# [R-sig-ME] Log likelihood of a glmer() binomial model .

Juho Kristian Ruohonen juho@kr|@t|@n@ruohonen @end|ng |rom gm@||@com
Sat Apr 20 09:42:16 CEST 2019

Rolf: Forgive my ignorance, but isn't the relevant log-likelihood here the
log-likelihood of the observed responses in the validation set given the
model-predicted probabilities? I.e. sum(dbinom(VS$y, size = VS$n, prob =
predict(fit, newdata = VS, type = "response"), log = TRUE))? Even this
would be somewhat off because dbinom() isn't aware of the random-effects
integral business. But it looks to me like your current call is calculating
the log-sum of the predicted probabilities of y = 1 in the validation set,
not the loglikelihood of the observed responses in the validation set.

la 20. huhtik. 2019 klo 8.51 Rolf Turner (r.turner using auckland.ac.nz)
kirjoitti:

>
> On 20/04/19 12:44 PM, Ben Bolker wrote:
>
> >    This seems wrong.
>
> Yeah, that figures.
>
> > The GLMM log-likelihood includes an integral over
> > the distribution of the random effects.
>
> I was aware of this.  I guess what I was naïvely expecting was that
> predict.merMod() would handle this.  I.e. that this predict method
> (with type = "response") would return, for each observed y_i in the
> (new) data set
>
>    Pr(Y = y_i) = \int_0 Pr(Y = y_i | R = r) f(r) dr
>
> where R is the vector of random effects and f(r) is its probability
> density function (multivariate normal, with mean 0 and some covariance
> matrix, which has been estimated in the fitting process.
>
> I guess that this is *not* what predict.merMod() returns --- but I don't
> understand why not.  It seems to me that this is what it "should" return.
>
> I'm probably misunderstanding something, possibly simple, possibly subtle.
>
> Apropos of nothing much, what *does* predict.merMod() return?
> Maybe Pr(Y = y_i | R = 0) ???
>
> <SNIP>
> >   Here is an **inefficient** method for computing the likelihood
> >
> >     coefs <- unlist(getME(fit,c("theta","beta"))
> >     newdev <- update(fit, data=VS, devFunOnly=TRUE)
> >     newdev(coefs)
> >
> > This is slow because it has to reconstruct all of the random-effects
> > matrices, do permutations to reorder the relevant matrices to be as
> > sparse as possible, etc. etc.
>
> Thanks for this.  I'll give it a go.  I think that the slowness may not
> be an overwhelming drawback.  Anyhow I shall try to test it out.
>
> Thanks again.
>
> cheers,
>
> Rolf
>
> --
> Honorary Research Fellow
> Department of Statistics
> University of Auckland
> Phone: +64-9-373-7599 ext. 88276
>
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