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In popular DNA ancestry tests you can read lines like "you have 2% neanderthal dna" or "you have 20% italian". How can you distinguish between a person having 20% italian vs italians having 20% of your dna.

Like if we bred with neanderthals, that means neanderthals would have some of our DNA, but that wouldn't mean that we would have some their neanderthal DNA. Is this reasoning true? Is it possible to distinguish and does this apply to modern groups of people aswell?

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The two cases are slightly different.

'Ancestry proportions', such as those given by 23andme, are given in the context of a reference panel. This means that they aggregate genetic data from individuals into particular countries, say Spain, Italy and France. They would then look at the genome of the individual they are testing, and find how much of the genome most closely matches individuals from any of the 3 reference populations. So an individual from the U.K. may share 70% of their genome most closely to the French reference individuals, 20% most closely to the Spanish and 10% most closely to the Italian. These proportions will change if we use different reference panels. In this case, the proportions only make sense in the context of which reference populations have been used. This information is available on here.

Humans have around 2% Neanderthal DNA because (we think) a small group of less than 10 Neanderthals mixed with the ancestors of all extant modern non-Africans. This Neanderthal ancestry was then propogated throughout all modern non-Africans today. The first human - Neanderthal hybrids would of course have been ~50% human and ~50% Neanderthal. However, because the effective population size of humans was much larger, the Neanderthal ancestry became diluted over time. If some early Neanderthal hybrids had then moved back into a population of only Neanderthals, they would have carried with them the human ancestry and we would say that Neanderthals contain x amount of human ancestry. However, as far as we know, this didn't happen and Neanderthals relatively quickly died out.

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