Thank you very much indeed for a very prompt and full reply. Joseph Knox (or a Joseph Knox) was still in Thorndale Avenue in 1918 (but at No 30) and the only Knox in Whitehead at the time was David B Knox. So it seems likely that Joseph and his family moved to Whitehead shortly after the directory was compiled meaning that Sarah Lewis Knox's residence in Whitehead was at the family home.
This reduces the chance of there being any connection with David Brown Knox and certainly none seems to exist from the information you supplied.
Thank you again.