Necessary and sufficient conditions for identifiability in the admixture model

TL;DR

This paper establishes precise conditions under which the ancestral allele frequencies and admixture proportions can be uniquely identified in the admixture model, correcting previous proofs and extending the theory.

Contribution

It provides a corrected proof of identifiability conditions, introduces new necessary and sufficient criteria, and extends results to non-admixed populations.

Findings

Anchor condition plus independence ensures identifiability.

Corrected proof of necessary conditions for identifiability.

Extended conditions to non-admixed populations.

Abstract

We consider M SNP data from N individuals who are an admixture of K unknown ancient populations. Let $\Pi_{si}$ be the frequency of the reference allele of individual i at SNP s. So the number of reference alleles at SNP s for a diploid individual is binomially distributed with parameters 2 and $\Pi_{si}$. We suppose $\Pi_{si}=\sum_{k=1}^KF_{sk}Q_{ki}$, where $F_{sk}$ is the allele frequency of SNP s in population k and $Q_{ki}$ is the proportion of population k in the ancestry of individual i. I am interested in the identifiability of F and Q, up to a relabelling of the ancient populations. Under what conditions, when $\Pi =F^1Q^1=F^2Q^2$ are $F^1$ and $F^2$ and $Q^1$ and $Q^2$ equal? I show that the anchor condition (Cabreros and Storey, 2019) on one matrix together with an independence condition on the other matrix is sufficient for identifiability. I will argue that the proof of the necessary condition in Cabreros and Storey, 2019 is incorrect, and I will provide a correct proof, which in addition does not require knowledge of the number of ancestral populations. I will also provide abstract necessary and sufficient conditions for identifiability. I will show that one cannot deviate substantially from the anchor condition without losing identifiability. Finally, I show necessary and sufficient conditions for identifiability for the non-admixed case.