. 2013 Nov 7;93(5):840–851. doi: 10.1016/j.ajhg.2013.09.014

Table 1.

Approximate IBD and Non-IBD Likelihoods

GT 1	GT 2	IBD Likelihood	Non-IBD Likelihood
AA	AA	$e_{0} p_{A}^{3} + 2 e_{1} p_{A}^{2} p_{B} + e_{2} p_{A} p_{B}$	$f_{A}^{4}$
AA	AB	$e_{0} p_{A}^{2} p_{B} + (e_{1} + 3 e_{2}) p_{A} p_{B} + 2 e_{1} p_{A}^{3}$	$2 f_{A}^{3} f_{B}$
AA	BB	$(e_{1} + e_{2} + e_{3}) p_{A} p_{B} + e_{2} (p_{A}^{3} + p_{B}^{3})$	$f_{A}^{2} f_{B}^{2}$
AB	AA	$e_{0} p_{A}^{2} p_{B} + (e_{1} + 3 e_{2}) p_{A} p_{B} + 2 e_{1} p_{A}^{3}$	$2 f_{A}^{3} f_{B}$
AB	AB	$(e_{0} + 4 e_{1} + 2 e_{2}) p_{A} p_{B} + 4 e_{2} p_{A}^{3} + 4 e_{2} p_{B}^{3}$	$4 f_{A}^{2} f_{B}^{2}$
AB	BB	$e_{0} p_{A} p_{B}^{2} + 2 e_{1} p_{B}^{3} + (e_{1} + 3 e_{2} + e_{3}) p_{A} p_{B} + 2 e_{3} p_{A}^{3}$	$2 f_{A} f_{B}^{3}$
BB	AA	$(e_{1} + e_{2} + e_{3}) p_{A} p_{B} + e_{2} (p_{A}^{3} + p_{B}^{3})$	$f_{A}^{2} f_{B}^{2}$
BB	AB	$e_{0} p_{A} p_{B}^{2} + 2 e_{1} p_{B}^{3} + (e_{1} + 3 e_{2} + e_{3}) p_{A} p_{B} + 2 e_{3} p_{A}^{3}$	$2 f_{A} f_{B}^{3}$
BB	BB	$e_{0} p_{B}^{3} + 2 e_{1} p_{A} p_{B}^{2} + e_{2} p_{A} p_{B} + 2 e_{3} p_{A}^{2} p_{B} + e_{4} p_{A}^{3}$	$f_{B}^{4}$

Probabilities for a pair of genotypes having either 1 or 0 alleles shared identical by descent. Allele errors are independent and have probability $ε \geq 0$ , and $e_{j} = ε^{j} {(1 - ε)}^{4 - j}$ for $0 \leq j \leq 4$ . Genotypes have major allele A, minor allele B, true allele frequencies p_A and p_B, and error-added allele frequencies f_A and f_B. We estimate f_A and f_B with the observed allele frequencies, and we estimate p_B and p_A = 1 − p_B using the relationship $p_{B} = (f_{B} - ε) / (1 - 2 ε)$ .