Optimality metrics for Select Optimal Design

Find definitions and interpretation guidance for every design optimality statistic.

Condition number

The condition number measures the collinearity among model terms. When you compare designs, a smaller condition number is better.

Interpretation

Use the condition number to compare different optimal designs or to compare the same design with different terms. A condition number of 1 indicates that the model terms are orthogonal. Larger values indicate more collinearity.

Most optimal designs are not orthogonal. Because terms in the model are not independent, the interpretation of non-orthogonal designs is less straightforward than the interpretation of orthogonal designs.

In these results, the condition number indicates that the data exhibit moderate to strong collinearity.

Optimal Design: Temperature, Copper, Endcap, Method

Factorial design selected according to D-optimality Number of candidate design points: 64 Number of design points in optimal design: 32 Model terms: A, B, C, D, AB, AC, AD, BC, BD, CD 50% of the points in initial design are generated randomly Remaining points added to initial design by Sequential method Initial design improved by Exchange method Number of design points exchanged is 1
Optimal Design Row number of selected design points: 46, 54, 36, 8, 44, 47, 31, 55, 30, 43, 38, 59, 62, 15, 56, 24, 42, 20, 32, 16, 6, 45, 19, 17, 25, 49, 64, 10, 37, 1, 39, 3 Condition number: 259.114 D-optimality (determinant of XTX): 7.92282E+28 A-optimality (trace of inv(XTX)): 12.1719 G-optimality (avg leverage/max leverage): 0.96875 V-optimality (average leverage): 0.96875 Maximum leverage: 1

D-optimality

D-optimality indicates the design's ability to obtain precise estimates or predictions. When you compare designs, a larger D-optimality value is better.

Interpretation

You can use optimality metrics to compare designs, but remember that the optimality of a given D-optimal design is model dependent. That is, optimality is defined for a fixed design size and for a particular model.

In these results, the first optimal design has 25 design points and the second optimal design has 20 design points. The first design has a higher D-optimality statistic than the second optimal design, which is expected with more runs.

Optimal Design: Blocks, A, B, C, D

Response surface design augmented according to D-optimality Number of candidate design points: 30 Number of design points to augment/improve: 20 Number of design points in optimal design: 25 Model terms: Block, A, B, C, D, AA, BB, CC, DD, AB, AC, AD, BC, BD, CD Initial design augmented by Sequential method Initial design improved by Exchange method Number of design points exchanged is 1 Optimal Design Row number of selected design points: 1, 3, 4, 6, 8, 9, 10, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 27, 28, 30, 2, 5, 14, 18, 20 Condition number: 8.53018 D-optimality (determinant of XTX): 3.73547E+20 A-optimality (trace of inv(XTX)): 1.99479 G-optimality (avg leverage/max leverage): 0.64 V-optimality (average leverage): 0.64 Maximum leverage: 1

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

A-optimality

A-optimality measures the average variance in the regression coefficients of the fitted model. When you compare designs, a smaller A-optimality value is better.

Interpretation

You can use optimality metrics to compare designs, but remember that the optimality of a given A-optimal design is model dependent. That is, optimality is defined for a fixed design size and for a particular model. Designs that are more D-optimal are not necessarily more A-optimal.

In these results, the first optimal design has 25 design points and the second optimal design has 20 design points. The first design has a lower A-optimality statistic than the second optimal design, which is expected with more runs.

Optimal Design: Blocks, A, B, C, D

Response surface design augmented according to D-optimality Number of candidate design points: 30 Number of design points to augment/improve: 20 Number of design points in optimal design: 25 Model terms: Block, A, B, C, D, AA, BB, CC, DD, AB, AC, AD, BC, BD, CD Initial design augmented by Sequential method Initial design improved by Exchange method Number of design points exchanged is 1 Optimal Design Row number of selected design points: 1, 3, 4, 6, 8, 9, 10, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 27, 28, 30, 2, 5, 14, 18, 20 Condition number: 8.53018 D-optimality (determinant of XTX): 3.73547E+20 A-optimality (trace of inv(XTX)): 1.99479 G-optimality (avg leverage/max leverage): 0.64 V-optimality (average leverage): 0.64 Maximum leverage: 1

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

G-optimality

G-optimality is the ratio of the average prediction variance to the maximum prediction variance over the design points. G-optimal designs minimize the denominator, while V-optimal designs minimize the numerator. Ideally, you want both the numerator and denominator to be smaller values.

Interpretation

You can use optimality metrics to compare designs, but remember that the optimality of a given G-optimal design is model dependent. That is, optimality is defined for a fixed design size and for a particular model. Designs that are more D-optimal are not necessarily more G-optimal.

In these results, the first optimal design has 25 design points and the second optimal design has 20 design points. The design with more points is less G-optimal than the design with more points, even though the larger design is more D-optimal.

Optimal Design: Blocks, A, B, C, D

Response surface design augmented according to D-optimality Number of candidate design points: 30 Number of design points to augment/improve: 20 Number of design points in optimal design: 25 Model terms: Block, A, B, C, D, AA, BB, CC, DD, AB, AC, AD, BC, BD, CD Initial design augmented by Sequential method Initial design improved by Exchange method Number of design points exchanged is 1 Optimal Design Row number of selected design points: 1, 3, 4, 6, 8, 9, 10, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 27, 28, 30, 2, 5, 14, 18, 20 Condition number: 8.53018 D-optimality (determinant of XTX): 3.73547E+20 A-optimality (trace of inv(XTX)): 1.99479 G-optimality (avg leverage/max leverage): 0.64 V-optimality (average leverage): 0.64 Maximum leverage: 1

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

V-optimality

V-optimality measures the average prediction variance over the set of design points. When you compare designs, a smaller V-optimality value is better.

Interpretation

You can use optimality metrics to compare designs, but remember that the optimality of a given V-optimal design is model dependent. That is, optimality is defined for a fixed design size and for a particular model. Designs that are more D-optimal are not necessarily more V-optimal.

In these results, the first optimal design has 25 design points and the second optimal design has 20 design points. The first design has a lower V-optimality statistic than the second optimal design, which is expected with more runs.

Optimal Design: Blocks, A, B, C, D

Response surface design augmented according to D-optimality Number of candidate design points: 30 Number of design points to augment/improve: 20 Number of design points in optimal design: 25 Model terms: Block, A, B, C, D, AA, BB, CC, DD, AB, AC, AD, BC, BD, CD Initial design augmented by Sequential method Initial design improved by Exchange method Number of design points exchanged is 1 Optimal Design Row number of selected design points: 1, 3, 4, 6, 8, 9, 10, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 27, 28, 30, 2, 5, 14, 18, 20 Condition number: 8.53018 D-optimality (determinant of XTX): 3.73547E+20 A-optimality (trace of inv(XTX)): 1.99479 G-optimality (avg leverage/max leverage): 0.64 V-optimality (average leverage): 0.64 Maximum leverage: 1

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

Maximum leverage

Maximum leverage indicates that a design has a highly influential point when the maximum leverage is much larger than V-optimality. Minitab uses this value in the denominator when calculating G-optimality.

Interpretation

Use maximum leverage to determine when a design contains at least one influential point. Designs that are more D-optimal can have influential points.

In these results, the maximum leverage is 1 and the V-optimality is 0.8. In this optimal design, none of the factor levels in row 2 are in any of the other points.

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

Largest and smallest distance between optimal points

Minitab displays the largest and smallest distances between the selected design points. This value is the Euclidean distance.

Interpretation

The difference between the largest and the smallest distance values indicates how uniformly the points are spread in the design space. You can use this information to compare designs.

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