Efficiency Measures for D- and A- Optimal Designs

The Summary box of both the Optimal Design Result: Mixture dialog box and the Optimal Design Result: Response Surface dialog box report different efficiency measures. G-efficiency measures are reported in the Summary of Efficiency Measures spreadsheet for mixture designs, available on the Quick tab and/or the Confounding tab of the Optimal Design Result: Mixture dialog box. The D- and A-efficiency measures are only available for Response Surface designs, see the Quick tab and/or the Confounding tab of the Optimal Design Result: Response Surface dialog box. Note that G-efficiency measures are also reported for response surface designs.

D-efficiency

This measure is related to the D-optimality criterion:

D-efficiency = 100 * (|X'X|1/p)/N)

Here, p is the number of factor effects in the design (columns in X), and N is the number of requested runs. This measure can be interpreted as the relative number of runs (in percent) that would be required by an orthogonal design to achieve the same value of the determinant |X'X|. However, remember that an orthogonal design may not be possible in many cases, that is, it is only a theoretical "yard-stick." Therefore, you should use this measure rather as a relative indicator of efficiency, to compare other designs of the same size, and constructed from the same design points candidate list.

A-efficiency

This measure is related to the A-optimality criterion:

A-efficiency = 100 * (p/trace(N*(X'X)-1)

Here, p stands for the number of factor effects in the design, N is the number of requested runs, and trace stands for the trace of (X'X)-1 (the inverse of X'X). This measure can be interpreted as the relative number of runs (in percent) that would be required by an orthogonal design to achieve the same value of the trace of (X'X)-1. However, again you should use this measure as a relative indicator of efficiency, to compare other designs of the same size, and constructed from the same design points candidate list.

G-efficiency

This measure is computed as:

G-efficiency = 100 * square root(p/N)/sM

Again, p stands for the number of factor effects in the design, and N is the number of requested runs; sM (sigmaM) stands for the maximum standard error for prediction across the list of candidate points (see Display design on the Optimal Design Result: Mixture (No Intercept) - Display design tab). This measure is related to the so-called G-optimality criterion; G-optimal designs are defined as those that will minimize the maximum value of the standard error of the predicted response.