# Test if one predictor's lack of a relationship with the DV is because of a negative relationship with another predictor correlated with the DV

I recently collected some survey data where people made a series of judgments about others' actions and their willingness to make some choice about that action.

Looking at the zero-order correlations, I see that some of the judgments are significantly correlated with the choices of the participant, whereas some others are not.

I am concerned that actual relationships may be hiding between the uncorrelated judgments and choices. Let me give a generalized example.

Imagine Predictor A and B are correlated with DV X, but Predictor C is not correlated with DV X. I was thinking that there could actually be a relationship between C and X, but that C may be negatively correlated with A or B resulting in a zero-order correlation between X and C coming out as a wash (and therefore not looking significant).

I imagine I should be able to test for this using regression, but I am not quite sure on the exact procedure. Would simply putting all A, B, and C predicting X in a regression showing that C remains insignificant be sufficient so show that it is not due to this inverse correlation?

Thank you very much for your suggestions.

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Have you ever heard of "mediation analysis"? – Macro Jun 17 '12 at 23:46

This is very close (or exact!) to partial correlation. It is a measure of the correlation between the independent and the dependent given all other variables are accounted for.

See the wikipedia page for details here.

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This is actually just a shortcut. In multivariate linear regression, the coefficients are proportional to the partial-correlation. So computing the regression, as Michael Bishop suggests, will give you a start. But the underlying idea is known as partial-correlation. – Cam.Davidson.Pilon Dec 15 '12 at 6:24

I think you're idea of a regression predicting X with A, B, and C, is a good place to start. Make sure to distinguish between a statistically insignificant effect and a precisely estimated effect of zero (or close to zero).

Also, consider the possibility of non-linear relationships. Throw squared or even cubed terms into the regression equation.

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If A is correlated with B or C, plugging them together into a linear regression will create colinearity and the results will be spurious (mean nothing).

There might be a more complex relationship between C and X than just a linear increasing one (for example, an inverse U shape relationship might look like zero correlation in the matrix).

One good way to test this (before regressions) is first to graph X vs A,B and C. Then you might see a relationship which is not trivial (such as a squared one etc.).

In addition, there might be interaction between A,B and C, that is, for example, people make a choice only when their judgment is of some kind. Suppose this is the relationship between B and C. To run this in a regression, multiply the values of B and C and add them as a new variable (call it D) to the regression.

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