For testing the effectiveness of a treatment on a binary outcome, a bewildering range of methods have been proposed. How similar are all these tests? What are their theoretical strengths and weaknesses? Which are to be recommended and what is a coherent basis for deciding? In this paper, we take seven standard but imperfect tests and apply three different methods of adjustment to ensure size control: maximization (M), restricted maximization (B) and bootstrap/estimation (E). Across a wide

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... ions, we compute exact size and power of the 7 basic and 21 adjusted tests. We devise two new measures of size bias and intrinsic power, and employ novel graphical tools to summarise a huge set of results. Amongst the 7 basic tests, Liebermeister's test best controls size but can still be conservative. Amongst the adjusted tests, E-tests clearly have the best power and results are very stable across different conditions.