Dear AllStaters,

Regarding my query on how to do metaanalysis of proportions, I got the following answers.
Thanks indeed to: Bharat Thakrar, Robert G. Newcombe, Paul T Seed, David W. Smith
and Jonathan Sterne
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On way forward is to do something like:
For each study estimate Log(prevelance) 
ie: LPi=Log(Pi), 
The variance of Log(Pi) is:
vi= Lpi(1-Lpi)/ni
(where pi is the prevelence for the i study, ni is the sample in the i study)

Do this for each study.
For each study define the inverse of the variance
ie: wi=1/vi

Now add the wi and call it W
ie: W=w1+w2 +...wn
Now define the weight for the i study as: ki= wi/W
define the pooled weighted  log(prevelence) as: k1*LP1 + k2*LP2 + ....kn*LPn
This is the overall estimate of prevelence.
The variance of this quantity is 1/W.
You can define 95% CI in the usual way, take the antilogs  and you are there.

Regards,
Bharat
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The variance imputed to an estimated proportion depends on two 
things, the sample size and the observed proportion.  It is maximal 
when the proportion is 0.5, and is zero when p=0 or 1.  This suggests 
there may be problems with weighting with the inverse of the 
variance.

The proportion estimated from each individual study should have a 
confidence interval attached, calculated using a good method.  The 
signature file that will attach to the foot of this message contains 
a pointer to my website, from which you can access SPSS and minitab 
macros to do so.

The next step is to carry out some sort of analysis of the degree of 
heterogeneity of the p's between the various studies.  This could 
simply be a chi-square test on a 2 by k table, comparing the k 
observed proportions, with k-1 df.  If numbers positive in some of 
the series are low, then you'll need something more sophisticated, 
for example from StatXact.

If the heterogeneity analysis suggests it is meaningful to pool the 
series together, then I suggest that you simply merge the k series 
together, so that your pooled proportion is sigma r / sigma n, where 
sigma denotes summation over series 1 to k.  Then get a CI for the 
pooled proportion, again preferably using a good method, not just 
using the asymptotic variance.  

Hope this helps.

Best wishes.
Robert Newcombe.

..........................................
Robert G. Newcombe, PhD, CStat, Hon MFPHM
Senior Lecturer in Medical Statistics
University of Wales College of Medicine
Heath Park
Cardiff CF14 4XN, UK.
Phone 01222 742329 or 742311
Fax 01222 743664
Email [log in to unmask]

Macros for good methods for confidence intervals 
for proportions and their differences available at
http://www.uwcm.ac.uk/uwcm/ms/Robert.html
____________________________________________________________________________

Sounds OK to me.  The usual points about fixed vs random effects apply.  In
particular, of there is evidence of heterogeneity, he should look for the
likely causes - major differences between the studies either in method of
measurement or in population.

Paul T Seed       ([log in to unmask])     
Department of Public Health Sciences,   
Guy's Kings and St. Thomas' School of Medicine, 
King's College London,
5th Floor, Capital House 42 Weston Street, London SE1 3QD
tel (44) (0) 171 955 5000 x 6223
fax (44) (0) 171 955 4877
____________________________________________________________________________

A transformation of each proportion, such as to stabilize the variance, is
likely to work better.  The arcsin-square root transformation is
traditional.  A logit transformation might also be
appropriate for prevalence data.

Use p' = arcsin [ sqrt {p} ].  If all the variation is binomial (an unlikely fact) 
then the variance of p' is about 821/n.  This doesn't depend on p or p', so it is a
variance-stabilizing transformation.

In order for this to work well, the  values of p should vary over a range of
0% to 30%.  If they vary from about 30% to 70%, there probably isn't any
point in using it.

I got all this from Snedecor and Cochran, Statistical Methods, eighth
edition, page 289.

I don't have a meta-analysis reference handy.  Snedecor and Cochran seem to
imply that the user would be combining proportions from different sources in their
discussion.

Cordially Yours,
David Smith
Associate Professor
Biostatistics and Epidemiology
College of Public Health - CHB 323
University of Oklahoma Health Sciences Center
PO Box 26901
Oklahoma City, OK 73190
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The short answer is that yes, you can use the meta command to make a
meta-analysis of proportions if you only have p and V(p). The basic 
syntax is: meta p varp, var

However you could run into problems if some studies had small number 
and therefore the approximation for V(p) was poor. Peter Sasieni
([log in to unmask]) presented a program, partly based on meta, at 
the 5th Stata UK users' group meeting which was designed to analyse 
such data directly using (if I remember correctly) exact binomial 
probabilities.

Jonathan Sterne



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