Example of selecting a D-optimal response surface design
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Suppose you want to conduct an experiment to maximize crystal growth. You have determined that four variables - time the crystals are exposed to a catalyst, temperature in the exposure chamber, pressure within the chamber, and percentage of the catalyst in the air inside the chamber - explain much of the variability in the rate of crystal growth.

You generate the default central composite design for four factors and two blocks (the blocks represent the two days you conduct the experiment). This design, which contains 30 design points, serves as the candidate set for the D-optimal design.

Available resources restrict the number of design points that you can include in your experiment to 20. You want to obtain a D-optimal design that reduces the number of design points.

1    Open the worksheet OPTDES.MTW.

2    Choose Stat > DOE > Response Surface > Select Optimal Design.

3    In Number of points in optimal design, type 20.

4    Click Terms. Click OK in each dialog box.

Session window output

Optimal Design: Blocks, A, B, C, D

 

 

Response surface design selected according to D-optimality

 

Number of candidate design points: 30

Number of design points in optimal design: 20

 

Model terms: Block, A, B, C, D, AA, BB, CC, DD, AB, AC, AD, BC, BD, CD

 

Initial design generated by Sequential method

Initial design improved by Exchange method

Number of design points exchanged is 1

 

 

Optimal Design

 

Row number of selected design points: 22, 23, 25, 27, 4, 8, 19, 2, 14, 15, 13, 6, 9, 3, 16,

                                      24, 28, 30, 26, 1

 

Condition number:                              10.2292

D-optimality (determinant of XTX):         2.73819E+18

A-optimality (trace of inv(XTX)):              2.50391

G-optimality (avg leverage/max leverage):          0.8

V-optimality (average leverage):                   0.8

Maximum leverage:                                    1

Interpreting the results

The Session window output contains the following five parts:

·    A summary of the D-optimal design. This design was obtained by selecting a subset of 20 points from a candidate set of 30 points.

·    The model terms. D-optimal designs are dependent on the specified model. In this example, the terms include:

Block A B C D AA BB CC DD AB AC AD BC BD CD

These are the full quadratic model terms that were the default in the Terms subdialog box. Remember, a design that is D-optimal for one model will most likely not be D-optimal for another model.

·    The method by which the initial design was generated and whether or not an improvement of the initial design was requested. In this example, the initial design was generated sequentially and the exchange method (using one design point) was used to improve the initial design.

·    The selected design points in the order they were chosen. The numbers shown identify the row of the design points in the original worksheet.

Note

The design points that are selected depend on the row order of the points in the candidate set. Therefore, Minitab may select a different optimal design from the same set of candidate points if they are in a different order. This can occur because there may be more than one D-optimal design for a given candidate set of points.

·    Minitab displays some variance-minimizing optimality measures. You can use this information to compare designs.