Is it possible to have a IV that is not correlated significantly with the DV but then in the multiple regression analysis this IV explain a significantly % of the unique variance of this DV? If so, how can we explain it?
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Yes it's possible. The most basic explanation is that the extra variable absorbs residual variance & increases power (see: How can adding a 2nd IV make the 1st IV significant?). It is also possible the variable is a suppressor (see: Suppression effect in regression: definition and visual explanation/depiction, & X and Y are not correlated, but X is significant predictor of Y in multiple regression. What does it mean?). – gung - Reinstate Monica Feb 26 '16 at 21:41
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I think you will find the information you need in the linked thread. Please read it. If it isn't what you want / you still have a question afterwards, come back here & edit your question to state what you learned & what you still need to know. Then we can provide the information you need without just duplicating material elsewhere that already didn't help you. – gung - Reinstate Monica Feb 26 '16 at 21:42
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Is it right to say that for testing the suppressor effect of a IV, this one must be significantly correlated to the other IV? – Audrey Feb 27 '16 at 18:09
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It must be correlated, but it may or may not be 'significant'. (I suppose for testing you want it to be significant, of course.) – gung - Reinstate Monica Feb 27 '16 at 18:43