The thought occurred to me to respond to a query from Lucy Sinkular on the Rootsweb Genealogy-DNA email list regarding matching autosomal chromosome segments to an adopted person, along with her known cousins. I thought I would mention something that I had put into my autosomal DNA spreadsheet to estimate cousin relationships. I used my "CousinCalc" equation (from my spreadsheet) to informed her that I got an estimation of 10th cousin for her genetic cousin, who was adopted.
I used my "CousinCalc" to estimate that her adopted genetic cousin was on the order of 10th cousin.
Back on September 28, 2011 Jared Roach, M.D., Ph.D. Senior Research Scientist Institute for Systems Biology posted a note on the Genealogy-DNA email list for the logic behind the prediction of cousin relationships. The theory is that the number of segments, when combined with the size of the segments, can be used to estimate distant relationships.
"Maximum-likelihood Estimation of Recent Shared Ancestry (ERSA)," Genome Res., May 21, 2011. ( see http://genome.cshlp.org/content/21/5/768/F1.expansion.html)
Long autosomal segments are unlikely to be from distant relationships, and short segments can either be from close or distant relationships. The equation given in the paper is based on an equation given by Thomas in 1994:
P(t) = e^
d = number of meiosis
t = length of segment in cM
For these past 20 years or so, the "number of meiosis" has been taken to be the number of segments. Terms have been introduced to define valid segments (Identical By Descent, or IBD) and invalid segments (Identical By State, or IBS). Segments are considered to be IBS if, in general, they are small (less than 5 to 7 cM, or less than 500 SNPs).
The above equation does not always work well, so a large number of probability distribution functions and Monte Carlo simulations have been invented in order to help make some reasonable estimates of relatedness between two matching individuals. The topic is popular because the predicting relatedness has a number of applications, from family history to medicine.
However, the thought that was nagging me was why were these scientists not using SNP's vs. cMs??
So, I thought I would try to find out. Upon investigating, I found that this equation worked for predicting my 4th and fifth cousins:
CousinCalc = (1,000 x ToTal_cMs)/ToTal_SNPs
When I ignore the concept of IBD and IBS (and just use the figures as given by Family Tree DNA in their Family Finder product), this equation works for the distant cousins that I knew about thus far. IBD and IBS at present are terms derived from the use of segment counts, the 'CousinCal' equation does not use segment counts, so my thoughts are that the current definition of IBD and IBS do not apply to the use of this equation.
But, the questions that bothered me was, 'is my sample too small??'
Would this be a statistically valid equation?
I don't have enough data to answer that, so I asked around.
I looked at Tim Janzen's autosomal segment matches to his mother, which has made publicly available. My "CousinCalc" came back with a cousin estimation of 6th cousin. Clearly, my "CousinCalc" equation does not work for cousins less than 1. But, to be fair, Tim Janzen does not list his IBS segments, so the sum of the IBS segments is an unknown part of the equation. Yet, in use of my equation, I do find that the sum of all segments delivers nearly the same result as the sum of IBD segments, so I presume that the expectation is valid that the "CousinCalc" equation will not to hold for cousins less than 1.
For more distant relationships, you should begin to see a departure from IBD reflecting the results of this equation. So, be careful about using IBD instead of sums.
Ann Turner had the most patience with this idea. However, she wasn't exactly warming up to the idea of using this new equation. She explained that 23AndMe uses segments vs. cMs, as in the article “Cryptic Distant Relatives are Common in both Isolated and Cosmopolitan Samples” and the chart (Fig, 3) is given on this page:
|Individual autosomal segments vs. Centimorgans (cMs)|
|When matching autosomal DNA segments are summed, the sums produce a chart that shows a direct relationship between total SNP's and total cMs. 1076 individual segments where SNP's and cM's have been summed per matching person.|
Elizabeth Harris wrote me to say that she did not use SNP's because they did not work for her. Basically, she tested with 23AndMe, and the calculation did not work for 4 of her cousins with matching segments on chromosome 15.The 'CousinCalc' equation came out to be 33rd cousins for that segment, and the math is rather tortuous if you want the equation to come out as fourth cousins for that particular segment on chromosome 15.
Ann Turner cited this chart from Rutgers' University:
The basic point there being that the marker position along chromosome 15 begins at about 20 MB. As does chromosome 13 and 14, but it is not a common phenomena among chromosome measurements.
However, I should point out that unfortunately, Elizabeth only gave the values for chromosome 15, and I did not get to see what 'CousinCalc' looks like for her 4th cousins using the sums across all matching chromosomes. Elizabeth did not provide data for any other matching chromosome, so I did not get to see what the data looks like from 23AndMe.
And finally, Ann Turner also pointed out that chromosome 15 has some poor regions being reported out, and a number of other chromosomes have the same problem. She cited Table 3 of this article: "Relationship Estimation from Whole-Genome Sequence Data," Hong Li, et. al. Jan 2014.
That Table 3 shows two segments on chromosome 15 - one between starting location of 20,967,673 and ending at 25,145,260 that show a length of 10.46 cMs and the other starting at 27,115,823 and ending at 30,295,750 with a length of 9.29 cMs. That's two to three times what you might expect to be the length in cMs.Other chromosomes showing this type of anomaly include chromosomes 1, 2, 6, 8, 9, 10, 16, 17, 21, and 22. A total of some 14 regions.
Finally, the GedMatch.com web site has a utility that plugs relationship calculations into some of their reports. However, I think that Tim Janzen mentioned that GedMatch has not yet converted to Build 37. Family Tree DNA is now at Build 37, so the results may look slightly different at GedMatch than what you may see at your vendor.
Having given an overview regarding why SNP's are not used vs. Centimorgans, here is a display of what the cousin calculator equation look like for a few of my cousins:
|Autosomal DNA 'CousinCalc' equation for fourth cousin using SNPs vs. cMs.|
|Autosomal DNA 'CousinCalc' equation for fourth cousin, once removed using SNPs vs. cMs.|
|Autosomal DNA 'CousinCalc' equation for fourth cousin using SNPs vs. cMs.|
|Autosomal DNA 'CousinCalc' equation for fifth cousin using SNPs vs. cMs.|
The next step should be to try to gather enough data regarding the results of this equation in order to determine if this is a valid calculation that I can use in my spreadsheet. If the statistics do not bear it out as valid, then the following step should be to determine if the problems mentioned above could be remedied (or avoided) by use of a program.
Updated 03/27/2014 - fix for math error in table for Frank, made hyperlinks active.
Update 08/25/2015 - I posted an improved equation to the Genealogy-DNA email list:
I have been playing around with my cousin calculator again, and have tried to have it work better at the extreme limits. That is, have it return a small result when the cMs are huge, and return a huge result when the cMs are small. I received some cMs, SNP counts, and relationships from Robert James Liguori, which includes some data on close relationships that I do not have.
If anyone would care to humor me and plug it into your spreadsheet from FTDNA, my cousin calculator version 2 equation now looks like this:
= SQRT(1/SQRT((F3*F3) / G3)*(0.75*SQRT(G3/F3))*(G3/(F3*600)))
F3 is the cM segment values from column 'F' on line 3
G3 is the SNP segment value from column 'G' on line 3
I would imagine that some academics would prefer exponential or natural logs in the equation, but this one was fairly straightforward for me to test. It certainly can be improved.
A few things I know at the moment:
- This works better on segments than it does on sums.
- It still does not work well for cousins less than 1, but this is an improvement.
- It does not handle "2nd Removed," etc. very well.
- It is mostly designed to work with values in excess of 3 cMs (use values greater than 3 cMs). Therefore, if you see the wrong value, please check the size in cMs first.
I designed it so that it could handle small values (large cousin values), just so I could see how large a cousin value it would return. The largest value that I have seen generated is around 100th cousin, which I would guess would be in the vicinity of 200 generations, or 5,000 years at 25 years per generation.
Thanks for the data from Robert James Liguori.