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.
orm.fit(x=NULL, y, family='logistic',
offset=0., initial, maxit=12L, eps=.005, tol=1e-7, trace=FALSE,
penalty.matrix=NULL, scale=FALSE)design matrix with no column for an intercept
response vector, numeric, factor, or character. The ordering of levels
is assumed from factor(y).
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.
optional numeric vector containing an offset on the logit scale
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.
maximum no. iterations (default=12).
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.
Singularity criterion. Default is 1e-7
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.
a self-contained ready-to-use penalty matrix - seelrm
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.
a list with the following components:
calling expression
table of frequencies for y in order of increasing y
vector of sorted unique values of 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 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.
set to TRUE if convergence failed (and maxit>1)
estimated parameters
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.
see orm
-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.
number of intercepts in model
the index of the middle (median) intercept used in
computing the linear predictor and var
the linear predictor using the first intercept
see above
see orm
# NOT RUN {
#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)
# }
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