Greetings All!
I have been encountering the so-called "Wald Test",
especially in regression analyses. I have a few queries.
Depending on context, this apparently might be defined as
square of (coeff estimate)/(SE of coeff estimate)
(Effect SS)/(Res SS/Res df) = F statistic times Numerator df
or similar expressions involving log-likelihoods or deviances.
The end-product is something to be referred to a chi-squared
distribution with df equal (in the above cases) to "numerator
degrees of freedom".
The above is necessarily vague, since it is distilled from
examples of use and discussion, and not derived from formal
definition. However, I'm trying to get to the bottom of it.
QUERIES:
1. Formal definition of "Wald Test"? Primary source thereof?
2. When did it start to really "come in"? Clearly, to bear the
great name of Wald it must be traceable back to something
which happened a long time ago, but its existence as a
"named test" in common use seems to be relatively recent.
3. Who can be credited with naming the "Wald Test" as commonly
used? And where?
4. I cannot see (from the above) that the "Wald test" can do
anything which cannot also be done (and probably better)
by other means. In particular, (estimate/SE) is basically
"t"; why necessarily square it? And the "chi-squared" amounts
to ignoring sampling variation of the SE: what merits could
this have? Likewise, the SS-ratio is basically F; why degrade
the distributional aspect by using chi-squared? And how does
it score over the usual deviance or likelihood-ratio tests?
5. In short: are there circumstances in which the "Wald Test"
enjoys definite theoretical advantages over other approaches
(t, t^2, F, likelihood-ratio, etc.)?
Does it perhaps have a robustness which may not be enjoyed by
the others? Clearly it has asymptotic validity, but so does
everything else.
POSTSCRIPT:
On the GENSTAT discussion list, on this topic, Nick Galwey
expressed a memory of hearing John Nelder, at a Conference,
proposing that Wald tests be banned; but not a memory of his
reasons.
If this happened, can anyone supply more details?
I'll happily summarise in due course.
With thanks,
Ted.
--------------------------------------------------------------------
E-Mail: (Ted Harding) <[log in to unmask]>
Fax-to-email: +44 (0)870 167 1972
Date: 10-Oct-01 Time: 20:12:25
------------------------------ XFMail ------------------------------
|