Mobility Effect Models: A Comment on Glenn, McRae, and Brody

Six geomnetric surfaces in three-dimensional space and their corresponding equations are recommnended as plausible linear or nonlinear models for explaining various discrepancy effects. Researchers are encouraged to compare such alternative models when developing hypotheses, rather than arbitrarily selecting a regression equation that lhappens to work. Glenn (1989) and McRae and Brody (1989) have exchanged views on how best to analyze data such as have been reported by McRae and Brody (1987). Glenn (1989) decries the seemingly arbitrary use of nonlinear terms to represent mobility, to which McRae and Brody (1989) reply that their use of these terms was somewhat arbitrary but in accordance with previous mobility theory. We would like to propose an alternative way to deal with situations in which theory leads one to expect an effect associated with a difference between two variables, as often occurs when one compares husband/wife scores, ideal versus actual scores, or origin and destination scores. Our central point is that one's initial theory about such possible effects should be clear enough to allow the selection of specific three-dimensional geometric models; in other words, what Glenn (1989) refers to as side information should be sufficient to permit researchers to predict a specific geometric shape in three-dimensional space. We recommend first that researchers be aware that equations used to predict a dependent variable from two independent variables and their nonlinear terms can be depicted in three-dimensional space. To clarify our point, a picture is worth a thousand words. The following equations yield the geometric surfaces depicted in figures 1 through 6: *Preparation of this mnanutscript was suipported in part by the Kansas Agricultural Experimnent Station, contributtion nutmnber 90-345-J. Direct correspondence to Walter R. Schuiynin at the Departinent of Huonan Developynent and Faynily Studies, Justi1t Hall, Kansas State University, Manlhattan, KS 66506. ? The University of North Carolina Press Social Forces, December 1990, 69(2):617-620 This content downloaded from 157.55.39.136 on Thu, 19 May 2016 05:05:58 UTC All use subject to http://about.jstor.org/terms 618 / Social Forces 69:2, December 1990 Figure Equation1 Description of Surface 1 Z constant + aX + bY A (flat) plane. Changing coefficients a or b change the position of the plane but not its shape. Deviations from flatness are what is meant by nonlinearity (actually, nonplanarity). 2 Z = constant + aX + bY A plane twisted in space. + cXY 3 Z = constant + aX + bY A rooftop or gable shaped surface. + cABS(X Y) The slope of the sides depends on the relative magnitudes of a, b, and c. ABS = absolute value of When X = Y, Z will have the lowest scores unless the constant is positive and a and b are negative, in which case Z will have the highest scores along the X = Y axis. 4 Z = constant + a(X Y) A quonset hut model. + b(X -y)2 5 Z = constant + aX + bY A complex model that + cXY2 + dYX2 (or other often resembles (in part) polynomial terms) a saddle. 6 Z = constant a(X _ c)2 A circumplex model in b(Y d)2 which the highest score is centered over the point X = c and Y = d (if the constant is positive). Next we suggest that researchers select a priori one of the above models (or an alternative one) from previous theory and research as their best estimate of the expected effect while simultaneously selecting plausible alternative models to compare against their primary model.2 Finally, we recommend that researchers use regression to test their main model and to compare it to their alternative models.3 For example, Walsh and Olson (1989:59) predict a shape similar to that of Figure 6 when describing their circumplex model of family functioning; but actual data might better fit regression models associated with figures 1 through 5. It is possible that all models (or none of them) may explain significant amounts of variance (R2). Since alternative models may explain similar amounts of variance, proving that one model is better than another may not be possible. In some cases, parsimonious explanation may be the best guide, i.e., accepting the simplest model if the results of both are comparable in terms of R2. In other cases, as suggested by one reviewer, it may be possible to "sample only from the range of X and Y where different models give the most discrepant predictions, acknowledging that data from those values of X and Y where the models are identical or nearly so are uninformative for current purposes." This content downloaded from 157.55.39.136 on Thu, 19 May 2016 05:05:58 UTC All use subject to http://about.jstor.org/terms Mobility Effect Models: A Comment / 619 F'IGURE 1: Six Geometric Surfaces in Three-Dimensional Space