Brant test in R In testing the parallel regression assumption in ordinal logistic regression I find there are several approaches. I've used both the graphical approach (as detailed in Harrell´s book) and the approach detailed using the  ordinal package in R. 
However I would also like to run the Brant test (from Stata) for both the individual variables and also for the total model. I've looked around but cannot find it implemented in R. 
Is there an implementation of the Brant test in R?
 A: Yes -- in fact the ordinal package that you linked can do it (although they don't call it the Brant test).  Take a look at pages 6 and 7 of your link, which demonstrate "a likelihood ratio test of the equal slopes or proportional odds assumption," which is exactly what you are looking for.
A: Some notes on the topic
The R package VGAM in the cumulative command (Ordinal Regression with Cumulative Probabilities) allows to change the proportional odds assumptions, with the option parallel=FALSE.
It is known to be a common problem (from the book: Regression Models for Categorical Dependent Variables Using Stata, Second Edition,  By J. Scott Long, Jeremy Freese)

"A Caveat regarding the parallel regression assumption: We find that the parallel regression assumption (PRA) is frequently violated. When this is rejected alternatives models that do not impose the constraint of parallel regressions should be considered. Violation of the  PRA is not rationale for usig OLS regression since the assumptions implied by the application of the LRM to ordinal data are even stronger. Alternative models that can be considered include models for nominal outcomes [...] Stereotype Logistic model or Stereotype ordered model; the Generalized Ordered Logit model; the continuation Ratio model, are alternatives" (page 221)

This paper goes in depth in this topic, being clear and well written, but it does not consider the VGAM package or the "cumulative" command:
Ordinal logistic regression in epidemiological studies
A: I implemented the brant test in R. The package and function is called brant and it's now available on CRAN. 
The brant test was defined by Rollin Brant to test the parallel regression assumption (Brant, R. (1990) Assessing proportionality in the proportional odds model for ordinal logistic regression. Biometrics, 46, 1171–1178). 
Here is a code example:
data = MASS::survey
data$Smoke = ordered(MASS::survey$Smoke, levels=c("Never","Occas","Regul","Heavy"))
model1 = MASS::polr(Smoke ~ Sex + Height, data=data, Hess=TRUE)
brant(model1)

In the example, the parallel regression assumption holds, because all p-values are above 0.05. The Omnibus is for the whole model, the rest for the indvidual coefficents.
A: This tutorial about ordinal logistic regression in R covers testing the proportional odds assumption.
