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Dear Dr. DOE, what are some tips on designing experiments for mixed models?

Dr. DOE will talk about DOE for mixed models with several examples.

 

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Hello, I'm Brad Jones. I'm talking to you in my role today as Dr. DOE, about designing optimal experiments for mixed models. Mixed models are models that have more than one source of variability.

 

The simplest kind of mixed model is a random block design, which is the first item in my JMP journal here. So, I’ll click to open this outline node. The picture you see is the cover of a textbook of about design of experiments that just came out in December with Doug Montgomery and me as authors. Clicking the link opens an experiment in that textbook. It's a filtration experiment and it has a filtration rate which is the response variable for the for the experiment. The four factors in the experiment are temperature, pressure, concentration, and stirring rate. The first column shows the settings of the random block factor. Half of the runs are done at one setting of the block factor and half of the runs are done in another setting. It might be the case that the random blocking variable in this case would be half the runs are run on day one and half of the runs are done on day two. Or it might be half the runs are done using one batch of material and the other half would be using another batch.

 

If I click on the DOE dialog link, it shows me how to set up an experiment with a random block. The first outline node shows the four factors that I talked about before. To add a random block, I just click the check box which is the first UI element in the Design Generation outline node. The number in the accompanying edit box is where I say how many runs the blocks should have. Now, all I need to do is click the Make Design button to get the same experiment that I just showed you the table. We now dealt with the simplest kind of mixed model.

 

My journal has five different examples that I want to show in this in this time spot. I just showed you the first example – design with a random block.

 

The second kind of experiment that I want to talk about is a split plot design. Split plot designs are designs that have a random blocking factor and one additional feature. That is, for each level of the random blocking factor, there is at least one factor that stays constant within that level. I put another picture of the cover of the book in the second outline. The link opens the table for split plot design has the factor, Temperature. Not that Temperature stays fixed for each setting of the Whole Plot column. The example appears in chapter 10 of the book which is about split plot designs.

 

Let's open the Graph Builder script – the first “green triangle” link at the left of the data table. We can see that the temperature is changing across each of four coatings that represent the four different kinds corrosion resistance that are being applied. Now, split plot designs require a different kind of analysis than ordinary least squares. Here, the appropriate analytical technique is called REML, or Restricted Maximum Likelihood, which takes explicit account of the fact that there are two components of variance rather than one.

 

Fortunately, when you when you make a split plot design in JMP, you get the correct analysis built into the data table that gets created once you have designed the experiment and clicked the Make Table button. So, JMP is protecting you from making an incorrect inference by doing an regression analysis using least squares instead of REML

 

The next kind of design, I want to talk about adds another feature of the split plot experiments. Suppose you have a hard to change factor and then another factor that's even harder to change. Now you have two random blocking factors with factors that are associated with them that are that are either Hard to Change or Very Hard to Change. The image in the Split-split-plot outline node shows the beginning of a paper that Peter Goos and I wrote on creating optimal designs for split-split-plot experiments that was published in Biometricka in 2009. Let's look at the cheese experiment, which is the link right below the image

 

We have two factors that are Very Hard to Change (w1 and w2), five factors that are Hard to Change (s1-s5), and finally, two factors that can be changed every run (Easy to Change, x1 and x2) . So, there are three kinds of factors here. Factors w1 and w2 change with every whole plot. For instance, the first whole plot has four runs. And you can see that both w1 and w2 stay constant for those four runs. The factors, s1 through s5 change with the subplots - they stay the same for each subplot of two runs. Clicking the DOE dialog on the left of the table brings up the UI for the Custom Design tool.

 

There are there three choices that you can make you can make. You can say that a factor is Very Hard to Change, which is true for w1 and w2. You can say that a factor is Hard to Change, which is true of s1 through s5. Finally, you can say that a factor is Easy to Change (x1 and x2). This is often a useful thing to do when you're dealing with a process that has more than one processing stage. The factors in the first stage of the process might be all very hard to change. The factors in the second stage could then be Hard to Change. In any further stage,  the factors would be Easy to Change.

 

The next kind of design we'll talk about is a strip-plot design. This design appears in this book that I wrote with Peter Goos from Belgium.  The cover image is the first thing under the Strip-plot Design outline node. We describe a battery life experiment. In this battery life experiment we have four factors (x1-x4) that relate to the creation of the batteries and then two other factors (x5 and x6) which control two factors designed to accelerate the failures of the batteries to test there reliability. You can see that the rows and columns are have different orderings. The randomization of the batteries for in processing is different from the randomization in stress testing. In a strip-plot design you have to be able to re-randomize going from one stage to another.

 

In this case, you make all the batteries first using one randomization scheme, and then then you test them in different rooms using another randomization scheme. To show you how this would be done in the Custom Design tool, let's say I'm going to have three factors. I'm going to make x2 be hard to change and x1 be very hard to change, then, and then I say how many, how many times I want to change the very hard to change factor and how many times I want to change the hard to change factor. And finally, how many total runs I'm willing to do. Then, if I click that the that says that the hard to change factors can vary independently of the very hard to change factors, I get a strip-plot design instead of a split-split-plot design.

 

You can see that with within each whole plot of 3 runs, I'm going to use three different subplot settings. Imagine I make I make the all these 16 different kinds of batteries first using process settings for x1-x4. Then I, re-randomize to divide them into six different rooms for stress testing. That's an example of a strip plot design, which is more statistically efficient than a split-split-plot design but also more logistically demanding.

 

The final example considers two random blocking factors. Given the Custom Design tool interface, it is not at all obvious how you would you would manage to do this. I wrote a blog on how to do this and the first text under the Design for Two Nested Factors outline is the URL for the blog.

 

Here's what you need to do for a design with two random blocking factors. You have to trick JMP into making such a design. You do that by creating two factors that you are not actually going to use.

 

To illustrate this, I click this Design Setup script that just creates the creates the situation I want. You can see that I've that I've created a fake whole plot factor and a fake subplot factor. I've set this up as if it were a split-split plot experiment. But now, I'm going to remove these two factors from the model. Let's make four whole plots and eight sub plots by filling in those edit boxes. I'll choose to do 32 runs and click the Make Design button. You can see that the, the, these, these fake factors are being set to just random quantities because they are not in the model. They're not important, because I am going to remove them from the JMP table later. What is important is that I have a whole plot column and a subplot column in my data table, or in my table in the in the designer. When I make the when I make the table, I can just select these two fake factors and delete those columns because I'm not going to use them.

 

In a JMP table there can be demographic information stored in the columns. The design role is one such piece of demographic information. The design role of the whole plot factor is that it's a random blocking factor. The design role of the subplot factor is also a random blocking factor. Now I've created a design that has two random blocking factors. But, I used the template for a split-split plot design to create this design for two random blocks, where one of the blocks is nested within the other block. I realize that this is a completely unobvious way to make this happen. That is why I wrote a blog about to show how it's done. One of the development goals for a future release of jump is to have a much more natural interface for creating designs for mix models that have more structure than any of the ones that I've talked about today. It would be nice particularly to have a natural interface for dealing with more than one random block as we've as we've done here.

 

That's my presentation for today. Thanks for joining.

 

 

 

 

 

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