# gIndex

##### 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`

.

- Keywords
- robust, univar, predictive accuracy

##### 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 gIndex
print(x, digits=4, abbrev=FALSE,
vnames=c("names","labels"), …)

# S3 method for 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.

##### See Also

##### Examples

```
# NOT RUN {
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)
f
anova(f)
z <- list()
for(type in c('terms','cterms','ccterms'))
{
zc <- predict(f, type=type)
cat('type:', type, '\n')
print(zc)
z[[type]] <- zc
}
zc <- z$cterms
GiniMd(zc[, 1])
GiniMd(zc[, 2])
GiniMd(zc[, 3])
GiniMd(f$linear.predictors)
g <- gIndex(f)
g
g['Total',]
gIndex(f, partials=FALSE)
gIndex(f, type='cterms')
gIndex(f, type='terms')
y <- y > .8
f <- lrm(y ~ x * w * u, x=TRUE, y=TRUE)
gIndex(f, fun=plogis, funlabel='Prob[y=1]')
# 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)
gIndex(f, type='cterms')
k <- coef(f)
b1 <- k[2]; b2 <- k[3]; b3 <- k[4]
n <- nrow(X)
( (a+b)*c*abs(b2) + (a+b)*d*abs(b2+b3) + c*d*abs(b3))/(n*(n-1)/2 )
# 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)
f
gIndex(f, type='cterms')
k <- coef(f)
b1 <- k[2]; b2 <- k[3]; b3 <- k[4]
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))
( 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))
# }
# NOT RUN {
# 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)
# }
# NOT RUN {
options(datadist=NULL)
# }
```

*Documentation reproduced from package rms, version 6.1-0, License: GPL (>= 2)*