Intersection of confidence regions
January 14, 2008
The question of producing a confidence interval by intersecting two or more confidence intervals calculated for the same unknown parameter, based on either the same data or different data, is occasionally of interest. This is a form of meta analysis – combining information from different sources – which is attractive since it can be done using only a minimal amount of information from each of the data sources, namely the confidence intervals alone, and since the calculation is easy and the result has an intuitive appeal.
It turns out that intersecting confidence regions (and specifically intersecting confidence intervals) does indeed produce confidence regions. This can be seen as a special case of the confidence region arithmetic.
Let C1 and C2 be level 1-p1 and 1-p2 confidence regions for a parameter θ. The probability that C1 ∩ C2 does not contain θ is the probability that either C1 does not contain θ or that C2 does not contain θ. This is less than or equal to the sum of the probabilities of those two events – p1 + p2. Therefore, C1 ∩ C2 is a level 1-p1-p2 confidence region for θ. Thus, a smaller region is attained, but at a reduced confidence level.
Note that an assumption of independence was not made in this calculation. If independence is assumed, then the level of the intersection confidence region can be tightened up to (1-p1)(1-p2). However, if p1 and p2 are small, this does not make much of a difference.