# Regression with dependent variable which ranges from -1 to 1

I performed a series of Pearson correlations which give me as expected values between -1 and 1 (actually very few below zero). I'd like now to see if some factors are linked to these correlation coefficients; therefore I'll run a multiple regression with various covariates which we want to test.

As my dependent variable is strictly bounded, would it be correct to use a simple linear regression? Or should I use some other model?

• The Fisher $z$ transformation $z = \text{atanh}(r)$ is one possibility. Regress $z$ as you please. But (1) using $r$ won't make such difference unless many correlations are near 1 (2) this seems a rather odd thing to do statistically. – Nick Cox Apr 21 '15 at 18:52
• @NickCox, I apologize for asking you this, but what is the difference between the current dependent variable restricted between -1 and 1 and the Fisher z transformation (restricted between -0.785 and 0.785)? In other words, why is this regression-ready and -1 to 1 isn't? Thanks! – Matt Reichenbach Apr 21 '15 at 19:07
• I see: you are confusing arc tangent atan() with inverse hyperbolic tangent atanh(). – Nick Cox Apr 21 '15 at 19:31
• atanh($r$) is just (1/2) logit((1 + $r$)/2) so logit modelling is also possible. – Nick Cox Apr 22 '15 at 2:42
• @MattReichenbach No need to apologise. This confusion is possible if anyone never studied hyperbolic functions or studied them years ago and does not use them routinely. It's useful to make explicit that atanh does not mean atan! – Nick Cox Apr 22 '15 at 14:11