In article <[EMAIL PROTECTED]>,
Bruce Weaver <[EMAIL PROTECTED]> wrote:
>On Thu, 30 Mar 2000, JohnPeters wrote:
>> Hi,
>> I was wondering if someone could help me. I am interested in combining
>> 2x2 tables from multiple studies. The test used is the McNemar's
>> chi-sq. I have the raw data from each of these studies. What is the
>> proper correction that should be used when combining the results.
>> Thanks!!!
>Meta-analysis is a common way to combine information from 2x2 tables, but
>I'm not sure how you would do this with McNemar's chi-square as your
>measure of "effect size" for each table. It might be possible if you
>are willing to use something else.
There are lots of ways to do meta-analysis, and one should
be careful as to what is being done, and why. Doing the
meta-analysis with initial raw data, not filtered through
the refereeing process, is very justified. After this
filter, this is unlikely to be the case. The quoted
material below is ONE way of doing this, and is relevant
in certain situations.
One very important thing to keep in mind is that the
chi-squared test of goodness of fit does NOT have the
chi-squared distribution. This is of small to moderate
importance in testing one table with large cell
frequencies, but can be quite deadly in combining them.
I can give you two examples where I was involved in this.
One was SUPPOSED to have the data on a 7-point scale in one
variable, and continuous in the other, with 26 (?) tables
of size 30. The 7-point scale never had more than 3 values
in any table, and the continuous variable was measured too
coarsely to be continuous. So what was done initially was
to reduce both of these to dichotomies, coming as close to
15 in each marginal division as the data permitted. As
long as one does not look at the joint distribution, this
is perfectly legitimate. The division in the "continuous"
variable was also noted, as this did vary quite a bit from
table to table.
Then the exact probabilities, plus use of random numbers,
was used to transform the tables in such a way that the
null hypothesis would produce uniform (0,1) random
variables. The set of such numbers produced was
insignificant using a Kolmogorov-Smirnov test, but it was
noted that the large values of the breaks in the continuous
scale corresponded to large numbers.
The other was more complicated, in that there were more
than 300 tables, with total sizes ranging from 3 to 9.
Here also, there was no ordering, as there was in the other
case. What was done here was to compute the likelihood
ratio against a reasonable Bayesian alternative, that which
made all the tables with the given marginals equally likely.
Then the means and variances were computed, and a classical
test was made of the sum, using the Central Limit Theorem
to approximate the probabilities. For those interested in
results, one of the sets came out at the 7 sigma level, and
the other sets were ordinary.
>It's Friday afternoon, and this is off the top of my head, but here goes
>anyway. I wonder if you could write the tables this way:
> Change
> Yes No
> - a b
>Before
> + c d
>Cell a: change from - to +
>Cell b: no change, - before and after
>Cell c: change from + to -
>Cell d: no change, + before and after
>Suppose we're talking about change in opinion after hearing a political
>speech. The odds ratio for this table would give you the odds of changing
>from a negative to a positive oppion over the odds of changing from
>positive to negative. If you're the speaker, you're hoping for an odds
>ratio greater than 1 (i.e., greater change in those who were negative
>before the speech). If the amount of change is similar in both groups,
>the odds ratio will be about 1.
>If this is a legitimate way to analyze the data for one such table, and I
>can't see why not, then you could pool the tables meta-analytically with
>ln(OR) as your measure of effect size. Here's a paper that describes how
>to go about it:
>Fleiss, JL. (1993). The statistical basis of meta-analysis. Statistical
>Methods in Medical Research, 2, 121-145.
>There are also free programs available for performing this kind of
>meta-analysis. I have links to some in the statistics section of my
>homepage.
>Hope this helps. Bruce
>Bruce Weaver
>[EMAIL PROTECTED]
>http://www.angelfire.com/wv/bwhomedir/
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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