The way I understood when to use a a robust linear fitting is for example when your variance is not constant (e.g. when you have heteroscedasticity as shown with a Breusch-Pagan test for example) or when your residuals are not normally distributed. Meaning that in the output of the robust linear fitting of your model there are no statistics such as the R² or residual standard error because the variance of the residuals is not constant.
Now, to my surprise, when I used a rlm() in R (to fit my linear model robustly) there was a residual standard error output (but no R²) but I don't understand why there is one if the residuals don't have a constant variance? Isn't the residual standard error invalid because the linear model's residual standard errors already showed heteroscedasticity?
Call: rlm(formula = y ~ x + I(x^2), data = datexperiment) Residuals: Min 1Q Median 3Q Max -2.43289 -0.48946 0.05967 0.54450 2.54706 Coefficients: Value Std. Error t value (Intercept) 26.4830 0.4326 61.2153 x 2.3973 0.2830 8.4702 I(x^2) -0.5107 0.0396 -12.9020 Residual standard error: 0.7989 on 75 degrees of freedom