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 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 = -Inf < zeta_1 < … < zeta_K = Inf

This leads to the model

logit P(Y <= 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.

### See Also

`optim`, `glm`, `multinom`.

### 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 7.3-45 Index]