```
require(ggplot2)
set.seed(1)
n <- 100
y <- round(runif(n), 2)
x1 <- sample(c(-1,0,1), n, TRUE)
x2 <- sample(c(-1,0,1), n, TRUE)
f <- lrm(y ~ x1 + x2, eps=1e-5)
g <- orm(y ~ x1 + x2, eps=1e-5)
max(abs(coef(g) - coef(f)))
w <- vcov(g, intercepts='all') / vcov(f) - 1
max(abs(w))
set.seed(1)
n <- 300
x1 <- c(rep(0,150), rep(1,150))
y <- rnorm(n) + 3*x1
g <- orm(y ~ x1)
g
k <- coef(g)
i <- num.intercepts(g)
h <- orm(y ~ x1, family=probit)
ll <- orm(y ~ x1, family=loglog)
cll <- orm(y ~ x1, family=cloglog)
cau <- orm(y ~ x1, family=cauchit)
x <- 1:i
z <- list(logistic=list(x=x, y=coef(g)[1:i]),
probit =list(x=x, y=coef(h)[1:i]),
loglog =list(x=x, y=coef(ll)[1:i]),
cloglog =list(x=x, y=coef(cll)[1:i]))
labcurve(z, pl=TRUE, col=1:4, ylab='Intercept')
tapply(y, x1, mean)
m <- Mean(g)
m(w <- k[1] + k['x1']*c(0,1))
mh <- Mean(h)
wh <- coef(h)[1] + coef(h)['x1']*c(0,1)
mh(wh)
qu <- Quantile(g)
# Compare model estimated and empirical quantiles
cq <- function(y) {
cat(qu(.1, w), tapply(y, x1, quantile, probs=.1), '\n')
cat(qu(.5, w), tapply(y, x1, quantile, probs=.5), '\n')
cat(qu(.9, w), tapply(y, x1, quantile, probs=.9), '\n')
}
cq(y)
# Try on log-normal model
g <- orm(exp(y) ~ x1)
g
k <- coef(g)
plot(k[1:i])
m <- Mean(g)
m(w <- k[1] + k['x1']*c(0,1))
tapply(exp(y), x1, mean)
qu <- Quantile(g)
cq(exp(y))
# Compare predicted mean with ols for a continuous x
set.seed(3)
n <- 200
x1 <- rnorm(n)
y <- x1 + rnorm(n)
dd <- datadist(x1); options(datadist='dd')
f <- ols(y ~ x1)
g <- orm(y ~ x1, family=probit)
h <- orm(y ~ x1, family=logistic)
w <- orm(y ~ x1, family=cloglog)
mg <- Mean(g); mh <- Mean(h); mw <- Mean(w)
r <- rbind(ols = Predict(f, conf.int=FALSE),
probit = Predict(g, conf.int=FALSE, fun=mg),
logistic = Predict(h, conf.int=FALSE, fun=mh),
cloglog = Predict(w, conf.int=FALSE, fun=mw))
plot(r, groups='.set.')
# Compare predicted 0.8 quantile with quantile regression
qu <- Quantile(g)
qu80 <- function(lp) qu(.8, lp)
f <- Rq(y ~ x1, tau=.8)
r <- rbind(probit = Predict(g, conf.int=FALSE, fun=qu80),
quantreg = Predict(f, conf.int=FALSE))
plot(r, groups='.set.')
# Verify transformation invariance of ordinal regression
ga <- orm(exp(y) ~ x1, family=probit)
qua <- Quantile(ga)
qua80 <- function(lp) log(qua(.8, lp))
r <- rbind(logprobit = Predict(ga, conf.int=FALSE, fun=qua80),
probit = Predict(g, conf.int=FALSE, fun=qu80))
plot(r, groups='.set.')
# Try the same with quantile regression. Need to transform x1
fa <- Rq(exp(y) ~ rcs(x1,5), tau=.8)
r <- rbind(qr = Predict(f, conf.int=FALSE),
logqr = Predict(fa, conf.int=FALSE, fun=log))
plot(r, groups='.set.')
# Make a plot of Pr(Y >= y) vs. a continuous covariate for 3 levels
# of y and also against a binary covariate
set.seed(1)
n <- 1000
age <- rnorm(n, 50, 15)
sex <- sample(c('m', 'f'), 1000, TRUE)
Y <- runif(n)
dd <- datadist(age, sex); options(datadist='dd')
f <- orm(Y ~ age + sex)
# Use ExProb function to derive an R function to compute
# P(Y >= y | X)
ex <- ExProb(f)
ex1 <- function(x) ex(x, y=0.25)
ex2 <- function(x) ex(x, y=0.5)
ex3 <- function(x) ex(x, y=0.75)
p1 <- Predict(f, age, sex, fun=ex1)
p2 <- Predict(f, age, sex, fun=ex2)
p3 <- Predict(f, age, sex, fun=ex3)
p <- rbind('P(Y >= 0.25)' = p1,
'P(Y >= 0.5)' = p2,
'P(Y >= 0.75)' = p3)
ggplot(p)
# Make plot with two curves (by sex) with y on the x-axis, and
# estimated P(Y >= y | sex, age=median) on the y-axis
ys <- seq(min(Y), max(Y), length=100)
g <- function(sx) as.vector(ex(y=ys, Predict(f, sex=sx)$yhat)$prob)
d <- rbind(data.frame(sex='m', y=ys, p=g('m')),
data.frame(sex='f', y=ys, p=g('f')))
ggplot(d, aes(x=y, y=p, color=sex)) + geom_line() +
ylab(expression(P(Y >= y))) +
guides(color=guide_legend(title='Sex')) +
theme(legend.position='bottom')
options(datadist=NULL)
if (FALSE) {
## Simulate power and type I error for orm logistic and probit regression
## for likelihood ratio, Wald, and score chi-square tests, and compare
## with t-test
require(rms)
set.seed(5)
nsim <- 2000
r <- NULL
for(beta in c(0, .4)) {
for(n in c(10, 50, 300)) {
cat('beta=', beta, ' n=', n, '\n\n')
plogistic <- pprobit <- plogistics <- pprobits <- plogisticw <-
pprobitw <- ptt <- numeric(nsim)
x <- c(rep(0, n/2), rep(1, n/2))
pb <- setPb(nsim, every=25, label=paste('beta=', beta, ' n=', n))
for(j in 1:nsim) {
pb(j)
y <- beta*x + rnorm(n)
tt <- t.test(y ~ x)
ptt[j] <- tt$p.value
f <- orm(y ~ x)
plogistic[j] <- f$stats['P']
plogistics[j] <- f$stats['Score P']
plogisticw[j] <- 1 - pchisq(coef(f)['x']^2 / vcov(f)[2,2], 1)
f <- orm(y ~ x, family=probit)
pprobit[j] <- f$stats['P']
pprobits[j] <- f$stats['Score P']
pprobitw[j] <- 1 - pchisq(coef(f)['x']^2 / vcov(f)[2,2], 1)
}
if(beta == 0) plot(ecdf(plogistic))
r <- rbind(r, data.frame(beta = beta, n=n,
ttest = mean(ptt < 0.05),
logisticlr = mean(plogistic < 0.05),
logisticscore= mean(plogistics < 0.05),
logisticwald = mean(plogisticw < 0.05),
probit = mean(pprobit < 0.05),
probitscore = mean(pprobits < 0.05),
probitwald = mean(pprobitw < 0.05)))
}
}
print(r)
# beta n ttest logisticlr logisticscore logisticwald probit probitscore probitwald
#1 0.0 10 0.0435 0.1060 0.0655 0.043 0.0920 0.0920 0.0820
#2 0.0 50 0.0515 0.0635 0.0615 0.060 0.0620 0.0620 0.0620
#3 0.0 300 0.0595 0.0595 0.0590 0.059 0.0605 0.0605 0.0605
#4 0.4 10 0.0755 0.1595 0.1070 0.074 0.1430 0.1430 0.1285
#5 0.4 50 0.2950 0.2960 0.2935 0.288 0.3120 0.3120 0.3120
#6 0.4 300 0.9240 0.9215 0.9205 0.920 0.9230 0.9230 0.9230
}
```

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