# Quantile regression ANOVA and ANCOVA [duplicate]

ANOVA posits that the mean of a particular group is equal to the overall mean plus some amount. If we find that amount to be significant, then we decide that the groups don't all have the same mean.

We can solve this in a regression format with $$\mathbb{E}\big[y\big] = \mu + \sum_j \beta_j X_j$$.

The $$X_j$$ are dummy variables encoding the group membership, and $$\mu$$ is the overall mean. If we want to include some covariates, we can stick those in the regression, too, and get ANCOVA.

$$\mathbb{E}\big[y\big] = \mu + \sum_j \beta_j X_j + \sum_{cov} \beta_{cov}X_{cov}$$

Assuming the standard assumptions, calculate the $$\beta_j$$ and p-values the usual way.

However, I'm suddenly much more interested in the $$0.75$$ quantile than the mean. Let's say that I don't have equal-variance normal distributions in all of the groups, so I can't conclude that quantiles $$0.75$$ are different just because the means are. I know how to run a quantile regression to predict a conditional quantile $$0.75$$, and there are ways of getting standard errors and p-values on the $$B_j$$. (The quantreg package in R has a couple of methods in summary.rq.)

I've never seen this done, so I'm hesitant to approach everything as analogous to ANOVA, but isn't it? Any pitfalls? Better methods?

(Ditto for a quantile-ANCOVA.)