According to the 1910 census, Roger Carroll arrived in the Unites States from Ireland in 1845. Despite the date provided through his death certificate,

According to Dr. D. G. Chadbourne, M.D., Roger Carroll died on August 24, 1917, of a "Fracture of hip, Fall on floor in home" with old age a contributing factor. His address is listed as 916 Willis Ave. (Syracuse, NY), Ward 9. The certificate noted that Roger had resided at his place of death for 35 years [since 1882?], was retired, and had lived in the State of New York for 60 years [since 1857?]. The information was provided by Roger Carroll of Syracuse, NY.

His date of birth is given as November 12, the year not provided. Age at death is provided as 67 years, 9 months, 12 days, which would put his year of birth at 1849. This conflicts with information in the 1880 Onondaga County, Geddes Village census information that would suggest 44-year old Roger's birthyear was 1836. His census entry listed that he was born in Ireland, but the death record of his daughter, Mary Agnes, indicates he was born in Syracuse (information on her record provided by Edward Capels).

If it's accurate that he had been in the state for 60 years, then he might have been 7 or 8 years old when his family arrived in upstate New York.

Obituary for Roger E. Carroll:
Roger E. CARROLL, a resident of the West End for many years, died Friday at his home, No. 916 Willis avenue. He had been ill some time. He is survived by his widow, Mrs. Catherine CARROLL; two daughters, Mrs. Loren CAPLES [Mary] and Josephine [sic - Johanna/Joanna] CARROLL, and four sons, John of Taunton, Mass.; Michael of West Monroe, Roger and Daniel CARROLL. The funeral will be held from the home at 9:45 o'clock this morning and half an hour later in St. Patrick's Church. Burial will be in St. Agnes's Cemetery.
[Source: Syracuse Post Standard, Monday, August 27, 1917; notes in brackets and surname capitalizations added by compiler VC.]

This obituary tells us that his son Thomas is no longer living in 1917.

• Married: ABT 1858

1. John E. Carroll b: ABT 1858 in New York
2. Edward Carroll b: ABT 1860 in New York
3. Roger Carroll b: ABT 1863
4. Mary Agnes "Mamie" Carroll b: 10 NOV 1864 in Syracuse, New York
5. Thomas Carroll b: ABT 1869
6. Michael E. Carroll b: 8 JAN 1869 in Syracuse, New York
7. Johanna Carroll b: ABT 1871
8. Daniel Carroll b: ABT 1874

1. Title: Death Certificate, Roger E. Carroll, issued 8/27/17, City of Syracuse, Onondaga County, New York.
   Repository:
   Media: Official Document
2. Title: Death Certificate, Loren Capels, 1/24/29
   Repository:
   Media: Official Document
3. Title: 1870 Federal Census, Town of Geddes, Onondaga County, New York, page 96
   Publication: Indexed by Ancestry.com; Data imaged from National Archives and Records Administration, Washington, D.C.: Roll M593-1060; Page 274; Image 550.
   Repository:
   Media: Book