Indeed, using PCA in order to handle collinearity is justified. But I rather keep the original variables instead of creating principal components because the original ones are more ‘meaningful’. Besides, keeping the original explanatory variables allows me to compare the obtained estimates with other from literature and these estimates have important biological meaning for my models (i.e. allometry of trees).

Thanks!

You may find this useful; I know I did

When Can You Safely Ignore Multicollinearity?

by Paul Allison

at http://www.statisticalhorizons.com/multicollinearity

both the original posting and the subsequent discussion.

Hello, if i understand you correctly you wanted a model without multicollinearity. you may transform all your variables with same mean and standaed deviation. then you compute for the accuracy ratio(AR) and do pairwise correlation tests. you may also try the Proc Varclus in SAS

Hello,

Thanks for your suggestion.

I would prefer to keep my variables untransformed (see my replay to Neeru Jaiswal).

Thanks for the SAS suggestion but I don’t have access to this software, therefore I am only familiar with R.

Is the relationship between explanatory variables linear, and is it collinearity between pair of variables, of between all variables?

You may try to write the model in a way that account for collinearity, by estimating parameters that are independent (a1, a2, etc.) and parameters that account for the collinearity (r1, r2, etc.).

If the relationship is linear, and all variables are correlated, this is something like:

Y~(X1^a1)*(X2^(r1* a1))*…*(Xn^(r^n-1*a1))

If the relationship is linear, and variables are correlated by pairs, this is something like:

Y~(X1^a1)*(X2^(r1* a1))*...*(X3^a3)*(X4^(r2* a3))*…*(Xn^(r^n-1*a1))

Well, I am not sure of the shape (whether a and r have to be multiplied or added) and actually you may need a bit more parameters, but do you see what I mean?

If the relationship is not linear, it is going to be more complex, but not impossible.

Of course then for model selection you have to start by deleting variables that are described as correlated to other variables...

Or I would rather suggest to make AIC comparisons rather than stepwise hierarchical selection, that would allow you to compare models where variables are assumed to be correlated, and models where they are not.

Everything is related with everything... - is about tree allometry (several diameters along stem, stem length & age are the explanatory variables - mostly non-linearly related). In this context I'm looking for the 'best' variables and simplify my models as much as possible (due to parsimony and practical reasons - some variables might be too costly to measure).

Thanks for you suggestions. I'm not sure if (over)-parameterizing my models will ease my work as I get convergence problems (I'm using R and nlme package)

I'm considering to use regression trees to 'detect' the 'best' variables, then build a full model and simplify based on significance levels and AIC differences.

Thanks!

Is there a theory or process that justifies the nonlinear model? You might consider simpler models in the first places, and then turn to pruning variables. Stepwise-regression-like approaches are notoriously bad, but you might consider more modern methods (I confess that I may not know the latest and greatest, but something in the family of lasso and elastic nets). And I wouldn't consider orthogonalizing variables just yet until these basic questions are addressed - best of luck!

There is a nice test outlined in "Applied regression with computing and graphics" by Cook & Weisberg. I've only used it in linear analyses, but it could easily be applied in some non-linear models.

You can use Ridge regression method and Modified Ridge Regression method to solve Multicollinearity Problem by avoiding the resulting large sampling variances of the ßs. This URL will be helpful for u.

http://www.m-hikari.com/ijcms-2011/9-12-2011/rashwanIJCMS9-12-2011.pdf