Exact Test Fisher’s Statistics

Exact Tests Favorable Unfavorable Total Test Control 10 2 4 12 6 12 6 10 2 4 12 6 12 6 18 A test treatment and a control are compared to determine whether the rates of favorable response are the same. The sample sizes requirements for the chi-square tests are not met by these data

if you can consider the margins (12, 6, 12, 6) to be fixed, then you can assume that the data are distributed hypergeometrically and write Pr(nij) = n1+!n2+!n+1!n+2!/n!n11!n12!n21!n22! p-value is the probability of the observed data or more extreme data occurring under the null hypothesis With Fisher’s exact test, determine the p-value for this table by summing the probabilities of the tables that are as likely or less likely, given the fixed margins.

The following table includes all possible table configurations and their associated probabilities. Table Cell (1,1) (1,2) (2,1) (2,2) Probabilities ---------------------------------------------------------------------------- 12 0 0 6 0.0001 11 1 1 5 0.0039 10 2 2 4 0.0533 9 3 3 3 0.2370 8 4 4 2 0.4000 7 5 5 1 0.2560 6 6 6 0 0.0498 To find the one-sided p-value, you sum the probabilities as small or smaller than those computed for the table observed, in the direction specified by the one-sided alternative. In this case, it would be those tables in which the Test treatment had the more favorable response, or p = 0.0533 + 0.0039 + 0.0001 = 0.0573

To find the two-sided p-value, you sum all of the probabilities that are as small or smaller than that observed, or p = 0.0533 + 0.0039 + 0.0001 + 0.0498 = 0.1071