Regression Models for Cylindrical Data in Psychology

Abstract

Cylindrical data are multivariate data which consist of a directional, in this paper circular, and a linear component. Examples of cylindrical data in psychology include human navigation (direction and distance of movement), eye-tracking research (direction and length of saccades) and data from an interpersonal circumplex (location and intensity on the IPC). In this paper we adapt four models for cylindrical data to include a regression of the circular and linear component onto a set of covariates. Subsequently, we illustrate how to fit these models and interpret their results on a dataset on the interpersonal behavior of teachers.

Keywords:

• Cylindrical data
• regression
• interpersonal behavior

Acknowledgements

The authors would like to thank Irene Klugkist 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 or the funding agencies is not intended and should not be inferred.

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: Jolien Cremers acknowledges financial support by Grant NWO 452‐12‐010 awarded to I. Klugkist from the Dutch Organization for Scientific Research (NWO). Christophe Ley acknowledges financial support by Grant 1510391N from the Research Foundation – Flanders (FWO). The data used in the motivating example was collected with support from Grant NWO/PROO 411-07-360 from the Dutch Organization for Scientific Research (NWO).

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 Irene Klugkist 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 or the funding agencies is not intended and should not be inferred.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 In the von Mises distribution, a common distribution for a circular outcome, we have a concentration parameter κ that is related to ρ as $\rho =A_1(\kappa ),$ where $A_1\left(\kappa \right)=I_1(\kappa )/I_0(\kappa )$ and $I_0()$ and $I_1()$ are modified Bessel functions of order 0 and 1, respectively.

2 Note that for the CL-GPN model the circular location parameter also depends on the variance-covariance matrix and the circular predicted values should be computed using numerical integration or Monte Carlo methods because a closed form expression for the mean direction is not available.

3 Note that for the GPN-SSN model the predicted circular component also depends on the variance-covariance matrix and the circular predicted values should be computed using numerical integration or Monte Carlo methods because a closed form expression for the mean direction is not available.

4 The selection of the origin in circumplex data depends on the scaling of the Agency and Communion scores. Their respective 0 scores form the origin. Although the scaling influences the average score and spread on the IPC and the intensity, the difference between the individual measurements will be retained (albeit them being different in size). With regards to the intensity, scaling has the same effect as in a standard linear regression. With regards to the location on the IPC (the circular component) rescaling to -1,1 affects the size of the circular (as well as bivariate linear) regression coefficients. This has a positive effect if the original Agency and Communion scores are far from the origin in bivariate space. In that case the locations on the IPC are very concentrated before scaling (small variance). Such data are hard to estimate using circular models as the circular regression coefficients may be very small (high risk of bias, see Cremers, Mainhard, & Klugkist, 2018a). Scaling will thus improve estimation. Finally note that scaling is only considered an issue in those instances where cylindrical data is derived from measurements in bivariate space.

5 $\text{atan}2({\beta }_0^{II},{\beta }_0^I)$ or $\text{atan}2({\beta }_{0_{s^{II}}},{\beta }_{0_{s^I}})$

6 Note that this is a linear approximation to the circular regression line representing the slope at a specific point. Therefore it is possible for the HPD interval to be wider than $2\pi .$ In this case the interval is much wider and covers 0, indicating there is no evidence for an effect.

7 Note that the difference in regression lines (and thus inflection point) seems to be influenced by an outlier with a low self-efficacy and IPC value of approximately 160${}^{°}$ = −200${}^{°}.$

8 similar to the standard analysis