On the Theory of Apportionment

If in an accepted sense, P is the probability that one method of treatment, T,, is better than a rival, T2, we may develop a system of apportionment such that the proportionate use of T1 is f(p), a monotolle increasing functioni, rather than make no discriminationi at all up to a certaini poinit and then finally entirely reject one or the other. The only paper * which has so far appeared in his field, as far as I am aware, is one by myself in a recent issue of Biometrilca. In this paper I have considered the case of choice between two such rival treatments,t and for symmetry suggested that f (QI f(p) where Q = 1 P. Then the riski of assignmenit to T1 when it is not the better is Q f(p), while the correspondinig risk for T2 is P f.(Q Accordingly, I suggested further that we set f(p) P, which is a necessary and sufficient conditioni that these two risks be equal. Their sum, the total qrisk, is then 2PQ. A special case was considered wherein the result of use of Ti at any given trial is either success or failure, the probability of failure being an unkniownl, pi, a priori (independenitly for i 1, , kc) equally likely to lie in either of any two equal intervals in the possible range, (0, 1). It is further assumed that for a given Ti we have an experience of exactly ni indepenidenit trials, the number of su?ccesses being si and of failu-res being ri = si-; and the probability of obtaininig such a sample is