Permutation Tests for the Partial Coefficient in a Multiple Linear Regression

Existing permutation tests for the partial coefficient in a multiple linear regression model are mostly stepwise algorithms which are difficult to be analyzed and compared theoretically. In this paper, we propose a unified framework for testing the partial coefficient in a multiple linear regression model. The unified framework provides a theoretical basis for constructing permutation tests for the partial coefficient as well as theoretical justifications why they work. As a result, several permutation tests are proposed based on the unified framework. The unified framework also enables us to compare the performances of the permutation tests theoretically. By showing that the existing permutation tests for the partial coefficient of a multiple linear regression model are special cases of the unified framework, we are able to identify redundant steps and provide simpler algorithms for the existing permutation tests. Recommendations on which methods to use under different situations are provided based on the theoretical and numerical comparisons we have conducted.