The hypothesis of "no difference" between two populations is the appropriat
e null hypothesis in studies intended to show that populations differ. In s
tudies intended to show that two populations are practically equivalent, th
e null hypothesis that a substantial difference between the populations exi
sts is more appropriate. We consider eight tests of the null hypothesis tha
t the absolute difference of two binomial random variables' success probabi
lities is at least a prespecified Delta > 0 versus the alternative that the
difference is less than Delta. The tests considered are: six forms of the
two one-sided test, a modified form of the Patel-Gupta test, and the likeli
hood ratio rest. The applicability of each test in a given setting depends
on how well the test maintains its nominal size, the power of the test, and
the ease with which it is implemented. Based on these criteria, we make re
commendations for choosing, among these tests.