Q: (Different slopes in different groups): I am fitting the same model to two samples -- for example, a sample from the US, and a sample from Sweden. It looks to me as though the slopes for the US are different from those for Sweden. But how I can I formally test this?

A: There are couple of popular approaches. These should give roughly the same result. I'd be interested in hearing about cases where they don't.

  1. One approach involves comparing the slopes directly. For a particular variable X, suppose that b1 and se1 are the estimated slope and standard error for the US, while b2 and se2 are the estimated slope and standard error for Sweden. The test statistic is Z = (b1-b2) / (se12+se22)1/2, which is very similar to the statistic for comparing two means. Under the null hypothesis of equal population slopes, Z has a standard normal distribution in large samples. So in large samples, you might reject the null hypothesis if |Z|>2 or so. [Note that some researchers have used a different, erroneous formula for Z (see Paternoster et al 1998 for a discussion).]
  2. Another approach involves interactions.Fit a model that combines both groups. In addition to variables from the separate models, the combined model will include a dummy variable representing group membership--for example, a dummy variable that is 1 in Sweden and 0 in the US. Also include interactions between this dummy variable and all the other variables in the model. This allows for the possibility that the slopes are different in the different groups. To test whether a particular slope is different, look at the p values for the interaction terms. To test whether any slopes are different, conduct a joint test of all the interactions at once. SAS and STATA have commands for such joint tests. In an OLS model, a joint test of interactions is equivalent to the "Chow test; this general approach to testing complex hypotheses in generalized linear models is discussed in chapter 4 of Long (1997).

References

Long, J.Scott. 1997. Regression Models for Categorical and Limited Dependent Variables. Thousand Oaks, CA: Sage.

Paternoster, R.; Brame, R.; Mazerolle, P.; Piquero, A. 1998. "Using the Correct Statistical Test for the Equality of Regression Coefficients." Criminology 36(1), 859-866.