Farrington-Manning in the Extreme Case

The Farrington-Manning method is a common method for evaluating equivalence and non-inferiority of independent proportions. It is implemented in various software, in particular SAS$^\circledR$ PROC FREQ, and the R$^\circledR$ function farrington.manning(), which is part of the DescrTab2 package. The equations for the estimated proportions can create numerical issues in case both sample proportions equal 1, and neither of these packages will yield an acceptable solution in this special case. In this note we demonstrate a closed form solution for the situation in this and other extreme cases.