Simple Counterexamples against the Conditionality Principle

The famous theorem of Birnbaum, stating that the likelihood principle follows from the conditionality principle together with the sufficiency principle, has caused much discussion among statisticians. Briefly, many writers dislike the consequences of the likelihood principle (among other things, confidence coefficients and levels of tests are dismissed as meaningless), but at the same time they feel that both the conditionality principle and the sufficiency principle are intuitively obvious. In the present article we give examples to show that the conditionality principle should not be taken to be of universal validity, and we discuss some consequences of these examples.