> > R in general tries hard to prohibit this behavior (i.e., including an
> > interaction but not the main effect). When removing a main effect and
> > leaving the interaction, the number of parameters is not reduced by
> > one (as would be expected) but stays the same, at least
> > when using model.matrix:
Surely this behaviour is less to do with a dislike of interactions without both
main effects (which we will necessarily use if we fit a simple two-factor
nested model) than the need to avoid non-uniqueness of a model fitted with too
many coefficients?
In a simple case, an intercept plus n coefficients for n factor levels gives us
n+1 coefficients to find, and we only have n independent groups to estimate
them from. In model matrix terms we would have one column that is a linear
combination of others. For OLS normal equations that generates a zero
determinant and for the numerical methods R uses the effect is the same; no
useful fit. To avoid that and allow least squares fitting, R sets up the model
matrix with only n-1 coefficients in addition to the intercept. As a result we
end up with fewer model coefficients than we might have expected (and that
annoyingly missing first level that always puzzles newcomers the first time we
look at a linear model summary), but we have exactly the number of coefficients
that we can estimate uniquely from the groups we have specified.
S
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