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implementation of boschloo's test: very slow execution


From   "Eva Poen" <eva.poen@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   implementation of boschloo's test: very slow execution
Date   Fri, 22 Feb 2008 14:27:38 +0000

Dear Statalisters,

I am currently trying to implement the unconditional test for
comparing two binomial proportions by Boschloo. The test is described
e.g. here: http://www.west.asu.edu/rlberge1/papers/comp2x2.pdf (page
3).

The problem with my implementation is that it takes a long time to
execute: for n1=148 and n2=132, it takes roughly 30 minutes on a
reasonably modern machine. I am using Stata 9.2 for this. I was
wondering if anyone could suggest improvements to make it faster.

Here is how the test works:
Say you have two samples of size n1 and n2. In sample 1, there are x1
successes whereas in sample 2 there are x2.
a) compute fisher's exact test for the two proportions and record the
two-sided p-value.

b) for every possible combination of successes in the two samples
(which is n1Xn2 combinations):
   b1) compute fisher's exact test
   b2) check whether the p-value is less than or equal to the p-value
from the original sample  in step a).
   b3) if it is <= the result from a), compute the product of the
binomial probability mass functions for this combination. Do this for
(a lot) of values for 0 <= p <= 1. If it is > the result from step a)
, do nothing.

c) The result of step b3) is added up over all n1Xn2 iterations.

d) Search for the maximum of c) over all values of p. This maximum is
the p-value for this test.

Here is what I do in every step.
I first create a variable that holds all probability values of p that
I want to do the calculations for:
    set obs 10001
    qui gen double p = (_n-1)/10001
The higher the "resolution" of p, the more accurate is the result going to be.

I also create a variable that holds the current calculations, and one
that holds the (running) sum over all iterations:
    qui gen double PH0sum = 0
    qui gen double current = .

a)
    qui tabi `=n1-x1' `=n2-x2' \ `=x1' `=x2', exact
    scalar fisher = r(p_exact)

b) and c)
    forvalues xx1 = 0/`=n1' {
        forvalues xx2 = 0/`=n2' {
            qui tabi `=n1-`xx1'' `=n2-`xx2'' \ `xx1' `xx2', exact
            scalar fishnew = r(p_exact)
            if fishnew <= fisher {
                qui replace current =
Binomial(n1,`xx1',theta)-Binomial(n1,`=`xx1'+1',theta))*(Binomial(n2,`xx2',theta)-Binomial(n2,`=`xx2'+1',theta))
                qui replace PH0sum = PH0sum + current
            }
        }
    }

d)
qui sum PH0sum
scalar p_boschloo = r(max)

********************************************************

I can replicate an example I found in a Biometrics(2003) paper by
Mehrotra et al., so I believe that the calculations are correct. Is
there a way to speed things up inside the nested loop? I'd be greatful
for any hints, as I have to run quite a lot of these tests; time does
matter here.

Best regards,
Eva
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