# Re: RE: st: RE: Shrinkage factor

 From Robert A Yaffee To statalist@hsphsun2.harvard.edu Subject Re: RE: st: RE: Shrinkage factor Date Wed, 14 Oct 2009 20:54:20 -0400

```prefer the use of "a pooling factor" in multilevel models to
indicate the the degree to which elements are pooled together.

They use the same formula for the residual intraclass coefficient that
is used for the shrinkage factor on population distribution a,
but refer to 1-B  as the pooling factor
when B = 1 - [ sigma^2/(sigma^2 + sigma_y^2/n_j)]

for them,   a_j (multilevel) =  B mu_a +  (1-B) ybar_j
where
ybar_j = avg of the y's within each group j
mu_a = average of the population

B = 0 when there is no pooling a_j=ybar_j
= 1 when there is complete pooling  a_j = mu_a

- Hope this helps.   This comes from Gelman and Hill  Data Analysis using regression and multilevel/hierarchical models, Cambridge University Press, p. 477.
Bob

Robert A. Yaffee, Ph.D.
Research Professor
Silver School of Social Work
New York University

Biosketch: http://homepages.nyu.edu/~ray1/Biosketch2009.pdf

CV:  http://homepages.nyu.edu/~ray1/vita.pdf

----- Original Message -----
From: Robert A Yaffee <bob.yaffee@nyu.edu>
Date: Wednesday, October 14, 2009 8:18 pm
Subject: Re: RE: st: RE: Shrinkage factor
To: statalist@hsphsun2.harvard.edu

> Elan, Evans,
>    Carlin and Lewis in their 3rd edition of Bayesian Methods for Data
> Analysis
> describe the Bayesian Shrinkage factor B = sigma^2/(sigma^2 + tau^2)
> where tau^2 would be the variance of the prior distribution  while sigma^2
> would be the  normal density  of the sample (or likelihood),  p. 17.
>
>    B is also used to compute the posterior mean =   B (mu) + (1-B)y
> a weighted average of the prior mean and that of the sample.
>     Regards,
>            Bob
>
>
> Robert A. Yaffee, Ph.D.
> Research Professor
> Silver School of Social Work
> New York University
>
> Biosketch: http://homepages.nyu.edu/~ray1/Biosketch2009.pdf
>
> CV:  http://homepages.nyu.edu/~ray1/vita.pdf
>
> ----- Original Message -----
> From: "Cohen, Elan" <cohened@upmc.edu>
> Date: Wednesday, October 14, 2009 1:40 pm
> Subject: RE: st: RE: Shrinkage factor
> To: "'statalist@hsphsun2.harvard.edu'" <statalist@hsphsun2.harvard.edu>
>
>
> > Just based on the index, the following book may be helpful:
> >
> > http://www.stata.com/bookstore/mlmus2.html
> >
> > - Elan
> >
> >
> > > -----Original Message-----
> > > From: owner-statalist@hsphsun2.harvard.edu
> > > [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of
> > > Sent: Wednesday, October 14, 2009 1:07 PM
> > > To: statalist@hsphsun2.harvard.edu
> > > Subject: Re: st: RE: Shrinkage factor
> > >
> > > Nick Cox wrote:
> > > > If there were, then a simple search would almost certainly find
> it.
> > > > -findit shrinkage- yields no hits. Did you try a Stata or
> > > >
> > > >
> > > > Nick
> > > > n.j.cox@durham.ac.uk
> > > >
> > > > Evans Jadotte
> > > >
> > > > I am estimating a three-level hierachical model using
> > > xtmixed and want
> > > > to get the 'shrinkage factor' (Rj) to help me with the
> > > calculation of
> > > > the variance for an empirical Bayes estimate. My model has many
>
> > > > covariates and clusters and this makes a manual calculation
> > > of the Rj
> > > > not malleable. Is there any user-written command to get the Rj?
> > > >
> > > > Hope my request is not too confusing and can receive some help.
> > > >
> > > > *
> > > > *   For searches and help try:
> > > > *   http://www.stata.com/help.cgi?search
> > > > *   http://www.stata.com/support/statalist/faq
> > > > *   http://www.ats.ucla.edu/stat/stata/
> > > >
> > > I tried under both findit shrinkage and findit reliability (this
> > last
> > > one took me to xtmepoisson but no further help), with no luck.
> > >
> > > Evans
> > > *
> > > *   For searches and help try:
> > > *   http://www.stata.com/help.cgi?search
> > > *   http://www.stata.com/support/statalist/faq
> > > *   http://www.ats.ucla.edu/stat/stata/
> > >
> > >
> > *
> > *   For searches and help try:
> > *   http://www.stata.com/help.cgi?search
> > *   http://www.stata.com/support/statalist/faq
> > *   http://www.ats.ucla.edu/stat/stata/
> *
> *   For searches and help try:
> *   http://www.stata.com/help.cgi?search
> *   http://www.stata.com/support/statalist/faq
> *   http://www.ats.ucla.edu/stat/stata/
*
*   For searches and help try:
*   http://www.stata.com/help.cgi?search
*   http://www.stata.com/support/statalist/faq
*   http://www.ats.ucla.edu/stat/stata/
```