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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

From: Approximated prediction of genomic selection accuracy when reference and candidate populations are related

\( {\mathcal{S}} \) \( {\mathcal{T}} \) \( {\text{E}}\left[ {g_{cls} g_{cmt} g_{ilu} g_{imv} |{\mathcal{S} }\,\& \text{ }\,T} \right] \)
\( {\mathcal{S}}_{ml} \) \( s = t \, and \, u = v \) \( \left( {1 - p_{m} } \right)\left( {1 - p_{l} } \right)p_{m} p_{l} + \Updelta_{lm} \left( {1 - 2p_{l} } \right)\left( {1 - 2p_{m} } \right) \)
\( s = t \, \, and \, \, u \ne v \) \( \left( {1 - p_{m} } \right)\left( {1 - p_{l} } \right)p_{m} p_{l} + \Updelta_{lm} \left( {1 - 2p_{l} } \right)\left( {1 - 2p_{m} } \right) \)
\( s \ne t \, and \, u = v \) \( \left( {1 - p_{m} } \right)\left( {1 - p_{l} } \right)p_{m} p_{l} + \Updelta_{lm} \left( {1 - 2p_{l} } \right)\left( {1 - 2p_{m} } \right) \)
\( s \ne t \, and \, u \ne v \) (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