[R-sig-ME] laplace deviance crossed factors glm

[Ben Pelzer]

Dear Douglas,

Thanks for your prompt reply, no nesting indeed! I now realize that a 
single variance term for both groups can lead to a better fit (smaller 
deviance) than two different variances, one for each group. At first 
thought this seemed counterintuitive to me. Regards,

Ben.



Op 22-2-2010 17:07, Douglas Bates schreef:
> On Mon, Feb 22, 2010 at 9:25 AM, Ben Pelzer<b.pelzer at maw.ru.nl>  wrote:
>> Dear all,
>>
>> Recently I ran into a counterintuitive result related to the Laplace
>> deviance value produced by lmer for a logistic regression model.
>>
>> For logistic regression models, one of the methods lmer can use to derive an
>> (approximate) deviance value, is
>> called 'Laplace approximation'. As explained in the lmer package's pdf, this
>> Laplace deviance is equal to the deviance that one would obtain when using
>> the 'adaptive Gaus-Hermite quadrature' (AGQ) with only 1 sample point.
>> However, with AGQ one can use more sample points to increase the precision
>> of the approximation to the true deviance. Hence, the Laplace deviance is
>> 'worse' than the AGQ deviance that uses, say, five sample points. So far so
>> good.
>>
>> My question now is, in how far it is safe to use the Laplace deviance to
>> compare the fit of two nested glm models. Realizing that the Laplace
>> deviance can be pretty rough, the Laplace deviance difference between two
>> nested models may not be as nicely chisquare distributed as we would like it
>> to be. Is there literature on this subject?
>>
>> This question was inspired by running a  model, in which two random factors
>> A and B are crossed, having 214 and 40 levels, respectively. For each (A,B)
>>   combination there is exactly 1 observation in my data, so random
>> interaction of A and B cannot be modeled. In total there 214 * 40 minus 1 =
>> 8559 observations, the "minus 1" caused by the fact that  for 1 particular
>> (A,B) combination the dependent Y is missing.
>>
>> The nice thing about the Laplace approximation method is that it can be
>> applied to such cross-factorial glm models, while AGQ (with more than 1
>> quadr. point) cannot. However, comparing the deviances of two nested
>> cross-factorial models, I discovered that the more restricted model has a
>> smaller (closer to zero) deviance, than the full model! This doesn't make
>> sense, of course. There appear to be no convergence problems, the models
>> being relatively simple.
>>
>> The restricted and full model differ as follows. There are two (independent)
>> groups of observations in the data, the indicator variables (0/1) gr1 and
>> gr2 indicating the group to which an observations belongs. In the full
>> model, the random effect variances of A and B are allowed to vary across the
>> two groups, so in total there are 4 variances to be estimated,  var(A) for
>> group 1, var(A) for group 2, var(B) for group 1 and var(B) group 2.  In the
>> restricted model, both groups have the same random effect variance for A,
>> but each group has a different variance for B, so there are three variances
>> to be estimated now, var(A), var(B) for group 1 and var(B) for group 2. In R
>> syntax:
>>
>> # full model; deviance = 7095.
>> M1<- lmer (Y ~  1  +  (gr1+0|A)  +  (gr2+0|A)  +  (gr1+0|B)  +  (gr2+0|B)
>>   +  gr1,
>>                    family=binomial(link="logit"), reml=FALSE)
>>
>> # restricted model; deviance = 7076.
>> M2<- lmer (Y ~ 1  +  (gr1+0|A)  +  (gr2+0|A)  +  (1|B)  +  gr1,
>>                     family=binomial(link="logit"), reml=FALSE)
>
> Those models don't appear to be nested.  The second model has a
> random-effects term (1|B) and I don't see such a term in the first
> model.
>
> (By the way, the argument name is "REML", not "reml" and it is
> unnecessary to provide it to a generalized linear mixed model.)
>> Now I'm wondering where the unexpected reduction of the deviance may come
>> from. Could this be related to the fact that, for comparing nested models,
>>   the Laplace deviance should not be used? Or am I simply overlooking
>> something which has nothing to do with the Laplace deviance at all? Any tip
>> would be greatly appreciated. Kind regards,
>>
>> Ben
>>
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>
>