Birth Record below says his mothers name was Elizabeth - Possibly true, as David Senior did not get married to Agnes until the following year (1861)
"Scotland Births and Baptisms, 1564-1950," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:XY1V-DH3 : 2 January 2015), David Grieve Fairbairn, 17 Jul 1860; citing CRAIL,FIFE,SCOTLAND, reference , index based upon data collected by the Genealogical Society of Utah, Salt Lake City; FHL microfilm 6,035,516.
It may be possible to confirm family relationships with David by comparing test results with other carriers of his ancestors' Y-chromosome or mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share DNA with David: