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 × Linear interaction of 2 quantitative predictors that yields point and interval estimates of 1 key parameter-the crossover 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 × Qualitative interactions, where the qualitative variable may have 2 or 3 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 National Institute of Child Health and Human Development Study of Early Child Care. The re-parameterized regression model had point and interval estimates of the crossover point that fell near the mean on the continuous environment measure. The disordinal form of the interaction supported 1 theoretical model-differential-susceptibility-over a competing model that predicted an ordinal interaction.

Figure 1

Predicted outcomes of GXE interaction under (A) diathesis-stress, and (B) differential susceptibility.

Figure 2

Plots of linear X linear interaction of two quantitative predictors X₁ and X₂: (A) ordinal interaction, and (B) disordinal interaction.

Figure 3

Predicted levels of social skills for the low-malleability and high-malleability groups as a function of childcare quality.