# Robust linear regression for group differences

I am trying to understand what it means when a p value for a grouping variable is < 0.05 after running robust linear regression.

Does this mean that the 2 groups significantly differ with respect to the dependent variable?

How is this method conceptually different from running a simpler anova?

• Q1 yes, Q2 not really. – mdewey Jun 7 at 13:48
• It will be helpful to write down exactly what the null and alternative hypotheses are, both for this and a regular ANOVA F-test (in regression form). – Dave Jun 7 at 13:50

If you run a regression where $$z$$ is a binary indicator for group membership, the the coefficient $$\beta_2$$ in the model
$$y_i = \beta_0 + \beta_1 x_i + \beta_2z_i + \varepsilon_i$$
represents the difference in the expectation conditioned on $$x$$.
In the case where the grouping variable has several groups and you have no other covariables to adjust for ($$x$$ in the model above) then the regression model is exactly an ANOVA.