Updates on the case:
Condom use and the prevention of AIDS
Subject: Alternative form of two-sample Binomial test
Keywords: Comparison of Binomial proportions
Date: Thursday, 9 November, 1995
From: Halina Frydman (hfrydman@stern.nyu.edu),
Jeffrey S. Simonoff (jsimonoff@stern.nyu.edu)
Organization: Department of Statistics and Operations Research,
New York University
The case considered two European studies conducted during
the period 1991-1993 that provided evidence about whether
condom use prevents the transmission of the HIV virus. Both
studies provided statistically significant evidence that
couples who use condoms regularly have less of a chance of
new HIV infection than couples that do not.
The tests that were used in the case are based on a
confidence interval construction, with the standard error
of the difference
- -
p - p
1 2
- -
estimated using both p and p . A different test, which is
1 2
based on constructing the test under the assumption that
the null hypothesis is true, also can be defined. This
alternative test estimates the desired standard error using
a pooled estimate of the common probability of new HIV
infection,
- - -
p = (n p + n p )/(n + n ) .
0 1 1 2 2 1 2
Then, the z-statistic has the form
- -
p - p
1 2
z = ------------------------------ .
- -
sqrt{p (1 - p )(1/n + 1/n )}
0 0 1 2
This test is analogous to the two-sample t-test that
assumes equality of variances, in the same way that the
test given in the case is analogous to the test that does
not assume equality of variances.
For the Saracco study, the pooled estimate of new infection is
-
p = 11/226=.0487, resulting in a z-statistic of
0
.1455 - .0175
z = -------------------------------- = 3.84,
sqrt{(.0487)(.9513)(1/55+1/171)}
which has a tail probability of .0001; for the DeVincenzi
-
study, the pooled estimate of new infection is p = 12/245=.0490,
0
resulting in z-statistic
.0984 - 0
z = --------------------------------- = 3.57,
sqrt{(.0490)(.9510)(1/122+1/123)}
which has a tail probability of .0004. Thus, both studies
still result in strong rejection of the null hypothesis.