The Alias Matrix

The Alias Matrix entries represent the degree of bias imparted to model parameters by the effects that you specified in the Alias Terms outline. Calculations for the Alias Matrix are based on the model matrix. See The Model Matrix.

Let X1 be the model matrix corresponding to the terms in the Model outline. Denote the matrix of model terms for the effects specified in the Alias Terms outline by X2.

The assumed model is given as follows:

Suppose that some of the alias terms are active and that the true model is given as follows:

The least squares estimator of β1 is given by:

Under the usual regression assumptions, the expected value of

is given by:

where

The matrix A is called the alias matrix.

Designs with Hard or Very Hard Factor Changes

For designs with hard-to-change or very-hard-to-change factors, the alias matrix is given as follows:

where V is the block diagonal covariance matrix of the responses.

Designs with If Possible Effects

For designs with If Possible effects (Bayesian D- and I-optimal designs), the alias matrix is given as follows:

where K is a diagonal matrix with these values:

•	k = 0 for Necessary terms

•	k = 1 for If Possible main effects, powers, and interactions involving a categorical factor with more than two levels

•	k = 4 for all other If Possible terms

In the Bayesian case, the alias matrix gives the aliasing of effects corresponding to a ridge regression with a prior variance of

. For additional detail on Bayesian designs, see Bayesian D-Optimality in Custom Designs and Bayesian I-Optimality in Custom Designs.

In the Custom Design platform, you can control the weights used for If Possible terms by selecting Advanced Options > Prior Parameter Variance from the red triangle menu. There you can set prior variances for all model terms by specifying the diagonal elements of K. The option updates to show the default weights when you click Make Design.