Proportional-hazards model

Topic of Categorical responses, part of Generalised linear models

Type	Topic
Taught in	Categorical responses
Part	Generalised linear models

A model in which covariates multiply the hazard by a constant factor at every time. In ordinal regression, replacing the logit by the complementary log-log in the cumulative model gives this form, and the coefficients are log hazard ratios, not log odds ratios.

In Categorical responses

From Categorical responses

Replacing the logit by the complementary log-log gives a proportional-hazards model, \log\bigl(-\log(1-\gamma_k)\bigr)=\beta_{0k}+{\boldsymbol{x}}^\top\boldsymbol{\beta}, with the same parallel-slope constraint and the same ordering of intercepts. The coefficient is a log hazard, not a log odds ratio. We can say what an odds ratio is. A hazard is a definition we can write down and a quantity we cannot usually explain. The call that forces the cumulative, reversed, parallel, complementary-log-log model is

need <- function(pkg) {
  if (!requireNamespace(pkg, quietly = TRUE)) {
    install.packages(pkg, repos = "https://cloud.r-project.org")
  }
  suppressPackageStartupMessages(library(pkg, character.only = TRUE))
}
need("VGAM")
data(pneumo, package = "VGAM")
pneumo$none <- pneumo$normal
pneumo$x <- log(pneumo$exposure.time)
fit <- vglm(cbind(none, mild, severe) ~ x,
            family = cumulative(reverse = TRUE, parallel = TRUE, link = "cloglog"),
            data = pneumo)
summary(fit)
deviance(fit)
logLik(fit)

and it returns intercepts -8.60 and -9.35, with slope 2.21 (z=7.06). The residual deviance is 6.65 against 5.03 for the proportional-odds fit, and the log-likelihood is -25.90 against -25.09. The odds model fits this table a little better. The two models are not nested, so that comparison is not a likelihood-ratio test. It is the same kind of absolute comparison as an AIC when the parameter count matches, which here it does.

Used in

Taught in Categorical responses
All 3 questions on Categorical responses

In Categorical responses

Used in

See also