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. The Alias Matrix is also used in defining alias optimality. See “Alias Optimality” in the “Custom Designs” chapter.
Calculations for the Alias Matrix are based on the model matrix. See “Model Matrix” in the “Custom Designs” chapter.
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:
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 the “Custom Designs” chapter and “Bayesian I-Optimality” in the “Custom Designs” chapter.
. For additional detail on Bayesian designs, see “Bayesian D-Optimality” in the “Custom Designs” chapter and “Bayesian I-Optimality” in the “Custom Designs” chapter.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.
Power Calculations
The Power Analysis report gives power calculations for single parameter values and, when the design includes a categorical factor with three or more levels, for whole effects. This section describes the calculations in the two cases:
Power for a Single Parameter
This section describes how power for the test of a single parameter is computed. Use the following notation:
X
The model matrix. See the Standard Least Squares chapter in the Fitting Linear Models book for information on the coding for nominal effects. Also, See “Model Matrix” in the “Custom Designs” chapter.
Note: You can view the model matrix by running Fit Model. Then select Save Columns > Save Coding Table from the red triangle menu for the main report.
βi
The parameter corresponding to the term of interest.
The least squares estimator of βi
The Anticipated Coefficient value. The difference you want to detect is 
.
.The variance of 
is given by the ith diagonal entry of 
, where σ2 is the error variance. Denote the ith diagonal entry of 
by 
.
is given by the ith diagonal entry of , where σ2 is the error variance. Denote the ith diagonal entry of by .The error variance, σ2, is estimated by the MSE, and has 
degrees of freedom, where n is the number of observations and p is the number of terms other than the intercept in the model.
degrees of freedom, where n is the number of observations and p is the number of terms other than the intercept in the model.The test of 
is given by:
is given by:
or equivalently by:
Under the null hypothesis, the test statistic F0 has an F distribution on 1 and 
degrees of freedom.
degrees of freedom.If the true value of 
is 
, then F0 has a noncentral F distribution with noncentrality parameter given by:
is , then F0 has a noncentral F distribution with noncentrality parameter given by:
To compute the power of the test, first solve for the α-level critical value Fc:
Then calculate the power as follows:
Power for a Categorical Effect
This section describes how power for the test for a whole categorical effect is computed. Use the following notation:
X
Model matrix. See “The Alias Matrix” in the “Technical Details” appendix.
Vector of parameters.
Least squares estimate of β.
Vector of Anticipated Coefficient values.
Matrix that defines the test for the categorical effect. The matrix L identifies the values of the parameters in β corresponding to the categorical effect and sets them equal to 0. The null hypothesis for the test of the categorical effect is given by:
r
Rank of L. Alternatively, r is the number of levels of the categorical effect minus one.
Note: You can view the design matrix by running Fit Model. Then select Save Columns > Save Coding Table from the red triangle menu for the main report.
The covariance matrix of 
is given by 
, where σ2 is the error variance.
is given by , where σ2 is the error variance.The error variance, σ2, is estimated by the MSE, and has 
degrees of freedom, where n is the number of observations and p is the number of terms other than the intercept in the model.
degrees of freedom, where n is the number of observations and p is the number of terms other than the intercept in the model.The test of 
is given by:
is given by:
Under the null hypothesis, the test statistic F0 has an F distribution on r and 
degrees of freedom.
degrees of freedom.If the true value of β is 
, then F0 has a noncentral F distribution with noncentrality parameter given by:
, then F0 has a noncentral F distribution with noncentrality parameter given by:
To compute the power of the test, first solve for the α-level critical value Fc:
Then calculate the power as follows:
Relative Prediction Variance
Consider the following notation:
X
Model matrix. See “Model Matrix” in the “Custom Designs” chapter. Custom designs provide a script that shows the model matrix. See “Save X Matrix” in the “Custom Designs” chapter.
σ2
Error variance.
Vector of least squares estimates of the parameters.
The ith row of X.
Using this notation, the predicted response for the ith row of X is given by:
The relative prediction variance at the settings defined by 
is given by:
is given by:
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