On the Ambiguity of Interaction and Nonlinear Main Effects in a Regime of Dependent Covariates

TL;DR

This paper investigates how nonlinear main effects and dependent covariates can cause ambiguous interactions in statistical models, proposing a new nonparametric method to clarify these effects and improve interpretation in experimental data analysis.

Contribution

It generalizes the analysis of covariate dependence from linear to nonlinear effects and introduces a novel nonparametric test for ambiguous interactions.

Findings

The proposed method successfully detects ambiguous interactions in simulations.

Reanalyses demonstrate the method's utility in real-world datasets.

Clarifying interactions aids in advancing theoretical understanding.

Abstract

The analysis of large experimental datasets frequently reveals significant interactions that are difficult to interpret within the theoretical framework guiding the research. Some of these interactions actually arise from the presence of unspecified nonlinear main effects and statistically dependent covariates in the statistical model. Importantly, such nonlinear main effects may be compatible (or, at least, not incompatible) with the current theoretical framework. In the present literature this issue has only been studied in terms of correlated (linearly dependent) covariates. Here we generalize to nonlinear main effects (i.e., main effects of arbitrary shape) and dependent covariates. We propose a novel nonparametric method to test for ambiguous interactions where present parametric methods fail. We illustrate the method with a set of simulations and with reanalyses (a) of effects of parental education on their children's educational expectations and (b) of effects of word properties on fixation locations during reading of natural sentences, specifically of effects of length and morphological complexity of the word to be fixated next. The resolution of such ambiguities facilitates theoretical progress.