🚀 go-pugleaf

[in an earlier note I (mcclella@yertle.colorado.edu) suggested
that one reason not to use stepwise regression was because it
"demonstrably suboptimal;" in particular, it is dominated by all
subsets regression which most computers can now do efficiently]

thompson@atlas.socsci.umn.edu (T. Scott Thompson) responds:

>Your criteria that we use only strategies that are not "demonstrably
>suboptimal" would rule out the use of least squares regression for
>estimating linear models in most cases, since least squares is not an
>admissible procedure for most of the statistical decision problems to
>which it is applied.  In fact least squares is dominated by at least
>one form of stepwise regression -- backwards elimination -- provided
>the cutoff values for elimination are chosen appropriately.  Backwards
>elimination is essentially a form of pretesting.  The discovery of
>pretest estimators (and of the closely related shrinkage estimators),
>which typically dominate their non-pretest cousins, showed that more
>conventional methods (e.g. ordinary least squares) typically are not
>admissible.

Scott rightfully takes me and others to task for not adequately
considering the implications of our loss functions (which are
often only implicit) for the statistical decision rules we adopt. 
We ought to do a better job of that.  However, for many types of
regression problems with which I deal backwards elimination 
definitely does _not_ dominate ordinary least squares.  This
class of problems invovles predictors which are products of other
predictors.  Unless one has ratio scales (a rarity in my areas of
substantive interest) it is very important to include each
component of a product in the model even if individual components
are not "statistically reliable."  For example, if one has the
component XZ^2 in a model, then it is important to also include
X, Z, XZ, and Z^2.  The problem with backwards elimination (or
any other steplike procedure) is that it isn't aware of such
constraints and often as not will not satisfy them.  Hierarchical
tests under the control of the data analyst are better.  For
this example, one would consider the model with X, Z, XZ, and Z^2
and then see if there is any improvement over and beyond that
model when adding XZ^2 as a predictor.

But I will follow Scott's advice to read:

>"Introduction to the theory and practice of econometrics" by George
>Judge et. al., chapter 20.

gary mcclelland
univ of colorado
mcclella@yertle.colorado.edu