# Method for determining which of two related continous variables has a stronger association with a third continous variable

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?

• 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?
– whuber
Commented May 2, 2023 at 17:29

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 $$x_2$$.

• 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. Commented May 2, 2023 at 17:29
• This null isn't quite right, because it ought to concern comparison of the absolute correlation coefficients.
– whuber
Commented May 2, 2023 at 17:30
• 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. Commented May 2, 2023 at 18:27
• 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 $n.$
– whuber
Commented May 2, 2023 at 19:03
• 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. Commented May 3, 2023 at 1:20