  Categorical regressor variables are usually handled by introducing a set of
indicator variables, and imposing a linear constraint to ensure identifiability
in the presence of an intercept, or equivalently, using one of various coding
schemes. As proposed in Yuan and Lin [J. R. Statist. Soc. B, 68 (2006), 49-67],
the group lasso is a natural and computationally convenient approach to perform
variable selection in settings with categorical covariates. As pointed out by
Simon and Tibshirani [Stat. Sin., 22 (2011), 983-1001], "standardization" by
means of block-wise orthonormalization of column submatrices each corresponding
to one group of variables can substantially boost performance. In this note, we
study the aspect of standardization for the special case of categorical
predictors in detail. The main result is that orthonormalization is not
required; column-wise scaling of the design matrix followed by re-scaling and
centering of the coefficients is shown to have exactly the same effect. Similar
reductions can be achieved in the case of interactions. The extension to the
so-called sparse group lasso, which additionally promotes within-group
sparsity, is considered as well. The importance of proper standardization is
illustrated via extensive simulations.
