[R-sig-ME] how reliable are inferences drawn from binomial models for small datasets fitted with lme4?
Roger Levy
rlevy at ling.ucsd.edu
Mon Jul 6 20:10:49 CEST 2009
On Jul 5, 2009, at 8:58 PM, Luca Borger wrote:
> Hello,
>
> but for m01 I think the model output:
>
> ####
>> print (m01 <- lmer(Response ~ Treatment + (Treatment - 1 | F1) +
> (1 | F2), dat, family=binomial))
> Generalized linear mixed model fit by the Laplace approximation
> Formula: Response ~ Treatment + (Treatment - 1 | F1) + (1 | F2)
> Data: dat
> AIC BIC logLik deviance
> 87.58 107.1 -37.79 75.58
> Random effects:
> Groups Name Variance Std.Dev. Corr
> F2 (Intercept) 1.6467e-12 1.2832e-06
> F1 Treatment1 9.0240e+00 3.0040e+00
> Treatment2 3.4832e+00 1.8663e+00 -1.000
> Number of obs: 190, groups: F2, 24; F1, 16
> ###
>
>
> indicates that there is not enough information to estimate that
> model (Corr=-1.00) and that the random effects part should be
> simplified notwithstanding the logLik value? Sorry fpr adding a
> question and hope this is relevant.
Dear Luca,
Many thanks for your thoughts on this -- I did notice the correlation
for F1 but in my thinking, this was sensible behavior by the model and
didn't necessarily indicate a need to simplify the F1 random effects
component. Let me explain my reasoning. Going back to the marginal
mean responses per F1/Treatment combination:
> with(dat,tapply(Response, list(F1, Treatment), mean))
1 2
1 1.0000000 1
2 0.5714286 1
3 0.8888889 1
4 1.0000000 NA
5 0.0000000 1
6 1.0000000 NA
7 0.6666667 1
8 1.0000000 1
9 1.0000000 1
10 1.0000000 1
11 0.8181818 1
12 0.6000000 1
13 1.0000000 NA
14 1.0000000 1
15 1.0000000 1
16 1.0000000 1
it seems to me that the only way a random effect can contribute to the
likelihood is by bringing down the linear predictor in Treatment for
levels 2, 3, 7, 11, and 12 of F1. So the random effects for these
levels in Treatment 1 must be negative. Now, given this constraint,
what choice random effects for these levels in Treatment 2 maximize
the likelihood? Since there are never any Treatment 2 failures, these
random effects should be as high as possible. This leads to an
inferred correlation of -1 for the random effects of F1.
The surprise to me isn't really that the random-effect correlation is
so extreme, but rather that the model log-likelihoods seem to indicate
that there's sufficient evidence to go with the model with more
complex random-effects structure. Which leads me to ask about how
reliable log-likelihood differences for mixed-effects logit models on
small datasets like this may be.
Many thanks once more!
Roger
--
Roger Levy Email: rlevy at ling.ucsd.edu
Assistant Professor Phone: 858-534-7219
Department of Linguistics Fax: 858-534-4789
UC San Diego Web: http://ling.ucsd.edu/~rlevy
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