Heteroscedasticity as a Basis of Direction Dependence in Reversible Linear Regression Models

ABSTRACT

Heteroscedasticity is a well-known issue in linear regression modeling. When heteroscedasticity is observed, researchers are advised to remedy possible model misspecification of the explanatory part of the model (e.g., considering alternative functional forms and/or omitted variables). The present contribution discusses another source of heteroscedasticity in observational data: Directional model misspecifications in the case of nonnormal variables. Directional misspecification refers to situations where alternative models are equally likely to explain the data-generating process (e.g., x → y versus y → x). It is shown that the homoscedasticity assumption is likely to be violated in models that erroneously treat true nonnormal predictors as response variables. Recently, Direction Dependence Analysis (DDA) has been proposed as a framework to empirically evaluate the direction of effects in linear models. The present study links the phenomenon of heteroscedasticity with DDA and describes visual diagnostics and nine homoscedasticity tests that can be used to make decisions concerning the direction of effects in linear models. Results of a Monte Carlo simulation that demonstrate the adequacy of the approach are presented. An empirical example is provided, and applicability of the methodology in cases of violated assumptions is discussed.

KEYWORDS:

• Direction dependence analysis
• heteroscedasticity
• linear regression model
• nonnormality

Article information

Conflict of Interest Disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.

Ethical Principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was not supported.

Role of the Funders/Sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: The authors would like to thank Phillip K. Wood, Shohei Shimizu, Stephen G. West, and the three anonymous reviewers for their comments on prior versions of this manuscript. The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors' institutions is not intended and should not be inferred.

Notes

1 It is important to note that notions of “true” and “false” models do not apply in practice because, ultimately, true statements about the exact data-generating mechanism are rarely possible (see Cudeck & Henly, 2003). Thus, both the proposed approach and DDA allow one to make statements concerning the model's capability to approximate an unobservable true model (which is, of course, not a unique feature of the proposed methodology, but rather concerns regression models in general).

2 Alternatively, it would also be possible to define the Type I error as identifying the misspecified model (i.e., rejecting H₀: Ω_{{x, z} → y} = I_n and retaining H₀: Ω_{{y, z} → x} = I_n) when no directional decision should be possible. Because the error term of the true model {x, z} → y was simulated to be homoscedastic and all variables were sampled from the normal distribution, homoscedasticity tests will reject the null hypothesis only by chance in both competing models. Thus, both Type I error definitions will show virtually identical results.

3 Although proportions of maximum scores are below commonly used cutoff criteria suggesting absence of a ceiling effect, we, in addition, re-ran all analyses excluding fourth-grade children. Overall, DDA based on the subgroup (n = 221) also suggested that AMC → AVC is more likely to reflect the underlying data-generating mechanism. Detailed results are discussed in the online supplement.

4 Note that including these offending data points did not affect overall substantive conclusions. Heteroscedasticity is more likely to be present in the model that treats AMC as the outcome variable.