# Testing whether two regression coefficients are significantly different

I'm hoping somebody can help me out with this question. For a study I did a path analysis, which looks like this:

````IV --> 
       Mediatior  --> DV
IV -->
````

So I have two IV's leading to my mediator. With SPSS I did two regression analysisses from the IV's to the Mediator.

Now I want to test whether the two regression coefficients are signifficantly different from each other using SPSS, but I have no idea how to do this. Except that I assume that I have to use a T-test, but don't know how.

I have the two regression coefficients, their standard deviation and my N (this n is the same for both regressions, because it's all from one sample).

I suspect that this is pretty basic for some, but my skills in statistics are not that great. So help would be greatly appreciated!

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1
Why do you want to do this? – Peter Flom Jun 26 '12 at 11:01

1
Because according to my regression coefficients IV(a) has more influence on the mediator than IV(b) does. And now I would want to know whether this difference between the regression coefficients is significant – user12208 Jun 26 '12 at 13:28

## 3 Answers

If you have the estimated covariance matrix for the coefficients, then you can construct the t-test as follows. Let the hypothesis, in its general form, be $R^T\beta = b$, and $\widehat{\Sigma} = \hat{\sigma}^2(X^TX)^{-1}$ be the estimated covariance matrix of the coefficients. In your case, assuming the test is that $\beta_2 = \beta_3$ and you have $K=3$ coefficients, $R^T = [0, 1, -1]$ and $b=0$. Then:

$T^* = \frac{R^T\beta - b}{\sqrt{R^T\widehat{\Sigma}R}}$

is distributed $t(N-K)$.

Source: Principles of Econometrics, Theil.

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Thanks for all of the answers! – user12208 Jun 30 '12 at 13:36
I have the same question as user1 but jbowman's response isn't clear to me. Is it possible to clarify?? – Candace White May 3 at 13:00
@Candace White I think you will need to be more specific than "isn't clear" if you would like a focused answer to your comment. – whuber♦ May 3 at 13:17

I believe the correct approach here is to compare the fit of a model where IV(a) and IV(b) are allowed to vary - that is, your present model - with the fit of a model where IV(a) and IV(b) are fit to the same value (in which case the mediator is just an average of the two). The two models can be compared using a Chi-Square difference test.

This is simple enough to be performed by hand - I am not quite sure how to do that last, final step in SPSS. But all of the requisite values for calculating the Chi-Square difference will be available to you in SPSS, and there are several online calculators that could be used for determining the value of this test statistic and its p-value. I hope that helps!

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CHCH's answer is interesting. But would it be simpler to use Fisher's Z transformation and the corresponding difference formula to compare the standardized regression coefficients of the two IVs?

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+1, in particular, I think the part about standardized is crucial--otherwise the whole thing is nonsense from the start. – gung Jun 27 '12 at 1:39
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