In many situations it is desirable or necessary to administer a set of tests to several different groups, and to ask if the results obtained in the different groups may be regarded as being essentially the same in some sense. In the case of two variables (one dependent and one independent) one may, for instance, ask if the errors of estimate and the regression lines may be regarded as being the same for the populations from which the different groups are drawn. For this case, the present article considers tests for three hypotheses regarding the populations from which the different groups are drawn: (a) HA, the hypothesis that all standard errors of estimate are equal; (b) HB, the hypothesis that all regression lines are parallel, (assuming HA); and (c) HC, the hypothesis that the regression lines are identical, (assuming HB). Test criteria for these three hypotheses and their sampling theory for large samples are presented. The results are extended to the case of several independent variables. An illustrative problem is presented for two groups, two independent and one dependent variable.