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

In multiple regression Y ~ β(0) + β(1)X(1) + β(2)X(2) + β(3)X(1) X(2) + ɛ., the interaction term is quantified as the product of X(1) and X(2). We developed fractional-power interaction regression (FPIR), using βX(1)(M) X(2)(N) as the interaction term. The rationale of FPIR is that the slopes of Y-X(1) regression along the X(2) gradient are modeled using the nonlinear function (Slope = β(1) + β(3)MX(1)(M-1) X(2)(N)), instead of the linear function (Slope = β(1) + β(3)X(2)) 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 ~ β(0) + β(1)X(1) + β(2)X(2) + β(3)X(1)(M) X(2)(N) + ɛ to replace the current regression model Y ~ β(0) + β(1)X(1) + β(2)X(2) + β(3)X(1) X(2) + 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.