lrm
Logistic Regression Model
Fit binary and proportional odds ordinal
logistic regression models using maximum likelihood estimation or
penalized maximum likelihood estimation. See cr.setup
for how to
fit forward continuation ratio models with lrm
.
Usage
lrm(formula, data, subset, na.action=na.delete, method="lrm.fit", model=FALSE, x=FALSE, y=FALSE, linear.predictors=TRUE, se.fit=FALSE, penalty=0, penalty.matrix, tol=1e7, strata.penalty=0, var.penalty=c('simple','sandwich'), weights, normwt, scale=FALSE, ...)
"print"(x, digits=4, strata.coefs=FALSE, coefs=TRUE, latex=FALSE, title='Logistic Regression Model', ...)
Arguments
 formula

a formula object. An
offset
term can be included. The offset causes fitting of a model such as $logit(Y=1) = X\beta + W$, where $W$ is the offset variable having no estimated coefficient. The response variable can be any data type;lrm
converts it in alphabetic or numeric order to an S factor variable and recodes it 0,1,2,... internally.  data
 data frame to use. Default is the current frame.
 subset
 logical expression or vector of subscripts defining a subset of observations to analyze
 na.action

function to handle
NA
s in the data. Default isna.delete
, which deletes any observation having response or predictor missing, while preserving the attributes of the predictors and maintaining frequencies of deletions due to each variable in the model. This is usually specified usingoptions(na.action="na.delete")
.  method

name of fitting function. Only allowable choice at present is
lrm.fit
.  model
 causes the model frame to be returned in the fit object
 x

causes the expanded design matrix (with missings excluded)
to be returned under the name
x
. Forprint
, an object created bylrm
.  y

causes the response variable (with missings excluded) to be returned
under the name
y
.  linear.predictors

causes the predicted X beta (with missings excluded) to be returned
under the name
linear.predictors
. When the response variable has more than two levels, the first intercept is used.  se.fit

causes the standard errors of the fitted values to be returned under
the name
se.fit
.  penalty

The penalty factor subtracted from the log likelihood is
$0.5 \beta' P \beta$, where $\beta$ is the vector of regression
coefficients other than intercept(s), and $P$ is
penalty factors * penalty.matrix
andpenalty.matrix
is defined below. The default ispenalty=0
implying that ordinary unpenalized maximum likelihood estimation is used. Ifpenalty
is a scalar, it is assumed to be a penalty factor that applies to all nonintercept parameters in the model. Alternatively, specify a list to penalize different types of model terms by differing amounts. The elements in this list are namedsimple, nonlinear, interaction
andnonlinear.interaction
. If you omit elements on the right of this series, values are inherited from elements on the left. Examples:penalty=list(simple=5, nonlinear=10)
uses a penalty factor of 10 for nonlinear or interaction terms.penalty=list(simple=0, nonlinear=2, nonlinear.interaction=4)
does not penalize linear main effects, uses a penalty factor of 2 for nonlinear or interaction effects (that are not both), and 4 for nonlinear interaction effects.  penalty.matrix

specifies the symmetric penalty matrix for nonintercept terms.
The default matrix for continuous predictors has
the variance of the columns of the design matrix in its diagonal elements
so that the penalty to the log likelhood is unitless. For main effects
for categorical predictors with $c$ categories, the rows and columns of
the matrix contain a $c1 \times c1$ submatrix that is used to
compute the
sum of squares about the mean of the $c$ parameter values (setting the
parameter to zero for the reference cell) as the penalty component
for that predictor. This makes the penalty independent of the choice of
the reference cell. If you specify
penalty.matrix
, you may set the rows and columns for certain parameters to zero so as to not penalize those parameters. Depending onpenalty
, some elements ofpenalty.matrix
may be overridden automatically by setting them to zero. The penalty matrix that is used in the actual fit is $penalty \times diag(pf) \times penalty.matrix \times diag(pf)$, where $pf$ is the vector of square roots of penalty factors computed frompenalty
byPenalty.setup
inrmsMisc
. If you specifypenalty.matrix
you must specify a nonzero value ofpenalty
or no penalization will be done.  tol
 singularity criterion (see
lrm.fit
)  strata.penalty
 scalar penalty factor for the stratification
factor, for the experimental
strat
variable  var.penalty

the type of variancecovariance matrix to be stored in the
var
component of the fit when penalization is used. The default is the inverse of the penalized information matrix. Specifyvar.penalty="sandwich"
to use the sandwich estimator (see below undervar
), which limited simulation studies have shown yields variances estimates that are too low.  weights

a vector (same length as
y
) of possibly fractional case weights  normwt

set to
TRUE
to scaleweights
so they sum to the length ofy
; useful for sample surveys as opposed to the default of frequency weighting  scale
 set to
TRUE
to subtract means and divide by standard deviations of columns of the design matrix before fitting, and to backsolve for the unnormalized covariance matrix and regression coefficients. This can sometimes make the model converge for very large sample sizes where for example spline or polynomial component variables create scaling problems leading to loss of precision when accumulating sums of squares and crossproducts.  ...
 arguments that are passed to
lrm.fit
, or fromprint
, toprModFit
 digits
 number of significant digits to use
 strata.coefs
 set to
TRUE
to print the (experimental) strata coefficients  coefs
 specify
coefs=FALSE
to suppress printing the table of model coefficients, standard errors, etc. Specifycoefs=n
to print only the firstn
regression coefficients in the model.  latex
 a logical value indicating whether information should be formatted as plain text or as LaTeX markup
 title
 a character string title to be passed to
prModFit
Value

The returned fit object of
 call
 calling expression
 freq

table of frequencies for
Y
in order of increasingY
 stats

vector with the following elements: number of observations used in the
fit, maximum absolute value of first
derivative of log likelihood, model likelihood ratio
$chisquare$, d.f.,
$P$value, $c$ index (area under ROC curve), Somers' $D_{xy}$,
GoodmanKruskal $gamma$, Kendall's $taua$ rank
correlations
between predicted probabilities and observed response, the
Nagelkerke $R^2$ index, the Brier score computed with respect to
$Y >$ its lowest level, the $g$index, $gr$ (the
$g$index on the odds ratio scale), and $gp$ (the $g$index
on the probability scale using the same cutoff used for the Brier
score). Probabilities are rounded to the nearest 0.002
in the computations or rank correlation indexes.
In the case of penalized estimation, the
"Model L.R."
is computed without the penalty factor, and"d.f."
is the effective d.f. from Gray's (1992) Equation 2.9. The $P$value uses this corrected model L.R. $chisquare$ and corrected d.f. The score chisquare statistic uses first derivatives which contain penalty components.  fail

set to
TRUE
if convergence failed (andmaxiter>1
)  coefficients
 estimated parameters
 var

estimated variancecovariance matrix (inverse of information matrix).
If
penalty>0
,var
is either the inverse of the penalized information matrix (the default, ifvar.penalty="simple"
) or the sandwichtype variance  covariance matrix estimate (Gray Eq. 2.6) ifvar.penalty="sandwich"
. For the latter case the simple informationmatrix  based variance matrix is returned under the namevar.from.info.matrix
.  effective.df.diagonal

is returned if
penalty>0
. It is the vector whose sum is the effective d.f. of the model (counting intercept terms).  u
 vector of first derivatives of loglikelihood
 deviance
 2 log likelihoods (counting penalty components) When an offset variable is present, three deviances are computed: for intercept(s) only, for intercepts+offset, and for intercepts+offset+predictors. When there is no offset variable, the vector contains deviances for the intercept(s)only model and the model with intercept(s) and predictors.
 est

vector of column numbers of
X
fitted (intercepts are not counted)  non.slopes
 number of intercepts in model
 penalty
 see above
 penalty.matrix
 the penalty matrix actually used in the estimation
lrm
contains the following components
in addition to the ones mentioned under the optional arguments.References
Le Cessie S, Van Houwelingen JC: Ridge estimators in logistic regression. Applied Statistics 41:191201, 1992.
Verweij PJM, Van Houwelingen JC: Penalized likelihood in Cox regression. Stat in Med 13:24272436, 1994.
Gray RJ: Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis. JASA 87:942951, 1992.
Shao J: Linear model selection by crossvalidation. JASA 88:486494, 1993.
Verweij PJM, Van Houwelingen JC: Crossvalidation in survival analysis. Stat in Med 12:23052314, 1993.
Harrell FE: Model uncertainty, penalization, and parsimony. ISCB Presentation on UVa Web page, 1998.
See Also
lrm.fit
, predict.lrm
,
rms.trans
, rms
, glm
,
latex.lrm
,
residuals.lrm
, na.delete
,
na.detail.response
,
pentrace
, rmsMisc
, vif
,
cr.setup
, predab.resample
,
validate.lrm
, calibrate
,
Mean.lrm
, gIndex
, prModFit
Examples
#Fit a logistic model containing predictors age, blood.pressure, sex
#and cholesterol, with age fitted with a smooth 5knot restricted cubic
#spline function and a different shape of the age relationship for males
#and females. As an intermediate step, predict mean cholesterol from
#age using a proportional odds ordinal logistic model
#
n < 1000 # define sample size
set.seed(17) # so can reproduce the results
age < rnorm(n, 50, 10)
blood.pressure < rnorm(n, 120, 15)
cholesterol < rnorm(n, 200, 25)
sex < factor(sample(c('female','male'), n,TRUE))
label(age) < 'Age' # label is in Hmisc
label(cholesterol) < 'Total Cholesterol'
label(blood.pressure) < 'Systolic Blood Pressure'
label(sex) < 'Sex'
units(cholesterol) < 'mg/dl' # uses units.default in Hmisc
units(blood.pressure) < 'mmHg'
#To use prop. odds model, avoid using a huge number of intercepts by
#grouping cholesterol into 40tiles
ch < cut2(cholesterol, g=40, levels.mean=TRUE) # use mean values in intervals
table(ch)
f < lrm(ch ~ age)
print(f, latex=TRUE, coefs=4)
m < Mean(f) # see help file for Mean.lrm
d < data.frame(age=seq(0,90,by=10))
m(predict(f, d))
# Repeat using ols
f < ols(cholesterol ~ age)
predict(f, d)
# Specify population model for log odds that Y=1
L < .4*(sex=='male') + .045*(age50) +
(log(cholesterol  10)5.2)*(2*(sex=='female') + 2*(sex=='male'))
# Simulate binary y to have Prob(y=1) = 1/[1+exp(L)]
y < ifelse(runif(n) < plogis(L), 1, 0)
cholesterol[1:3] < NA # 3 missings, at random
ddist < datadist(age, blood.pressure, cholesterol, sex)
options(datadist='ddist')
fit < lrm(y ~ blood.pressure + sex * (age + rcs(cholesterol,4)),
x=TRUE, y=TRUE)
# x=TRUE, y=TRUE allows use of resid(), which.influence below
# could define d < datadist(fit) after lrm(), but data distribution
# summary would not be stored with fit, so later uses of Predict
# or summary.rms would require access to the original dataset or
# d or specifying all variable values to summary, Predict, nomogram
anova(fit)
p < Predict(fit, age, sex)
ggplot(p) # or plot()
ggplot(Predict(fit, age=20:70, sex="male")) # need if datadist not used
print(cbind(resid(fit,"dfbetas"), resid(fit,"dffits"))[1:20,])
which.influence(fit, .3)
# latex(fit) #print nice statement of fitted model
#
#Repeat this fit using penalized MLE, penalizing complex terms
#(for nonlinear or interaction effects)
#
fitp < update(fit, penalty=list(simple=0,nonlinear=10), x=TRUE, y=TRUE)
effective.df(fitp)
# or lrm(y ~ \dots, penalty=\dots)
#Get fits for a variety of penalties and assess predictive accuracy
#in a new data set. Program efficiently so that complex design
#matrices are only created once.
set.seed(201)
x1 < rnorm(500)
x2 < rnorm(500)
x3 < sample(0:1,500,rep=TRUE)
L < x1+abs(x2)+x3
y < ifelse(runif(500)<=plogis(L), 1, 0)
new.data < data.frame(x1,x2,x3,y)[301:500,]
#
for(penlty in seq(0,.15,by=.005)) {
if(penlty==0) {
f < lrm(y ~ rcs(x1,4)+rcs(x2,6)*x3, subset=1:300, x=TRUE, y=TRUE)
# True model is linear in x1 and has no interaction
X < f$x # saves time for future runs  don't have to use rcs etc.
Y < f$y # this also deletes rows with NAs (if there were any)
penalty.matrix < diag(diag(var(X)))
Xnew < predict(f, new.data, type="x")
# expand design matrix for new data
Ynew < new.data$y
} else f < lrm.fit(X,Y, penalty.matrix=penlty*penalty.matrix)
#
cat("\nPenalty :",penlty,"\n")
pred.logit < f$coef[1] + (Xnew %*% f$coef[1])
pred < plogis(pred.logit)
C.index < somers2(pred, Ynew)["C"]
Brier < mean((predYnew)^2)
Deviance< 2*sum( Ynew*log(pred) + (1Ynew)*log(1pred) )
cat("ROC area:",format(C.index)," Brier score:",format(Brier),
" 2 Log L:",format(Deviance),"\n")
}
#penalty=0.045 gave lowest 2 Log L, Brier, ROC in test sample for S+
#
#Use bootstrap validation to estimate predictive accuracy of
#logistic models with various penalties
#To see how noisy crossvalidation estimates can be, change the
#validate(f, \dots) to validate(f, method="cross", B=10) for example.
#You will see tremendous variation in accuracy with minute changes in
#the penalty. This comes from the error inherent in using 10fold
#cross validation but also because we are not fixing the splits.
#20fold cross validation was even worse for some
#indexes because of the small test sample size. Stability would be
#obtained by using the same sample splits for all penalty values
#(see above), but then we wouldn't be sure that the choice of the
#best penalty is not specific to how the sample was split. This
#problem is addressed in the last example.
#
penalties < seq(0,.7,length=3) # really use by=.02
index < matrix(NA, nrow=length(penalties), ncol=11,
dimnames=list(format(penalties),
c("Dxy","R2","Intercept","Slope","Emax","D","U","Q","B","g","gp")))
i < 0
for(penlty in penalties)
{
cat(penlty, "")
i < i+1
if(penlty==0)
{
f < lrm(y ~ rcs(x1,4)+rcs(x2,6)*x3, x=TRUE, y=TRUE) # fit whole sample
X < f$x
Y < f$y
penalty.matrix < diag(diag(var(X))) # save time  only do once
}
else
f < lrm(Y ~ X, penalty=penlty,
penalty.matrix=penalty.matrix, x=TRUE,y=TRUE)
val < validate(f, method="boot", B=20) # use larger B in practice
index[i,] < val[,"index.corrected"]
}
par(mfrow=c(3,3))
for(i in 1:9)
{
plot(penalties, index[,i],
xlab="Penalty", ylab=dimnames(index)[[2]][i])
lines(lowess(penalties, index[,i]))
}
options(datadist=NULL)
# Example of weighted analysis
x < 1:5
y < c(0,1,0,1,0)
reps < c(1,2,3,2,1)
lrm(y ~ x, weights=reps)
x < rep(x, reps)
y < rep(y, reps)
lrm(y ~ x) # same as above
#
#Study performance of a modified AIC which uses the effective d.f.
#See Verweij and Van Houwelingen (1994) Eq. (6). Here AIC=chisq2*df.
#Also try as effective d.f. equation (4) of the previous reference.
#Also study performance of Shao's crossvalidation technique (which was
#designed to pick the "right" set of variables, and uses a much smaller
#training sample than most methods). Compare crossvalidated deviance
#vs. penalty to the gold standard accuracy on a 7500 observation dataset.
#Note that if you only want to get AIC or Schwarz Bayesian information
#criterion, all you need is to invoke the pentrace function.
#NOTE: the effective.df( ) function is used in practice
#
## Not run:
# for(seed in c(339,777,22,111,3)){
# # study performance for several datasets
# set.seed(seed)
# n < 175; p < 8
# X < matrix(rnorm(n*p), ncol=p) # p normal(0,1) predictors
# Coef < c(.1,.2,.3,.4,.5,.6,.65,.7) # true population coefficients
# L < X %*% Coef # intercept is zero
# Y < ifelse(runif(n)<=plogis(L), 1, 0)
# pm < diag(diag(var(X)))
# #Generate a large validation sample to use as a gold standard
# n.val < 7500
# X.val < matrix(rnorm(n.val*p), ncol=p)
# L.val < X.val %*% Coef
# Y.val < ifelse(runif(n.val)<=plogis(L.val), 1, 0)
# #
# Penalty < seq(0,30,by=1)
# reps < length(Penalty)
# effective.df < effective.df2 < aic < aic2 < deviance.val <
# Lpenalty < single(reps)
# n.t < round(n^.75)
# ncv < c(10,20,30,40) # try various no. of reps in crossval.
# deviance < matrix(NA,nrow=reps,ncol=length(ncv))
# #If model were complex, could have started things off by getting X, Y
# #penalty.matrix from an initial lrm fit to save time
# #
# for(i in 1:reps) {
# pen < Penalty[i]
# cat(format(pen),"")
# f.full < lrm.fit(X, Y, penalty.matrix=pen*pm)
# Lpenalty[i] < pen* t(f.full$coef[1]) %*% pm %*% f.full$coef[1]
# f.full.nopenalty < lrm.fit(X, Y, initial=f.full$coef, maxit=1)
# info.matrix.unpenalized < solve(f.full.nopenalty$var)
# effective.df[i] < sum(diag(info.matrix.unpenalized %*% f.full$var))  1
# lrchisq < f.full.nopenalty$stats["Model L.R."]
# # lrm does all this penalty adjustment automatically (for var, d.f.,
# # chisquare)
# aic[i] < lrchisq  2*effective.df[i]
# #
# pred < plogis(f.full$linear.predictors)
# score.matrix < cbind(1,X) * (Y  pred)
# sum.u.uprime < t(score.matrix) %*% score.matrix
# effective.df2[i] < sum(diag(f.full$var %*% sum.u.uprime))
# aic2[i] < lrchisq  2*effective.df2[i]
# #
# #Shao suggested averaging 2*n crossvalidations, but let's do only 40
# #and stop along the way to see if fewer is OK
# dev < 0
# for(j in 1:max(ncv)) {
# s < sample(1:n, n.t)
# cof < lrm.fit(X[s,],Y[s],
# penalty.matrix=pen*pm)$coef
# pred < cof[1] + (X[s,] %*% cof[1])
# dev < dev 2*sum(Y[s]*pred + log(1plogis(pred)))
# for(k in 1:length(ncv)) if(j==ncv[k]) deviance[i,k] < dev/j
# }
# #
# pred.val < f.full$coef[1] + (X.val %*% f.full$coef[1])
# prob.val < plogis(pred.val)
# deviance.val[i] < 2*sum(Y.val*pred.val + log(1prob.val))
# }
# postscript(hor=TRUE) # along with graphics.off() below, allow plots
# par(mfrow=c(2,4)) # to be printed as they are finished
# plot(Penalty, effective.df, type="l")
# lines(Penalty, effective.df2, lty=2)
# plot(Penalty, Lpenalty, type="l")
# title("Penalty on 2 log L")
# plot(Penalty, aic, type="l")
# lines(Penalty, aic2, lty=2)
# for(k in 1:length(ncv)) {
# plot(Penalty, deviance[,k], ylab="deviance")
# title(paste(ncv[k],"reps"))
# lines(supsmu(Penalty, deviance[,k]))
# }
# plot(Penalty, deviance.val, type="l")
# title("Gold Standard (n=7500)")
# title(sub=format(seed),adj=1,cex=.5)
# graphics.off()
# }
# ## End(Not run)
#The results showed that to obtain a clear picture of the penalty
#accuracy relationship one needs 30 or 40 reps in the crossvalidation.
#For 4 of 5 samples, though, the super smoother was able to detect
#an accurate penalty giving the best (lowest) deviance using 10fold
#crossvalidation. Crossvalidation would have worked better had
#the same splits been used for all penalties.
#The AIC methods worked just as well and are much quicker to compute.
#The first AIC based on the effective d.f. in Gray's Eq. 2.9
#(Verweij and Van Houwelingen (1994) Eq. 5 (note typo)) worked best.