Mobility Effect Models: The Fundamental Difference Between Linear and Nonlinear Discrepancy Effects: A Reply

Thtough the approach recommended by Schtumrn et al. should be useful in estimating discrepanicy effects, it is not a general solution, since it is not applicable when any important component of an estimated effect is linear. Linear and nonlinear discrepancy effects are fundamentally different and must be approached in fundamiientally different ways. This comment is an addendum rather than a rejoinder to the comment by Schumm et al., and its main purpose is to elaborate on a crucial issue only incidentally treated by these authors. The procedures they recommend may prove to be quite useful and, as the authors argue, should be superior to analysis of variance or dummy variable regression analysis for estimating nonlinear discrepancy effects. Although such effects are reflected as interactions between the two defining variables, not all such interactions should be considered evidence for discrepancy effects, and thus the specific form of the predicted interaction should be incorporated into the models to the greatest extent possible. My only major reservation about the comment is that it is potentially confusing to readers who lack a very high level of understanding of the identification problem and the logic of estimating discrepancy effects. Such readers probably include most of those who occasionally deal with possible discrepancy effects. For instance, Schumm et al. seem to present their strategy as an all-purpose solution to the problems faced in "situations in which theory leads one to expect an effect associated with a difference between two variables," but their methods are not applicable when any important component of the predicted discrepancy effect is linear.' The potential for misunderstanding is increased by the authors' reference in their abstract to "plausible linear or *Direct correspondence to the autthor at the Departmnent of Sociologiy, University of Texas, Autstin, TX 78712. i The University of North Carolina Press Social Forces, December 1990, 69(2):621-623 This content downloaded from 157.55.39.235 on Fri, 07 Oct 2016 06:05:43 UTC All use subject to http://about.jstor.org/terms 622 / Social Forces 69:2, December 1990 nonlinear models." Although the referenced linear models are those with no discrepancy effects, the reference may lead some readers to believe that the proposed methods are useful in testing for both linear and nonlinear discrepancy effects. Persons who learn to apply the recommended techniques will discover that they can be used only to estimate nonlinear effects, but those whose theories predict only linear effects should be saved the trouble of trying to learn. Furthermore, some readers who do not try to apply the techniques may get the false impression that statistical comparisons of competing plausible models are useful when the predicted discrepancy effects are linear. Linear and nonlinear discrepancy effects are fundamentally different and must be treated in fundamentally different ways in research. This point is elementary, but it cannot be overemphasized, since failure to understand it has probably been the most common source of mistakes in discrepancy effects research. The potential for capturing evidence for discrepancy effects through statistical-model testing varies directly with the extent to which the effects are nonlinear. To the extent that they are linear, they are confounded with any linear effects of the defining variables and cannot be statistically separated from them. In note 2, Schumm et al. seem essentially to dismiss concern with linear discrepancy effects, and I have considerable sympathy for their position. Unlike them, I do not think that parsimony is an appropriate ground for rejecting explanations that include the effects, yet there often is not a clear and sharp conceptual distinction between the posited linear discrepancy effects and the effects of the defining variables. But the importance of discussing and understanding the special properties of linear effects depends not on what I or Schumm et al. think of the utility of conceiving of such effects but on the fact that they have been frequently predicted in the social science literature.2 For instance, effects with at least a strong linear component are predicted when opposite-signed effects of upward and downward mobility or of husband-high and wife-high differences (say, in earnings) are expected. Or, to give a specific example, if education-high-earnings-low status inconsistency is expected to incline persons to be politically liberal, whereas education-low-earnings-high inconsistency is expected to incline them to be conservative, the predicted effect is monotonic with at least a large linear component. An essential first step in discrepancy effects research is to identify the predicted effects as linear or nonlinear. If both or either of these kinds of effects are expected to be present, one must realize that no one approach is adequate for dealing with both.3 If the predicted effects are nonlinear, the approach recommended by Schumm et al. should be seriously considered, my tentative opinion being that it is better than most other methods of estimating nonlinear discrepancy effects. If the predicted effects are linear, no statistical analysis will provide unambiguous evidence for them, and the best one can usually to do is to estimate models including the two defining variables and to consider, on the basis of theory and side information, that some or all of the effects attributed to those variables may be discrepancy effects. One should not use an arbitrary (or nonarbitrary but weak) identifying restriction so that the discrepancy can be included as a variable in the model, since doing so will result in an arbitrary or largely arbitrary allocation of the estimated effects. This content downloaded from 157.55.39.235 on Fri, 07 Oct 2016 06:05:43 UTC All use subject to http://about.jstor.org/terms Mobility Effect Models: A Reply / 623