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". When using html with Quarto or RMarkdown,
results='asis' need not be written in the chunk header.
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",
family=c("logistic", "probit", "loglog", "cloglog", "cauchit"),
model=FALSE, x=FALSE, y=FALSE, linear.predictors=TRUE, se.fit=FALSE,
penalty=0, penalty.matrix,
var.penalty=c('simple','sandwich'), scale=FALSE,
maxit=30, weights, normwt=FALSE, ...)# S3 method for orm
print(x, digits=4, r2=c(0,2,4), coefs=TRUE, pg=FALSE,
intercepts=x$non.slopes < 10, title, ...)
# S3 method for orm
Quantile(object, codes=FALSE, ...)
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, \(R^2\) indexes computed by
R2Measures, 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.
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.
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 frame to use. Default is the current frame.
logical expression or vector of subscripts defining a subset of observations to analyze
function to handle NAs 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").
name of fitting function. Only allowable choice at present is orm.fit.
character value specifying the distribution family, which is one of the following:
"logistic", "probit", "loglog", "cloglog", "cauchit", corresponding to logistic (the
default), Gaussian, Cauchy, Gumbel maximum (\(exp(-exp(-x))\);
extreme value type I), and Gumbel minimum
(\(1-exp(-exp(x))\)) distributions. These are the cumulative
distribution functions assumed for \(Prob[Y \ge y | X]\). The default is "logistic".
causes the model frame to be returned in the fit object
causes the expanded design matrix (with missings excluded)
to be returned under the name x. For print, an object
created by orm.
causes the response variable (with missings excluded) to be returned
under the name y.
causes the predicted X beta (with missings excluded) to be returned
under the name linear.predictors. The first intercept is used.
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.
see lrm
see lrm
see lrm
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.
maximum number of iterations to allow in orm.fit
a vector (same length as y) of possibly fractional case weights
set to TRUE to scale weights so they sum to the length of
y; useful for sample surveys as opposed to the default of
frequency weighting
arguments that are passed to orm.fit, or from
print, to prModFit. Ignored for
Quantile. One of the most important arguments is family.
number of significant digits to use
vector of integers specifying which R^2 measures to print,
with 0 for Nagelkerke R^2 and 1:4 corresponding to the 4 measures
computed by R2Measures. Default is to print
Nagelkerke (labeled R2) and second and fourth R2Measures
which are the measures adjusted for the number of predictors, first
for the raw sample size then for the effective sample size, which
here is from the formula for the approximate variance of a log odds
ratio in a proportional odds model.
set to TRUE to print g-indexes
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.
By default, intercepts are only printed if there are
fewer than 10 of them. Otherwise this is controlled by specifying
intercepts=FALSE or TRUE.
a character string title to be passed to prModFit.
Default is constructed from the name of the distribution family.
an object created by orm
if TRUE, uses the integer codes \(1,2,\ldots,k\)
for the \(k\)-level response in computing the predicted quantile
Frank Harrell
Department of Biostatistics, Vanderbilt University
fh@fharrell.com
For the Quantile function:
Qi Liu and Shengxin Tu
Department of Biostatistics, Vanderbilt University
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 https://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