When looking for surname relationships, analysis of the DNA of the Y chromosome, which is unique to males and is passed on through the male line only, can be useful where no paper evidence exist, or where paper evidence is inconsistent. Unlike the kind of DNA testing used in criminal cases, it is not possible to prove a relationship beyond any reasonable doubt, though it is quite possible to prove that there is no relationship.
Y-DNA is not passed on completely unchanged from generation to generation, being changed on a random basis by mutation. It is even possible, though quite unlikely, that the Y-DNA of male siblings might not match exactly. The farther back you go in generations, the greater the likelihood of changes in the genetic pattern of individuals in the same family line.
DNA analysis is complex and most of the information about it is full of jargon. Worse, the tests offered by different laboratories vary in what is tested and the results are quoted in a different format. I have had my Y-DNA tested by Family Tree DNA, and have results for 67 markers or Short Tandem Repeats [STR]s. The number of repeats of a particular pattern at a particular location in the DNA strand is counted - in my case for 67 selected locations. My brothers should have the same results but my late father's may or may not have been identical. My father had no brothers so I have to go back a generation. My paternal grandfather's results may or may not have been identical but are more likely than my fathers to be different. He had one brother whose results should have been the same as his. The chance of mutations, which change the number of repeats, increases generation by generation, but it is a random process and comparison of results from two individuals gives only a probability that they share a common ancestor within so many generations.
There is a Downie/Downey Y-DNA project hosted by Family Tree DNA and run by Damien Downie in Australia. As well as Downie and Downie, Downs and Downes are included in the results but none of these is known to be of Scottish or Irish origin. Most with the surname Downey are of Irish or unknown origin.
The largest group is identified as having the haplogroup R-M222+. Basically, if you assume that we came from a single male ancestor, haplogroups formed when there was a significant mutation in the Y-DNA from one generation to the next. You can think of R-M222+ as being a collection of people with the same ancestry. This haplogroup is very common amongst people with Irish and Scottish ancestry. M222+ signifies that these Downies tested positive for the single nucleotide polymorphism (SNP) labelled M222. The ISOGG Wiki gives more details about SNPs. Damien belongs to this group and his website provides more information about the origins of Downies with the R-M222+ haplotype.
In addition to having the 67 marker STR test, I have had my DNA tested against the M222 SNP, and it is negative. The only other Downie currently in the project whose DNA falls into this category is Ron Downie, who is one of the many descendants of David Downie and Janet Greig who married in Auchterderran, Fife, in 1754. We have so far been unable to trace David back before this so there is no paper connection with my Downies who lived twenty miles or so East of Auchterderran. Out of 67 STRs, Ron and I match on 63, making our genetic distance 4. FT DNA reports that individuals with 63 or 64 matching STRs are likely to share a common ancestor within the genealogical time frame (during which surnames came into general use). All Downies seem to have disappeared from the Elie/Kilconquhar/Newburn area in the middle of the 18th century, my family being among the last to leave, between 1767 and 1770. I've never so far succeeded in tracing any of the other Downies who were born or had children there in the first half of the 18th Century.