Identifying the Maximum Safe Dose MaxSD

All of the preceding discussion extends naturally to the problem of identifying the MaxSD in toxicity studies with a few minor changes as we note below. In order to keep the forms of the hypotheses (11.6) and the test statistics in Eq. (11.10) the same, and also to conform with the past literature, we will assume that lower implies a more toxic (less safe) dose. Toxicity generally increases with dose level and the zero dose has the least toxicity. Therefore the /¿i's are generally decreasing and the threshold X < 1. Thus, dose i with > X^0 is regarded as safe, while dose i with < X^0 is regarded as unsafe. For example, X = 0.90 means that a 10% decrease in safety level (increase in toxicity) is regarded as clinically unsafe.

The maximum safe dose (MaxSD) for specified X < 1 is defined as

Analogous to the discussion of the MinED, there could be two definitions of the MaxSD. However, we assume monotonicity of the toxicity response so that the definitions are identical. The hypotheses are the same as in Eq. (11.6) (where now Hi states that the ith dose is unsafe). If

MaxSD = max{i : Hj is rejected V j < i} denotes the estimated MaxSD then we want to guarantee that

Since the goal is now to find the MaxSD, both SD1PC and SD2PC start by testing H1 and proceed to testing H2 if H1 is rejected (dose 1 is declared safe) and so on. If H1 is not rejected then all Hi are accepted without further tests and all doses are declared unsafe, i.e., there is no MaxSD. SD1PC rejects Hi using the representation Hi = P|kj=i Hj if

Ti,max = max Tj > tg}Rr i < j <j t where I = j — i + 1 and Rt = {pij}, while SD2PC rejects Hi if Ti > tva. For details see Tamhane et al.