"Standardization" is nice concept but it only goes so far. As an
example, you may want to have a measure of effect size for the
categorical case, but unfortunately there is no unambiguous measure of
effect size in contingency tables. Phi is one of a family of measures,
none of which are fully satisfactory; another well known measure is
the odds-ratio which is, well, odd. Almost all of these statistics
depend on the marginals, which more or less disqualifies them for any
general use, let alone as a suitable definition of effect sizes; the
ones that aren't sensitive to the marginals rely on funny assumptions
like that the categories result from a cut on a continuum, which means
their applicability varies from case to case with the implication they
can't be applied uniformly either. In my view, the situation with
effect sizes 2x2 tables is so dramatic that there is good reason to
doubt whether we at all have a clear idea of what is meant by "effect
size" in that context (it may very well be an inappropriate
generalization of the statistical default mode of thinking in terms of
linear relations between normal distributions). The situation for
multiway tables is fittingly a multiple as bad. If you insist on
having something that counts as an effect size, as far as I'm
concerned you might as well divide the chi-square by N. But the point
is: you can't have a uniform style of reporting because the problems
and their solutions aren't uniform.
Here are some good references on the topic.
Warrens MJ, On similarity coefficients for 2x2 tables and correction
for chance. Psychometrika 73:487-502.
Warrens MJ, On association coefficients for 2x2 tables and properties
that do not depend on the marginal distributions. Psychometrika
73:777-789.
The part of statistics that most people get to see in their practical
research work is a very orderly and preprocessed tip of an otherwise
chaotic and very un-uniform iceberg. Statistics is a jungle, really.
Best
d
--
Denny Borsboom
Department of Psychology
University of Amsterdam
Weesperplein 4
1018 XA Amsterdam
The Netherlands
+31 20 525 6882
d.bor...@uva.nl
http://sites.google.com/site/borsboomdenny/dennyborsboom