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Understanding the Alias-Optimal design algorithm, optimization of a fixed block effect

Not_A_Student

I was wondering: when one adds a fixed block effect to the alias-optimal design, how is it optimized?

For instance, following the explications by Jones and Nachtsheim(2011): is it optimized in the first objective being D-optimality optimization for the model terms, or, is it optimized in the secondary objective, being optimizing the orthogonality between Alias term and the model terms?

.. or something else?

Accepted answers

Mark_Bailey

The fixed effect of X5 (blocking factor) is in X1, not in X2.

All comments

Mark_Bailey

The algorithm is the same: coordinate exchange. The criterion is different. It is the minimum of the trace of A'A, where A is the alias matrix as described in the Help.

Not_A_Student

Thank you Mark,

One point that remains confusing for me:

Is the Fixed Block effect placed in the Model (the X1) or in the matrix of model terms for the alias effects the (X2)?

See picture:

Reason for asking is that Following Jones and Nachtsheim (2011), i saw we minimize a mix of the trace of the Alias matrix and the D-efficiency with a weight factor to balance out. More formally, we minimize the objective function:

Where:

This let's me wonder:

If the block effect becomes a model term of design matrix X1 i guess it would optimize predictability (predictive variance) of the blocks as strongly as other model terms. If the block effect was in the model terms for matrix for the alias X2 we would optimize it's orthogonality against the X1 terms but it's predictive variance would be greater and estimated block effect sizes would not necessarily be orthogonal against potentially aliased effects.

Thank you for helping me understanding

Mark_Bailey

The fixed effect of X5 (blocking factor) is in X1, not in X2.