In the sourced records, John Self is listed only as "John". There is no evidence of a middle name "Joseph".
Sources
1820, 1830, and 1840 Natchitoches Parish, LA Census
L&D-108: John Self sold all his property to his wife, Nancy W. Spade for $230 on 11 Apr 1850. This was the same day as the James Tyler to Thomas Tyler instrument was filed. Sabine Parish Library File. aka: Nancy W. SPADE Self.
Natchitoches Parish Conveyences. Deed (D. Cons. Par. Pro.), John Self to Nancy Spade Self. Bk A, p. 433. Dated 11 Apr 1850. (From Helen Brumley files)
"United States Census, 1850," database with images, FamilySearch (https://www.familysearch.org/ark:/61903/1:1:MCJD-Q5F : 22 December 2020), John Self, Sabine, Louisiana, United States; citing family , NARA microfilm publication (Washington, D.C.: National Archives and Records Administration, n.d.).
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DNA Connections
It may be possible to confirm family relationships with John by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with John:
I am this 3rd great grandaughter. I can say with certainty that his father has not yet been determined. He came to Natchitoches, La. in 1816 per his own testimony in the Rio Hondo land claim of John Litton. Another Rio Hondo claimant testified that John Litton had been previously in the Red River settlement of Pecan Point. That is where James Tyler and Sam Wiley had been too. He was very young when he got to Louisiana, which causes me to believe that he was orphaned. Also, all of the Creek nation Selfs who came to Natchitoches lived in the same area (about 1830), and this John is buried at Vowel's Mill in the same cemetary as some of the Creek Selfs who came to Natchitoches. I believe that he was part Creek, although I have no proof. The coincidental evidence is very strong.
Patrick, I think you may have a problem with this one; You have a man born 1798 to a father who died in the early 1700s. Let me know if you need help detaching them.