I apologize that my question was not better explained. If I may use your candidate as an example, Thomas Rogers was a Mayflower passenger, while his wife Alice was not, because she remained in Holland when Mayflower sailed. Their son, Joseph, would have been half-Mayflower ancestry had he remained with his mother, but is a Mayflower passenger because he sailed with his father. Joseph's son, James would be half-Mayflower from his father, and a full-blooded Mayflower descendant if his mother, Hannah, had sailed aboard the Mayflower or been born to parents who sailed aboard the Mayflower. Hannah, however, is unfortunately among the many early colonial women whose ancestry is uncertain, and is generally assumed to have arrived later descended from parents who were not Mayflower passengers. Even had Hannah been a Mayflower descendant, James wife, Mary Paine, was one-quarter Mayflower ancestry from her maternal grandmother, Constance (Hopkins) Snow, so their son James would be 5/8 Mayflower ancestry if Hannah was a Mayflower descendant or only 3/8 Mayflower ancestry if Hannah was not.