Two-population coalescent model for imputation reference panel selection. (A) Two populations, labeled 1 and 2, of sizes

*N*_{1} and

*N*_{2} diploid individuals, diverge from an ancestral population of size

*N*_{A} at time

*t*_{D}. A single haplotype

*T* for which genotypes at untyped markers are to be imputed is sampled from population 1. We consider two possible reference panels for imputing

*T*: an internal reference panel of

*n*_{1} haplotypes sampled from population 1 and an external reference panel of

*n*_{2} haplotypes sampled from population 2. If

*T* first coalesces with a type 1 lineage (blue), then the internal panel is optimal for imputing

*T* (event

*C*_{1}). The external panel is optimal (event

*C*_{2}) if

*T* first coalesces with a lineage of type 2 (red). Finally, if

*T* first coalesces with a type 1–2 lineage (orange), then the two reference panels are equivalent (event

*C*_{12}). (B) To compute the probability of optimality for each reference panel, we condition on

(the event that

*T* coalesces before the divergence), the quantities

*i*_{D} and

*j*_{D} (the numbers of lineages originating in populations 1 and 2, respectively, that remain at the time of divergence), and

*i*_{C},

*j*_{C}, and

*k*_{C} (the numbers of type 1, type 2, and type 1–2 lineages remaining at the instant when

*T* first coalesces). In the realization pictured,

*T* does not coalesce before the divergence time (event

^{c}) and

*i*_{D} = 3,

*j*_{D} = 2,

*i*_{C} = 2, and

*j*_{C} =

*k*_{C} = 1. Because

*T* first coalesces with a type 1–2 lineage (event

*C*_{12}), the two reference panels are equivalent for imputing

*T*.

## PubMed Commons