The val.prob function is useful for validating
predicted probabilities against binary events.
Given a set of predicted probabilities p or predicted log odds
logit, and a vector of binary outcomes y that were not
used in developing the predictions p or logit,
val.prob computes the following indexes and statistics: Somers'
\(D_{xy}\) rank correlation between p and y
[\(2(C-.5)\), \(C\)=ROC area], Nagelkerke-Cox-Snell-Maddala-Magee
R-squared index, Discrimination index D [ (Logistic model
L.R. \(\chi^2\) - 1)/n], L.R. \(\chi^2\),
its \(P\)-value, Unreliability index \(U\), \(\chi^2\)
with 2 d.f. for testing unreliability (H0: intercept=0, slope=1), its
\(P\)-value, the quality index \(Q\), Brier score (average
squared difference in p and y), Intercept, and
Slope, \(E_{max}\)=maximum absolute difference in predicted
and loess-calibrated probabilities, Eavg, the average in same,
E90, the 0.9 quantile of same, the Spiegelhalter \(Z\)-test for
calibration accuracy, and its two-tailed \(P\)-value. If
pl=TRUE, plots fitted logistic
calibration curve and optionally a smooth nonparametric fit using
lowess(p,y,iter=0) and grouped proportions vs. mean predicted
probability in group. If the predicted probabilities or logits are
constant, the statistics are returned and no plot is made.
Eavg, Emax, E90 were from linear logistic calibration before
rms 4.5-1.
When group is present, different statistics are computed,
different graphs are made, and the object returned by val.prob is
different. group specifies a stratification variable.
Validations are done separately by levels of group and overall. A
print method prints summary statistics and several quantiles of
predicted probabilities, and a plot method plots calibration
curves with summary statistics superimposed, along with selected
quantiles of the predicted probabilities (shown as tick marks on
calibration curves). Only the lowess calibration curve is
estimated. The statistics computed are the average predicted
probability, the observed proportion of events, a 1 d.f. chi-square
statistic for testing for overall mis-calibration (i.e., a test of the
observed vs. the overall average predicted probability of the event)
(ChiSq), and a 2 d.f. chi-square statistic for testing
simultaneously that the intercept of a linear logistic calibration curve
is zero and the slope is one (ChiSq2), average absolute
calibration error (average absolute difference between the
lowess-estimated calibration curve and the line of identity,
labeled Eavg), Eavg divided by the difference between the
0.95 and 0.05 quantiles of predictive probabilities (Eavg/P90), a
"median odds ratio", i.e., the anti-log of the median absolute
difference between predicted and calibrated predicted log odds of the
event (Med OR), the C-index (ROC area), the Brier quadratic error
score (B), a chi-square test of goodness of fit based on the
Brier score (B ChiSq), and the Brier score computed on calibrated rather than raw
predicted probabilities (B cal). The first chi-square test is a
test of overall calibration accuracy ("calibration in the large"), and
the second will also detect errors such as slope shrinkage caused by
overfitting or regression to the mean. See Cox (1970) for both of these
score tests. The goodness of fit test based on the (uncalibrated) Brier
score is due to Hilden, Habbema, and Bjerregaard (1978) and is discussed
in Spiegelhalter (1986). When group is present you can also
specify sampling weights (usually frequencies), to obtained
weighted calibration curves.
To get the behavior that results from a grouping variable being present
without having a grouping variable, use group=TRUE. In the
plot method, calibration curves are drawn and labeled by default
where they are maximally separated using the labcurve function.
The following parameters do not apply when group is present:
pl, smooth, logistic.cal, m, g,
cuts, emax.lim, legendloc, riskdist,
mkh, connect.group, connect.smooth. The following
parameters apply to the plot method but not to val.prob:
xlab, ylab, lim, statloc, cex.
val.prob(p, y, logit, group, weights=rep(1,length(y)), normwt=FALSE,
pl=TRUE, smooth=TRUE, logistic.cal=TRUE,
xlab="Predicted Probability", ylab="Actual Probability",
lim=c(0, 1), m, g, cuts, emax.lim=c(0,1),
legendloc=lim[1] + c(0.55 * diff(lim), 0.27 * diff(lim)),
statloc=c(0,0.99), riskdist=c("predicted", "calibrated"),
cex=.7, mkh=.02,
connect.group=FALSE, connect.smooth=TRUE, g.group=4,
evaluate=100, nmin=0)# S3 method for val.prob
print(x, ...)
# S3 method for val.prob
plot(x, xlab="Predicted Probability",
ylab="Actual Probability",
lim=c(0,1), statloc=lim, stats=1:12, cex=.5,
lwd.overall=4, quantiles=c(.05,.95), flag, ...)
val.prob without group returns a vector with the following named
elements: Dxy, R2, D, D:Chi-sq, D:p,
U, U:Chi-sq, U:p, Q, Brier,
Intercept, Slope, S:z, S:p, Emax.
When group is present val.prob returns an object of class
val.prob containing a list with summary statistics and calibration
curves for all the strata plus "Overall".
predicted probability
vector of binary outcomes
predicted log odds of outcome. Specify either p or logit.
a grouping variable. If numeric this variable is grouped into
g.group quantile groups (default is quartiles). Set group=TRUE to
use the group algorithm but with a single stratum for val.prob.
an optional numeric vector of per-observation weights (usually frequencies),
used only if group is given.
set to TRUE to make weights sum to the number of non-missing observations.
TRUE to plot calibration curves and optionally statistics
plot smooth fit to (p,y) using lowess(p,y,iter=0)
plot linear logistic calibration fit to (p,y)
x-axis label, default is "Predicted Probability" for val.prob.
y-axis label, default is "Actual Probability" for
val.prob.
limits for both x and y axes
If grouped proportions are desired, average no. observations per group
If grouped proportions are desired, number of quantile groups
If grouped proportions are desired, actual cut points for constructing
intervals, e.g. c(0,.1,.8,.9,1) or seq(0,1,by=.2)
Vector containing lowest and highest predicted probability over which to
compute Emax.
If pl=TRUE, list with components x,y or vector c(x,y) for upper left corner
of legend for curves and points. Default is c(.55, .27) scaled to
lim. Use locator(1) to use the mouse, FALSE to suppress legend.
\(D_{xy}\), \(C\), \(R^2\), \(D\), \(U\), \(Q\), Brier score, Intercept, Slope, and \(E_{max}\)
will be added to plot, using
statloc as the upper left corner of a box (default is c(0,.9)).
You can specify a list or a vector. Use locator(1)
for the mouse, FALSE to suppress statistics.
This is plotted after the curve legends.
Use "calibrated" to plot the relative frequency distribution of
calibrated probabilities after dividing into 101 bins from lim[1] to
lim[2].
Set to "predicted" (the default as of rms 4.5-1) to use raw assigned risk, FALSE to omit risk distribution.
Values are scaled so that highest bar is 0.15*(lim[2]-lim[1]).
Character size for legend or for table of statistics when group is given
Size of symbols for legend. Default is 0.02 (see par()).
Defaults to FALSE to only represent group fractions as triangles.
Set to TRUE to also connect with a solid line.
Defaults to TRUE to draw smoothed estimates using a dashed line.
Set to FALSE to instead use dots at individual estimates.
number of quantile groups to use when group is given and variable is
numeric.
number of points at which to store the lowess-calibration curve.
Default is 100. If there are more than evaluate unique predicted
probabilities, evaluate equally-spaced quantiles of the unique
predicted probabilities, with linearly interpolated calibrated values,
are retained for plotting (and stored in the object returned by
val.prob.
applies when group is given. When nmin \(> 0\), val.prob will not
store coordinates of smoothed calibration curves in the outer tails,
where there are fewer than nmin raw observations represented in
those tails. If for example nmin=50, the plot function will only
plot the estimated calibration curve from \(a\) to \(b\), where there are
50 subjects with predicted probabilities \(< a\) and \(> b\).
nmin is ignored when computing accuracy statistics.
result of val.prob (with group in effect)
optional arguments for labcurve (through plot). Commonly used
options are col (vector of colors for the strata plus overall) and
lty. Ignored for print.
vector of column numbers of statistical indexes to write on plot
line width for plotting the overall calibration curve
a vector listing which quantiles should be indicated on each
calibration curve using tick marks. The values in quantiles can be
any number of values from the following: .01, .025, .05, .1, .25, .5, .75, .9, .95, .975, .99.
By default the 0.05 and 0.95 quantiles are indicated.
a function of the matrix of statistics (rows representing groups)
returning a vector of character strings (one value for each group, including
"Overall"). plot.val.prob will print this vector of character
values to the left of the statistics. The flag function
can refer to columns of the matrix used as input to the function by
their names given in the description above. The default function
returns "*" if either ChiSq2 or B ChiSq is
significant at the 0.01 level and " " otherwise.
Frank Harrell
Department of Biostatistics, Vanderbilt University
fh@fharrell.com
The 2 d.f. \(\chi^2\) test and Med OR exclude predicted or
calibrated predicted probabilities \(\leq 0\) to zero or \(\geq 1\),
adjusting the sample size as needed.
Harrell FE, Lee KL, Mark DB (1996): Multivariable prognostic models: Issues in developing models, evaluating assumptions and adequacy, and measuring and reducing errors. Stat in Med 15:361--387.
Harrell FE, Lee KL (1987): Using logistic calibration to assess the accuracy of probability predictions (Technical Report).
Miller ME, Hui SL, Tierney WM (1991): Validation techniques for logistic regression models. Stat in Med 10:1213--1226.
Stallard N (2009): Simple tests for the external validation of mortality prediction scores. Stat in Med 28:377--388.
Harrell FE, Lee KL (1985): A comparison of the discrimination of discriminant analysis and logistic regression under multivariate normality. In Biostatistics: Statistics in Biomedical, Public Health, and Environmental Sciences. The Bernard G. Greenberg Volume, ed. PK Sen. New York: North-Holland, p. 333--343.
Cox DR (1970): The Analysis of Binary Data, 1st edition, section 4.4. London: Methuen.
Spiegelhalter DJ (1986):Probabilistic prediction in patient management. Stat in Med 5:421--433.
Rufibach K (2010):Use of Brier score to assess binary predictions. J Clin Epi 63:938-939
Tjur T (2009):Coefficients of determination in logistic regression models-A new proposal:The coefficient of discrimination. Am Statist 63:366--372.
# Fit logistic model on 100 observations simulated from the actual
# model given by Prob(Y=1 given X1, X2, X3) = 1/(1+exp[-(-1 + 2X1)]),
# where X1 is a random uniform [0,1] variable. Hence X2 and X3 are
# irrelevant. After fitting a linear additive model in X1, X2,
# and X3, the coefficients are used to predict Prob(Y=1) on a
# separate sample of 100 observations. Note that data splitting is
# an inefficient validation method unless n > 20,000.
set.seed(1)
n <- 200
x1 <- runif(n)
x2 <- runif(n)
x3 <- runif(n)
logit <- 2*(x1-.5)
P <- 1/(1+exp(-logit))
y <- ifelse(runif(n)<=P, 1, 0)
d <- data.frame(x1,x2,x3,y)
f <- lrm(y ~ x1 + x2 + x3, subset=1:100)
pred.logit <- predict(f, d[101:200,])
phat <- 1/(1+exp(-pred.logit))
val.prob(phat, y[101:200], m=20, cex=.5) # subgroups of 20 obs.
# Validate predictions more stringently by stratifying on whether
# x1 is above or below the median
v <- val.prob(phat, y[101:200], group=x1[101:200], g.group=2)
v
plot(v)
plot(v, flag=function(stats) ifelse(
stats[,'ChiSq2'] > qchisq(.95,2) |
stats[,'B ChiSq'] > qchisq(.95,1), '*', ' ') )
# Stars rows of statistics in plot corresponding to significant
# mis-calibration at the 0.05 level instead of the default, 0.01
plot(val.prob(phat, y[101:200], group=x1[101:200], g.group=2),
col=1:3) # 3 colors (1 for overall)
# Weighted calibration curves
# plot(val.prob(pred, y, group=age, weights=freqs))
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