Deeply digging the interaction effect in multiple linear regressions using a fractional-power interaction term

Abstract

In multiple regression Y ~ β0 + β1X1 + β2X2 + β3X1 X2 + ɛ., the interaction term is quantified as the product of X1 and X2. We developed fractional-power interaction regression (FPIR), using βX1M X2N as the interaction term. The rationale of FPIR is that the slopes of Y-X1 regression along the X2 gradient are modeled using the nonlinear function (Slope = β1 + β3MX1M-1 X2N), instead of the linear function (Slope = β1 + β3X2) that regular regressions normally implement. The ranges of M and N are from -56 to 56 with 550 candidate values, respectively. We applied FPIR using a well-studied dataset, nest sites of the crested ibis (Nipponia nippon).We further tested FPIR by other 4692 regression models. FPIRs have lower AIC values (-302 ± 5003.5) than regular regressions (-168.4 ± 4561.6), and the effect size of AIC values between FPIR and regular regression is 0.07 (95% CI: 0.04–0.10). We also compared FPIR with complex models such as polynomial regression, generalized additive model, and random forest. FPIR is flexible and interpretable, using a minimum number of degrees of freedom to maximize variance explained. We have provided a new R package, interactionFPIR, to estimate the values of M and N, and suggest using FPIR whenever the interaction term is likely to be significant.• Introduced fractional-power interaction regression (FPIR) as Y ~ β0 + β1X1 + β2X2 + β3X1M X2N + ɛ to replace the current regression model Y ~ β0 + β1X1 + β2X2 + β3X1 X2 + ɛ;• Clarified the rationale of FPIR, and compared it with regular regression model, polynomial regression, generalized additive model, and random forest using regression models for 4692 species;• Provided an R package, interactionFPIR, to calculate the values of M and N, and other model parameters.