(Edited after I found the correct citation)
In my younger days, we learned from Bill Venable’s Exegeses (still available on the Internet) never to use interactions without the marginal terms, so weight_loss = bmi + bmi*sex was definitively not what the teacher expects.
I checked projpred today, and it happily returned such lonely interactions.
Any reasons?
From Venables/Ripley MASS, page 184 third edition:
A two-factor interaction a:b is marginal to any higher order interaction that contains a and b. Fitting a model such as a +a:b leasts to a model matrix where the columns corresponding to :b are extended to compensate for the absent marginal terms, b, and the fitte values are the same as if it were present. Fitting models with marginal terms removed such as with a*b-b generates a model whith no readily understood statistical meaning… In other words removing marginal factors terms from a fitted model is either statistically meangingless or futile in the sense that the model simply changes its parametrization to something equivalent.