# Compare regression coefficients within the same model

I ran a regression model with around 10 regression coefficients.

I now want to test wether these regression coefficients are significantly different from one another.

How do I do this? Is a Wald test appropriate here?

Also, I would appreciate some reference to how to implement the respective solution (if available) in SAS, as I am required to use SAS to conduct my analysis.

EDIT: My first guess was to conduct multiple t-tests. It was pointed out that this is not correct, which is why I edited my question.

• These comparisons only make sense if all of the corresponding predictors are on the same scale. A distance predictor expressed in miles will have a much larger regression coefficient than the same distance expressed in millimeters. Is this the case with your data? And what is the purpose of this comparison?
– EdM
Jul 22, 2020 at 16:42
• Yes, this is the case with my data. The purpose of this is to simply understand wether the coefficients differ in their impact on the dependent variable. Put differently, my null hypothesis is H0: b1 = b2 = b3 =b4 = b5 = b6 = b7 = b8 = b9 = b10. I want to derive a statistically sound statistic that tells me wether I can reject this null hypothesis or accept it. Jul 22, 2020 at 16:46

With all of your predictors on the same scale, it could make sense to test the hypothesis that all coefficients are equal with a Wald test. That's a standard way to make tests involving groups or combinations of parameter estimates. You will, however, need to be careful in how you set this up.

A Wald test is frequently set up to test the null hypothesis that all coefficients are equal to 0. That is not your hypothesis. Presumably you need to test whether any of the coefficients differs from the mean coefficient value. So be sure to read the manual before you just call for a Wald test in statistical software.

Note that there might be some extra variability here as you will be estimating the mean value from the data rather than specifying a pre-determined value. I'm not sure how much of a difference that will make, as the test only holds asymptotically in any event.

• Alright, this reassures me in the assumption that a Wald test is the appropriate method here. Thanks. Jul 23, 2020 at 8:37
• @shenflow if you go on to test particular coefficients against each other, be sure to use the rules for the variance of sums of correlated variables together with the covariance matrix of the coefficient estimates to get the error estimates, and correct appropriately for multiple comparisons.
– EdM
Jul 23, 2020 at 13:43

It is not only impractical to do t-tests for each, but most importantly wrong. Model selection based on t test is misleading. Your problem is a question of model selection. First way would be to use information criteria like AIC and BIC to compare different models which consist of different subset of variables. Regarding doing this in SAS see here.

• How does my issue translate into a model selection problem? I want to compare wether different regression coefficients have a different impact on a dependent variable, i.e. I want to test wether they differ in value. I dont want to drop or add any variables or compare different models with different variables. Jul 22, 2020 at 15:46
• For example, one may apply the Wald test to test multiple hypotheses jointly. Similar to the ones I am concerned with. This has nothing to do with model selection in my opinion. Jul 22, 2020 at 15:49
• @shenflow The problem is that the regression coefficients might be correlated to each other. You can't just put in all regression coefficients into a model and then use t test results to check e.g. if they are significantly different from zero. There might be the case that the t test tells you that a regression coefficient is significantly different from zero, but this is actually not the case, this is just due to distortion. Same result could be vice versa. Same if you want to check if they differ in value. How can you tell, if you have no information about the correlation? Jul 22, 2020 at 15:49
• I understand. However, arent there ways to jointly test my issue in one hypothesis? I.e. H0: b1 = b2 = b3 = b4 = b5, for example? Isnt a Wald test appropriate in that case? Jul 22, 2020 at 16:16