Using the correct statistical test for the equality of regression coefficients

Criminologists are often interested in examining interactive effects within a regression context. For example, "holding other relevant factors constant, is the effect of delinquent peers on one's own delinquent conduct the same for males and females?" or "is the effect of a given treatment program comparable between first-time and repeat offenders?" A frequent strategy in examining such interactive effects is to test for the difference between two regression coefficients across independent samples. That is, does b(1) = b(2)? Traditionally, criminologists have employed a t or z test for the difference between slopes in making these coefficient comparisons. While there is considerable consensus as to the appropriateness of this strategy, there has been some confusion in the criminological literature as to the correct estimator of the standard error of the difference, the standard deviation of the sampling distribution of coefficient differences, in the t or z formula. Criminologists have employed two different estimators of this standard deviation in their empirical work. In this note, we point out that one of these estimators is correct while the other is incorrect. The incorrect estimator biases one's hypothesis test in favor of rejecting the null hypothesis that b(1) = b(2). Unfortunately, the use of this incorrect estimator of the standard error of the difference has been fairly widespread in criminology. We provide the formula for the correct statistical test and illustrate with two examples from the literature how the biased estimator can lead to incorrect conclusions.