Paired binary sample size calculation ($n\simeq \{z_{1-\alpha /2}\sqrt{q}+z_{1-\beta }\sqrt{q-{\delta }^2}{\}}^2/{\delta }^2$)

For a paired binary difference $D=C-M$, set $\delta =P(C=1)-P(M=1)$ and $q=P(C\ne M)$. Then $\mathbb{E}D=\delta$ and $\mathrm{Var}(D)=q-{\delta }^2$. Under equal marginal probabilities the variance is $q$. The McNemar test has approximate upper rejection boundary $z_{1-\alpha /2}\sqrt{nq}$ for the sum of differences. Requiring that the alternative mean $n\delta$ exceed this boundary by $z_{1-\beta }\sqrt{n(q-{\delta }^2)}$ gives the displayed planning size for a positive alternative, neglecting its very small lower-tail rejection probability. Use $z_{1-\alpha }$ for a pre-specified one-sided test. The marginal probabilities alone do not determine $q$: co-occurrence of positive outcomes must be specified or bounded. Discreteness and clustered data can change achieved statistical power.