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I have a dataset with 299 observations, 35 independent and 141 dependent variables. This is a vegetation dataset, the IVs are environment variables, the DVs are coverage of 141 species (of course many of them sparse, with few occurences). All variables are continuous. My aim is to select those IVs which are relevant in explaining the species composition, so the 141 DVs. I am confused which methods are the most appropriate. The authors of the dataset suggested using redundancy analysis with a stepwise variable selection. However I read convincing materials that the stepwise selection best to be avoided. And I am curious if indeed this is the best and state-of-the-art method for this type of analysis. Regularized regression, particularly LASSO, are better in variable selection, but suitable only for one DV. I heard ideas to fit one model for each DV. But then, how to come to a consensus between independent variable selections? Or, make a PCA on the DVs, and fit model and do variable selection on PC1. Then, there are the multivariate regressions. For example with PLS I could fit a model, but then how to select the variables? I see that there are many approaches, too many to try them all. I decided to ask a new question, because other questions are either generic, or about different type of data. I am wondering for this type of data which methods do you suggest.

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