3. general considerations for the analysis of case-control ... - IARC

134 BRESLOW & DAY
tained if the mean and variance of the exact conditional distribution are substituted in
(4.17) for A and Var; however, this requires solution of polynomial equations of an
even higher order.
Logit confidence limits
A more easily calculated set of confidence limits may be derived from the normal
approximation to the distribution of log & (Woolf, 1955). This has mean log q and a
large sample variance which may be estimated by the sum of the reciprocals of the
cell entries
1 1 1 1
VarClog I)) = - + - + - + - .
a b c d
Consequently, approximate lOO(1-a) % confidence limits for log q are
which may be exponentiated to yield q, and qU. Gart and Thomas (1972) find that
such limits are generally too narrow, especially when calculated from small samples.
Since log $3 is the difference between two logit transformations (see Chapter 5), the
limits obtained in this fashion are known as logit limits.
Test- based confidence limits
Miettinen (1976) has provided an even simpler and rather ingenious method for
constructing confidence limits using only the point estimate and x2 test statistic. Instead
of using (4.18), he solves
for the variance of log(@), arguing that both left and right side provide roughly
equivalent statistics for testing the null hypothesis I,L~ = 1. This technique is of even
greater value in complex situations where significance tests may be fairly simple to
calculate but precise estimates for the variance require more effort.
Substituting the test-based variance estimate into (4.19) yields the approximate
limits
where is raised to the power (1 + Zalzlx). Whether q~~ corresponds to the - sign in
this expression and qu to the + sign, or vice versa, will depend on the relative magnitude
of and x. The x2 statistic (4.16), however, should be calculated without
the continuity correction especially when I) is close to unity, since otherwise the variance
may be overestimated and the limits too wide. In those rare cases where $J is exactly
equal to unity, the uncorrected x2 is equal to zero and the test-based limits are consequently
undefined.