Calculate Total and Partial g-indexes for an rms Fit

gIndex computes the total \(g\)-index for a model based on the vector of linear predictors, and the partial \(g\)-index for each predictor in a model. The latter is computed by summing all the terms involving each variable, weighted by their regression coefficients, then computing Gini's mean difference on this sum. For example, a regression model having age and sex and age*sex on the right hand side, with corresponding regression coefficients \(b_{1}, b_{2}, b_{3}\) will have the \(g\)-index for age computed from Gini's mean difference on the product of age \(\times (b_{1} + b_{3}w)\) where \(w\) is an indicator set to one for observations with sex not equal to the reference value. When there are nonlinear terms associated with a predictor, these terms will also be combined.

A print method is defined, and there is a plot method for displaying \(g\)-indexes using a dot chart.

These functions use Hmisc::GiniMd.

Usage

gIndex(object, partials=TRUE, type=c('ccterms', 'cterms', 'terms'),
           lplabel=if(length(object$scale) && is.character(object$scale))
           object$scale[1] else 'X*Beta',
           fun, funlabel=if(missing(fun)) character(0) else
           deparse(substitute(fun)),
           postfun=if(length(object$scale)==2) exp else NULL,
           postlabel=if(length(postfun))
           ifelse(missing(postfun),
                  if((length(object$scale) > 1) &&
                     is.character(object$scale)) object$scale[2] else
                     'Anti-log',
                     deparse(substitute(postfun))) else character(0),
           ...)

# S3 method for class 'gIndex'
print(x, digits=4, abbrev=FALSE,
 vnames=c("names","labels"), ...)

# S3 method for class 'gIndex'
plot(x, what=c('pre', 'post'),
 xlab=NULL, pch=16, rm.totals=FALSE,
sort=c('descending', 'ascending', 'none'), ...)

Arguments

object: result of an rms fitting function
partials: set to FALSE to suppress computation of partial \(g\)s
type: defaults to 'ccterms' which causes partial discrimination indexes to be computed after maximally combining all related main effects and interactions. The is usually the only way that makes sense when considering partial linear predictors. Specify type='cterms' to only combine a main effect with interactions containing it, not also with other main effects connected through interactions. Use type='terms' to separate interactions into their own effects.
lplabel: a replacement for default values such as "X*Beta" or "log odds"/
fun: an optional function to transform the linear predictors before computing the total (only) \(g\). When this is present, a new component gtrans is added to the attributes of the object resulting from gIndex.
funlabel: a character string label for fun, otherwise taken from the function name itself
postfun: a function to transform \(g\) such as exp (anti-log), which is the default for certain models such as the logistic and Cox models
postlabel: a label for postfun
...: For gIndex, passed to predict.rms. Ignored for print. Passed to dotchart2 for plot.
x: an object created by gIndex (for print or plot)
digits: causes rounding to the digits decimal place
abbrev: set to TRUE to abbreviate labels if vname="labels"
vnames: set to "labels" to print predictor labels instead of names
what: set to "post" to plot the transformed \(g\)-index if there is one (e.g., ratio scale)
xlab: \(x\)-axis label; constructed by default
pch: plotting character for point
rm.totals: set to TRUE to remove the total \(g\)-index when plotting
sort: specifies how to sort predictors by \(g\)-index; default is in descending order going down the dot chart

Details

For stratification factors in a Cox proportional hazards model, there is no contribution of variation towards computing a partial \(g\) except from terms that interact with the stratification variable.

Value

gIndex returns a matrix of class "gIndex" with auxiliary information stored as attributes, such as variable labels. GiniMd returns a scalar.

References

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

Author

Frank Harrell
Department of Biostatistics
Vanderbilt University
fh@fharrell.com

Examples

set.seed(1)
n <- 40
x <- 1:n
w <- factor(sample(c('a','b'), n, TRUE))
u <- factor(sample(c('A','B'), n, TRUE))
y <- .01*x + .2*(w=='b') + .3*(u=='B') + .2*(w=='b' & u=='B') + rnorm(n)/5
dd <- datadist(x,w,u); options(datadist='dd')
f <- ols(y ~ x*w*u, x=TRUE, y=TRUE)
#> Error in Design(X, formula = formula): dataset dd not found for options(datadist=)
f
#> Error: object 'f' not found
anova(f)
#> Error: object 'f' not found
z <- list()
for(type in c('terms','cterms','ccterms'))
  {
    zc <- predict(f, type=type)
    cat('type:', type, '\n')
    print(zc)
    z[[type]] <- zc
  }
#> Error: object 'f' not found

zc <- z$cterms
GiniMd(zc[, 1])
#> [1] NA
GiniMd(zc[, 2])
#> [1] NA
GiniMd(zc[, 3])
#> [1] NA
GiniMd(f$linear.predictors)
#> Error: object 'f' not found
g <- gIndex(f)
#> Error: object 'f' not found
g
#> Error: object 'g' not found
g['Total',]
#> Error: object 'g' not found
gIndex(f, partials=FALSE)
#> Error: object 'f' not found
gIndex(f, type='cterms')
#> Error: object 'f' not found
gIndex(f, type='terms')
#> Error: object 'f' not found

y <- y > .8
f <- lrm(y ~ x * w * u, x=TRUE, y=TRUE, reltol=1e-5)
#> Error in Design(data, formula = formula): dataset dd not found for options(datadist=)

gIndex(f, fun=plogis, funlabel='Prob[y=1]')
#> Error: object 'f' not found

# Manual calculation of combined main effect + interaction effort of
# sex in a 2x2 design with treatments A B, sexes F M,
# model -.1 + .3*(treat=='B') + .5*(sex=='M') + .4*(treat=='B' & sex=='M')

set.seed(1)
X <- expand.grid(treat=c('A','B'), sex=c('F', 'M'))
a <- 3; b <- 7; c <- 13; d <- 5
X <- rbind(X[rep(1, a),], X[rep(2, b),], X[rep(3, c),], X[rep(4, d),])
y <- with(X, -.1 + .3*(treat=='B') + .5*(sex=='M') + .4*(treat=='B' & sex=='M'))
f <- ols(y ~ treat*sex, data=X, x=TRUE)
#> Error in Design(X, formula = formula): dataset dd not found for options(datadist=)
gIndex(f, type='cterms')
#> Error: object 'f' not found
k <- coef(f)
#> Error: object 'f' not found
b1 <- k[2]; b2 <- k[3]; b3 <- k[4]
#> Error: object 'k' not found
n <- nrow(X)
( (a+b)*c*abs(b2) + (a+b)*d*abs(b2+b3) + c*d*abs(b3))/(n*(n-1)/2 )
#> Error: object 'b2' not found

# Manual calculation for combined age effect in a model with sex,
# age, and age*sex interaction

a <- 13; b <- 7
sex <- c(rep('female',a), rep('male',b))
agef <- round(runif(a, 20, 30))
agem <- round(runif(b, 20, 40))
age  <- c(agef, agem)
y <- (sex=='male') + age/10 - (sex=='male')*age/20
f <- ols(y ~ sex*age, x=TRUE)
#> Error in Design(X, formula = formula): dataset dd not found for options(datadist=)
f
#> Error: object 'f' not found
gIndex(f, type='cterms')
#> Error: object 'f' not found
k <- coef(f)
#> Error: object 'f' not found
b1 <- k[2]; b2 <- k[3]; b3 <- k[4]
#> Error: object 'k' not found
n <- a + b
sp <- function(w, z=w) sum(outer(w, z, function(u, v) abs(u-v)))

( abs(b2)*sp(agef) + abs(b2+b3)*sp(agem) + 2*sp(b2*agef, (b2+b3)*agem) ) / (n*(n-1))
#> Error: object 'b2' not found

( abs(b2)*GiniMd(agef)*a*(a-1) + abs(b2+b3)*GiniMd(agem)*b*(b-1) +
  2*sp(b2*agef, (b2+b3)*agem) ) / (n*(n-1))
#> Error: object 'b2' not found

if (FALSE) { # \dontrun{
# Compare partial and total g-indexes over many random fits
plot(NA, NA, xlim=c(0,3), ylim=c(0,3), xlab='Global',
     ylab='x1 (black)  x2 (red)  x3 (green)  x4 (blue)')
abline(a=0, b=1, col=gray(.9))
big <- integer(3)
n <- 50   # try with n=7 - see lots of exceptions esp. for interacting var
for(i in 1:100) {
   x1 <- runif(n)
   x2 <- runif(n)
   x3 <- runif(n)
   x4 <- runif(n)
   y  <- x1 + x2 + x3 + x4 + 2*runif(n)
   f <- ols(y ~ x1*x2+x3+x4, x=TRUE)
   # f <- ols(y ~ x1+x2+x3+x4, x=TRUE)   # also try this
   w <- gIndex(f)[,1]
   gt <- w['Total']
   points(gt, w['x1, x2'])
   points(gt, w['x3'], col='green')
   points(gt, w['x4'], col='blue')
   big[1] <- big[1] + (w['x1, x2'] > gt)
   big[2] <- big[2] + (w['x3'] > gt)
   big[3] <- big[3] + (w['x4'] > gt)
   }
print(big)
} # }

options(datadist=NULL)