# 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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