#Fit a logistic model containing predictors age, blood.pressure, sex
#and cholesterol, with age fitted with a smooth 5-knot 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
#
require(ggplot2)
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'
# Group cholesterol unnecessarily into 40-tiles
ch <- cut2(cholesterol, g=40, levels.mean=TRUE) # use mean values in intervals
table(ch)
f <- lrm(ch ~ age)
# options(prType='latex')
print(f) # write latex code to console if prType='latex' is in effect
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*(age-50) +
(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$coef[-1])
pred <- plogis(pred.logit)
C.index <- somers2(pred, Ynew)["C"]
Brier <- mean((pred-Ynew)^2)
Deviance<- -2*sum( Ynew*log(pred) + (1-Ynew)*log(1-pred) )
cat("ROC area:",format(C.index)," Brier score:",format(Brier),
" -2 Log L:",format(Deviance),"\n")
}
#penalty=0.045 gave lowest -2 Log L, Brier, ROC in test sample for S+
#
#Use bootstrap validation to estimate predictive accuracy of
#logistic models with various penalties
#To see how noisy cross-validation estimates can be, change the
#validate(f, \dots) to validate(f, method="cross", B=10) for example.
#You will see tremendous variation in accuracy with minute changes in
#the penalty. This comes from the error inherent in using 10-fold
#cross validation but also because we are not fixing the splits.
#20-fold cross validation was even worse for some
#indexes because of the small test sample size. Stability would be
#obtained by using the same sample splits for all penalty values
#(see above), but then we wouldn't be sure that the choice of the
#best penalty is not specific to how the sample was split. This
#problem is addressed in the last example.
#
penalties <- seq(0,.7,length=3) # really use by=.02
index <- matrix(NA, nrow=length(penalties), ncol=11,
dimnames=list(format(penalties),
c("Dxy","R2","Intercept","Slope","Emax","D","U","Q","B","g","gp")))
i <- 0
for(penlty in penalties)
{
cat(penlty, "")
i <- i+1
if(penlty==0)
{
f <- lrm(y ~ rcs(x1,4)+rcs(x2,6)*x3, x=TRUE, y=TRUE) # fit whole sample
X <- f$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=chisq-2*df.
#Also try as effective d.f. equation (4) of the previous reference.
#Also study performance of Shao's cross-validation technique (which was
#designed to pick the "right" set of variables, and uses a much smaller
#training sample than most methods). Compare cross-validated 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
#
if (FALSE) {
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 cross-val.
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$stats["Model L.R."]
# lrm does all this penalty adjustment automatically (for var, d.f.,
# chi-square)
aic[i] <- lrchisq - 2*effective.df[i]
#
pred <- plogis(f.full$linear.predictors)
score.matrix <- cbind(1,X) * (Y - pred)
sum.u.uprime <- t(score.matrix) %*% score.matrix
effective.df2[i] <- sum(diag(f.full$var %*% sum.u.uprime))
aic2[i] <- lrchisq - 2*effective.df2[i]
#
#Shao suggested averaging 2*n cross-validations, but let's do only 40
#and stop along the way to see if fewer is OK
dev <- 0
for(j in 1:max(ncv)) {
s <- sample(1:n, n.t)
cof <- lrm.fit(X[s,],Y[s],
penalty.matrix=pen*pm)$coef
pred <- cof[1] + (X[-s,] %*% cof[-1])
dev <- dev -2*sum(Y[-s]*pred + log(1-plogis(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(1-prob.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()
}
}
#The results showed that to obtain a clear picture of the penalty-
#accuracy relationship one needs 30 or 40 reps in the cross-validation.
#For 4 of 5 samples, though, the super smoother was able to detect
#an accurate penalty giving the best (lowest) deviance using 10-fold
#cross-validation. Cross-validation 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.
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