[R-sig-ME] F and Wald chi-square tests in mixed-effects models

[Helios de Rosario]

[I sent this question to the R-help list, but following Ben Bolker's
advice, I re-send it here. Please excuse me for the duplication.]

I know that, except in specific cases like well-balanced designs, the F
statistic that is usually calculated in ANOVA tables may be far from
being distributed as a known, exact F distribution, and that's the
reason why the anova method on "mer" objects (calculated by lmer) does
not calculate the denominator df nor a p-value. --- See for instance
Douglas Bates' long post on this topic, in:
https://stat.ethz.ch/pipermail/r-help/2006-May/094765.html 

However, Anova (car package) does calculate p-values from Wald
chi-square tests for fixed terms in "mer" objects (as well as in "lme"
objects). I suppose that the key to understand the logic for this is in
Fox & Weisberg's commentary in "An R Companion to Applied Regression"
(2nd edition, p. 272), where they say: "Likelihood ratio tests and F
tests require fitting more than one model to the data, while Wald tests
do not."

Unfortunately, that's too brief a commentary for me to understand why
and how the Wald test can overcome the deficiencies of F-tests in
mixed-effects models. The online appendix of "An R Companion..." about
mixed-effects models does not comment on hypothesis tests either.

I would appreciate if someone can give some clues or references to read
about this issue.

Thanks,
Helios

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