DOE: how is Sum of Squares of each effects calculated?

[#1]

Hi,

How is sum of squares (SS) of each effects calculated in DOE parameter estimate table?

any simple 2-3 factors DOE example to share?

Thanks

[#2]

The answer to your question can be found in any basic regression textbook. There are some fairly simple calculations that can be performed by hand if you happen to have only a single factor. Otherwise the calculations will involve matrix algebra.

You might want to consider enrolling in the free online statistics course titled "Statistical Thinking for Industrial Problem Solving". Based on the questions you have been asking, you might want to concentrate on the hypothesis testing, regression, and design of experiments portions of that course.

[#3]

This post is a couple years old, but I too would like to know how the sum of squares is calculated.  I refer specifically to the Effect Tests report.

I went to a basic regression textbook to learn.  The book is called "Design and Analysis of Experiments" by Douglas C. Montgomery, 6th Edition.  In that book there is an example calculation. Example 5-1 is a full factorial experiment with two factors and four replications.  I have entered the data into Excel, Minitab and JMP and have compared the results to Montgomery.  Minitab and Excel Analysis Toolpak match Montgomery, JMP does not.

The calculations from the text are attached as an image.  So are images from Minitab and Excel.  Also JMP, with and without the interaction term.  The JMP project files is attached, with data table and Fit Least Squares report.

Across all the analyses, the sums of squares match except for JMP's Sum of Squares for Material Type in Effects Test.  Therefore, its p-value is also different, making it appear as if the factor is insignificant.  How is this particular SS calculated?

In the JMP output, Sum of Squares for Model in the Analysis of Variance matches the sum of SS from the model terms in Montgomery, Minitab, and Excel.  It is not equal to the sum of SS for the model terms in JMP's Effect Test report.  Why is this?

When the interaction is removed from the model, then the Sum of Squares for Material Type is the same as in Montgomery.  Why is that?  Why does the inclusion of the interaction term affect the sum of squares for that particular main effect?  The sum of Sum of Squares in the Effect Test is equal to the Sum of Squares for the Model term in the ANOVA table.  Why does it work if the interaction is not in the model?

According to Montgomery's analysis, all three terms are significant (two main effects and interactions).  According to JMP, one main effect and the interaction are significant.  I really need to understand why JMP gets a different result for the significance of Material Type.

[#4]

I didn't see anything in the JMP project.  Would you mind attaching the data table so we can take a closer look.

[#5]

Here is the data table.

[#6]

I've only taken a brief glance at this, but it would appear that to get the same results that you get from the other quotes sources you need to perform a Type I ANOVA.  This is turned on from the red triangle:

Estimates> Sequential Tests

This I think, explains the difference in the outputs. It raises the question of which type of ANOVA is appropriate.  Perhaps others can contribute to that discussion!

[#7]

Thank you for that, using Sequential Tests did change the result and it makes it align with the other methods.  Here's what Montgomery's supplemental material says about the topic:

"Type 1 sums of squares refer to a sequential or “effects-added-in- order” decomposition of the overall regression or model sum of squares.  In sequencing the factors, interactions should be entered only after all of the corresponding main effects, and nested factors should be entered in the order of their nesting."

Also,

"For balanced experimental design data, Types 1, 2, 3, and 4 sums of squares are identical."

Now my questions are:

- What kind of Sums of Squares does JMP calculate by default (without my choosing Sequential Tests)?  

- My data structure is balanced, why are the sums of squared different between the default and the Sequential Tests?

AHA: When I change Temperature to data type Nominal the result is the same as Montgomery!  Something in JMP is using knowledge of the ordinality of Temperature and it changes the calculation for Sums of Square.

I sure do wish some information existed somewhere that explained how JMP calculated sums of squares.  I'd now especially like to know how the model changes depending on Nominal versus Ordinal factors.  Telling customers to refer to basic regression textbooks did not provide the answer.

[#8]

By default JMP uses the Type III Sums of Squares. This makes the results the same regardless of the order the terms appear in the model. Many people prefer Type III sums of squares for this reason. But Type I sums of squares are easier to calculate by hand, which is why many texts take that route.

As you noticed, Ordinal factors are certainly different than Nominal factors. For most ANOVA examples, you want a nominal modeling type (but this is problem specific). Most notably, how they are coded and the interpretations of the tests are different. For details, refer to the JMP help.

Ordinal factors: https://www.jmp.com/support/help/en/17.0/index.shtml#page/jmp/ordinal-factors.shtml#ww96069

Nominal factors: https://www.jmp.com/support/help/en/17.0/index.shtml#page/jmp/nominal-factors.shtml#ww65535

Or you can just search for "sums of squares for ordinal effects" and read the first paragraph to get an overview. 

[#9]

Maybe these will give some information (JMP documentation):

• https://www.jmp.com/support/help/en/16.2/#page/jmp/the-factor-models.shtml#
• https://www.jmp.com/support/help/en/16.2/#page/jmp/continuous-factors.shtml#
• https://www.jmp.com/support/help/en/16.2/#page/jmp/nominal-factors.shtml# (especially this)
• https://www.jmp.com/support/help/en/16.2/#page/jmp/ordinal-factors.shtml# (especially this)

[#10]

It's not realy explained how the sums of squares are calculated for ordinal factors. I get it for nominal and continuous.  A nominal model is one where subgroup means deviate from a grand mean, as in Montgomery's battery life example ANOVA. The continuous model has predictors and a grand mean and the sums of squares for the predictors are based on the difference between grand mean and predicted value at the limits of the model.  

Ordinal models are some sort of build-up of effects starting from the lowest level and incrementally working up to the highest level.  For those it's still not clear how Sum of Squares for factors are calculated, but it seems to have something to do with amount or variation of group means relative to the mean of a reference level.