R: Ordered Logistic or Probit Regression

polr {MASS}

R Documentation

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"))

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

\zeta_0 = -\infty < \zeta_1 < \cdots < \zeta_K = \infty

This leads to the model

\mbox{logit} P(Y \le k | x) = \zeta_k - \eta

with logit replaced by probit for a normal latent variable, and \eta 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 \eta (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) = -log(-log(p)) and F^{-1}(p) = log(-log(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.

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).

Note

The vcov method uses the approximate Hessian: for reliable results the model matrix should be sensibly scaled with all columns having range the order of one.

Prior to version 7.3-32, method = "cloglog" confusingly gave the log-log link, implicitly assuming the first response level was the ‘best’.

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

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)

[Package MASS version 7.3-61 Index]