LASSO regression shrinks coefficients towards zero, thus providing effectively model selection. I believe that in my data there are meaningful interactions between nominal and continuous covariates. Not necessarily, however, are the 'main effects' of the true model meaningful (non-zero). Of course I do not know this since the true model is unknown. My objectives are to find the true model and predict the outcome as closely as possible.
I have learned that the classical approach to model building would always include a main effect before an interaction is included. Thus there cannot be a model without a main effect of two covariates $X$ and $Z$ if there is an interaction of the covariates $X*Z$ in the same model. The
step function in
R consequently carefully selects model terms (e.g. based on backward or forward AIC) abiding to this rule.
LASSO seems to work differently. Since all parameters are penalized it may without doubt happen that a main effect is shrunk to zero whereas the interaction of the best (e.g. cross-validated) model is non-zero. This I find in particular for my data when using
I received criticism based on the first rule quoted above, i.e. my final cross-validated Lasso model does not include the corresponding main effect terms of some non-zero interaction. However this rule seems somewhat strange in this context. What it comes down to is the question whether the parameter in the true model is zero. Let's assume it is but the interaction is non-zero, then LASSO will identify this perhaps, thus finding the correct model. In fact it seems predictions from this model will be more precise because the model does not contain the true-zero main effect, which is effectively a noise variable.
May I refute the criticism based on this ground or should I take pre-cautions somehow that LASSO does include the main effect before the interaction term?