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Penalized regression approaches are standard tools in quantitative genetics.It is known that the fit of an \emph{ordinary least squares} (OLS) regressionis independent of certain transformations of the coding of the predictorvariables, and that the standard mixed model \emph{ridge regression best linearunbiased prediction} (RRBLUP) is neither affected by translations of thevariable coding, nor by global scaling. However, it has been reported that anextended version of this mixed model, which incorporates interactions byproducts of markers as additional predictor variables is affected bytranslations of the marker coding. In this work, we identify the cause of thisloss of invariance in a general context of penalized regression on polynomialsin the predictor variables. We show that in most cases, translating the codingof the predictor variables has an impact on effect estimates, with an exceptionwhen only the size of the coefficients of monomials of highest total degree arepenalized. The invariance of RRBLUP can thus be considered as a special case ofthis setting, with a polynomial of total degree 1, where the size of the fixedeffect (total degree 0) is not penalized but all coefficients of monomials oftotal degree 1 are. The extended RRBLUP, which includes interactions, is notinvariant to translations because it does not only penalize interactions (totaldegree 2), but also additive effects (total degree 1). Our observations are notrestricted to ridge regression, but generally valid for penalized regressions,for instance also for the $\ell_1$ penalty of LASSO.

Lost in translation: On the impact of data coding on penalized regression with interactions