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