[R-sig-ME] Replicating type III anova tests for glmer/GLMM

[Fox, John]

Dear Emmanuel,

First, the relevant linear hypothesis is for several coefficients simultaneously -- for example, all 3 coefficients for the contrasts representing a 4-level factor -- not for a single contrast. Although it's true that any linear combination of parameters that are 0 is 0, the converse isn't true. Second, for a GLMM, we really should be talking about type-III tests not type-III sums of squares.

Type-III tests are dependent on coding in the full-rank parametrization of linear (and similar) models used in R, to make the tests correspond to reasonable hypotheses. The invariance of type-II tests with respect to coding is attractive, but shouldn't distract from the fundamental issues, which are the hypotheses that are tested and the power of the tests. 

Best,
 John

> -----Original Message-----
> From: Emmanuel Curis [mailto:emmanuel.curis at parisdescartes.fr]
> Sent: February 23, 2016 11:50 AM
> To: Fox, John <jfox at mcmaster.ca>
> Cc: Francesco Romano <francescobryanromano at gmail.com>; r-sig-mixed-
> models at r-project.org
> Subject: Re: [R-sig-ME] Replicating type III anova tests for glmer/GLMM
> 
> Dear Pr Fox,
> 
> Thanks for your precision. But to summarize this test of, let's say 3
> parameters to 0 for a 4-levels factor, by a single value with its SE, as
> mentionned in Francesco's mail, the linear combination of these parameters
> that is practically tested by this sum of square is needed, isn't it ?
> 
> I mean, if really the parameters are all 0, whatever linear combination could
> do the job, but type III sum of square just tests one of all possible linear
> combinations, right?
> 
> By the way, I was always very annoyed by the fact that Type III sum of
> squares are so dependent on coding, but that's another debate...
> 
> Best regards,
> 
> On Tue, Feb 23, 2016 at 04:15:02PM +0000, Fox, John wrote:
> < Dear Emmanuel,
> <
> < With proper contrast coding (i.e., a coding that's orthogonal in the *basis*
> of the design, such as provided by contr.sum() ), a "type-III" test is just a test
> that the corresponding parameters are 0. The models in question are
> generalized linear (mixed) models and so sums of squares aren't really
> involved, but one could do the corresponding Wald (like car::Anova) or LR
> test. The Wald test is what you'd get with multcomp:glht or
> car:linearHypothesis. BTW, I don't think that it would be hard for car::Anova
> to be extended to provide LR tests in this case.
> <
> < Best,
> <  John
> 
> --
>                                 Emmanuel CURIS
>                                 emmanuel.curis at parisdescartes.fr
> 
> Page WWW: http://emmanuel.curis.online.fr/index.html