polr: Ordered Logistic or Probit Regression

Description

Fits a logistic or probit regression model to an ordered factor response. The default logistic case is proportional odds logistic regression, after which the function is named.

Usage

polr(formula, data, weights, start, ..., subset, na.action,
     contrasts = NULL, Hess = FALSE, model = TRUE,
     method = c("logistic", "probit", "loglog", "cloglog", "cauchit"))

Value

A object of class "polr". This has components

coefficients: the coefficients of the linear predictor, which has no intercept.
zeta: the intercepts for the class boundaries.
deviance: the residual deviance.
fitted.values: a matrix, with a column for each level of the response.
lev: the names of the response levels.
terms: the terms structure describing the model.
df.residual: the number of residual degrees of freedoms, calculated using the weights.
edf: the (effective) number of degrees of freedom used by the model
n, nobs: the (effective) number of observations, calculated using the weights. (nobs is for use by stepAIC.
call: the matched call.
method: the matched method used.
convergence: the convergence code returned by optim.
niter: the number of function and gradient evaluations used by optim.
lp: the linear predictor (including any offset).
Hessian: (if Hess is true). Note that this is a numerical approximation derived from the optimization proces.
model: (if model is true).

Arguments

formula: a formula expression as for regression models, of the form response ~ predictors. The response should be a factor (preferably an ordered factor), which will be interpreted as an ordinal response, with levels ordered as in the factor. The model must have an intercept: attempts to remove one will lead to a warning and be ignored. An offset may be used. See the documentation of formula for other details.
data: an optional data frame, list or environment in which to interpret the variables occurring in formula.
weights: optional case weights in fitting. Default to 1.
start: initial values for the parameters. This is in the format c(coefficients, zeta): see the Values section.
...: additional arguments to be passed to optim, most often a control argument.
subset: expression saying which subset of the rows of the data should be used in the fit. All observations are included by default.
na.action: a function to filter missing data.
contrasts: a list of contrasts to be used for some or all of the factors appearing as variables in the model formula.
Hess: logical for whether the Hessian (the observed information matrix) should be returned. Use this if you intend to call summary or vcov on the fit.
model: logical for whether the model matrix should be returned.
method: logistic or probit or (complementary) log-log or cauchit (corresponding to a Cauchy latent variable).

Details

This model is what Agresti (2002) calls a cumulative link model. The basic interpretation is as a coarsened version of a latent variable $Y_{i}$ which has a logistic or normal or extreme-value or Cauchy distribution with scale parameter one and a linear model for the mean. The ordered factor which is observed is which bin $Y_{i}$ falls into with breakpoints $ζ_{0} = - \infty < ζ_{1} < \dots < ζ_{K} = \infty$ This leads to the model $logit P (Y \leq k | x) = ζ_{k} - η$ with logit replaced by probit for a normal latent variable, and $η$ being the linear predictor, a linear function of the explanatory variables (with no intercept). Note that it is quite common for other software to use the opposite sign for $η$ (and hence the coefficients beta).

In the logistic case, the left-hand side of the last display is the log odds of category $k$ or less, and since these are log odds which differ only by a constant for different $k$ , the odds are proportional. Hence the term proportional odds logistic regression.

The log-log and complementary log-log links are the increasing functions $F^{- 1} (p) = - l o g (- l o g (p))$ and $F^{- 1} (p) = l o g (- l o g (1 - p))$ ; some call the first the ‘negative log-log’ link. These correspond to a latent variable with the extreme-value distribution for the maximum and minimum respectively.

A proportional hazards model for grouped survival times can be obtained by using the complementary log-log link with grouping ordered by increasing times.

predict, summary, vcov, anova, model.frame and an extractAIC method for use with stepAIC (and step). There are also profile and confint methods.

References

Agresti, A. (2002) Categorical Data. Second edition. Wiley.

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

Examples

Run this code

options(contrasts = c("contr.treatment", "contr.poly"))
house.plr <- polr(Sat ~ Infl + Type + Cont, weights = Freq, data = housing)
house.plr
summary(house.plr, digits = 3)
## slightly worse fit from
summary(update(house.plr, method = "probit", Hess = TRUE), digits = 3)
## although it is not really appropriate, can fit
summary(update(house.plr, method = "loglog", Hess = TRUE), digits = 3)
summary(update(house.plr, method = "cloglog", Hess = TRUE), digits = 3)

predict(house.plr, housing, type = "p")
addterm(house.plr, ~.^2, test = "Chisq")
house.plr2 <- stepAIC(house.plr, ~.^2)
house.plr2$anova
anova(house.plr, house.plr2)

house.plr <- update(house.plr, Hess=TRUE)
pr <- profile(house.plr)
confint(pr)
plot(pr)
pairs(pr)

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