What test can I use to find out which bivariate regression is *significantly* better?

#1
Test for which bivariate reg. is sig. better

I have three variables: y, x1 and x2 (n=130). I want to know whether one of the x variables is a significantly better bivariate predictor than the other.

When I regress y on x1 only, I get a t-stat of 4.93 and R^2=0.16
When I regress y on x2 only, I get a t-stat of 6.83 and R^2=0.26

Clearly, x2 provides a better fit. But what test might I use to check if the fit is significantly better?

Addendum: in this case, the fit can be improved further if I run a multivariate regression (i.e. regress y on x1 and x2). When I do this, the x variables are both individually significant. But I don't want to do this. I want to imagine that I am only allowed to use one predictor and I want to know if I can say that x2 is significantly better than x1.

Thanks for reading.
 
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#2
Re: What test can I use to find out which bivariate regression is *significantly* bet

Hi,

Model building is a pretty personal preference in my area (medicine) often relying on what you actually want to know rather than statistical significance. However, you need to be careful because the more variables you add in, the more you R^2 will increase (or stay the same, it wont decrease).

If statistical significance is what you are interested in then perhaps doing something like a stepwise regression is the best approach (only sensible if you are using some kind of program though), however, this is usually used for huge amounts of variables, and doesnt require a lot of thought.

To actually test the R^2 values after fitting the models I think you can use a Vuong test but I'm afraid I've never actually used that, but maybe someone else can offer some insight there.

I'll be interested to hear others opinions on this too.
 

Mean Joe

TS Contributor
#3
Re: What test can I use to find out which bivariate regression is *significantly* bet

On the contrary, I would say x1 provides the better fit, because I think R^2 measures the goodness-of-fit. As for if the fit is statistically better, I can't say.

Having a higher t-stat, x2 is "more significantly different from 0", although really don't want to infer something like that by comparing statistical test values. It could mean that x2 has a larger coefficient than x1, but that doesn't mean better fit.
 
#4
Re: What test can I use to find out which bivariate regression is *significantly* bet

Joe, yes, sorry: I reversed the R^2 numbers when I put them in. They're fixed now.