Shift-invariant target in allocation problems

We provide a template for finding target allocation proportions in optimal allocation designs where the target will be invariant for both shifts in location and scale of the response distributions. One possible application of such target allocation proportions is to carry out a response-adaptive allocation. While most of the existing designs are invariant for any change in scale of the underlying distributions, they are not location invariant in most of the cases. First, we indicate this serious flaw in the existing literature and illustrate how this lack of location invariance makes the performance of the designs very poor in terms of allocation for any drastic change in location, such as the changes from degrees centigrade to degrees Fahrenheit. We illustrate that unless a target allocation is location invariant, it might lead to a completely irrelevant and useless target for allocation. Then we discuss how such location invariance can be achieved for general continuous responses. We illustrate the proposed method using some real clinical trial data. We also indicate the possible extension of the procedure for more than two treatments at hand and in the presence of covariates.