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Compositional explanatory variables

Sep 12, 2016 5:39 PM
(2992 views)

Hello

My question is how to implement compositional explanatory variable in a linear regression model using JMP Pro. Say we have one continuous dependent variable Y and we have d explanatory variables X1, ...,Xd that are represented in percentages and are characterized by a constant sum constraint (100%). Aitchison in 1986 introduced the log-contrast to address this issue but there are still problems with the interpretation of the parameters. Is there any model in JMP Pro that can handle the issue of Compositional explanatory variables ? I read about using centered log-ratio transformation (clr) or isometric log-ratio transformation (ilr) but they are not available in JMP

Thank you

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Re: Compositional explanatory variables

You might want to interpret your problem as a mixture analysis: Fitting Mixture Designs

I don't know if that makes sense for your application, but it would be my first shot.

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Re: Compositional explanatory variables

Hello Sebastian,

Thank you for your response. I don't need to design any experiments. I already have my data and need to fit them into a model. My response is a continuous variable and my explanatory variables are mixture of continuous and categorical variables . Four of my explanatory variables are continuous compositional variables that their sum is 100% . Compositional variables are singular and I read that compositional explanatory variables should not be directly used in a linear regression model because any inference statistic can become misleading

I looked at the "Fitting Mixture Designs" in JMP Pro but my understanding is that their model types are for designing models rather than fitting models.

Best

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Re: Compositional explanatory variables

Sorry for me being lazy. DanO's answer is basically what I wanted to say!

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Re: Compositional explanatory variables

When you are building your model in Fit Model look under the Attributes red triangle towards the bottom of the dialog. In there you will find a "mixture effect." Select the columns (they should already be in the Model Effects area of the dialog) that you want to be part of your mixture and select the Mixture Effect attribute.

Best,

M

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Re: Compositional explanatory variables

Hello Micheal,

First thank you for your response. I tried your approach to this problem but the result is somehow confusing for me since adding "Mixture Effect" attributes to my compositional explanatory variables only suppressed the intercept in the Standard Least Squares regression model. In other words , the result of the "Mixture Effect" is the same if I had the standard least squares without "Mixture Effect" and "No Intercept" . So my understanding is that there are no transformations of any kind .

Do I need to choose specific personality if I want to use "Mixture Effect" ? Also do I need to replace all zeros in my compositional explanatory variables with some number like 0.0001? My third concern is whether out of d compostional explanatory variables , I have to use d-1 in order to avoid multicolinearity.

I really appreciate your response.

S

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Re: Compositional explanatory variables

S,

Short answer is that YES, JMP Pro (and even regular JMP) can analyze this situation appropriately.

Longer, more detailed answer:

Your situation could certainly be considered a mixture model. A mixture model is a model where the "factors" (X1 to Xd in your case) are constrained that they add up to 1 or 100%. Because of this constraint, you cannot use standard regression techniques (you have a singular matrix that cannot be inverted). There are several different possible "tricks" that could be used to turn this into a regression problem. The most common is to use what is called a Scheffe model. This model will put the intercept into each of the main effects. This will "remove" the intercept term as being estimated separately. This is the approach that MIchael.Anderson proposed. If you look carefully, the resulting analysis of variance table will have a line that says "Testing Against Reduced Model Y = Mean". This is proof that it is a Scheffe model, not a no-intercept model. A no-intercept model would have the line "Testing Against Reduced Model Y = 0".

This is by far the easiest way to analyze your situation, but Scheffe models are interpreted differently from a typical regression model (in fact, all of the models for these "compositional variables" will need to be interpreted differently because you cannot use typical regression). The interpretation would take some time to go through here as it takes about 30 minutes in our class on mixture analysis. The interpretation would also depend on the order of the model that you are fitting. Hopefully this will give you some insights into what is going on and places to start to learn about how to properly analyze this situation.

Dan Obermiller

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Re: Compositional explanatory variables

I need to modify my answer slightly. I noticed that if your X variables add up to 1, JMP will automatically switch to a Scheffe model. It looks like you cannot fit a no-intercept model to compositional variables. So my previous note is important in case the variables do not add up to 1 exactly. Thanks to a Jian Cao for pointing out this automatic fitting of a Scheffe model in this situation even if the mixture attribute is not set.

Dan Obermiller