Can someone explain what the Brant test in R does? I’m using R to run proportional odds models through the polr function in the MASS package. I want to see if the proportional odds or parallel lines assumption holds and I have come across the brant package. Can someone explain/break down what the brant test from the brant package actually does?
 A: Update: The previous answer described Brant's test as he proposed it.  The test that's implemented in the brant package turns out to be different; it looks for all possible differences between $\beta$s at different cutpoints, not at the particular alternative Brant suggested.  This all-possible-departures test is described in the original paper by Brant, but he argues that it wouldn't be very helpful
Previous answer
More seriously, if your outcome variable $Y$ is ordinal, you can dichotomise it at $k$ and get a set of $k-1$ logistic models
$$\mathrm{logit} P[Y>k] = x\beta_k-\alpha_k$$
It is not possible for the $\alpha_k$ to be the same for different $k$, but it is possible for the $\beta_k$ to be the same. The proportional odds model comes from the assumption that they are the same.
Since the $\beta_k$ are the same under the proportional odds model, any $\hat\beta_k -\hat\beta_j$ is a reasonable test statistic for the proportional odds assumption.  The Brant test computes the variance-covariance matrix of all the $\hat\beta_k$, so that any departure from proportional odds could in principle be examined.  However, as Brant noted, a test that tries to detect everything will have poor power for any specific alternative. For this reason, the test focuses on the alternative $\beta_k=\phi_k\beta$, which will tend to arise either if the true link function is not the logit or if there is misclassification in $Y$.
So, the Brant test is a test of $\beta_k=\beta$ vs $\beta_k=\phi_k\beta$, and it is a Wald-type test, based on differences in $\hat\beta_k$ and their estimated variances.
