Comparing the slopes of two independent regression lines when there is complete heteroscedasticity

For two independent groups, let (X-1j, Y-1j),...,(Xn(j)j, Yn(j)j) be a random sample from some bivariate distribution corresponding to the jth group, and assume that Y-ij = beta(1j)X(ij) + beta(0j) + epsilon(ij). Type I heteroscedasticity is defined to be a situation where for fixed j,epsilon(ij) does not have a common variance. That is, the variance of epsilon(ij) depends on the value of X. Type II heteroscedasticity is taken to mean that the variance differs between groups. That is, the variance of epsilon(i1) is not equal to the variance of epsilon(i2) even when X-i1 = X-i2 Complete heteroscedasticity refers to a situation where there is both type I and II heteroscedasticity. This paper considers the problem of computing a confidence interval for beta(11) - beta(12), the difference between the slopes, when there is complete heteroscedasticity. Some results on computing a confidence interval for the difference between the intercepts are reported as well. The one method found to give good results is used to reanalyse some data from a study dealing with pygmalion in the classroom.