10963_en-US

liliconcom

estimates in multipule regression<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />

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

Re: estimates in multipule regression

Hi @Ressel : To follow up, here is my take (in no particular order):

1. Keep in mind that the Arrhenius equation (page 12)  shows the relationship between the slope and the temperature. So, k is the slope at a given temperature. As you can see on page 7, the slope (k) gets steeper with increasing temperature.

2. Generally speaking (and speaking as a statistician), it's rare that you would not include the lower order terms when there are interactions which include those terms. However, this is one of those rare cases where you would not include those terms. That is because they are using a Taylor series (actually a Maclaurin series, which is a special case of a Taylor series) approximation for k; they did not have to do this; in principle, they could have used the first equation below (from page 16) directly. But that is a nonlinear model and would be more cumbersome. The red circle below really should encompass the entire exponential portion of expression for k (as shown by my green circle).  The term in brackets is then the Taylor series approximation for k/A (it actually goes on infinitely, but the idea is that the first few terms will get you close enough). Using this approximation they can use the all the benefits of a linear model. If you look carefully at the first equation below, it is just the equation of a straight line, with the slope, k, replaced by the Arrhenius equation.

3. On page 17 they recommend centering Arh(T); this is a good idea for interpretation of the coefficient A; i.e., create a variable, Arh(T) - Arh(T0), to use in your model (rather than Arh(T) directly).

 

4. There is more to talk through perhaps, but I'll leave it here for now. The link below may be of some help (in particular, see the last example in the Example section).

https://en.wikipedia.org/wiki/Taylor_series

        .   

Re: estimates in multipule regression

@MRB3855, I am amazed at this coincidence. Very basic question for you, very difficult for me: If I add the terms you suggest, I will have an approximation of the Arrhenius equation? Also, what other information in this presentation is too incomplete to allow direct use by a novice model fitter?

Re: estimates in multipule regression

Hi @Ressel  : This is the presentation I assume.

https://community.jmp.com/t5/Discovery-Summit-Europe-2019/Taking-Accelerated-Stability-Assessment-to-the-Next-Level-in-JMP/ta-p/156431

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..

Re: estimates in multipule regression

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.)

Re: estimates in multipule regression

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

Re: estimates in multipule regression

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 )}
 )
);

Re: estimates in multipule regression

Hi @Judd,

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

Re: estimates in multipule regressionIs 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!Re: estimates in multipule regression

Great explanation on centering polynomials!

Re: estimates in multipule regression

Thanks Julian for the immensely detailed and helpful answer. I consider the problem solved now.

Also thanks mpb for the helpful tip

Re: estimates in multipule regression

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

 

 

Re: estimates in multipule regression

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.