# Design details for Select Optimal Design

Find definitions and interpretation guidance for every statistic that is provided with the an optimal design.

## Design selected or augmented according to criterion

Minitab displays the criterion and indicates whether the design was selected or augmented.

For example, the following are examples of different designs (factorial, response surface, or mixture), different tasks (select or augment), and different criterion (D-optimality or distance-based).
• Factorial design selected according to D-optimality
• Response surface design selected using distance-based optimality
• Mixture design augmented according to D-optimality

For factorial designs, D-optimality is the only criterion that Minitab provides.

## Number of candidate design points

The number of candidate design points shows how many design points (worksheet rows) are considered in the search for the optimal design. A design point is an experimental condition or factor level combination at which responses are measured. Each point corresponds to a row in the worksheet that contains the candidate points.

## Number of design points to augment/improve

The number of design points to augment/improve shows how many experimental runs are in the design before the augmentation or improvement is complete.

### Interpretation

Use the number of design points to see the number of points in the initial design. A point is an experimental condition or factor level combination at which responses are measured. The initial design can have replicated points, so the number of design points to augment/improve can exceed the number of candidate design points.

## Number of design points in optimal design

The number of optimal design points shows how many experimental runs are in the final optimal design.

### Interpretation

Use the number of optimal design points to see how many points are in the final design. A point is an experimental condition or factor level combination at which responses are measured. If you store the optimal design, each point corresponds to a row in the worksheet.

## Model terms

The list shows the letters that represent the terms in the model. Higher order terms are represented by multiple letters. For example, the first factor is A and the second factor is B. The interaction between the first two factors in the worksheet is AB. The number of terms must be less than the number of design points in the optimal design.

The degrees of freedom for all the terms in the model must be less than the number of design points in the optimal design. For terms with only continuous variables, the degrees of freedom that the terms use is the same as the number of terms. For categorical terms, the degrees of freedom depend on the number of levels for the categorical factors or process variables.

### Interpretation

Use the results to see the terms that Minitab uses to calculate the optimality criteria. Because D-optimality depends on the terms, a design that is D-optimal for one set of terms will most likely not be D-optimal for another set of terms.

## Number of factors, components, or process variables

When using distance-based optimality, Minitab spreads the design points uniformly over the design space. For a response surface design, you can include all the factors or you can use a subset of the factors. For a mixture design, you must include all the components in the design. You can also add process variables for a mixture design.

### Interpretation

For a response surface design, Minitab indicates the number of factors in the design. For a mixture design, Minitab indicates the number of components in the mixture, and the number of process variables in the design.

## Method for generating the initial design

Minitab displays whether the algorithm selects all of the design points sequentially or whether some percentage of the points were selected randomly.
Sequential selection
Sequential selection means that all of the points in the initial design were added in an order that provided the maximum increase in D-optimality. If you repeat the design selection and the runs that are in the candidate set are in the same order, the algorithm will find the same solution.
Random selection
In purely random selection, the algorithm assigns the points to the design at random. If you repeat the design selection, the algorithm can find different solutions. Because the algorithm can find different solutions, you can select to use between 1 and 25 initial designs as starting points for the algorithm. More initial designs increases the time to select an optimal design, but also increases the possibility that the final design will be close to the most D-optimal design possible.
With pure random selection, the algorithm will sometimes choose rank deficient matrices, so the algorithm allows a combination of random and sequential selection. In a combination of random and sequential selection, you can specify between 10% and 100% of the points to select at random in increments of 10. The random points enter the initial design first. The more points the algorithm selects at random, the more variation is likely in the different initial designs.

### Interpretation

For example you compare the results using an all sequential selection and the results using a combination of sequential and random selection for the same design.

The first set of results uses the default sequential selection method.

### 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 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: 18, 61, 1, 24, 30, 42, 6, 56, 15, 44, 7, 58, 64, 41, 27, 39, 25, 32, 51, 13, 53, 3, 59, 34, 8, 40, 17, 22, 5, 2, 46, 49 Condition number: 223.585 D-optimality (determinant of XTX): 6.43729E+28 A-optimality (trace of inv(XTX)): 11.4062 G-optimality (avg leverage/max leverage): 0.96875 V-optimality (average leverage): 0.96875 Maximum leverage: 1
The second set of results uses a combination of sequential and random selection, where 50% of the points are random.

### 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

In these results, by trying different starting points, Minitab found a more D-optimal design by using the combination method with different initial designs.

## Method for improving the design

Minitab displays whether the algorithm improves the initial design with the exchange method, the Fedorov method, or not at all.
Exchange method
In the exchange method, you can select whether to exchange from 1 to 5 points at a time. Minitab first adds the points that increase the D-optimality the most. Then, Minitab drops the points that contribute the least to D-optimality. The exchange continues until the D-optimality of the design does not improve.
Fedorov's method
In Fedorov's method, Minitab simultaneously switches a pair of points from the candidate set and the current design. The switch leads to the greatest improvement in D-optimality. The switches continue until the D-optimality of the design does not improve.

### Interpretation

Compare the results for the exchange method and the Fedorov method. The first set of results uses the exchange method. The second set of results uses the Fedorov method.

In these results, the algorithm found a more D-optimal design with Fedorov's method. Larger D-optimality values indicate a more optimal design.

### 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 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: 18, 61, 1, 24, 30, 42, 6, 56, 15, 44, 7, 58, 64, 41, 27, 39, 25, 32, 51, 13, 53, 3, 59, 34, 8, 40, 17, 22, 5, 2, 46, 49 Condition number: 223.585 D-optimality (determinant of XTX): 6.43729E+28 A-optimality (trace of inv(XTX)): 11.4062 G-optimality (avg leverage/max leverage): 0.96875 V-optimality (average leverage): 0.96875 Maximum leverage: 1

### 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 Initial design generated by Sequential method Initial design improved by Fedorov method Optimal Design Row number of selected design points: 18, 61, 1, 24, 30, 42, 6, 56, 15, 44, 7, 58, 20, 64, 41, 27, 39, 25, 32, 51, 13, 53, 3, 59, 34, 8, 40, 17, 22, 5, 46, 33 Condition number: 213.875 D-optimality (determinant of XTX): 8.91317E+28 A-optimality (trace of inv(XTX)): 11.1267 G-optimality (avg leverage/max leverage): 0.96875 V-optimality (average leverage): 0.96875 Maximum leverage: 1

## Row number of selected design points

The list shows the row numbers of the points in the candidate set in the order that the algorithm adds the points to the design.

### Interpretation

Use the list so that you can identify the optimal points in the candidate set. The order corresponds to rows, not to the standard order or run order columns. The order of the points in the candidate set affects how the algorithm proceeds, so if the worksheet order changes then the sequential algorithm will most likely find a different optimal solution.

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