# Table 4 Expectations of products of four allelic values received by two individuals at two loci depending on the IBD status and parental origins of the alleles

$${\mathcal{S}}$$ $${\mathcal{T}}$$ $${\text{E}}\left[ {g_{cls} g_{cmt} g_{ilu} g_{imv} |{\mathcal{S} }\,\& \text{ }\,T} \right]$$
$${\mathcal{S}}_{ml}$$ $$s = t \, and \, u = v$$ $$\left( {1 - p_{m} } \right)\left( {1 - p_{l} } \right)p_{m} p_{l} + \Updelta_{lm} \left( {1 - 2p_{l} } \right)\left( {1 - 2p_{m} } \right)$$
$$s = t \, \, and \, \, u \ne v$$ $$\left( {1 - p_{m} } \right)\left( {1 - p_{l} } \right)p_{m} p_{l} + \Updelta_{lm} \left( {1 - 2p_{l} } \right)\left( {1 - 2p_{m} } \right)$$
$$s \ne t \, and \, u = v$$ $$\left( {1 - p_{m} } \right)\left( {1 - p_{l} } \right)p_{m} p_{l} + \Updelta_{lm} \left( {1 - 2p_{l} } \right)\left( {1 - 2p_{m} } \right)$$
$$s \ne t \, and \, u \ne v$$ (1 − p m ) (1 − p l )p m p l
$$s = t \, and \, u = v$$ $$\Updelta_{lm}^{2} \times {{\left[ {p_{m}^{3} + \left( {1 - p_{m} } \right)^{3} } \right]} \mathord{\left/ {\vphantom {{\left[ {p_{m}^{3} + \left( {1 - p_{m} } \right)^{3} } \right]} {\left[ {p_{m} \left( {1 - p_{m} } \right)} \right]}}} \right. \kern-0pt} {\left[ {p_{m} \left( {1 - p_{m} } \right)} \right]}}$$
$$s = t \, and \, u = v$$ $$\Updelta_{lm}^{2} \times {{\left[ {p_{l}^{3} + \left( {1 - p_{l} } \right)^{3} } \right]} \mathord{\left/ {\vphantom {{\left[ {p_{l}^{3} + \left( {1 - p_{l} } \right)^{3} } \right]} {\left[ {p_{l} \left( {1 - p_{l} } \right)} \right]}}} \right. \kern-0pt} {\left[ {p_{l} \left( {1 - p_{l} } \right)} \right]}}$$
$$s = t \, and \, u = v$$ $$\Updelta_{lm}^{2} \times \left( {1 - 2p_{m} } \right)\left( {1 - 2p_{l} } \right)$$
1. Only non-null terms are given
2. p m and p l are the frequencies of the most frequent alleles at loci m and l. $$\Updelta_{lm}$$ is the linkage disequilibrium measure between m and l
3. $$g_{cls} = \left( {0 \,or\, 1} \right) - p_{l}$$ is the allelic value the candidate received from its parent $$s$$ at locus l etc
4. $${\mathcal{S}}_{ml}$$ means c and i genes are IBD at m and l, only at m etc