Testing for the Equality of EC50 Values in the Presence of Unequal Slopes With Application to Toxicity of Selenium Types

The likelihood ratio test (LRT) for the equality of EC50 values using a probit model that has parallel slopes is implemented in a variety of software packages. A preliminary LRT can be used to ascertain the plausibility of parallel slopes. Testing for equal EC50 values is not as straightforward if the preliminary test rejects that the slopes are equal or, equivalently, if a practitioner would rather not deal with the implications on the size of the test in the presence of the preliminary test. An LRT for testing equal EC50 values is not available in software packages for the case of arbitrary slopes. In this article, we describe a simple and effective algorithm for implementing the LRT procedure in this case. We also derive a quadratic form test procedure for the same hypothesis and compare the two tests (size and power) in the context of our application that deals with comparing the toxicity of four different types of selenium. The R-code is available as supplemental material online.