HI
An other problem I am having now is,
Orthogonal Polynomials (Complete ANOVA)
Book <Fundamental concepts in the design of experiments, C. Hicks 1982>
Page 156,
shows the following figure.
Raw data are
I had tried to create the SAME results as above in JMP, failed.
JMP's result is surely wrong since Sum of the SS does not equal to the Model SS. (they should be same.)
So here is my question.
To create the exact same results as the Book, how would I run the JMP??
In above, both C and T were designated as Continuous.
Please inform me. thanks.
Accepted Solutions
Although the results do not match, JMP is not wrong.
Textbooks will often present results that you can match with by-hand calculations. In this case, the results in the book are using what are called Type I Sums of Squares. These sums of squares are additive and easy to calculate by hand. However, the model results will depend on the order that the terms are listed in the model. What this means is that the model Y=intercept + B1*C + B2*T + B12*C*T will be different than the model Y=intercept + B2*T + B1*C + B12*C*T. Since the order that you specify your model can provide different results, Type I Sums of Squares are not often used. This is the root of your issue. Your JMP model was not specified in the same order as the book model. But there is more.
By default, JMP uses what are called Type III Sums of Squares. Type III Sums of Squares calculate the sums of squares for each model term as if it were the last term entered into the model. That means that you will get the same sums of squares and statistical test results regardless of the order that you enter the terms. But, this also means that the sums of squares will no longer be additive. This is why your JMP results do not add up as you would have expected.
So how do you get your results to match? First, specify the same model order as the book. That means your Fit Model dialog should look like this:
When you run this model, the sums of squares still will not match. However, if you go to the red triangle at the top of the model results and select Estimates > Sequential Tests you will get this report:
These results match the book results.
Although the results do not match, JMP is not wrong.
Textbooks will often present results that you can match with by-hand calculations. In this case, the results in the book are using what are called Type I Sums of Squares. These sums of squares are additive and easy to calculate by hand. However, the model results will depend on the order that the terms are listed in the model. What this means is that the model Y=intercept + B1*C + B2*T + B12*C*T will be different than the model Y=intercept + B2*T + B1*C + B12*C*T. Since the order that you specify your model can provide different results, Type I Sums of Squares are not often used. This is the root of your issue. Your JMP model was not specified in the same order as the book model. But there is more.
By default, JMP uses what are called Type III Sums of Squares. Type III Sums of Squares calculate the sums of squares for each model term as if it were the last term entered into the model. That means that you will get the same sums of squares and statistical test results regardless of the order that you enter the terms. But, this also means that the sums of squares will no longer be additive. This is why your JMP results do not add up as you would have expected.
So how do you get your results to match? First, specify the same model order as the book. That means your Fit Model dialog should look like this:
When you run this model, the sums of squares still will not match. However, if you go to the red triangle at the top of the model results and select Estimates > Sequential Tests you will get this report:
These results match the book results.
This is not an issue about which is to be more believed. They are just different ways of thinking about the problem. If you have factors A, B, and C, are you okay with the model Y=A+B+C being different than Y=C+A+B? then Type I Sums of Squares are just fine. In some cases Type I makes sense -- think of model hierarchy. Y=A+A^2+A^3 would be a good example where Type I might make sense. Remember that Type III SS would just have each effect listed last in the model order. So, Y=A+B+C would give you the C sums of squares. Y=A+C+B would give you the B sums of squares, and Y=C+B+A would give you the A sums of squares. So the actual mathematics are not really different. Type III is just telling you how much variance that factor explains given that all other terms are in the model. Another way of wording it: once all other factors are considered, how much additional variance does this factor explain? The Type I SS are telling you how much variance that factor explains given the preceding terms in the model (but not the remaining model terms).
Now, not to confuse the issue, but it is important to note that if you truly have orthogonal terms in your model then Type I SS will equal Type III SS, so the difference doesn't matter. You can see this with your example data. Just fit a model of Y=C+T+C*T.
You saw differences in your initial analysis because the quadratic and cubic terms are NOT orthogonal to the other terms in the model. That is the situation where the Type I and Type III sums of squares will not match.
Introductory texts do not want to confuse the issue on the different types of sums of squares because they are introductory. To avoid the issue they will typically use balanced data with orthogonal model terms so the sums of squares would match. This particular example is somewhat surprising because they have balanced data, but added the non-orthogonal quadratic and cubic terms.
I should be more precise on my wording. You asked what situations have Y=A+B is not the same as Y=B+A. Remember, we are talking about sums of squares which leads to the statistical testing. So your book example already shows this situation. However, here is a simpler example that I just made.
I fit the model Y=A+B+A*B and Y=B+A+A*B. Here are the results for Y=A+B+A*B:
and Y=B+A+A*B
Although the parameter estimates are the same, the statistical testing results are NOT the same when using the Type 1 SS.
Also, remember that statistical models are ESTIMATING the parameter estimates. They are NOT known exact quantities. There is uncertainty in the parameters. But this issue of the statistical testing being order dependent is why Type I Sums of Squares are not often used. People dislike the fact that the order of the model terms can lead to different statistical conclusions and different interpretations of what we think the correct model is. It sounds like you are starting to see why the default in JMP is to use the Type III sums of squares because the order of the model does not matter then.
Finally, factorial designs are only one type of design. There are response surface designs like central composite designs or box-behnken designs. There are fractional factorial designs, There are Plackett-Burman designs. There are optimal designs which have different types such as D, I, A, and G. There are space-filling designs. And so on. So although most people learn about factorial designs first, it is not the only type and I would guess that it is likely not the most commonly used design since there are so many other choices.