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Let's say I want to investigate if there is any correlation between the response (continuous) variable individual_fish_size and the three explanatory (continuous) variables Depth, substrate and Temperature. But as my fishes come from several different species and the distribution of these species along the three variables is certainly not random, I'd like to include the categorical variable Fish.species to get rid of the variance due to difference of size along fish.species.

Once I ran such a model, does it make sense to do a selection (a backward selection for example) of the best formula by AIC (This can be realized with the function step in R)? Or would we do better to avoid such a thing because we might lose the part of the variance that is explained by fish.species.

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    $\begingroup$ See here. Reducing a model is fraught with problems & you should only consider it when (1) your primary concern is predictive accuracy, & (2) you can demonstrate poor performance owing to over-fitting: even then stepwise methods are not generally a very good solution. $\endgroup$ – Scortchi - Reinstate Monica Nov 23 '13 at 13:43
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You would do better to avoid such a thing because it (very rarely) makes sense.

What would make sense in your situation? Well, since you have only three continuous IVs and one categorical one, unless your N is very small indeed, there is little problem with keeping all the variables.

Also, you wrote you want to investigate whether there is "any correlation". There is some correlation, guaranteed without even looking at the data. What you want to know is how big that correlation is; more precisely, it seems you don't actually want correlations exactly, but regression coefficients. That is, you want to explore the relationships among the IVs and the DV, controlling for species.

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