Gini's Mean Difference

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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 2np(1-p)/(n-1). 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 2n[ab|A-B|+ac|A-C|+bc|B-C|]/(n-1).

Usage

1
GiniMd(x, na.rm=FALSE)

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.

Value

a scalar numeric

Author(s)

Frank Harrell
Department of Biostatistics
Vanderbilt University
f.harrell@vanderbilt.edu

References

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

Examples

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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)

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