topic Re: Linear and quadratic contrasts with unequally spaced treatments in Discussions

Linear and quadratic contrasts with unequally spaced treatments

thayermo07 — Fri, 09 Jun 2023 01:00:56 GMT

I am using the contrasts specification window with + and - buttons. However, when treatments are unequally spaced, the simple + and - isn't working. Here's my specific example:

Treatment doses: 0, 1, 2, 4

These doses are NOT equally spaced so we can not use the typical 4 level linear and quadratic contrasts. I calculated appropriate coefficients as shows in this link.

https://real-statistics.com/one-way-analysis-of-variance-anova/trend-analysis-polynomial-contrast-coefficients/trend-analysis-unequal-intervals/

From my calculations, the linear coefficients should be -7, -3, 1, 9. In JMP, -0.7, -0.3, 0.1, and 0.9 satisfy the JMP requirements of adding up to 0 and absolute value less than 2.

My issue is with quadratic coefficients. From my calculations, the quadratic coefficients should be 224, -128, -256, 160. How do I get these quadratic coefficients into JMP 16 Pro?

Thank you for any and all advice!!

Re: Linear and quadratic contrasts with unequally spaced treatments

Ross_Metusalem — Mon, 24 Apr 2023 20:12:57 GMT

You may be interested in the Custom Test tool in Fit Model (top red triangle > Estimates > Custom Test), which uses manual entry of contrast coefficients.

It looks like a similar question was asked and answered a few years ago: https://community.jmp.com/t5/Discussions/How-to-specify-contrast-coefficients-with-quot-Contrast/td-p/337578. You may want to review that thread.

Re: Linear and quadratic contrasts with unequally spaced treatments

thayermo07 — Tue, 25 Apr 2023 12:09:32 GMT

Thank you for sharing this thread. It looks like at the end of it, the same question is asked that I am asking here (how to conduct a contrast with unequal spacing between treatments).

When using the Custom Test tool, do the same rules apply in regards to adding up to 0 and absolute value less than 2?

Re: Linear and quadratic contrasts with unequally spaced treatments

MRB3855 — Tue, 25 Apr 2023 12:45:49 GMT

No such rules apply here; for each column in the Custom Test, you can test/estimate any contrast of interest...

Re: Linear and quadratic contrasts with unequally spaced treatments

Mark_Bailey — Tue, 25 Apr 2023 17:44:57 GMT

You can see in the example that appears in the other cited discussion that JMP automatically normalizes the coefficients used to compute the contrasts.

Re: Linear and quadratic contrasts with unequally spaced treatments

thayermo07 — Tue, 25 Apr 2023 18:14:12 GMT

Mark,

Can you explain what you mean by normalizes the coefficients?

Are you saying that I can more or less ignore the calculation of coefficients for unequally spaced treatments and act like they are evenly spaced?

Thanks for your help!

Re: Linear and quadratic contrasts with unequally spaced treatments

Mark_Bailey — Tue, 25 Apr 2023 18:55:29 GMT

Use the link in the first reply from Ross to see the previous discussion. Look at the first picture, Table A.2. My example is for the case where V = 4. I entered the contrast coefficients -3, -1, +1, and +3 from Table A.2 using the + and - buttons in the JMP interface. The results Contrast > Test Detail in the screenshot below Table A.2 showed that JMP used the linear contrast coefficients of 0.75, 0.25, -0.25, and -0.75. Note in the original discussion, I entered the coefficients in reverse order by mistake. The Test Details would have shown as -0.75, -0.25, 0.25, and 0.75 if I entered them correctly.

I do not know what you mean by 'even spaced.' That is not a requirement for the Analysis of Variance. The levels are modeled as nominal values, and the tests are for group means or combinations of group means.

Re: Linear and quadratic contrasts with unequally spaced treatments

thayermo07 — Tue, 25 Apr 2023 19:18:04 GMT

I agree that the contrast coefficients you assigned of -0.75, -0.25, 0.25, and 0.75 would be correct if the treatments were 0, 50, 100, 150 (evenly spaced). Because the last treatment is actually 200, that's what I mean by unequally spaced.

I am interested to see if responses have a linear, quadratic, or cubic trend, not just comparing the means of one treatment group to another treatment group. Here is the link to calculate trend analysis for unequal intervals for one-way ANOVA: https://real-statistics.com/one-way-analysis-of-variance-anova/trend-analysis-polynomial-contrast-coefficients/trend-analysis-unequal-intervals/

Re: Linear and quadratic contrasts with unequally spaced treatments

MRB3855 — Wed, 26 Apr 2023 09:57:08 GMT

Not knowing the details of your experiment makes this challenging, but can you treat dose as a continuous factor and model it via ANCOVA? Yes, you only have 4 levels (0, 1,2, 4) but it may not be a bad way to at least visualize what is going on.

Re: Linear and quadratic contrasts with unequally spaced treatments

MRB3855 — Thu, 27 Apr 2023 07:33:27 GMT

@thayermo07 , here is how ya do it via Custom Test (I have batch where you will have dose).

First, do a custom test to find the coefficients you need in the Custom Test dialog. You say you need this contrast: 224, -128, -256, 160.

Ok, so put those in four columns in the custom test like this. Batch D is that way because of the way the model (via Fit Model platform) is parameterized.

Then add across columns to get the following.

Then, finally, do another custom test to get your final contrast of interest. i.e., the contrast shown below is your desired quadratic contrast. It is important to understand how the model (via the Fit Model platform) is parameterized to use the Custom Test dialog correctly.

Re: Linear and quadratic contrasts with unequally spaced treatments

thayermo07 — Thu, 27 Apr 2023 12:49:50 GMT

This is a bit confusing to me. Would you mind sharing a link to material explaining this? Just at first glance, why is column 5 160 and then negative 160 for the cells below it?

Thanks for your help!

Re: Linear and quadratic contrasts with unequally spaced treatments

MRB3855 — Sat, 29 Apr 2023 20:48:40 GMT

Yes, it can be confusing. It is how the model is parameterized. Yes, it looks funny...but once you realize how the model is parameterized it's just addition to get the contrasts/comparisons/estimates/etc. of interest.

Here is some detail: https://www.jmp.com/support/help/en/17.1/index.shtml#page/jmp/nominal-factors.shtml#ww65535.

Now, you can also see another parameterization if you look at the "indicator function parameterization" results in the Fit Model platform (compare those to the "parameter estimates"). They look different, but they are equivalent. That, however, won't change the parameterization in the Custom Test dialog. I'd take some time looking at your parameter estimates to make sure you understand them, and their respective p-values (i.e., what the null hypotheses are). One way to help is to select "Show Prediction Expression" to see how the model actually looks.