Here's a couple of entries from the 1880 census:

Myron E. LAWRENCE, self, married, male, white, 32, born NY, farmer, father born NY, mother born NY
Philinda L. LAWRENCE, wife, married, female, white, 27, OH, keeping house, NY, VT
William E. LAWRENCE, son, single, male, white, 5, WI, at school, NY, OH
Walter A. LAWRENCE, son, single, male, white, 1, WI, NY, OH
Source Information:
Census Place Adams, Adams, Wisconsin
Family History Library Film 1255417
NA Film Number T9-1417
Page Number 3B



George LAWRENCE, self, married, male, white, 34, born England, huckstor, father born ENG, mother born ENG
Ann LAWRENCE, wife, married, female, white, 40, ENG, keeping house, ENG, ENG
George LAWRENCE, son, single, male, white, 14, ENG, ENG, ENG
Thomas LAWRENCE, son, single, male, white, 12, ENG, ENG, ENG
Laura LAWRENCE, dau, single, female, white, 10, ENG, ENG, ENG
William LAWRENCE, son, single, male, white, 7, ENG, ENG, ENG
Maude LAWRENCE, dau, single, female, white, 5, WI, ENG, ENG
Bertie LAWRENCE, son, single, male, white, 1, WI, ENG, ENG
Source Information:
Census Place Princeton, Green Lake, Wisconsin
Family History Library Film 1255428
NA Film Number T9-1428
Page Number 372A


I'm not finding a census entry that includes a Rose, with a George and a William. Perhaps Rose is a nickname. Someone would need to obtain William's death certificate, to see if the parents name are included.