orm.fit: Ordinal Regression Model Fitter

Description

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 (yielding the proportional odds model). Penalized estimation will be implemented in the future. Weights are not implemented. The optimization method is Newton-Raphson with step-halving. Execution time is linear in the number of intercepts.

Usage

orm.fit(x=NULL, y, family='logistic',
        offset=0., initial,  maxit=12L, eps=.005, tol=1e-7, trace=FALSE,
        penalty.matrix=NULL, scale=FALSE)

Value

a list with the following components:

call: calling expression
freq: table of frequencies for y in order of increasing y
yunique: vector of sorted unique values of y
stats: vector with the following elements: number of observations used in the fit, number of unique y values, median y from among the observations used in the fit, maximum absolute value of first derivative of log likelihood, model likelihood ratio chi-square, d.f., P-value, score chi-square and its P-value, Spearman's \(\rho\) rank correlation between linear predictor and y, the Nagelkerke \(R^2\) index, the \(g\)-index, \(gr\) (the \(g\)-index on the ratio scale), and \(pdm\) (the mean absolute difference between 0.5 and the estimated probability that \(y\geq\) the marginal median). When penalty.matrix is present, the \(\chi^2\), d.f., and P-value are not corrected for the effective d.f.
fail: set to TRUE if convergence failed (and maxit>1)
coefficients: estimated parameters
var: estimated variance-covariance matrix (inverse of information matrix). Note that in the case of penalized estimation, var is not the improved sandwich-type estimator (which lrm does compute). The only intercept parameter included in the stored object is the middle intercept.
family, trans: see orm
deviance: -2 log likelihoods. 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
linear.predictors: the linear predictor using the first intercept
penalty.matrix: see above
info.matrix: see orm

Arguments

x

design matrix with no column for an intercept

y

response vector, numeric, factor, or character. The ordering of levels is assumed from factor(y).

family

the distribution family, 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 family argument can be an unquoted or a quoted string, e.g. family=loglog or family="loglog". To use a built-in family, the string must be one of the following corresponding to the previous list: logistic, probit, loglog, cloglog, cauchit. The user can also provide her own customized family by setting family to a list with elements cumprob, inverse, deriv, deriv2; see the body of orm.fit for examples. An additional element, name must be given, which is a character string used to name the family for print and latex.

offset

optional numeric vector containing an offset on the logit scale

initial

vector of initial parameter estimates, beginning with the intercepts. If initial is not specified, the function computes the overall score \(\chi^2\) test for the global null hypothesis of no regression.

maxit

maximum no. iterations (default=12).

eps

difference in \(-2 log\) likelihood for declaring convergence. Default is .005. If the \(-2 log\) likelihood gets worse by eps/10 while the maximum absolute first derivative of

-2 log

likelihood is below 1E-9, convergence is still declared. This handles the case where the initial estimates are MLEs, to prevent endless step-halving.

tol

Singularity criterion. Default is 1e-7

trace

set to TRUE to print -2 log likelihood, step-halving fraction, change in -2 log likelihood, and maximum absolute value of first derivative at each iteration.

penalty.matrix

a self-contained ready-to-use penalty matrix - seelrm

scale

set to TRUE to subtract column means and divide by column standard deviations of x 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.

Author

Frank Harrell
Department of Biostatistics, Vanderbilt University
fh@fharrell.com

Examples

Run this code

#Fit an additive logistic model containing numeric predictors age, 
#blood.pressure, and sex, assumed to be already properly coded and 
#transformed
#
# fit <- orm.fit(cbind(age,blood.pressure,sex), death)