Fits ordinal cumulative probability models for continuous or ordinal
response variables, efficiently allowing for a large number of
intercepts by capitalizing on the information matrix being sparse.
Five different distribution functions are implemented, with the
default being the logistic (i.e., the proportional odds
model). The ordinal cumulative probability models are stated in terms
of exceedance probabilities (\(Prob[Y \ge y | X]\)) so that as with
OLS larger predicted values are associated with larger `Y`

. This is
important to note for the asymmetric distributions given by the
log-log and complementary log-log families, for which negating the
linear predictor does not result in \(Prob[Y < y | X]\). The
`family`

argument is defined in `orm.fit`

. The model
assumes that the inverse of the assumed cumulative distribution
function, when applied to one minus the true cumulative distribution function
and plotted on the \(y\)-axis (with the original \(y\) on the
\(x\)-axis) yields parallel curves (though not necessarily linear).
This can be checked by plotting the inverse cumulative probability
function of one minus the empirical distribution function, stratified
by `X`

, and assessing parallelism. Note that parametric
regression models make the much stronger assumption of linearity of
such inverse functions.

For the `print`

method, format of output is controlled by the
user previously running `options(prType="lang")`

where
`lang`

is `"plain"`

(the default), `"latex"`

, or
`"html"`

.

`Quantile.orm`

creates an R function that computes an estimate of
a given quantile for a given value of the linear predictor (which was
assumed to use thefirst intercept). It uses a linear
interpolation method by default, but you can override that to use a
discrete method by specifying `method="discrete"`

when calling
the function generated by `Quantile`

.
Optionally a normal approximation for a confidence
interval for quantiles will be computed using the delta method, if
`conf.int > 0`

is specified to the function generated from calling
`Quantile`

and you specify `X`

. In that case, a
`"lims"`

attribute is included
in the result computed by the derived quantile function.

```
orm(formula, data=environment(formula),
subset, na.action=na.delete, method="orm.fit",
model=FALSE, x=FALSE, y=FALSE, linear.predictors=TRUE, se.fit=FALSE,
penalty=0, penalty.matrix, tol=1e-7, eps=0.005,
var.penalty=c('simple','sandwich'), scale=FALSE, …)
```# S3 method for orm
print(x, digits=4, coefs=TRUE,
intercepts=x$non.slopes < 10, title, …)

# S3 method for orm
Quantile(object, codes=FALSE, …)

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; `orm`

converts it
in alphabetic or numeric order to a factor variable and
recodes it 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 is `na.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 using `options(na.action="na.delete")`

.

method

name of fitting function. Only allowable choice at present is `orm.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`

. For `print`

, an object
created by `orm`

.

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`

. The first intercept is used.

se.fit

causes the standard errors of the fitted values (on the linear predictor
scale) to be returned under the name `se.fit`

. The middle
intercept is used.

penalty

see `lrm`

penalty.matrix

see `lrm`

tol

singularity criterion (see `orm.fit`

)

eps

difference in \(-2 log\) likelihood for declaring convergence

var.penalty

see `lrm`

scale

set to `TRUE`

to subtract column means and divide by
column standard deviations of the design matrix
before fitting, and to back-solve for the un-normalized 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 `orm.fit`

, or from
`print`

, to `prModFit`

. Ignored for
`Quantile`

. One of the most important arguments is `family`

.

digits

number of significant digits to use

coefs

specify `coefs=FALSE`

to suppress printing the table
of model coefficients, standard errors, etc. Specify `coefs=n`

to print only the first `n`

regression coefficients in the
model.

intercepts

By default, intercepts are only printed if there are
fewer than 10 of them. Otherwise this is controlled by specifying
`intercepts=FALSE`

or `TRUE`

.

title

a character string title to be passed to `prModFit`

.
Default is constructed from the name of the distribution family.

object

an object created by `orm`

codes

if `TRUE`

, uses the integer codes \(1,2,\ldots,k\)
for the \(k\)-level response in computing the predicted quantile

The returned fit object of `orm`

contains the following components
in addition to the ones mentioned under the optional arguments.

calling expression

table of frequencies for `Y`

in order of increasing `Y`

vector with the following elements: number of observations used in the
fit, number of unique `Y`

values, median `Y`

from among the
observations used int he fit, maximum absolute value of first
derivative of log likelihood, model likelihood ratio
\(\chi^2\), d.f., \(P\)-value, score \(\chi^2\)
statistic (if no initial values given), \(P\)-value, Spearman's
\(\rho\) rank correlation between the linear predictor and `Y`

,
the Nagelkerke \(R^2\) index, the \(g\)-index, \(gr\) (the
\(g\)-index on the odds ratio scale), and \(pdm\) (the mean absolute
difference between 0.5 and the predicted probability that \(Y\geq\)
the marginal median).
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. \(\chi^2\) and corrected d.f.
The score chi-square statistic uses first derivatives which contain
penalty components.

set to `TRUE`

if convergence failed (and `maxiter>1`

) or if a
singular information matrix is encountered

estimated parameters

estimated variance-covariance matrix (inverse of information matrix)
for the middle intercept and regression coefficients. See
`lrm`

for details if penalization is used.

see `lrm`

the character string for `family`

. If `family`

was a user-customized list, it must have had an element named
`name`

, which is taken as the return value for `family`

here.

a list of functions for the choice of `family`

, with
elements `cumprob`

(the cumulative probability distribution
function), `inverse`

(inverse of `cumprob`

), `deriv`

(first derivative of `cumprob`

), and `deriv2`

(second
derivative of `cumprob`

)

-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.

number of intercepts in model

the index of the middle (median) intercept used in
computing the linear predictor and `var`

see `lrm`

the penalty matrix actually used in the estimation

a sparse matrix representation of type
`matrix.csr`

from the `SparseM`

package. This allows the
full information matrix with all intercepts to be stored efficiently,
and matrix operations using the Cholesky decomposition to be fast.
`link{vcov.orm}`

uses this information to compute the covariance
matrix for intercepts other than the middle one.

Sall J: A monotone regression smoother based on ordinal cumulative logistic regression, 1991.

Le Cessie S, Van Houwelingen JC: Ridge estimators in logistic regression. Applied Statistics 41:191--201, 1992.

Verweij PJM, Van Houwelingen JC: Penalized likelihood in Cox regression. Stat in Med 13:2427--2436, 1994.

Gray RJ: Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis. JASA 87:942--951, 1992.

Shao J: Linear model selection by cross-validation. JASA 88:486--494, 1993.

Verweij PJM, Van Houwelingen JC: Crossvalidation in survival analysis. Stat in Med 12:2305--2314, 1993.

Harrell FE: Model uncertainty, penalization, and parsimony. Available from http://hbiostat.org/talks/iscb98.pdf.

`orm.fit`

, `predict.orm`

, `solve`

,
`rms.trans`

, `rms`

, `polr`

,
`latex.orm`

, `vcov.orm`

,
`num.intercepts`

,
`residuals.orm`

, `na.delete`

,
`na.detail.response`

,
`pentrace`

, `rmsMisc`

, `vif`

,
`predab.resample`

,
`validate.orm`

, `calibrate`

,
`Mean.orm`

, `gIndex`

, `prModFit`

# NOT RUN { 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.') options(datadist=NULL) # } # NOT RUN { ## 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 # }