This appears to be him with Martha and children in census data. 1870 says born in MO? Don't know if searching for the children's records would lead to any more info. Did Martha apply for a widow's pension, as an Lawrence W Rogers served in the Civil War?
U S Census, 1870 Pleasant Gap, Bates, MO
Citing this Record: "United States Census, 1870," index and images, FamilySearch (https://familysearch.org/pal:/MM9.1.1/M4XW-2XZ), L W Rogers, Missouri, United States; citing p. 32, family 233, NARA microfilm publication M593, National Archives and Records Administration, Washington D.C.; FHL microfilm 552,257.
L W Rogers M 27 Missouri
Martha Rogers F 26 Missouri
Wm Rogers M 7 Missouri
Thomas Rogers M 3 Missouri
Jessee Rogers M 1 Missouri
U S Census, 1880 Shawnee, Bates, MO
Citing this Record: "United States Census, 1880," index and images, FamilySearch (https://familysearch.org/pal:/MM9.1.1/M668-5VF), Martha J Rogers, Shawnee, Bates, Missouri, United States, 154; citing sheet 133A, film number 0673, NARA microfilm publication T9, National Archives and Records Administration, Washington D.C.; FHL microfilm 1,254,673.
Martha J Rogers Self F 36 Missouri widow
Jas T Rogers Son M 13 Kansas
Jesse F Rogers Son M 11 Missouri
Albert Rogers Son M 6 Missouri
Wm L Rogers Son M 17 Missouri