Distinguishing ordinal and disordinal interactions.

Abstract

Re-parameterized regression models may enable tests of crucial theoretical predictions involving interactive effects of predictors that cannot be tested directly using standard approaches. First, we present a re-parameterized regression model for the linear X linear interaction of two quantitative predictors that yields point and interval estimates of one key parameter – the cross-over point of predicted values – and leaves certain other parameters unchanged. We explain how resulting parameter estimates provide direct evidence for distinguishing ordinal from disordinal interactions. We generalize the re-parameterized model to linear X qualitative interactions, where the qualitative variable may have two or three categories, and then describe how to modify the re-parameterized model to test moderating effects. To illustrate our new approach, we fit alternate models to social skills data on 438 participants in the NICHD Study of Early Child Care. The re-parameterized regression model had point and interval estimates of the cross-over point that fell near the mean on the continuous environment measure. The disordinal form of the interaction supported one theoretical model – differential susceptibility – over a competing model that predicted an ordinal interaction.

The BDNF Val66Met Polymorphism Interacts with Maternal Parenting Influencing Adolescent Depressive Symptoms: Evidence of Differential Susceptibility Model

“…The model presented in Eqs. 3 and 4 is consistent with what Widaman et al (2012) and refers to as the ''weak differential susceptibility'' model where (a) the crossover point falls within the range of environmental measurement and (b) all allelic groups prove susceptible to environmental influence to some extent (i.e., estimates of slopes all different from zero), though one is more so than the other. In contrast, by constraining B 1 = 0 in Eq.…”

“…If C falls within the range of values of X 1 with a relatively narrow confidence interval (CI), the G 9 E effect conforms to the disordinal form, supporting the differential susceptibility model. Conversely, if C approaches or goes beyond the extreme point on X 1 , or has a rather wide confidence interval, then the G 9 E effect conforms to the ordinal form, supporting the diathesisstress model (Widaman et al 2012). The model presented in Eqs.…”

“…If the crossover point and its CI fall within the range of values of the (parenting) predictor, the G 9 E effect conforms to the disordinal form, supporting the differential susceptibility model. Conversely, if the crossover point and its confidence interval (CI) approaches or goes beyond the extreme point of the (parenting) predictor, the G 9 E effect conforms to the ordinal form, supporting the diathesis-stress model (Widaman et al 2012). Thus, the location of the crossover point is the crucial parameter that distinguishes the two competing G 9 E models.…”

“…Finally, we implemented the Widaman et al (2012)'s approach, which involves a re-parameterization of standard multiple linear equation (Eq. 2) by centering predictor X 1 at C, the crossover point on X 1 (see Eq.…”

“…Both AIC and BIC penalize for model complexity; therefore, adding unnecessary parameters results in increased AIC and BIC values. Additional statistical details can be found in Widaman et al (2012) and .…”