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Rmd fbcb572 Matthew Stephens 2020-03-12 workflowr::wflow_publish(“sparse_structure.Rmd”)

Introduction

The structure/admixture model is a model for multinomial genotype data, originally introduced and fit by MCMC in the “Structure” paper (Pritchard, Stephens and Donnelly; 2000), and subsequently fit by maximum likelihood in the “Admixture” paper (Alexander et al).

The basic idea of the model is to model each sample \(i\) as having some degree of membership \(q_{i1},\dots,q_{iK}\) in each of \(K\) populations, with allele frequencies \(p_k\).

Here we explore the possibility to regularize this model by introducing the idea that the allele frequencies \(p_k\) are similar – or even identical – for most populations \(k\) at most sites \(l\).

Model

We consider the haploid version of the model, where each indidividual \(i\) has one observation at each locus \(l\).

Notationally, it is helpful to represent the data in each individual at each locus as indicator vectors: \(X_{il\cdot}= X_{il1},\dots,X_{ilJ}\) where \(X_{ilj}=1\) if individual \(i\) is of type \(j\) at locus \(l\), and \(X_{ilj}=0\) otherwise.

Then the admixture model is \[X_{ilj} \sim \text{Mult}(1;\pi_{il\cdot})\] \[\pi_{ilj} = \sum_k q_{ik} p_{klj}\]

where Mult(n,\(\pi\)) denotes a (discrete) categorical distribution on \(1,...,J\) with probabilities \(\pi=(\pi_1,\dots,\pi_J)\), \(i\) indexes individuals, \(l\) indexes sites (loci), and \(j\) indexes alleles.

Writing it this way, one can see the connection with factor models, or ``low-rank" models, which sum over a small number of factors \(k=1,\dots,K\). The model for \(\pi_{ilj}\) is “low rank”.

For computation it is helpful to introduce latent variables to indicate the population of origin of each individual at each locus. Let \[Z_{il\cdot} \sim \text{Mult}(1; q_{i1},\dots,q_{iK}).\] So \(Z_{il\cdot}\) is a binary indicator vector (length \(K\)), with \(Z_{ilk}=1\) indicating that the population of origin of individual \(i\) at locus \(l\) is \(k\). We can then write \[X_{il\cdot} | p \sim \sum_k Z_{ilk} \text{Mult}(1; p_{kl\cdot})\]

Modification

Now we introduce a prior to allow that the \(p_k\) may be similar to one another at many loci/sites. This is similar to the idea of the ``correlated allele frequencies model" in Pritchard et al, but with more of a focus on sparsity (meaning exact or almost-exact equality of the allele frequencies for different \(k\)) and an eye on potential maximimum likelihood approach.

The proposed prior is: \[p_{kl\cdot} | \alpha, \mu \sim \text{Dir}(\alpha_{kl} \mu_{l\cdot})\] where \(\alpha_{kl}\) is a scalar that controls how similar \(p_k\) is to the mean \(\mu_{l\cdot}\) at locus \(l\), and \(\mu_{l\cdot}\) is the mean of the Dirichlet prior. Intuitively \(\mu_{l\cdot}\) represents the allele frequencies at locus \(l\), that are shared across populations. The idea of a separate \(\alpha\) for each \(k,l\) seems related to the so-called “automatic relevance determination prior”, and also the Sparse Factor Analysis method of Engelhardt and Stephens.

In the limit \(\alpha_{kl} \rightarrow \infty\) the \(p_{kl\cdot} = \mu_{kl\cdot}\) with probability 1.

Under this prior the distribution on the \(X\)s becomes a mixture of Dirichlet-Multinomials: \[X_{il\cdot} | \alpha, \mu \sim \sum_k Z_{ilk} \text{Dir-Mult}(1; \alpha_{kl}\mu_{kl\cdot}).\]

I suspect there is an EM algorithm for fitting this model, which here means estimating \(\alpha,\mu,q\). The estimated \(\mu\) should be approximately the site-specific marginal frequencies, and \(alpha_{kl}\) will be small when the factors \(k\) deviate from this mean at locus \(l\). I found this paper on fitting mixtures of Dirichlet multinomials by EM, which seems relevant here. Indeed, the above is basically extending theis mixture model to the “grade of membership” idea, but with the addition of more parameters \(\alpha\).

We may want to put a hyper-prior on \(\alpha\) that encourages it to be big at most loci in most factors…