I may be looking at incorrect information, but it appears the family are in these census records, but it does imply that they moved around an awful lot.....
"United States Census, 1900," database with images, FamilySearch(https://familysearch.org/ark:/61903/1:1:M9W6-54M : accessed 9 September 2015), Lee Gibson, Magisterial District 3, Cannonsburg, Boyd, Kentucky, United States; citing sheet 6B, family 108, NARA microfilm publication T623 (Washington, D.C.: National Archives and Records Administration, n.d.); FHL microfilm 1,240,509.
The 1910 is interesting as it has a possible Father for Robert
"United States Census, 1910," database with images, FamilySearch(https://familysearch.org/ark:/61903/1:1:M2C9-8NL : accessed 9 September 2015), Lee Gibson, Pittsburg Ward 4, Crawford, Kansas, United States; citing enumeration district (ED) 101, sheet 7A, family 116, NARA microfilm publication T624 (Washington, D.C.: National Archives and Records Administration, n.d.); FHL microfilm 1,374,449.
If the 1920 Census is for your family, then the following WW1 Draft Registration would be for Robert Lee.
"United States Census, 1920," database with images, FamilySearch(https://familysearch.org/ark:/61903/1:1:MDRF-WF4 : accessed 9 September 2015), Robert L Gibson, Portsmouth Ward 4, Scioto, Ohio, United States; citing sheet 1A, family 7, NARA microfilm publication T625 (Washington D.C.: National Archives and Records Administration, n.d.); FHL microfilm 1,821,433.
"United States World War I Draft Registration Cards, 1917-1918," database with images, FamilySearch(https://familysearch.org/ark:/61903/1:1:K6N7-SCX : accessed 9 September 2015), Robert Lee Gibson, 1917-1918; citing Scioto County, Ohio, United States, NARA microfilm publication M1509 (Washington D.C.: National Archives and Records Administration, n.d.); FHL microfilm 1,851,091.
"United States Census, 1930," database with images, FamilySearch(https://familysearch.org/ark:/61903/1:1:X7M8-1JN : accessed 9 September 2015), Robert Gibson, Owosso, Shiawassee, Michigan, United States; citing enumeration district (ED) 0014, sheet 2A, family 45, line 49, NARA microfilm publication T626 (Washington D.C.: National Archives and Records Administration, 2002), roll 1026; FHL microfilm 2,340,761.
Not exactly the information you were after, but a little bit more ammunition.