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For affected offspring and heterozygous parents at a diallelic locus, let $b$ count transmissions of allele 1 with allele 2 untransmitted and $c$ count the reverse. Under the fair-transmission null, conditional on $b+c$, $b$ is binomial with probability $1/2$. The asymptotic statistic is $(b-c{)}^2/(b+c)$, with a one-degree-of-freedom chi-squared null distribution. Homozygous-parent transmissions give no contrast. Conditioning within families protects this test from ordinary population stratification.

Within-family Mendelian randomization uses genetic contrasts within families, such as differences between siblings. Conditioning on shared parental background can reduce population stratification and family-level sources of confounding, although the exclusion restriction and sufficient instrument relevance remain necessary.