The DNA matching services (I've used Ancestry.com, MyHeritage.com, and 23andMe.com) analyze DNA, compare it with others' DNA samples and identify which segments of you have in common with the others. From this, they quantify quantity of common DNA and the length of the segments in common and apply their proprietary algorithms to estimate your relationships with others. They present you with a list (about 1000 from 23andMe, about 3000 from MyHeritage; more is not necessarily better) of persons with DNA matches in default order of how closely related they predict you are. There are some capabilities for attaching family trees or lists of surnames or family origins, etc. to allow for some searching.
Contacts
So the first, and arguably most important, use of DNA analysis is that you get a list of likely related persons and a means to contact them (usually internal messaging). You can sometimes take advantage of searching through family surnames and trees, but I find that most people don't contribute this information. It's easy to contact your new-found relatives and ask about connections, though less than half seem to reply. Still, the first use of DNA analysis is to identify relatives with whom you might collaborate and find common relatives.
DNA map
The second use requires some calibration. By identifying your closest relatives you can begin to associate segments of DNA with ancestral lines of your family. This is so much easier to do with first and second cousins. If you've done lots of research and have a well-developed family tree, third cousins can be found without too much trouble. With more distant relatives, if you can find the connection, the segments in common can be used to add a great deal of precision to your map of inherited DNA. But these more distant connections are generally very hard to establish. In any case, the second use of DNA analysis is identifying DNA segments that belong to ancestors. This in turn helps to focus your search for a common ancestor on certain families, avoiding impossible amounts of time spent researching and eliminating family lines.
By the way, to keep track of DNA segments and compare them and to keep track of the status of my investigations, I use a third party software called Genome Mate Pro ( https://www.getgmp.com/ ). Doesn't have the flash and polish of a Microsoft product, but it serves it's very basic (but complicated) purpose very well. You export your DNA analysis from the testing service, then import it into GMP. Results from different services and individuals can be displayed side by side, as if they all came from one service. I'm a novice at this, am still trying to figure out the best way to use the DNA information, and have not yet reached any huge insights or made any huge discoveries. So far it seems to be a lot of work, just using different tools and modified processes.
Relationship estimates
The third use is a confidence measure. Each DNA match is labeled with an estimate of the relationship, or a range of relationships. It's useful to understand some basic DNA math. Find "DNA" in the word cloud on this blog to read more about that. The most important thing is to understand all the halves and halve nots. Off the top of my head, some are:
1) When referring to the DNA of a common ancestor, each subsequent generation inherits about half as much DNA as their parent has. So a great-grandchild has half as much as a grandchild. Your first cousin once removed (child of your first cousin) has only about half as shared DNA in common with you as your first cousin. When you put these two examples together, second cousins (each of whom has just 1/2 half as much of their ancestors DNA as their parents) have about 1/4 as much shared DNA as their first cousin parents. Etc.
2) For this same reason, each "removed" in a relationship cuts the amount of shared DNA by 1/2.
3) You inherit about 1/2 of your DNA from each parent. About 1/2 of my DNA is. from my mom; about 1/2 is from my dad. This is important for step- or 1/2- relationships. Suppose your maternal grandparents divorced and remarried. Consider a first cousin who is a grandchild, like you, of your maternal grandfather, but your maternal grandmother's were different. Since any DNA you share with this 1/2 first cousin came only from your maternal grandfather, but none is shared by you from your maternal grandmothers, a 1/2 first cousin will only share half as much DNA from grandparents as first cousins who share both their maternal grandmother and grandfather.
4) Siblings are a little trickier. You and your sibling each inherit 1/2 of your DNA from each parent. But it isn't usually the same half. So even though you both get your genes from the same source, on average you share about 1/2 of the same DNA as a sibling.
A bag of marbles, anyone?
This can be confusing, especially if you're math-phobic. Do I need to say anymore? If you're lost, think of DNA like a bag of marbles. Each of the marbles - cat-eye? - has a different color. One bag of marbles is your mom's DNA, one is your dad's. To get your DNA mix, you scoop out half of your dad's marbles and half of your mom's and put them in your marble bag. About half (maybe exactly in this case) of your marbles come form mom, half from dad. You do the same thing for your sister. (But first you have to replace all the marbles you took out.) After you scoop half of mom's marbles and half of dad's into your sister's bag, you'll find that half of your sister's marbles are from you mom and half are from your dad. But if you compare which marbles you each got from your parents, you'll find they're not the same marbles. You'll find that about 1/2 of the marbles you got from your mom are the same as ones your sister got, but that about 1/2 of what you got does not match what your sister got. In fact, if you have several siblings, you'll find that collection of marbles you have are all different. Some of the marbles are the same, some are different, and the mix is different for each sibling. The exception is for identical twins. There you each got the exact same marbles from each parent, and you share 100% of your DNA with your identical twin. That is my "halve not":
5) Identical twins/triplets/etc. have the same DNA.
Relationship estimates continued
This brings me to my fourth use of DNA analysis. Sometimes you come up with the relatives that make up a new branch of your family tree. I'll call this "my claimed relationship" (as opposed to the DNA service's "estimated relationship"). You can compare shared DNA for your claimed relationship with the amount shared in the estimated relationship. It should be approximately the same. An example: Ancestry.com claimed a relationship of 4th cousin for a recent pairing. When I came up with what I thought was the new family tree, my genealogy software told me my claimed relationship was 1/2 third cousin once removed. Ouch! Start from 3rd cousin. Rule 3, above, says cut shared DNA in half because of the "1/2", meaning that the two persons were descended from step siblings. Rule 2 says cut it in half again for the "once removed". Now I've cut shared DNA to 1/4 compared to third cousins. Rule 1, above, says 4th cousins share 1/4 as much DNA from a common ancestor as their third cousin parents. In other words, 4th cousins share the same amount of DNA as 1/2 3rd cousins once removed, so Ancestry's estimate supports my claim.
Recap
Articles about DNA case studies may be a little wordier to show what's involved in conducting DNA-assisted genealogy. I've so far found DNA analysis to be useful in (1) identifying relatives which whom I can collaborate to find our common ancestor, (2) helping associate DNA segments with specific grandparents to help focus our efforts in finding a common ancestor, (3) and providing a relationship estimate that might be used to validate the plausibility of my claimed relationship.
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