportion of ones equal to \(p\) and sample size \(n\), Gini's mean difference is \(2\frac{n}{n-1}p(1-p)\). For a trinomial variable (e.g., predicted values for a 3-level categorical predictor using two dummy variables) having (predicted) values \(A, B, C\) with corresponding proportions \(a, b, c\), Gini's mean difference is \(2\frac{n}{n-1}[ab|A-B|+ac|A-C|+bc|B-C|]\)

Usage

GiniMd(x, na.rm=FALSE)

Value

a scalar numeric

Arguments

x

a numeric vector (for GiniMd)

na.rm

set to TRUE if you suspect there may be NAs in x; these will then be removed. Otherwise an error will result.

Author

Frank Harrell
Department of Biostatistics
Vanderbilt University

References

David HA (1968): Gini's mean difference rediscovered. Biometrika 55:573--575.

Examples

set.seed(1)
x <- rnorm(40)
# Test GiniMd against a brute-force solution
gmd <- function(x) {
  n <- length(x)
  sum(outer(x, x, function(a, b) abs(a - b))) / n / (n - 1)
  }
GiniMd(x)
gmd(x)

z <- c(rep(0,17), rep(1,6))
n <- length(z)
GiniMd(z)
2*mean(z)*(1-mean(z))*n/(n-1)

a <- 12; b <- 13; c <- 7; n <- a + b + c
A <- -.123; B <- -.707; C <- 0.523
xx <- c(rep(A, a), rep(B, b), rep(C, c))
GiniMd(xx)
2*(a*b*abs(A-B) + a*c*abs(A-C) + b*c*abs(B-C))/n/(n-1)