I really thought that Dienekes was smart enough to know when he had blundered.

Clearly I overestimated him.

Instead of putting down his shovel, he digs his hole deeper in his first post to comment on my work. The title of that post is “Amerindian-like admixture in northern Europe is real”. This idea has become an article of faith for Dienekes. There’s zero evidence to support it, and ample evidence against it, but none of that matters to Dienekes. He just *knows* that it *must* be true.

In his post Dienekes takes the phylogeny I used for running the F4 ratio estimation program and shows that it won’t work for f3 statistics.

No kidding.

I didn’t use a phylogeny with that structure because I thought it would be appropriate for f3 statistics, I used it because the F4 ratio estimation program *requires* that structure for the phylogeny. It fixes the structure, and you get to pick the five populations.

Filling the structure with the populations I used does produce something that partially reflects reality, but of course it’s a simplification of reality, a simplification required to run the program.

Dienekes was only stating the obvious when he said that you can never get negative f3 statistics for population B in the F4 ratio estimation phylogeny. Population B in that phylogeny is unmixed, and *of course* you’re never going to get negative f3 statistics for a population based on a phylogeny in which that population is *postulated* to be unmixed.

The fact that F3(Europeans; Sardinians, Amerindians) is negative means that Europeans must be mixed. And of course they *are* mixed: they’re all mixes of Nordics and Mediterraneans.

Sardinians have more of the Mediterranean element than any other Europeans, but they too are a Nordic-Mediterranean mix.

And Amerindians are a mixture of Mongoloids, Nordics, and proto-Caucasoids.

The reason that such large negative f3 statistics occur for Europeans as mixtures of Sardinians and Amerindians is that, as the K = 17 admixture analysis of Amerindians and the K = 16 admixture analysis of Mongoloids showed, the European DNA in Amerindians is more purely Nordic than the DNA of any European alive today. The strong Nordic-Mediterranean contrast between Amerindians and Sardinians produces some of the largest negative f3 statistics for the Nordic-Mediterranean mixture of Europeans. However, as I pointed out in this post, the negative Z-score for French as a mixture of Sardinians and Russians is larger than the negative Z-score for French as a mixture of Sardinians and Karitiana.

Since Europeans, Sardinians, and Amerindians are all mixed, it would obviously be dumb to do what Dienekes did and use the F4 ratio estimation phylogeny for f3 statistics.

A sensible phylogeny for F3(Europeans; Sardinians, Amerindians) is below.

V is Veddoids, C is Caucasoids, M is Mediterraneans, N is Nordics, G is Mongoloids, S is Sardinians, E is Europeans, and A is Amerindians.

Of course even this phylogeny is a simplification of reality: There are other Caucasoid subracial elements in Europeans besides Nordics and Mediterraneans. South Europeans have Negroid admixture. Some Eastern European populations *really do* have Mongoloid admixture. Mongoloids and Amerindians have proto-Caucasoid admixture. Mongoloids have Caucasoid admixture. And so on. But the phylogeny captures the most important relationships between the races and populations included, and it is therefore useful for calculating the expected values of f3 statistics for Europeans.

There are four paths from Europeans to Sardinians and four paths from Europeans to Amerindians, so there are sixteen path combinations. For ten of these path combinations there is no overlap along the drift edges, so the corresponding terms in the equation for the expected values of the f3 statistics drop out. What we’re left with is

f3(E; S, A) = α(1 − β)αγ(CM + CN) + α(1 − β)α(1 − γ)CM − α(1 − β)(1 − α)(1 − γ)CN − (1 − α)βαγ(CM + CN) − (1 − α)βα(1 − γ)CM + (1 − α)β(1 − α)(1 − γ)CN

Simplifying, we get

f3(E; S, A) = α²CN + α²CM − αβCN − αβCM + αγCN − αCN − βγCN + βCN

Now let’s try plugging some numbers into this formula and seeing what we get. We’ll see that, except for Russians, the expected values are similar to the empirical f3 statistics. I’ll be using admixture proportions and FST values from Dienekes’ globe13 ADMIXTURE analysis. The empirical values below are from the ADMIXTOOLS paper.

The FST between the Nordic component and the Mediterranean component is 0.048. The FST between the Nordic component and the Veddoid component is 0.075, and the FST between the Mediterranean component and the Veddoid component is 0.087. So we’ll set CN = 0.048 / 2 − (0.087 − 0.075) / 2 = 0.018 and CM = 0.048 / 2 + (0.087 − 0.075) / 2 = 0.03.

The Nordic admixture proportion in the Karitiana was determined by the F4 ratio estimation program to be 0.182781, by the K = 17 admixture analysis of Amerindians to be 0.1216, and by the K = 16 admixture analysis of Mongoloids to be 0.1596. So for the Karitiana we’ll set γ = (0.182781 + 0.1216 + 0.1596) / 3 = 0.1546603.

The Nordic admixture proportion in the Pima was determined by the F4 ratio estimation program to be 0.186089, by the K = 17 admixture analysis of Amerindians to be 0.1437, and by the K = 16 admixture analysis of Mongoloids to be 0.1771. So for the Pima we’ll set γ = (0.186089 + 0.1437 + 0.1771) / 3 = 0.168963.

The Nordic admixture proportion in Sardinians is 0.161, so we’ll set β = 1 − 0.161 = 0.839.

The Nordic admixture proportion in Russians is 0.685, so we’ll set α = 1 − 0.685 = 0.315. The expected value of f3(Russians; Sardinians, Karitiana) is calculated to be 0.00005036. The empirical value is −0.0086. The large discrepancy is of course due to the fact that Russians *really do* have Mongoloid admixture.

The Nordic admixture proportion in the French is 0.476, so we’ll set α = 1 − 0.476 = 0.524. The expected value of f3(French; Sardinians, Karitiana) is calculated to be −0.003130. The empirical value is −0.0060.

The Nordic admixture proportion in Tuscans is 0.265, so we’ll set α = 1 − 0.265 = 0.735. The expected value of f3(Tuscans; Sardinians, Karitiana) is calculated to be −0.002087. The empirical value is −0.0052.

The Nordic admixture proportion in North Italians is 0.314, so we’ll set α = 1 − 0.314 = 0.686. The expected value of f3(North Italians; Sardinians, Pima) is calculated to be −0.002749. The empirical value is −0.0045.

The Nordic admixture proportion in Orcadians is 0.588, so we’ll set α = 1 − 0.588 = 0.412. The expected value of f3(Orcadians; Sardinians, Karitiana) is calculated to be −0.001947. The empirical value is −0.0019.

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