McNemar's test ($X^2=(b-c{)}^2/(b+c)$)

For paired binary observations $(C_i,M_i)$, let $b$ count pairs $(1,0)$ and $c$ pairs $(0,1)$. Under the null hypothesis of equal marginal event probabilities, these discordant outcomes have equal probabilities. Conditional on $b+c$, $b$ has the binomial distribution $\mathrm{Bin}(b+c,1/2)$, giving an exact test. For enough discordant pairs, $(b-c)/\sqrt{b+c}$ has an approximate standard normal distribution, or its square has an approximate one-degree-of-freedom chi-squared distribution. Concordant pairs provide no information about the direction of the marginal difference. If there are no discordant pairs, there is no directional information and the divided statistic is not computed. The pairing and independence between pairs are essential; the usual independent-two-sample formula is not a paired test.

Let a paired binary table have cells $a,b,c,d$ and $n=a+b+c+d$. The uncorrected McNemar test is $(b-c{)}^2/(b+c)$, while the ordinary independent-sample chi-squared test on the two marginal allele columns is $2n(b-c{)}^2/[(2a+b+c)(2d+b+c)]$. For nonzero difference and positive denominators, they coincide exactly when $(2a+b+c)(2d+b+c)=2n(b+c)$. Such numerical coincidence does not justify discarding pairing; altering concordant counts can change the unpaired statistic while leaving McNemar's statistic unchanged.

// Target: biology.bigb

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.