Ordinal Regression Model

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.

Usage

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, lpe=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 class 'orm'
print(x, digits=4, r2=c(0,2,4), coefs=TRUE, pg=FALSE,
    intercepts=x$non.slopes < 10, title, ...)

# S3 method for class 'orm'
Quantile(object, codes=FALSE, ...)

Note

When creating ordinal levels for non-integer numeric Y, this function does not use the unique function because it was found to be non-reproducible across hardware platforms. Instead, intercepts are mapped to integers obtained from rounding original Y levels after multiplying by 1e7 (by default). When using the ExProb or Survival functions to compute exceedance probabilities or survival curves, when the user wishes to obtain probabilities at select non-integer y values, the only way to guarantee the correct mapping to intercepts is to request estimates at the "official" levels of Y stored in the yunique object found in the fit object. Users may opt instead to replace original data with e.g. 1e-7 * round(Y * 1e7) before model fitting.

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; 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 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").
method: name of fitting function. Only allowable choice at present is orm.fit.
family: 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".
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.
lpe: set lpe=TRUE to store the vector of likelihood probability elements in the fit object in a list element named lpe. This will enable the ordESS function to summarize the effective sample sizes of any censored observations.
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
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.
maxit: maximum number of iterations to allow in orm.fit
weights: a vector (same length as y) of possibly fractional case weights
normwt: 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.
digits: number of significant digits to use
r2: 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.
pg: set to TRUE to print g-indexes
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

Value

The returned fit object of orm contains the following components in addition to the ones mentioned under the optional arguments.

call: calling expression
freq: table of frequencies for Y in order of increasing Y
stats: vector with the following elements: number of observations used in the fit, effective sample size ESS, 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.
fail: set to TRUE if convergence failed (and maxiter>1) or if a singular information matrix is encountered
coefficients: estimated parameters
var: estimated variance-covariance matrix (inverse of information matrix) for the middle intercept and regression coefficients. See lrm for details if penalization is used.
effective.df.diagonal: see lrm
family: the character string for family.
famfunctions: a vector of expressions containing functions for the cumulative probability, inverse cumulative probability, derivative, second derivative, and derivative as a function of only x
trans: 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)
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.
non.slopes: number of intercepts in model
interceptRef: the index of the middle (median) intercept used in computing the linear predictor and var
penalty: see lrm
penalty.matrix: the penalty matrix actually used in the estimation
info.matrix: 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.

Author

Frank Harrell
Department of Biostatistics, Vanderbilt University
fh@fharrell.com
For the Quantile function:
Qi Liu and Shengxin Tu
Department of Biostatistics, Vanderbilt University

References

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.