John Anthony the immigrant was given a pedigree involving armigerous persons in 1904. This vaguely resembled information in the Visitations of London and Bedfordshire, and some subsequent genealogies have made corrections to the line based on the visitations.
The author of the 1904 genealogy states in the preface: “Our line of connection between this ancient family and John Anthony, the emigrant, has not been established and fortified by complete and satisfactory evidence but numerous circumstances point in this one direction."
A 1644 will for Richard Arnold names the children of John Anthony MD (d. 1655), including sons Samuel and John. The language of the will implies that John and his siblings are present in London. John the emigrant was in Rhode Island before 1644.
In addition John Anthony MD (d. 1655) left a will in which he does not mention son John but mentions his other 4 children. And his gravestone says he left only one son and 3 daughters. His child John had obviously died in the interim.
There is more detail on the profile. Anderson also calls his origin Unknown in the GM Directory.