betaDelta and betaSandwich: Confidence Intervals for Standardized Regression Coefficients in R

Abstract

The multivariate delta method was used by Yuan and Chan to estimate standard errors and confidence intervals for standardized regression coefficients. Jones and Waller extended the earlier work to situations where data are nonnormal by utilizing Browne’s asymptotic distribution-free (ADF) theory. Furthermore, Dudgeon developed standard errors and confidence intervals, employing heteroskedasticity-consistent (HC) estimators, that are robust to nonnormality with better performance in smaller sample sizes compared to Jones and Waller’s ADF technique. Despite these advancements, empirical research has been slow to adopt these methodologies. This can be a result of the dearth of user-friendly software programs to put these techniques to use. We present the betaDelta and the betaSandwich packages in the R statistical software environment in this manuscript. Both the normal-theory approach and the ADF approach put forth by Yuan and Chan and Jones and Waller are implemented by the betaDelta package. The HC approach proposed by Dudgeon is implemented by the betaSandwich package. The use of the packages is demonstrated with an empirical example. We think the packages will enable applied researchers to accurately assess the sampling variability of standardized regression coefficients.

Keywords:

• Standardized regression coefficients
• confidence intervals
• delta method standard errors
• heteroskedasticity-consistent standard errors
• R package

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 funded.

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 Professor Niels Waller for his comments on prior versions of this manuscript. The ideas and opinions expressed herein are those of the authors alone, and endorsement by the author’s institutions is not intended and should not be inferred.

Open Scholarship: This article has earned the Center for Open Science badges for Open Data through Open Practices Disclosure. The data and materials are openly accessible at https://osf.io/8hy7w/. To obtain the author's disclosure form, please contact the Editor.

Notes

1 The betaSandwich package also has functions that use normal-theory (BetaN) and ADF (BetaADF) asymptotic sampling covariance matrices. These functions apply the SEM approach described by Dudgeon (2017) but without the HC adjustment. While both functions use different calculations compared to the delta method, they generate numerically equivalent results to betaDelta::BetaDelta(object, type = "mvn") and betaDelta::BetaDelta(object, type = "adf"), respectively.