[R] About lmer output

[Douglas Bates]

On 1/25/06, Juan Pablo Sánchez <juansan at dca.upv.es> wrote:
> Dear R users:
> I am using lmer fo fit binomial data with a probit link function:
>
> > fer_lmer_PQL<-lmer(fer ~ gae + ctipo + (1|perm) -1,
> +                family = binomial(link="probit"),
> +                method = 'PQL',
> +                data = FERTILIDAD,
> +                msVerbose= True)
>
> The output look like this:
> > fer_lmer_PQL
> Generalized linear mixed model fit using PQL
> Formula: fer ~ gae + ctipo + (1 | perm) - 1
>    Data: FERTILIDAD
>  Family: binomial(probit link)
>       AIC      BIC    logLik deviance
>  2728.086 2918.104 -1332.043 2664.086
> Random effects:
>      Groups        Name    Variance    Std.Dev.
>        perm (Intercept)     0.28256     0.53156
> # of obs: 2802, groups: perm, 529
>
> Estimated scale (compare to 1)  0.8958656
>
> My question is about the meaning of  "Estimated scale (compare to 1)  0.8958656 "
>
> I think that the scale would be 0.28256+1.0, Isn´t it?

The estimated scale is what would be the estimate of the scale
parameter in the GLM family if there was a scale parameter.  For the
binomial and Poisson families there is no scale parameter but the
Iteratively Reweighted Least Squares (IRLS) algorithm still produces
an estimate of one.  If the data are neither overdispersed nor
underdispersed then that estimate should be close to 1.

It can provide a diagnostic for the model.  A value that is
substantially different from 1 indicates model failure or
over-modeling the data.  I would say that the value of 0.896 is close
to an indication of underdispersion.  Frequently this is caused be
including random effects associated with groups that have few
observations in each group.