Boschloo's test

Boschloo's test is a statistical hypothesis test for analysing 2x2 contingency tables. It examines the association of two Bernoulli distributed random variables and is a uniformly more powerful alternative to Fisher's exact test. It was proposed in 1970 by R. D. Boschloo. A 2x2 contingency table visualizes independent observations of two binary variables and : The probability distribution of such tables can be classified into three distinct cases. The row sums and column sums are fixed in advance and not random. Then all are determined by . If and are independent, follows a hypergeometric distribution with parameters : . The row sums are fixed in advance but the column sums are not. Then all random parameters are determined by and and follow a binomial distribution with probabilities : Only the total number is fixed but the row sums and the column sums are not. Then the random vector follows a multinomial distribution with probability vector . Fisher's exact test is designed for the first case and therefore an exact conditional test (because it conditions on the column sums). The typical example of such a case is the Lady tasting tea: A lady tastes 8 cups of tea with milk. In 4 of those cups the milk is poured in before the tea. In the other 4 cups the tea is poured in first. The lady tries to assign the cups to the two categories. Following our notation, the random variable represents the used method (1 = milk first, 0 = milk last) and represents the lady's guesses (1 = milk first guessed, 0 = milk last guessed). Then the row sums are the fixed numbers of cups prepared with each method: . The lady knows that there are 4 cups in each category, so will assign 4 cups to each method. Thus, the column sums are also fixed in advance: . If she is not able to tell the difference, and are independent and the number of correctly classified cups with milk first follows the hypergeometric distribution . Boschloo's test is designed for the second case and therefore an exact unconditional test.