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Regression is a way to model the relationship between the conditional expected value and the predictors. But in the robust regression we don't have the expected value (say, arithmetic mean for Gaussian response), but rather an M estimator - truncated mean, winsorized mean. So, when it comes to interpret the coefficients in a linear regression, it's simple: "a unit change in X1, holding constant all others, causes, in AVERAGE, beta change in the response". Now, how to replace the "AVERAGE" in robust regression?

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  • $\begingroup$ The model is predicting a summary of your response, conditional on predictors, where the summary is defined by the algorithm or recipe used. The wording need not be -- arguably cannot be -- a single short word or term. The estimand is defined by the estimator. $\endgroup$ – Nick Cox Mar 3 at 20:47
  • $\begingroup$ @Nick I don't follow your final thought, because if it were true, then why wouldn't all estimators be unbiased? If an estimator can be biased, then how exactly does an estimator determine its estimand? $\endgroup$ – whuber Mar 3 at 21:37
  • $\begingroup$ We’ve had this discussion before. What does a trimmed mean estimate? $\endgroup$ – Nick Cox Mar 3 at 21:44
  • $\begingroup$ The allusion is to this 2013 exchange: stats.stackexchange.com/questions/63386/… $\endgroup$ – Nick Cox Mar 4 at 15:12
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With robust regression you are still estimating the conditional mean function $E(Y|X) $ so that the interpretation that you suggest carries over. The difference is that robust estimators require fewer assumptions on the error and the covariates than the typical least squares estimator.

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  • $\begingroup$ Thank you very much for the answer, Sir. I guess this has some price - more data are necessary, or the calculation process may face computational issues, am I right? I am asking because it hit me that robust methods seem to be not as common as non-parametric, bootstrap or other methods, like GEE. $\endgroup$ – kazzzo Mar 3 at 21:36
  • $\begingroup$ @kazzzo There are nonparametric robust methods and there is also bootstrap for robust regression. The price you have to pay is in the efficiency of the robust estimators in clean data-they are outperformed by least squares methods there-and as you point out, they are less easy to compute. $\endgroup$ – JohnK Mar 4 at 4:46

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