According to Gedmatch's Are your parents related? tool, my grandmother's DNA shows several runs of homozygosity (ROH):

Chr Start Location  End Location    Centimorgans (cM)   SNPs
3   180690286       192749897       23.5                2151
8   11855773        21377758        17                  2555
9   22131034        30815690        11.3                1612
10  36514828        63439845        16.6                3005
14  93668370        106176088       28.4                2381

Gedmatch gives the conclusion:

This analysis indicates that your parents are probably related within the last several generations.

My grandmother's parents were known to be second cousins, so this is no surprise.

However, I am wondering how this data, in particular the number and length of ROH can help determine how closely one's parents are related.

This is my grandmother's family tree as determined via traditional genealogical methods, showing her inter-related ancestors:

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As you can see, her parents were not only second cousins, but also double third cousins.

I have no doubt that the two great grandmothers were sisters, but proving the generation above (i.e. the fact that the two great great grandmothers were also sisters) has been challenging with paper records alone. Does this ROH data provide evidence in support of this genealogy? Or in other words, how does one interpret the number and length of ROH in terms of degree of relationship between the parents?

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    Harry rightly pointed out a flaw in the logic of an answer I had posted earlier today. Since I don't at the moment know how to address the issue raised, I've deleted that posting. – RobertShaw Nov 12 '18 at 7:39

From your diagram, you are showing that your grandmother's paternal grandfather's side is related to her materal grandfather's side.

Approximately half of your grandmother's paternal chromosomes would have come from her paternal grandfather. As well, approximately half of your grandmother's maternal chromosomes would have come from her maternal grandfather. That results in approximately 1/4 of your grandmother's genome having her father's chromosome from her paternal grandfather and her mother's chromosome from her maternal grandmother.

Your grandmother's 5 homozygous segments total 96.8 cM. If they are all from the above specified sides, then those 96.8 cM are occurring over 1/4 of her genome, and statistically she would have had 4 times as much homozygosity if she got her father's and mother's chromosomes all from her two grandfathers. Of course that's impossible, but if that would have happened, there would be about 387 cM of homozygous segments.

According to the Shared cM Project 4.0, a 2nd Cousin should share on average 229 cM. A third cousin should share on average 73 cM.

If your grandmother's parents were 2nd cousins and double 3rd cousins, they should share on average 229 + 73 + 73 = 375 cM between them. Each would have passed about half of what was shared to your grandmother resulting in her ending up with about 1/4 of it, i.e. 94 cM. And in fact, she is homozygous over 97 cM.

In most cases, DNA doesn't get passed down this cleanly and work out this well. In your case it does happen to, so it does add evidence that the relationships you've determined for your grandmother's parents might be correct.

If you want to learn more about Runs of Homozygosity (ROH), I would recommend the International Society of Genetic Genealogy (ISOGG)'s wiki page on Homozygosity.

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