The analysis of data from matched pairs binary experiments, often performed with McNemar's test, presents a unique experimental design challenge in dealing with the effect of the discordance probability, p. Most approaches for determining size and power use point estimates or maximization, but this fails to account for the considerable variability across values of the nuisance parameter that occur for all common tests. We recommend viewing the size and power functions across the full range of possible discordance probability values, which gives a complete picture of the behavior of a test for any given sample size. This method also allows us to compare the behavior of different hypothesis tests. We present exact power and size functions for several tests, including the original McNemar's test and its most common variants, and compare their properties. This analysis reveals that, in general, McNemar's test comes closest to the nominal size and has the highest power. We also demonstrate our technique using the transmission/disequilibrium test (TDT) to check for linkage between schizophrenia and a locus related to the D(3) dopamine receptor, and on a hypnosis pain management data set.