Another post on how to test for curvature in two-level factorial with center points, including discussion of centerpoint effect

[#1]

A test for curvature in a factorial experiment is the p-value of the centerpoint effect.  It is calculated by getting the statistical significance of the difference between the centerpoint value and the average of the modeled corner points.  I'd like a way to see this in an ANOVA table or a lack of fit table.  I'd like a way for the model fit to exclude the center points.  I'd like the model to not include effects that I'm pooling into the error.  In other words, I'd like to get JMP to do what Minitab does.

This doesn't seem to be an option in the standard features of JMP 16.  It I'm wrong about this, please point me to how to do it.

If I'm right, I still want to do it in JMP.  My question is whether it would be a possible for a script to do it, or whether this would be impossible and I should just give up and do it by hand.    The main challenge to me is I don't know how to use a script to extract sums of squares and degrees of freedom from a model report.

I could create a JMP script that would do this:

- For the centerpoint:

-- Calculate Ybar-center, the average response for rows that have a "0" in the "Pattern" column (center points)

-- Calculate Ybar-corner, the average response for rows that don't have a "0" in the "Pattern" column (corner points)

-- Determine the number of center points and corner points to get degrees of freedom, DFcenter = Number of centerpoints - 1

-- Calculate the sums of squares of error of the center point replication

--- SScenter-error = (sums of squares of differences between centerpoint values and Ybar-center)

-- Calculate the sums of squares for curvature effect

--- SScurvature = Ncorner * Ncenter / (Ncorner + Ncenter) * (Ybar-center - Ybar-corner)^2

-- Mean square of curvature effect

--- MScurvature = SScurvature / 1 degree of freedom.

Now I need the total mean square error.  This error should include the pure error of any replicated center points as well as the error of replicated corner points and of the contribution of unmodelled effects. 

- To get the sums of squares of error in the model and the degrees of freedom in the error, excluding all centerpoints, the script does this:

-- Exclude rows that have a "0" in the "Pattern" column

-- Show the model dialog so the user can select which effects to include

-- Create the model

-- Somehow extract the sums of square of error

--- SSmodel-error = (SSpure-error + SSlack-of-fit) and the degrees of freedom for error DFmodel-error

-- Add the sums of squares of error from the centerpoints to the sum of squares of error of the model.  

--- SSerror-total = SScenter-error + SSmodel-error.

-- Then MSerror-total = SSerror-total / ( DFcenter-error + DFmodel-error)

-- Calculate F ratio for curvature effect = MScurvature / MEerror-total.

-- Then get p-value from the F distribution.

Will it work?  What guidance could you give about the approach or the scripting?  Any amount of canned automation would be welcome?

Thanks,

Mike

[#2 · Accepted solution]

You are correct that this not an option in our standard model report.

We don't recommend 2-level designs with multiple centre-points. I can provide more information on the reasons for that if you like.

However, it is not difficult to test for curvature if you would like to. Have you read Mike Anderson's blog post on this?

Regards,

Phil

[#3]

@Phil_Kay 

Thanks for the link to the blog post.  It leads to a solution.  

The TL;DR is this: Create an attribute column having two levels to distinguish corner points from center points.  Then include that column as an effect in the model, along with any other terms that are under investigation.  Its effect and significance will appear in the Parameter Estimates.  

This is much better than my method of doing manually it by scripting.  In fact I could probably write a script that would create that column by recoding the "Pattern" column.

I understand that the JMP community and SAS don't recommend various things.  I'd like more information about why you don't recommend it.

My question wasn't about whether running replicated centerpoints is a good idea, it was about analyzing them once they exist. I have an application where replicated centerpoints is a good idea. I can provide more information on the reasons for that if you like.

Mike

[#4]

Hi Mike,

I am glad that you found Mike Anderson's advice useful.

I mentioned that JMP do not recommend 2-level designs with replicated centre-points to provide context for why there is no test for curvature in the analysis report. By the way, you can always make requests for features in the JMP Wish List.  

The problems with 2-level designs with replicated center points have been discussed by Bradley Jones before in different blog posts and articles:

1. Repeat centre points are not a good estimate of the noise of the system because they only give us information about variance at the centre, which is likely to be less than in other regions of the factor space.
2. We need to get maximum information from the minimum number of runs so placing multiple repeat runs in the least interesting part of the factor space is a poor use of experimental resource. 
3. If curvature is found to be active, all quadratic terms are confounded so we need to do an expensive augmentation to determine the active quadratic effects.

In most situations where you might traditionally use a 2-level design a Definitive Screening Design would be a much better choice. This blog post is a good source for explaining the benefit of DSDs. For a thorough discussion I would recommend this article and the discussion articles and rejoinder.

I hope that helps.

Phil

[#5]

I'll share my thoughts, which, of course, can be completely ignored.  There is no one way or right way to experiment.  Certainly the methodology you use can vary in its efficiency and/or effectiveness, but this must be weighed against resource allocation. Regarding replicates of the center point:  

1. If the center points are the current settings for the process, then replicates at this conditions might in fact give a fairly good estimate of the random errors (natural variation) in the process to be used as the basis for statistical tests (F-test).  

2. If those center points are run randomly over the course of the experiment (particularly if the experiment takes some time), then when time ordered, they provide some evaluation of stability over the experiment region.

3. Again if they are the current conditions, then it allows for bolder level setting for the factor increasing the likelihood of exposing factor effects.

4. You can certainly have a degree of freedom to test for non-linear relationships (just add the quadratic for any one term in the model), though the non-linear effect is not specific.  This is akin to fractional factorials, so in the right sequence of your experiment iterations, it is quite economical and useful.  In fact, Daniel suggests there are benefits to center points that are preferred over three-level designs:

• As he points out in his book, The general shape and orientation of the response surface is more heavily influenced by the center point (vs. center edge points),
• Three-level designs lack radial symmetry and these designs are not rotatable, this issue does not exist with center points.

Daniel, Cuthbert (1976) “Applications of Statistics to Industrial Experiments” Wiley (ISBN 0-471-19469-7), P. 34

[#6]

Thanks for your excellent summary of the value of center points.  Your points are very practical.

I have used center points successfully for years as part of sequential experimentation plans.  This approach has produced great value for many of the points you have laid out as well as the value for augmenting the efficient factorial designs when we desire to extend the factorial for modeling purposes.  The use of center points is also very understandable by novices in DOE and further supports their acceptance.