I did the Ancestry.com DNA test and found out that I had traces of Native American. I myself am from Sweden but have American ancestry.
How can I trace this back further?
I am a beginner.
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Before you put a lot of work in chasing what might be a shadow, learn more about ethnicity estimates and about how DNA works in general.
Ethnicity estimates are a guess -- they are made by comparing test subjects' DNA to a selected group of people called a "reference population" or "reference panel".
Suppose you are setting up your own site to study people from Sweden. You want to find other people whose family came from your family's hometown, so you choose a reference population of people whose families have been in that town for a specific number of generations. You use that group of people as your baseline to compare the test-takers to.
As you can guess, this method has limitations. The estimate can only tell you so much, and unless you are a DNA data geek who is intimately familiar with all the companies' DNA tests and their reference populations, you can only learn so much from it.
Judy G. Russell, who blogs as The Legal Genealogist, has written often about the difficulties involved in using ethnicity estimates. In her post from Apr 16, 2017, Still Not Soup, she posted seven different ethnicity estimates and asked: "How many people are represented by the DNA results shown below?"
I highly recommend that you read all of her posts about ethnicity estimates, but especially "Still Not Soup" (the 'soup' reference is about a US TV commercial with a running joke about whether dinner is ready or not). She posts in clear, commonsense language about DNA and posts about changes in different companies' algorithms and estimates in a way that non-scientists can understand.
Under the hood
If you do want to dive into the science behind Ancestry's estimate, their support site has a set of documents called "White Papers" that explain what they are doing. Ancestry's support pages move a lot, so if this link goes bad, search for "AncestryDNA white papers" to find it again. AncestryDNA's White Paper on Ethnicity Estimates
As you educate yourself by reading about genetic genealogy, especially ethnicity estimates, keep in mind that the field is changing rapidly. Any article or webinar needs to be read/watched while noting the date it was produced, with the awareness that something might have changed since the content was made. Companies change testing methods (microarray chips) all the time, reference populations get tweeked, etc.
Tribal belonging via DNA? It's complicated.
There is a common misconception that you can use DNA testing to prove you are Native American. But just like the rest of genealogy, where DNA results must be used in context and combined with paper-trail research and information from the real world, you can't use DNA results alone and declare yourself the member of a Native American nation. Kim Tallbear, an anthropologist at the University of Texas, Austin, is the author of Native American DNA: Tribal belonging and the false promise of genetic science (University of Minnesota Press) -- she discussed some of the issues in New Scientist: 'There is no DNA test to prove you're Native American'. Her work is especially valuable when you're trying to put these estimates into their proper context.
The TL;dr? AncestryDNA could change their reference panel again, and anything that has trace amounts (the small slices on the pie chart) could disappear. Use your ethnicity estimates for clues to further paper-trail research, not as proof of anything. (And just as I said that -- look, they've updated again! See the update from October 21, 2019 in Further Reading.)
Start by writing down everything you know about your 'American ancestry'. Interview older relatives who may know something about this side of your family. Begin with the most recent generation and get names, birth/marriage/death dates for every name, and then double-check those with actual BMD certificates from relevant states and cities in the USA. Once you have an ancestor's name, age, and approximate place of residence in the States, you can start looking for them and their birth families in US Census records:
and get back to older generations that way. Remember that it is crucial to cross-check all of your information from relatives, online family histories, etc. with actual primary-source documents like vital records (BMD certs), census records, city directories, land and court records, etc. This is not done quickly, and depending on how much Native American ancestry you have, you may have to go back many generations to find it.
Is the percentage given so low as to be possibly within the margin of error?
Have any of your other relatives taken a DNA test - what did they say? This might allow you to at least work out what side of the family you should be looking on.
How far back do you "know" your ancestry? Do you have documentation for this - if not try to find proof for everything you believe to be true. Are there any gaps in this? If there is something you don't know about or that doesn't add up properly look there.