I've been reading more stuff about empirical bayes methods in meta-analysis
recently and realised that what I posted last month in reply to Aurelio
Atobias's query was completely wrong, so I think I better set the record
straight:
The form of empirical Bayes used in -meta- is simply another way of looking
at the random-effects model. The EB combined estimate that Aurelio was after
is exactly the same as the random-effects pooled estimate that is always
given in the output of -meta-. The second pass through -meta- I suggested
before is utter nonsense.
As an apology for confusing instead of clarifying things before, here's some
further explanation of what's going on as I now understand it, together with
a couple of references I've found particularly useful:
The individual EB estimates are identical to what are known in the REML
approach to mixed models as Best Linear Unbiased Predictors (or BLUPs for
short) of the random effects. Both approaches can explain the 'shrinkage' of
these estimates towards the mean.
The concept of BLUPs has caused some confusion, which may be why calling
them empirical Bayes estimates is more usual in most areas (the term BLUP
seems to have caught on mainly in animal breeding - a major application of
REML). Robinson (1991) attempts to clarify what BLUPs are, explains that
BLUPs are identical to the individual posterior estimates under the most
commonly used Empirical Bayes model, and essentially attempts a
reconciliation between these two ways of looking at things. I read this a
while ago when I was coming at things from the other direction (i.e. an
interest in REML rather than empirical Bayes), but somehow managed to get
forget all about it last month.
(Note that the above applies only to *empirical* Bayes. True Bayes is a
different matter.)
The Empirical Bayes framework has the advantage of perhaps making it clearer
that the CI for the random-effects pooled estimate is the CI for the mean of
a distribution, and is therefore not a measure of the width of the whole
distribution. Carlin (1992) expresses concern that
"..commonly used [random-effects] analyses tend to place undue emphasis on
inference for the overall mean effect. Uncertainty about the probable
treatment effect in a particular population where a study has not been
performed (...) might be more reasonably represented by inference for a new
study effect ... rather than for the overall mean. ... In this case,
uncertainty is of course much greater."
Hope that makes things clearer. Having confused things totally in my
previous post I thought I better take a bit more trouble this time.
Roger.
Robinson, G. K. (1991). That BLUP is a good thing: the estimation of random
effects. Statistical Science, 6, 15-51.
Carlin, J.B. (1992). Meta-analysis for 2x2 tables: a Bayesian approach.
Statistics in Medicine, 11, 141-158.
On Fri, 26 Jul 2002 11:37:03 +0100 Roger Harbord <[email protected]>
wrote:
> > Date: Thu, 25 Jul 2002 11:08:04 +0200
> > From: atobias <[email protected]>
> > Subject: st: Empirical Bayes using meta ado
> >
> > Dear Statalisters,
> >
> > I am trying to carry out a Empirical Bayes meta-analysis (EB) using
> > the "meta" ado file (by Sharp & Sterne). But ... although the "meta"
> > command computes in a right way the EB estimates for each individual
> > study, the combined estimate --by using either fixed or random effects
> > model-- is not based in such EB individual estimates (!).
> >
> > I will be grateful if anyone could clarify me if there is a bug (?).
> > Thanks indeed
> >
> I'll try to answer this, as I know Sterne is highly unlikely to have
> time at the moment (can't speak for Sharp though :-)
>
> >From the -meta- help file:
>
> "-ebayes- creates two new variables in the dataset: ebest contains
> empirical Bayes estimate for each study, and ebse the corresponding
> standard errors."
>
> In fact, this is *all* the -ebayes- option does when used on its own -
> it changes nothing in the output displayed in the results window. If
> you want to compute combined estimates (under both random and fixed
> effects models) based on the EB individual estimates, follow your
> initial "meta ..., ebayes" command by "meta ebest ebse".
>
> Note that "meta ..., graph(e)" automatically requests the -ebayes-
> option and gives a graph of the EB individual estimates plus a
> combined estimate produced by combining those EB individual estimates
> under a random effects model. However, it doesn't display this
> combined estimate in the results window, so if you want to know its
> numerical value, you still need to do "meta ebest ebse" afterwards.
>
> I'd agree that this is a little confusing and arguably the -ebayes-
> option should do this for you automatically. However, this two-stage
> procedure does make it clearer how the estimates are being produced, and
> the -ebayes- option seems unlikely to be of interest to casual users.
>
<snip>
>
> Roger.
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