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I have two continuous variables and would like to determine which of these is more strongly associated with a third continuous variable.

I realize I could perhaps do a multiple regression and see if both remain significant in this model but I fear my two independent variables are rather correlated to each other and collinearity would be a problem. Is there any other statistical way I can determine a difference? Can you compare two single regressions somehow? Or any other completely different test?

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  • $\begingroup$ Welcome to CV, Carl. I would be reluctant to use correlation (or the equivalent, like $R^2$) to assess strength of association without first inspecting the scatterplots for outliers and high-leverage points. Regardless, do you really need a test? Why wouldn't it suffice to pick the variable with the largest (absolute) correlation with the third variable? $\endgroup$
    – whuber
    Commented May 2, 2023 at 17:29

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Here is the information for a null hypothesis statistical test (NHST) a single-sample test comparing 2 correlations (this comes from Kleinbaum, Kupper, Nizam & Rosenberg, 5th ed.).

If you wish to test the claim that two single-sample correlations (same variable correlated with 2 other variables) are different from each other, $$H_o : \rho_{y,x_1} = \rho_{y,x_2}$$ we can using the following test statistic (for sufficiently large $n$): $$Z = \frac{(r_1-r_2)\times\sqrt{n}}{\sqrt{(1-r_1^2)^2+(1-r_2^2)^2-2r_{12}^3-(2r_{12}-r_1r_2)(1-r_1^2-r_2^2-r_{12}^2)}}$$ where $r_i$ is the correlation of $y$ and $x_i$ and $r_{12}$ is the correlation of $x_1$ and $x_2$.

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  • $\begingroup$ From experience (personally, and watching many students make this mistake)...be sure to note the cubed correlation in the denominator...it is easy to mistype it as just another squared value. $\endgroup$
    – Gregg H
    Commented May 2, 2023 at 17:29
  • $\begingroup$ This null isn't quite right, because it ought to concern comparison of the absolute correlation coefficients. $\endgroup$
    – whuber
    Commented May 2, 2023 at 17:30
  • $\begingroup$ If the null is associated with a 2-tailed test, the test statistic can be positive or negative...I don't see what the issue is. The sign (or order of the subtraction) would be more important if it were a 1-tailed/directional NHST. $\endgroup$
    – Gregg H
    Commented May 2, 2023 at 18:27
  • $\begingroup$ The issue is that a correlation of $-1/2$ and a correlation of $+1/2$ are identically strong, whereas your test would likely declare them to be significantly different even for small $n.$ $\endgroup$
    – whuber
    Commented May 2, 2023 at 19:03
  • $\begingroup$ I will leave this to the OP to clarify...they indicate at the first about strength of association, and later in the post they ask about determining a difference (of regressions)...which I took to mean the correlations. The reference I cite addresses this later query. $\endgroup$
    – Gregg H
    Commented May 3, 2023 at 1:20

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