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We have a simple unvariate linear model of woodpecker abundance vs elevation:

woodpeckers ~ elevation

The model reports significantly positive slope for elevation. But if we add another variable - area of forests, which is positively correlated with elevation:

woodpeckers ~ elevation + forest

then suddenly, the elevation is no longer significant; only the forest is (significantly positive).

Now, we are writing a book for general public, who doesn't know statistics, and we want to present this result in a simple, yet still correct way. The most correct way would be to say:

Positive relationship with elevation is no longer significant after taking forest area into account.

I came up with this simplification, which could be more understandable to general public:

The relationship with elevation is positive only due to the preference of forests.

But can we really interpret the result like this? Isn't there any pitfal? I see that the "due to" might suggest causality which is not tested by regression. Or is there any other serious problem? How would you present this result?

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    $\begingroup$ Because regression contains no concept of "preference," nor does it model it in this instance, it's hard to justify your last interpretation. Indeed, what would you say if you had also tried the univariate regression woodpeckers ~ forest (which likely is significant and shows a positive association)? Consider describing what the statistical procedure actually does instead of imputing meaning that isn't there. As far as reporting the results goes, consider what information this procedure might have added to your initial knowledge and tell your readers that. $\endgroup$
    – whuber
    Aug 6 at 18:22
  • $\begingroup$ Thanks @whuber! The woodpeckers ~ forest is significant and positive as you guessed. 1) The problem is that describing what the statistics actually does, in bivariate case, isn't easy in language accessible to general public. 2) I thought that the last interpretation is what was added to my knowledge, but with the added info of the limits of the statistical method - which again I don't know if it's easily explainable to non-statisticians. Anyway this "added info" advice is really good, thanks. $\endgroup$
    – Tomas
    Aug 6 at 19:32
  • $\begingroup$ Use graphics! see for instance stats.stackexchange.com/questions/89747/… and stats.stackexchange.com/questions/73320/… $\endgroup$ Aug 7 at 0:30

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