On a Measure of Aliasing Due to Fitting an Incomplete Model

Abstract

This paper proposes and develops a method for selecting a design to estimate a set of linear parametric functions in cases wherein the adequacy of the preliminary linear model is in doubt. The proposed method relies on the norm of the aliasing matrix. If the expected value of the estimator $\hat{\psi}$ of a set of linear functions $\psi = L_1\theta_1$ using a design $\Gamma$, under the true model is $E\lbrack\hat{\psi}\rbrack = \psi + A_\Gamma \theta_2$, then the norm $\|A_\Gamma\| =$ (trace $A_\Gamma' A_\Gamma)^{\frac{1}{2}}$ is presented as a measure for use in determining "alias balance" and "alias goodness." Therefore, $\|A_\Gamma\|$ may be used in the selection of a design for experimentation, and its behaviour under various operations is discussed. Some theorems concerning aliases of rank equivalent and complementary designs in certain settings are obtained.