[R-sig-ME] calculating number of parameters for AICc

[Ben Bolker]

Jude Phillips <birdlists at ...> writes:

> 
> Hi Listers,
> 
> I am wondering about how 'K' (number of parameters in AICc equation) is
> calculated for mixed effects models. I am using aictab in the AICcmodavg
> library, to produce an AICc table for a set of models.  In the table, K
> seems to equal number of fixed effects + number of random effects + 2.  Is
> that the correct calculation?  In Vaida and Blanchard (2005), they suggest
> that for the 'population' focus (ie when you are interested in the fixed
> effects, not the random effects, which is the case for my study), K is the
> number of fixed parameters counting mean parameters and variance
> components.  I'm not sure how you calculate the number of variance
> components, so I'm not sure if this gives the same value for K as is
> supplied in aictab?  So I'm wondering if the K given in aictab is really
> correct for a 'population focus', and if not, how should K be calculated?

   Well, you can just look in the AICc.mer() function:

K <- attr(logLik(mod), "df")

this leads you (obscurely, I admit) to getMethods("logLik",sig="mer"):

  attr(val, "df") <- dims[["p"]] + dims[["np"]] + 
         as.logical(dims[["useSc"]])

To get past here, you would have to dig quite a bit deeper, but
basically the answer is that dims[["p"]] is the same as
length(fixef(model)) -- i.e., the number of fixed-effect coefficients --
and dims[["np"]] is the same as length(getME(model,"theta")) -- the
number of variance parameters.  For each random term in the model with
q components, it has q*(q+1)/2 parameters -- for example, a term of
the form (slope|group) has 3 parameters (intercept variance, slope
variance, correlation between intercept and slope).  The last term
says whether the model uses a scale parameter or not (yes for
linear mixed models, no for typical GLMMs like binomial or Poisson).

Your statement of "number of fixed effects + number of random effects + 2"
doesn't seem correct, but perhaps if you gave an example ...

Whether to add nuisance parameters or not, such as the residual
variance parameter that is estimated based on the residual variance,
is as far as I know an open question.  In the classic AIC context
it doesn't matter as long as one is consistent.  In the AICc context,
I don't think anyone really knows the answer ... adding +1 for
the residual variance parameter (as lme4 does) would make the
model selection process slightly more conservative.