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Table 3 Unordered genotypes, their genotype dosages, frequencies, and expectations of genotype dosages, and squares and cross products of genotype dosages, for locus M and N, in the crossbred offspring from random mating between lines A and B

From: Comparison of linkage disequilibrium estimated from genotypes versus haplotypes for crossbred populations

Genotype dosage Frequency Expectations
\({M}_{g}\) \({N}_{g}\) \(f\) \(E\left({M}_{g}^{2}\right)\)a \(E\left({M}_{g}\right)\) \(E\left({N}_{g}^{2}\right)\) \(E\left({N}_{g}\right)\) \(E\left({{M}_{g}N}_{g}\right)\)
0 0 \(rr{^{\prime}}\) 0 0 0 0 0
0 1 \(r{^{\prime}}s+rs{^{\prime}}\) 0 0 \({r}{^{\prime}}s+r{s}{^{\prime}}\) \({r}{^{\prime}}s+r{s}{^{\prime}}\) 0
0 2 \(s{^{\prime}}s\) 0 0 \(4s{^{\prime}}s\) \(2s{^{\prime}}s\) 0
1 0 \(r{^{\prime}}t+rt{^{\prime}}\) \(r{^{\prime}}t+r{t}{^{\prime}}\) \({r}{^{\prime}}t+rt{^{\prime}}\) 0 0 0
1 1 \({r}{^{\prime}}u+ru{^{\prime}}+{s}{^{\prime}}t+st{^{\prime}}\) \(r{^{\prime}}u+ru{^{\prime}}+s{^{\prime}}t+st{^{\prime}}\) \(r{^{\prime}}u+ru{^{\prime}}+s{^{\prime}}t+s{t}{^{\prime}}\) \(r{^{\prime}}u+ru{^{\prime}}+{s}{^{\prime}}t+st{^{\prime}}\) \({r}{^{\prime}}u+r{u}{^{\prime}}+s{^{\prime}}t+s{t}{^{\prime}}\) \(r{^{\prime}}u+r{u}{^{\prime}}+s{^{\prime}}t+st{^{\prime}}\)
1 2 \(s{^{\prime}}u+su{^{\prime}}\) \(s{^{\prime}}u+su{^{\prime}}\) \(s{^{\prime}}u+s{u}{^{\prime}}\) \(4s{^{\prime}}u+4su{^{\prime}}\) \(2{s}{^{\prime}}u+2s{u}{^{\prime}}\) \(2s{^{\prime}}u+2su{^{\prime}}\)
2 0 \(t{^{\prime}}t\) \(4t{^{\prime}}t\) \(2t{^{\prime}}t\) 0 0 0
2 1 \(t{^{\prime}}u+t{u}{^{\prime}}\) \({4t}{^{\prime}}u+4t{u}{^{\prime}}\) \(2t{^{\prime}}u+2tu{^{\prime}}\) \(t{^{\prime}}u+tu{^{\prime}}\) \({t}{^{\prime}}u+tu{^{\prime}}\) \({2t}{^{\prime}}u+2t{u}{^{\prime}}\)
2 2 \(u{^{\prime}}u\) \({4u}{^{\prime}}u\) \({2u}{^{\prime}}u\) \(4u{^{\prime}}u\) \(2u{^{\prime}}u\) \(4{u}{^{\prime}}u\)
  1 \(\left(t+u\right)+\left({t}{^{\prime}}+{u}{^{\prime}}\right)+2\left({t}{^{\prime}}+{u}{^{\prime}}\right)\left(t+u\right)\) \(\left(t+u\right)+\left({t}{^{\prime}}+{u}{^{\prime}}\right)\) \(\left(s+u\right)+\left({s}{^{\prime}}+u{^{\prime}}\right)+2\left({s}{^{\prime}}+{u}{^{\prime}}\right)\left(s+u\right)\) \(\left(s+u\right)+\left({s}{^{\prime}}+{u}{^{\prime}}\right)\) \(u{^{\prime}}\left(1+t+u\right)+u\left(1+{t}{^{\prime}}+{u}{^{\prime}}\right)+s{^{\prime}}\left(t+u\right)+s\left({t}{^{\prime}}+u{^{\prime}}\right)\) w
  1. a\(E\left({M}_{g}^{2}\right)\) refers to the expectation of the squared genotype dosage for locus\(M\), and similar definitions apply for the\(E\left({M}_{g}\right)\),\(E\left({N}_{g}^{2}\right) ,E\left({N}_{g}\right)\),\(E\left({{M}_{g}N}_{g}\right)\). The derivation of the expectations of genotype dosages is given in Appendix 2