This is Part 2 of a four part series. The full list of posts are:
- Part 1 – How many Irish cousins: according to Tim Urban
- Part 2 – How many Irish cousins: according to 23 and Me
- Part 3 – How many Irish cousins: according to AncestryDNA
- Part 4 – How many Irish cousins: the impact of endogamy
Yesterday I wrote a blog post based on an entertaining article from Tim Urban to calculate our number of cousins at various distances using birth rate statistics. Today’s post is based on an academic study by researchers from 23andMe (Henn et al) which covers a wealth of complex analysis that includes a table of “expected number of cousins” at degree of cousinship. This table is reproduced in several websites without the detail of the formula that arrived at the very specific numbers.
It took me a few reads to realize that the formula used by Henn et al is exactly the same as what Tim Urban devised. The formula is kinda buried near the end of the article under a section titled “Calculation of expected number of individuals sharing DNA IBD. Yep, that translates to “how many cousins do we have.”
Critically, this section specifies that a birth rate of 2.5 is used to produce the cousin numbers in the table. I’ll come to that later, as it may be more appropriate to the United States than to Irish people over a certain age.
Now for the Math – but it’s easy to follow
I actually found the academic explanation easier to follow than Tim’s post. So now I’ll try to explain the formula as opposed to using it blindly to produce numbers. The formula is expressed using i as the degree of cousinship and z as birth rate. :
Why 2 to the power of i? For our first cousins, we share two sets of grandparents, therefore two couples. For our secound cousins we share four sets of great grandparents, therefore four couples. And so on up the generations, where the number of couples is equal to 2 to the power of the degrees of cousinship.
Why (z-1)? If one set of grandparents have four children, one of those children is our parent and that person’s children are ourselves and our average of three siblings so we need to remove 1 from the birth rate.
z to the power of i is the total number of non-ancestral cousins that stem from a particular ancestral generation. Thus the number of cousins from one set of ancestors is the product of (z-1) and z^i. The full total is then achieved by multiplying by the number of ancestral couples for the degree of cousinship.
The table below has the totals for three different fertility rates in recent Irish history. The fourth line uses the fertility rate chosen by Denn et all for their numbers.
As befits a scholarly study, Denn et al also specify the assumptions that they made to simplify calculations. “Perfect survivorship” jumps out for me i.e. that every offsping at every generation lives to produce the average number of children. Given that the user base of 23andMe is predominantly American and with disposable income for personal DNA testing, this might be a reasonable assumption for their purposes. I do think we Irish need to pay some attention to the impact of our mortality rates up to the 1970s.
They also list other assumptions which I’d like to consider. But I’ll save that for another post in this “how many Irish cousins” series. In my next post I want to look at the quoted numbers from AncestryDNA. (The list of blog posts in this series are at the top of this post).