Abstract
In the thesis, we consider a linear model with a continuous response $Y$ and two 2-level predictors $A$ and $B$, where the two levels are labeled by 0 and 1. Denote by $Y_{00}$, $Y_{01}$, $Y_{10}$, and $Y_{11}$ respectively, the responses obtained on the four level combinations of $A$ and $B$ (i.e., (0, 0), (0, 1), (1, 0), and (1, 1)), and assume that they have equal variance. The two predictors $A$ and $B$ are said to have a synergistic (or antagonistic) interaction on the response $Y$ if $Y_{00},$ $Y_{01},$ and $Y_{10}$ are normally distributed with almost identical means, but $Y_{11}$ follows a normal distribution with a significantly larger (or smaller) mean. To identify whether a synergistic or antagonistic interaction (or a synergistic interaction, or an antagonistic interaction) exists, Lin (2015) suggested a linear model based on Helmert coding. In the thesis, we adopt the model to develop tests for the presence of the interactions under the assumption that the variance of $Y$ is known. We formulate the test problem as an intersection-union test (IUT), which is composed of three individual tests: two equivalence tests and one two-sided (or one-sided) test. The rejection region of an IUT is the intersection of the rejection regions of its individual tests. Berger (1982) gave a condition, under which an IUT is a size-$\alpha$ test if its individual tests are of size $\alpha$. However, the condition does not hold in our case so that the IUT for the interactions based on Berger's method is not a level-$\alpha$ test. Furthermore, Berger's method does not consider the correlations between the test statistics of the individuals tests. To address the issues, we study the maximum probability of type I error of the IUT for the interactions and obtain some theorems. Based on the theorems, we propose a new method to construct a size-$\alpha$ IUT for the interactions. The new IUT has higher power than the one based on Berger's method. We also discuss some questions about the new IUT, including the choice of equivalence margins, and why our method cannot be directly generalized to the case of unknown variance. In the end, we use a computer simulation to study the performance of the new IUT, and compare it with the one based on Berger's method.