Using numerical methods to find the least favorable configuration when comparing k test treatments with both positive and negative controls

Traditionally, comparisons to control problems dealt with comparing k test treatments to either a positive control or a negative control. However, in certain situations, it is necessary to include both types of control in the study. When comparing k test treatments to both positive and negative controls, the null hypothesis is composite and finding the least favorable configuration (LFC) under the null analytically is difficult. We propose a numerical solution for this problem. Using our method, we show that the LFC under the null is when half of the test treatment means are equal to the mean of the negative control and the other half of the test treatment means are equal to the mean of the positive control. We calculate critical points at the LFC for k =2-6. This work was motivated by real life problems that we discuss and illustrate with an example.