Igor, can give more information? Instead of a random effects model you could try to use a SEM approach where you can model the collinearity. In this model you can have multiple layers of dependent variables. How many lakes do you have? With a random effects model you need to have 30+ lakes to use it in modelling. You could also use a multilevel SEM model, but if that can be done depends on the amount of lakes and the total size of your study. If you want to explore the relations between your variable you could also try to use data mining.

Thank you Peter for your interest and suggestions! There are 8 lakes, 10 predictors and 152 observations. Some problematic issues are that: (1) Several predictors are strongly positively or negatively correlated to each other (Pearson correlation coefficients >0.8 or

I believe in listening to a Salford Systems webinar a while back they indicated that gradient boosting methods were able to get around multicollinearity or more precisely collinearity does not affect tree based models.  Of course, the models are only appropriate for prediction not explanation. The Salford manuals for CART,etc are available of the web so you might check there.  Best wishes, David  BTW the attached file by Dr Shih may be helpful as is the one by Dr Elith et al

Anatoly Gordinsky, Israel

Dear Igor,

If you have reliable a priori information about the coefficients (their signs, possible absolute values, response properties, etc.), the approach considered in the article

"Gordinsky, A. (2016) New Facts in the Regression Estimation Under Conditions of Multicollinearity. Open Journal of Statistics, 6, 842-861 .htp: //dx.doi.org/10.4236/ojs.2016.6507" may be useful.

Igor, Your sample size is not that large and it will not be able to have many patterns. 130 is a little larger then 128 (= 2 **7). So 7 binary variables could span the whole 130 cases, If they were orthogonal to each other. It does not give much room to perform data-mining as well. The tree might not be very deep. You will need to set the n for the end notes well below what normally is used (a minimum of 50 in the end notes). You might want to have the minimum value at 10 and the above value at 25 instead of 100. If you have reliable estimates to use in a SEM model, you could all fix the parameters and do no estimation at all, just look at the fit measures and some of the modification indexes. For a normal use of SEM models you just have a too small sample. It is recommended to have at least 200 cases even for simple models, but with 10 variables it is not 'simple' compared to the models that were used for these robustness test (See Boomsma http://www.rug.nl/research/portal/en/publications/on-the-robustness-of-lisrel-maximum-likelihood-estimation-against-small-sample-size-and-nonnormality(13671dbd-9743-4780-be47-d8bb51909b28).html).  

Thank you everyone for your suggestions and interest! After attempting gradient boosting on these data, only two highly correlated predictors turned out to be problematic. I checked this by performing the following manipulation: (1) replace the outcome variable by a linear function of a selected predictor; (2) run the gradient boosting; (3) compare relative influence scores for all predictors. Ideally, the relative influence for the "true" predictor (from which the outcome was calculated) should be dramatically higher than those of all other predictors (irrelevant ones in this case). However, I found that for the two correlated variables this is not the case: if only one is the "true" predictor, the other one always achieves a comparable relative influence. Based on this, I am thinking of combining these two variables. Both are on a log scale, but of course with different means and ranges. What is a good way to combine them?

Suppose the remaining two predictors do not compensate each other, contrary to, for example, insulin and glucose. It looks plausible on the basis of your explanations. Then you can avoid any transformations, using the estimator (37) from the article mentioned earlier, the notations are in the same paper. This estimator is not biased and has significantly less risk than LS estimator. Moreover, you improve other estimates. Proofs are in my previous article "A Dual estimator as a tool for solving regression problems (Electronic Journal of Statistics, Vol. 7 (2013), 2372-2394, ISSN: 1935-7524)". Both articles are on the ResearchGate website.