GiniMd function

Gini's Mean Difference

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

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.

Returns

a scalar numeric

References

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

Author(s)

Frank Harrell

Department of Biostatistics

Vanderbilt University

fh@fharrell.com

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)