
lrm
or orm
Fit
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 $Y \geq$ 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.
Score residuals are not yet implemented for orm
fits when
family
is not "logistic"
.
## 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(\dots, labels, center=FALSE, ylim)
lrm
or orm
residuals
, applies to type="partial"
when pl
is not FALSE
. These are extra arguments passed to the smoothing
function. Can also be used to pass extra arguments to boxplot
for ty
type="lp1"
to get approximate leave-out-1
linear predictors, derived by subtracting the dffit
from the original
linear predictor values.type="partial"
, "score"
, and
"score.binary"
. For score residuals in an ordinal model, set
pl=TRUE
to get means and approximate 0.95 confidence bars
vs. $Y$, separately for each $X$. Altscore
or score.binary
or
range of all smoothed curves for partial
if pl=TRUE
, or
0.1 and 0.9 quantilesli.shepherd
,
partial
, score
, or score.binary
, specifies
the intercept (and the cutoff of $Y$) to use for the calculations.
Specifying kint=2
, foFALSE
to suppress curve labels when type="partial"
.
The default, TRUE
, causes labcurve
to be invoked to label
curves where they are most separated. label.curves
can be a list
containtype="partial","score","score.binary"
.plot.lrm.partial
this specifies a vector of character strings
providing labels for the list of binary fits. By default, the names of
the fit objects are used as labels. The labcurve
function is used
to label the curve withplot.lrm.partial
this causes partial residuals for every
model to have a mean of zero before smoothing and plottingtype="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.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.
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.
lrm
, orm
,
naresid
, which.influence
,
loess
, supsmu
, lowess
,
boxplot
, labcurve
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)</nPartial>
par(mfrow=c(1,1))
# 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</nBinary>
<keyword>models</keyword>
<keyword>regression</keyword>
<concept>logistic regression model</concept>
<concept>model validation</concept></nOrdinal></nSlopes>
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