It examines the association of two Bernoulli distributed random variables and is a uniformly more powerful alternative to Fisher's exact test.
: The probability distribution of such tables can be classified into three distinct cases.
Then the row sums are the fixed numbers of cups prepared with each method:
of correctly classified cups with milk first follows the hypergeometric distribution
Examples of such a case are often found in medical research, where a binary endpoint is compared between two patient groups.
Then the row sums equal the group sizes and are usually fixed in advance.
The column sums are the total number of cures respectively disease continuations and not fixed in advance.
This third scenario describes most observational studies or "field-observations", where data is collected as-available in an uncontrolled environment.
If its sex also requires close examination of the butterfly, that also is independently binomially random.
That means that because of the experimental design, the column sums are unconstrained just like the rows are: Neither the count for either of species, nor count of the sex of the captured butterflies in each species is predetermined by the process of observation, and neither total constrains the other.
The only possible constraint is the grand total of all butterflies captured, and even that could itself be unconstrained, depending on how the collector decides to stop.
But since one cannot reliably know beforehand for any one particular day in any one particular meadow how successful one's pursuit might be during the time available for collection, even the grand total might be unconstrained: It depends on whether the constraint on data collected is the time available to catch butterflies, or some predetermined total to be collected, perhaps to ensure adequately significant statistics.
Each of the cells of the contingency table is a separate binomial probability and neither Fisher's fully constrained 'exact' test nor Boschloo's partly-constrained test are based on the statistics arising from the experimental design.
The null hypothesis of Boschloo's one-tailed test (high values of
favor the alternative hypothesis): The null hypothesis of the two-tailed test is: There is no universal definition of the two-tailed version of Fisher's exact test.
as fixed in advance, Fisher's exact test can also be applied to the second case.
The true size of the test then depends on the nuisance parameters
Boschloo proposed to use Fisher's exact test with a greater nominal level
This made performing Boschloo's test computationally easy.
The decision rule of Boschloo's approach is based on Fisher's exact test.
Fisher's p-value is calculated from the hypergeometric distribution (for ease of notation we write
is equal to the nominal level of Boschloo's original approach.
The Berger & Boos procedure takes a different approach by maximizing
[6] All exact tests hold the specified significance level but can have varying power in different situations.
Mehrotra et al. compared the power of some exact tests in different situations.
For small sample sizes (e.g. 10 per group) the power difference is large, ranging from 16 to 20 percentage points in the regarded cases.
The power difference is smaller for greater sample sizes.
-Pooled test has greater power, with differences mostly ranging from 1 to 5 percentage points.
-Unpooled test has greater power, with differences ranging from 1 to 5 percentage points.
However, in some other cases, Boschloo's test has noticeably greater power, with differences up to 68 percentage points.