Re: st: -predict , reffects- after -xtmelogit-

[Jeph Herrin <stata@spandrel.net>]

Bobby,

This is very helpful, thanks. I understand shrinkage, but it didn't
click when I read the documentation that the distinction was made here.

So is there a way to get the mle random effects? I tried

 predict ymu, mu
 predict yxb, xb
 gen yrwe=logit(ymu)
 gen re_cons=ywre-yfix

but this doesn't agree with either sd(_cons) nor with the result
of -predict, reffects-

thanks,
Jeph


On 12/20/2010 1:09 PM, Roberto G. Gutierrez, StataCorp wrote:
> Jeph Herrin<stata@spandrel.net>  asks:
>
> > I am using -xtmelogit- to estimate a random effects model, and am wondering
> > about what is being predicted by -predict, reffects-.
>
> > Example:
>
> >     clear
> >     use http://www.stata-press.com/data/r11/bangladesh
> >     xtmelogit c_use || district:
> >     predict re_cons, reffects
>
> When you use -predict, reffects- after -xtmelogit-, you obtain estimates of
> the modes of the posterior distribution of the random effects given the data
> and estimated parameters; see pg. 277 of [XT] xtmelogit postestimation for a
> complete discussion.
>
> > Now, I would expect the standard deviation of the random effect reported by
> > the model:
>
> > --------------------------------------------------------
> >     Random-effects Parameters  |   Estimate   Std. Err.
> > -----------------------------+--------------------------
> > district: Identity           |
> >                      sd(_cons) |   .4995265   .0798953
> > --------------------------------------------------------
>
> > To be approximately the standard error of the predicted randome effects, at
> > the district level:
>
> >     bys district : gen tolist = _n==1
> >     sum re_cons if tolist
>
> >       Variable |       Obs        Mean    Std. Dev.       Min        Max
> > -------------+--------------------------------------------------------
> >        re_cons |        60    .0069783    .3787135  -.9584643   .9257698
>
> > But it seems very different, 0.4995 vs .37871. I must be missing something
> > obvious, but what?
>
> The phenomenon you are seeing is known as "shrinkage".  Predictions based on
> the random-effects posterior distribution tend to be closer in magnitude to
> zero because they are incorporating the prior information that the random
> effects have mean zero.  That is, if you have a relatively small cluster size
> the prior information that the random effect should be zero tends to dominate.
> The estimate of sd(_cons) is, in contrast, based on maximum likelihood where
> all the clusters are considered jointly.  Thus, prior information tends to not
> dominate as much because all clusters are pooling what they have to say about
> the random-effects standard deviation.
>
> Shrinkage dimishes as cluster size gets larger.  To see this, try
>
>      . clear
>      . set seed 1234
>      . set obs 100				// 100 clusters
>      . gen u = sqrt(2)*invnorm(uniform())	// random effects
>      . gen id = _n
>      . expand 1000				// cluster size is 1000
>      . gen e = log(1/runiform() - 1)             // logistic errors
>      . gen y = (e + u)>  0			// binary response
>      . xtmelogit y || id:
>      . predict r, reffects
>      . bysort id: gen tolist = _n==1
>      . sum r if tolist
>
> The standard deviations match much more closely -- having a cluster size of
> 1,000 helps!
>
> --Bobby
> rgutierrez@stata.com
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