estimates in multipule regression

[#1]

Hello,

I have an issue in JMP pro10.

I am performing a multiple regression with 3 continuous variables (x1,x2,x3) and an interaction between x2*x3

The estimates that I get for the interaction are usually in the following form:

(x1 - "number")*(x2-"number"), I do not understand this report method, but the estimates worked well for predicting new values.

I wanted to report these estimates without the "number" subtracted. I went to r and performed the same analysis and I got the exact same R2 so I know that It was the same analysis. The estimates that I got were different but it work exactly as the function JMP gave me, only without the "number".

I searched here and found a post ( Need help understanding interaction variable) that explain how to get the estimates without the "numbers" subtracted in JMP. After using the suggestion, the estimates were exactly the same as in r.

Now things start to be messy:

The t values (and P values) are not exactly the same between the two outputs that I have (JMP and r).

x1   and   x2*x3  -> has exactly the same t values (and P values)

x2, x3,               -> as a standalone are very different between the softwares.

Can anyone tell me what may be the reason for these differences? or perhaps something about JMP model assumption that may influence the t values?

Thanks

[#2 · Accepted solution]

Try this:

When filling in the Fit Model dialog, in the upper left corner of the dialog window there is a red triangle right next to the words "Model Specification". Click on that red triangle to bring up a context menu which includes as its first entry "Center Polynomials". Click it to uncheck it. Then run your model. The resulting report should agree with R.

[#3]

Hi abra,

Center Polynomials is the default option in JMP for situations in which you are fitting powers or interactions between variables, and for some good reasons. This process centers each variable (subtracts the mean from each observation) before operating on it (through powers or cross-products with other variables) so that the lower order terms are unconfounded with higher-order terms, and it also maintains an easy (and often more useful) interpretation of the coefficients: the "average" effect of a variable assuming other variables are held constant at their mean.

If you do not center variables, the interpretation of the lower order terms is different: the coefficients represent the increase in Y for each unit change of the variable when all other variables involved in higher-order terms with that variable are held constant at 0. This is why the test statistics and p-values for your X2 and X3 variables are different. The test of X2 in your model without centering is a test of the partial regression slope of Y|X2 when X3 is 0, and the test of X3 is a test of the partial regression slope of Y|X3 when X2 is 0. This happens because of that interaction term, which is capturing the degree to which the level of X2 affects the relationship between Y and X3, or alternatively and equivalently, the degree to which the level of X3 affects the relationship between Y and X2. All multiple regressions involve partial regression coefficients, which represent the effects of variables if we are to hold constant the levels of other variables. Where "constant" is, numerically, depends on that centering. With centering "constant" will be at the mean of other variables, otherwise "constant" will be at 0 of the other variables. (It's worth noting that without any interactions this choice is immaterial, since the slope of X2 and X3 are fit to be constant, thus their slopes are the same at 0 and the mean of the other variables, hence no effect of centering).

X1 is not involved in any higher order terms so the interpretation of it is unchanged by centering.

This is a great question and in the interest of making is as clear as possible I recorded a quick video using some of the profiling tools in JMP to drive home the main points and included it below.

I hope this helps!

julian

[#4]

Great explanation on centering polynomials!

[#5]

Is this video still around? I don't see a link to it. Also, if I uncheck the "Center Polynomials" option, will I be essentially testing the difference between the intercepts? If so, when making pairwise comparisons, does that then compare the intercepts (instead of the slopes)? Thanks!

[#6]

Hi @Judd,

I've replaced the video - it seems that was lost at some point in the past.

[#7]

Probably a stupid question, but when I run the same model once without and once with "center polynomials" ticked, the report output looks totally different. I am trying to model relative degradation (t=0=100%) of a chemical compound over three different temperatures and in a multitude of different batches, following a recipe I received from a peer recently.

Copying the model windows to a script window shows that this is indeed the only difference. What am I missing?

Fit Model(
	Y( Log( :"QCparameter_2_anon2 [%rel.]"n ) ),
	Effects(
		:BatchID, :TimePoint, :TimePoint * :"Arrhenius[StorageTemp]"n,
		:TimePoint * :"Arrhenius[StorageTemp]"n * :"Arrhenius[StorageTemp]"n
	),
	Personality( "Standard Least Squares" ),
	Emphasis( "Minimal Report" ),
	Run(
		:"QCparameter_2_anon2 [%rel.]"n << {Summary of Fit( 1 ),
		Analysis of Variance( 1 ), Parameter Estimates( 1 ), Lack of Fit( 0 ),
		Scaled Estimates( 0 ), Plot Actual by Predicted( 0 ), Plot Regression( 0 ),
		Plot Residual by Predicted( 0 ), Plot Studentized Residuals( 0 ),
		Plot Effect Leverage( 0 ), Plot Residual by Normal Quantiles( 0 ),
		Box Cox Y Transformation( 0 )}
	)
);

Fit Model(
	Y( Log( :"QCparameter_2_anon2 [%rel.]"n ) ),
	Effects(
		:BatchID, :TimePoint, :TimePoint * :"Arrhenius[StorageTemp]"n,
		:TimePoint * :"Arrhenius[StorageTemp]"n * :"Arrhenius[StorageTemp]"n
	),
	Center Polynomials( 0 ),
	Personality( "Standard Least Squares" ),
	Emphasis( "Minimal Report" ),
	Run(
		:"QCparameter_2_anon2 [%rel.]"n << {Summary of Fit( 1 ),
		Analysis of Variance( 1 ), Parameter Estimates( 1 ), Lack of Fit( 0 ),
		Scaled Estimates( 0 ), Plot Actual by Predicted( 0 ), Plot Regression( 0 ),
		Plot Residual by Predicted( 0 ), Plot Studentized Residuals( 0 ),
		Plot Effect Leverage( 0 ), Plot Residual by Normal Quantiles( 0 ),
		Box Cox Y Transformation( 0 )}
	)
);

[#8]

Hi @Ressel : Why are you not including Arrhenius[StorageTemp] and Arrhenius[StorageTemp]^2 in your model? 

[#9]

I am sheepishly, following a recipe. Below is a screenshot from a presentation I received and I am trying to replicate.

I was informed, that the below corresponds with a Taylor series, where TP = timepoint and Arrh(T) is the unspecific part of the Arrhenius equation per JMP documentation.

Question #1: Is this really a Taylor expansion?

Question #2: How many temperature settings do I need to use a 3rd degree Taylor expansion?

Question #3: Any hints on where to find literature that explains for dummies what the basic constraints and applications of the Taylor series in model fitting are?

Apologies for opening Pandora's box. (At the very least, people can now say that I had courage for exposing myself here.)

[#10]

Hi @Ressel  : This is the presentation I assume.

https://community.jmp.com/t5/Discovery-Summit-Europe-2019/Taking-Accelerated-Stability-Assessment-to...

If you don't include the other terms I mentioned, then you have two different models...that is why the stat details are so different. It's easy to see if you expand out the centered factors, collect like-terms etc..