# How does one deal with correlated predictor variables in SDM studies?

I utilize climate and terrain data with GLM for species distribution models (SDM). I know that the predictor variables have to be independent. But when I check correlations within a correlation matrix and a pairs plot (in R using pairs() and cor()) nearly all of my variables seem to be correlated. After Dormann et a. 2013 the collinearity issue cannot be solved and one logical step is to exclude one of the variables and keep the ecological relevant predictor. This is tricky in my case, because if I sort out variables with a correlation coefficient >0.7 only two remain.

Most of the climate variables are strongly correlated by nature (e.g. altitude and temperature). How does one deal with this issue in SDM studies? Is collinearity such a problem? I am asking this because my distribution maps look quite well. Also AIC scores are around 0.9.

Thanks in advance, Patrick

## 1 Answer

The correlation between response variables, such as repeated measurement, needs to be considered when the model is fit. If predictor variables have COMPLETE collinearity, you need to exclude some of them, such that complete collinearity disappears, otherwise model fitting will fail. Some software, such as SAS, will do it automatically.

For incomplete collinearity among predictors, it was the problem 30~ years ago because computer just kept few digits, so even incomplete collinearity could result in the stop of the algorithm. Currently, because the increase of the precision of computer, incomplete collinearity generally does not bring any problem on computation.

"the predictor variables have to be independent." I do not think this independence is the condition of fitting model.

Of course, the collinearity can result in the instability of the model (for example, after excluding one predictor, the results may have tremendous change such that you cannot believe it), and the difficulty on the explanation of the results. So if possible and reasonable, it is good to exclude some predictors to decrease the degree of collinearity in practice.

• @ a_statistician, thank you for answering. What does COMPLETE collinearity mean? A correlation coefficient =1? – parallax May 4 '17 at 19:34
• Yes. In math, if $a_1x_1 + a_2x_2 + ... + a_kx_k=0$ for some $a_i \ne 0$ is called complete colinearity. Or from the values of k-1 of $x_i$, you can DETERMINE the value of $x_i$ that is not included in that k-1. – user158565 May 4 '17 at 19:50