3. Linkage model

This is essentially a generalization of the admixture model to deal with ``admixture linkage disequilibrium''-i.e., the correlations that arise between linked markers in recently admixed populations. There is a manuscript in preparation (Falush, Stephens and Pritchard) that describes the model, and computations in more detail. The basic model is that, $t$ generations in the past, there was an admixture event that mixed the $K$ populations. If you consider an individual chromosome, it is composed of a series of ``chunks'' that are inherited as discrete units from ancestors at the time of the admixture. Admixture LD arises because linked alleles are often on the same chunk, and therefore come from the same ancestral population. The sizes of the chunks are assumed to be independent exponential random variables with mean length $1/t$ (in Morgans). In practice we estimate a ``recombination rate'' $r$ from the data that corresponds to the rate of switching from the present chunk to a new chunk.⁵ Each chunk in individual $i$ is derived independently from population $k$ with probability $q^{(i)}_k$, where $q^{(i)}_k$ is the proportion of that individual's ancestry from population $k$. Overall, the new model retains the main elements of the admixture model, but all the alleles that are on a single chunk have to come from the same population. The new MCMC algorithm integrates over the possible chunk sizes and break points. It reports the overall ancestry for each individual, taking account of the linkage, and can also report the probability of origin of each bit of chromosome, if desired by the user. This new model performs better than the original admixture model when using linked loci to study admixed populations. It achieves more accurate estimates of the ancestry vector, and can extract more information from the data. It should be useful for admixture mapping. Clearly, this model is a big simplification of the complex realities of most real admixed populations. However, the major effect of admixture is to create long-range correlation among linked markers, and so our aim here is to encapsulate that feature within a fairly simple model. The computations are a bit slower than for the admixture model, especially with large $K$ and unphased data. Nonetheless, they are practical for hundreds of sites and individuals and multiple populations. The model can only be used if there is information about the relative positions of the markers (usually a genetic map).