Residuals from an lrm or orm Fit
For a binary logistic model fit, computes the following residuals, letting P denote the predicted probability of the higher category of Y, X denote the design matrix (with a column of 1s for the intercept), and L denote the logit or linear predictors: ordinary or Li-Shepherd (Y-P), score (X (Y-P)), pearson ((Y-P)/√{P(1-P)}), deviance (for Y=0 is -√{2|\log(1-P)|}, for Y=1 is √{2|\log(P)|}, pseudo dependent variable used in influence statistics (L + (Y-P)/(P(1-P))), and partial (X_{i}β_{i} + (Y-P)/(P(1-P))).
Will compute all these residuals for an ordinal logistic model, using
as temporary binary responses dichotomizations of Y, along with
the corresponding P, the probability that Y ≥q cutoff. For
type="partial", all
possible dichotomizations are used, and for type="score", the actual
components of the first derivative of the log likelihood are used for
an ordinal model. For type="li.shepherd" the residual is
Pr(W < Y) - Pr(W > Y) where Y is the observed response and W is a
random variable from the fitted distribution.
Alternatively, specify type="score.binary"
to use binary model score residuals but for all cutpoints of Y
(plotted only, not returned). The score.binary,
partial, and perhaps score residuals are useful for
checking the proportional odds assumption.
If the option pl=TRUE is used to plot the score or
score.binary residuals, a score residual plot is made for each
column of the design (predictor) matrix, with Y cutoffs on the
x-axis and the mean +- 1.96 standard errors of the score residuals on
the y-axis. You can instead use a box plot to display these residuals,
for both score.binary and score.
Proportional odds dictates a horizontal score.binary plot. Partial
residual plots use smooth nonparametric estimates, separately for each
cutoff of Y. One examines that plot for parallelism of the curves
to check the proportional odds assumption, as well as to see if the
predictor behaves linearly.
Also computes a variety of influence statistics and the le Cessie - van Houwelingen - Copas - Hosmer unweighted sum of squares test for global goodness of fit, done separately for each cutoff of Y in the case of an ordinal model.
The plot.lrm.partial function computes partial residuals for a series
of binary logistic model fits that all used the same predictors and that
specified x=TRUE, y=TRUE. It then computes smoothed partial residual
relationships (using lowess with iter=0) and plots them separately
for each predictor, with residual plots from all model fits shown on the
same plot for that predictor.
## S3 method for class 'lrm'
residuals(object, type=c("li.shepherd","ordinary",
"score", "score.binary", "pearson", "deviance", "pseudo.dep",
"partial", "dfbeta", "dfbetas", "dffit", "dffits", "hat", "gof", "lp1"),
pl=FALSE, xlim, ylim, kint, label.curves=TRUE, which, ...)
## S3 method for class 'orm'
residuals(object, type=c("li.shepherd","ordinary",
"score", "score.binary", "pearson", "deviance", "pseudo.dep",
"partial", "dfbeta", "dfbetas", "dffit", "dffits", "hat", "gof", "lp1"),
pl=FALSE, xlim, ylim, kint, label.curves=TRUE, which, ...)
## S3 method for class 'lrm.partial'
plot(..., labels, center=FALSE, ylim)object |
object created by |
... |
for |
type |
type of residual desired. Use |
pl |
applies only to |
xlim |
plotting range for x-axis (default = whole range of predictor) |
ylim |
plotting range for y-axis (default = whole range of residuals, range of
all confidence intervals for |
kint |
for an ordinal model for residuals other than |
label.curves |
set to |
which |
a vector of integers specifying column numbers of the design matrix for
which to compute or plot residuals, for
|
labels |
for |
center |
for |
For the goodness-of-fit test, the le Cessie-van Houwelingen normal test statistic for the unweighted sum of squared errors (Brier score times n) is used. For an ordinal response variable, the test for predicting the probability that Y≥q j is done separately for all j (except the first). Note that the test statistic can have strange behavior (i.e., it is far too large) if the model has no predictive value.
For most of the values of type, you must have specified
x=TRUE, y=TRUE to lrm or orm.
There is yet no literature on interpreting score residual plots for the ordinal model. Simulations when proportional odds is satisfied have still shown a U-shaped residual plot. The series of binary model score residuals for all cutoffs of Y seems to better check the assumptions. See the examples.
The li.shepherd residual is a single value per observation on the probability scale and can be useful for examining linearity, checking for outliers, and measuring residual correlation.
a matrix (type="partial","dfbeta","dfbetas","score"),
test statistic (type="gof"), or a vector otherwise.
For partial residuals from an ordinal
model, the returned object is a 3-way array (rows of X by columns
of X by cutoffs of Y), and NAs deleted during the fit
are not re-inserted into the residuals. For score.binary, nothing
is returned.
Frank Harrell
Department of Biostatistics
Vanderbilt University
fh@fharrell.com
Landwehr, Pregibon, Shoemaker. JASA 79:61–83, 1984.
le Cessie S, van Houwelingen JC. Biometrics 47:1267–1282, 1991.
Hosmer DW, Hosmer T, Lemeshow S, le Cessie S, Lemeshow S. A comparison of goodness-of-fit tests for the logistic regression model. Stat in Med 16:965–980, 1997.
Copas JB. Applied Statistics 38:71–80, 1989.
Li C, Shepherd BE. Biometrika 99:473-480, 2012.
set.seed(1)
x1 <- runif(200, -1, 1)
x2 <- runif(200, -1, 1)
L <- x1^2 - .5 + x2
y <- ifelse(runif(200) <= plogis(L), 1, 0)
f <- lrm(y ~ x1 + x2, x=TRUE, y=TRUE)
resid(f) #add rows for NAs back to data
resid(f, "score") #also adds back rows
r <- resid(f, "partial") #for checking transformations of X's
par(mfrow=c(1,2))
for(i in 1:2) {
xx <- if(i==1)x1 else x2
plot(xx, r[,i], xlab=c('x1','x2')[i])
lines(lowess(xx,r[,i]))
}
resid(f, "partial", pl="loess") #same as last 3 lines
resid(f, "partial", pl=TRUE) #plots for all columns of X using supsmu
resid(f, "gof") #global test of goodness of fit
lp1 <- resid(f, "lp1") #approx. leave-out-1 linear predictors
-2*sum(y*lp1 + log(1-plogis(lp1))) #approx leave-out-1 deviance
#formula assumes y is binary
# Simulate data from a population proportional odds model
set.seed(1)
n <- 400
age <- rnorm(n, 50, 10)
blood.pressure <- rnorm(n, 120, 15)
L <- .05*(age-50) + .03*(blood.pressure-120)
p12 <- plogis(L) # Pr(Y>=1)
p2 <- plogis(L-1) # Pr(Y=2)
p <- cbind(1-p12, p12-p2, p2) # individual class probabilites
# Cumulative probabilities:
cp <- matrix(cumsum(t(p)) - rep(0:(n-1), rep(3,n)), byrow=TRUE, ncol=3)
# simulate multinomial with varying probs:
y <- (cp < runif(n)) %*% rep(1,3)
y <- as.vector(y)
# Thanks to Dave Krantz for this trick
f <- lrm(y ~ age + blood.pressure, x=TRUE, y=TRUE)
par(mfrow=c(2,2))
resid(f, 'score.binary', pl=TRUE) #plot score residuals
resid(f, 'partial', pl=TRUE) #plot partial residuals
resid(f, 'gof') #test GOF for each level separately
# Show use of Li-Shepherd residuals
f.wrong <- lrm(y ~ blood.pressure, x=TRUE, y=TRUE)
par(mfrow=c(2,1))
# li.shepherd residuals from model without age
plot(age, resid(f.wrong, type="li.shepherd"),
ylab="li.shepherd residual")
lines(lowess(age, resid(f.wrong, type="li.shepherd")))
# li.shepherd residuals from model including age
plot(age, resid(f, type="li.shepherd"),
ylab="li.shepherd residual")
lines(lowess(age, resid(f, type="li.shepherd")))
# Make a series of binary fits and draw 2 partial residual plots
#
f1 <- lrm(y>=1 ~ age + blood.pressure, x=TRUE, y=TRUE)
f2 <- update(f1, y==2 ~.)
par(mfrow=c(2,1))
plot.lrm.partial(f1, f2)
# Simulate data from both a proportional odds and a non-proportional
# odds population model. Check how 3 kinds of residuals detect
# non-prop. odds
set.seed(71)
n <- 400
x <- rnorm(n)
par(mfrow=c(2,3))
for(j in 1:2) { # 1: prop.odds 2: non-prop. odds
if(j==1)
L <- matrix(c(1.4,.4,-.1,-.5,-.9),
nrow=n, ncol=5, byrow=TRUE) + x / 2
else {
# Slopes and intercepts for cutoffs of 1:5 :
slopes <- c(.7,.5,.3,.3,0)
ints <- c(2.5,1.2,0,-1.2,-2.5)
L <- matrix(ints, nrow=n, ncol=5, byrow=TRUE) +
matrix(slopes, nrow=n, ncol=5, byrow=TRUE) * x
}
p <- plogis(L)
# Cell probabilities
p <- cbind(1-p[,1],p[,1]-p[,2],p[,2]-p[,3],p[,3]-p[,4],p[,4]-p[,5],p[,5])
# Cumulative probabilities from left to right
cp <- matrix(cumsum(t(p)) - rep(0:(n-1), rep(6,n)), byrow=TRUE, ncol=6)
y <- (cp < runif(n)) %*% rep(1,6)
f <- lrm(y ~ x, x=TRUE, y=TRUE)
for(cutoff in 1:5) print(lrm(y >= cutoff ~ x)$coef)
print(resid(f,'gof'))
resid(f, 'score', pl=TRUE)
# Note that full ordinal model score residuals exhibit a
# U-shaped pattern even under prop. odds
ti <- if(j==2) 'Non-Proportional Odds\nSlopes=.7 .5 .3 .3 0' else
'True Proportional Odds\nOrdinal Model Score Residuals'
title(ti)
resid(f, 'score.binary', pl=TRUE)
if(j==1) ti <- 'True Proportional Odds\nBinary Score Residuals'
title(ti)
resid(f, 'partial', pl=TRUE)
if(j==1) ti <- 'True Proportional Odds\nPartial Residuals'
title(ti)
}
par(mfrow=c(1,1))
# Shepherd-Li residuals from orm. Thanks: Qi Liu
set.seed(3)
n <- 100
x1 <- rnorm(n)
y <- x1 + rnorm(n)
g <- orm(y ~ x1, family=probit, x=TRUE, y=TRUE)
g.resid <- resid(g)
plot(x1, g.resid, cex=0.4); lines(lowess(x1, g.resid)); abline(h=0, col=2,lty=2)
set.seed(3)
n <- 100
x1 <- rnorm(n)
y <- x1 + x1^2 +rnorm(n)
# model misspecification, the square term is left out in the model
g <- orm(y ~ x1, family=probit, x=TRUE, y=TRUE)
g.resid <- resid(g)
plot(x1, g.resid, cex=0.4); lines(lowess(x1, g.resid)); abline(h=0, col=2,lty=2)
## Not run:
# Get data used in Hosmer et al. paper and reproduce their calculations
v <- Cs(id, low, age, lwt, race, smoke, ptl, ht, ui, ftv, bwt)
d <- read.table("http://www.umass.edu/statdata/statdata/data/lowbwt.dat",
skip=6, col.names=v)
d <- upData(d, race=factor(race,1:3,c('white','black','other')))
f <- lrm(low ~ age + lwt + race + smoke, data=d, x=TRUE,y=TRUE)
f
resid(f, 'gof')
# Their Table 7 Line 2 found sum of squared errors=36.91, expected
# value under H0=36.45, variance=.065, P=.071
# We got 36.90, 36.45, SD=.26055 (var=.068), P=.085
# Note that two logistic regression coefficients differed a bit
# from their Table 1
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.