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Level I

## Augmented design with diffrent optimality criteria

Dear all,

I have 4 continuous factors. I made an custom Design with recommended optimality which is d for a model containing main effects, 2nd order interractions and 2nd order powers. I run 80 Experiments and use least square estimate for linear model. Then i see that i have to include 3rd order powers to my model and i made augmented Design with I optimality by including 3rd order powers to my model.

Now i wonder that augmented Design with I optimality can be use With custom Design with d optimality? Does this combination work well?
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Staff

## Re: Augmented design with diffrent optimality criteria

This approach will work fine. But I think a few clarifications are in order. I do have questions, such as how did you determine that you needed cubic terms? And do the cubic terms make sense from a physical science/process point of view? To name just a few. But I trust that you had good reasons to add the cubic terms to the model, so I will set those questions aside and try to answer yours.

D-optimal designs are optimal for estimating the parameters of your model. In other words, they will minimize the standard errors of the parameter estimates.

I-optimal designs are optimal for prediction. They will minimize the integrated variance of the prediction error.

Although D-Optimal designs are best at estimating parameter estimates, they still predict very well. The prediction errors will only be slightly higher than an I-optimal design. These designs do not guarantee orthogonality as that is not part of the optimization criteria. They are the default designs in JMP UNLESS you click the RSM button to build the model. When you click that, JMP will switch to I-Optimality. Notice that if you build the RSM model by clicking the cross button and the powers button rather than RSM, the default will remain D-optimal.

Although I-Optimal designs are best at predicting, they still do a good job of estimating the parameters. The parameter errors will only be slightly larger than those from a D-optimal design. The I-optimal designs do not guarantee orthogonality either as that is not part of the optimization criteria.

So all of this means that the user should choose the optimality that best meets their needs. There is no such thing as "the best" design. The "best" design will depend on the researcher's objectives and wants/needs from the design. Those are determined by people, not statistics. Therefore, different opinions can lead to different designs and design approaches. After all, someone could say that you should have only created one design and not augmented the original design. I don't believe that, but I hope you see the point. The design creation process utilizes the science knowledge of the experimenter as much or more than it utilizes the statistical/mathematical properties of a design.

So is the approach that you outline of a D-Optimal augmented with I-optimal a valid approach? Absolutely. Is it the best? That will depend on YOUR needs. I can tell you that it is a fine approach and you should get a design with good mathematical properties. The resulting design would not have as good of a D-optimality criteria as if you had created the entire design using D-optimality. The resulting design would likely not have as good of an I-optimality criteria as if you had created the entire design using I-optimality. But that is okay, as long as the design meets your purposes and allows you to build a good model that answers your questions.

Dan Obermiller
2 REPLIES 2
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Staff

## Re: Augmented design with diffrent optimality criteria

This approach will work fine. But I think a few clarifications are in order. I do have questions, such as how did you determine that you needed cubic terms? And do the cubic terms make sense from a physical science/process point of view? To name just a few. But I trust that you had good reasons to add the cubic terms to the model, so I will set those questions aside and try to answer yours.

D-optimal designs are optimal for estimating the parameters of your model. In other words, they will minimize the standard errors of the parameter estimates.

I-optimal designs are optimal for prediction. They will minimize the integrated variance of the prediction error.

Although D-Optimal designs are best at estimating parameter estimates, they still predict very well. The prediction errors will only be slightly higher than an I-optimal design. These designs do not guarantee orthogonality as that is not part of the optimization criteria. They are the default designs in JMP UNLESS you click the RSM button to build the model. When you click that, JMP will switch to I-Optimality. Notice that if you build the RSM model by clicking the cross button and the powers button rather than RSM, the default will remain D-optimal.

Although I-Optimal designs are best at predicting, they still do a good job of estimating the parameters. The parameter errors will only be slightly larger than those from a D-optimal design. The I-optimal designs do not guarantee orthogonality either as that is not part of the optimization criteria.

So all of this means that the user should choose the optimality that best meets their needs. There is no such thing as "the best" design. The "best" design will depend on the researcher's objectives and wants/needs from the design. Those are determined by people, not statistics. Therefore, different opinions can lead to different designs and design approaches. After all, someone could say that you should have only created one design and not augmented the original design. I don't believe that, but I hope you see the point. The design creation process utilizes the science knowledge of the experimenter as much or more than it utilizes the statistical/mathematical properties of a design.

So is the approach that you outline of a D-Optimal augmented with I-optimal a valid approach? Absolutely. Is it the best? That will depend on YOUR needs. I can tell you that it is a fine approach and you should get a design with good mathematical properties. The resulting design would not have as good of a D-optimality criteria as if you had created the entire design using D-optimality. The resulting design would likely not have as good of an I-optimality criteria as if you had created the entire design using I-optimality. But that is okay, as long as the design meets your purposes and allows you to build a good model that answers your questions.

Dan Obermiller
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Level I

## Re: Augmented design with diffrent optimality criteria

Mr. Dan,

I really appriciate for your detail explanation.

I want to state that i suspect model needs 3rd order power because of the fluctuations in the plot of std residuals vs predicted values. What do you think about it, is it a sufficient clue to include third order power to the model?

Beside this, your explanation about I and D optimal designs is very clear but i still do not understand some points. This may be because of my lack of thoretical knowledge but i want to learn.

You said below that "I-Optimal designs are best at predicting."

One needs best model estimation for best predictions. That means that we need a best design at estimating parameters which is D-Optimal designs. So, i am little bit confused that which optimality criteria should be used for the study which aims to construct a good models and then make optimization by using profiler to find the best factor levels for targeted response values.

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