I believe that it is possible that your Alexander Allardyce MAY be related to Elizabeth Allardyce Clark, of Wayne, Wayne, MI, but have no proof.Here is what I do have.
Elizabeth Allardyce was b. Feb 1806-1807 in Scotland and married James Clark, very likely in Scotland.There is a marriage of an Elizabeth Allardyce and James Clark on 14 Jun 1836 at Rhynie And Essie, Aberdeen, Scotland, which is very likely them.They would have immigrated to MI between 1836 and 1843.I don't recall if James Clark was in Wayne, MI in 1840 and have misplaced the census records on him.He died in 1846.
Their eldest son, James Clark (Jr.), was born in 1838, but I don't know where.He died in 1843 and is buried at Glenwood Cemetery in Wayne, Wayne, MI.James Clark (Sr.)died in 1846 as did an infant son, Alexander, and buried at Glenwood Cemetery in Wayne, MI.
James Clark (Sr.) and Elizabeth Allardyce had three children that survived childhood:William, Isabella, and Elizabeth.All born in Wayne County, in or near Nankin. William and Elizabeth married and had families.Elizabeth married James Jamieson. I'm told that there is a street or streets in Wayne, named after the Clark or Clark-Jamieson family.
The Jamiesons (Scottish Church) came from Northumberland, England, with family in Scotland, since they were so near the border.I've often wondered if the Clark and Jamieson families knew each other in the "old country."