Are SE of Total effect in Generalized Regression scaled?

[#1]

It is my understanding that Total Effects as determined using Generalized Regression: Standard Least Squares: Prediction Profiler: Variable Importance: Independent Resampled Inputs, are scaled to sum to 1.  Are the SE of each total effect also scaled?

I am using JMP Pro 17.

Thanks for your help

[#2 · Accepted solution]

Hi @olddabbler,

Welcome in the Community !

If you look at the JMP Help page Statistical Details for Assess Variable Importance and at litterature/articles about Sobol indices like Sensitivity analysis using Sobol’ indices — OpenTURNS 1.24 documentation, you can find several clues to answer your question :

• The formula used in JMP to calculate effects importance for a factor is a ratio between different variance components (response variance due to the specific factor (main effect and any effect related to this factor) / total variance of the response),
• There is an adjustment done in calculations to make sure Total Effect estimates are equal or bigger than Main Effect estimates (but can't be lower) : this is quite normal as total effect sensitivity indices values do include main effect sensitivity indice, so Total effect should be equal or bigger than Main effect.
• If you look at the documentation on OpenTURNS (particularly the sections Sobol’ decomposition and equation of Total sensitivity index of a variable), you can see that Total Effect sensitivity values for each variable are bound in the range [0,1], but there are no constraints on the sum of the Total Effect sensitivity values for all variables.
  Note that interaction effects are integrated in the calculations several time independantly : for example interaction X1*X2 will be incorporated in the calculation of Total Effect sensitivity indices both for variables X1 and X2, which may explain why the sum of all Total Effect sensitivity values can be higher than 1 (even if all individual sensitivity values are ratio of response variance from a variable to total response variance).  

The values for terms/factors behind Assess Variable Importance are different than the coefficients you could get from your regression model, as this method is independant of the model tested, and the model is only used to predict values that will be used to compute sensitivity values for Main and Total Effects of each variable.

Hope this response will clarify your understanding,

PS: If you want to dive deeper into the maths behind, the same paper listed in the JMP References are freely available in the ressources for OpenTURNS documentation :
[saltelli2002] Saltelli, A. Making best use of model evaluations to compute sensitivity indices. Computer Physics Communication, 2002, 145, 580-297. pdf

[sobol1993] Sobol, I. M. Sensitivity analysis for non-linear mathematical model Math. Modelling Comput. Exp., 1993, 1, 407-414. pdf

[#3]

Hi Victor,

Thanks very much for this very helpful information.  I will dive deeper into the sources you cite.  I expect that I will learn a lot by doing that.  If I have any questions after doing my homework, I assume that I can continue this thread.