Integrating Methods to Optimize Circumplex Description and Comparison of Groups

Abstract

Using the interpersonal circumplex as an exemplar, this article serves as a methodological primer for integrating techniques of group description and comparison when employing circumplex-based assessment instruments. Circular statistics (Mardia & Jupp, 1999) and the structural summary method (Gurtman & Balakrishnan, 1998) each offer unique and incrementally useful information when applied to group-level data on circumplex measures. Circular statistics offer a set of parameters that are conceptually similar to their linear equivalents (i.e., mean, variance, and confidence intervals). In interpersonal circumplex models, these parameters each provide specific information regarding substantive theme and group homogeneity and allow for the statistical comparison of groups based on the geometry of the circular model. In a similar fashion, the structural summary method for circumplex data provides a set of parameters that complement circular statistics by offering measures of the interpersonal prototypicality of the group profile, levels of profile differentiation and elevation, and a weighted measure of substantive theme. Used in conjunction, these methods offer more information than is available using either in isolation. We provide 4 examples to demonstrate the complementary information the 2 methods provide for assessments employing interpersonal circumplex measures. These examples will allow investigators to generalize the methods to other personality assessment domains in which circumplex models are utilized, such as emotion and vocational preference.

[Supplementary materials are available for this article. Go to the publisher's online edition of the Journal of Personality Assessment for the following free supplemental resources: an Excel file that calculates the circular statistics and structural summary information described in this article using manually entered octant scores from up to 500 participants.]

Acknowledgments

A portion of this article was presented at the Society for Personality Assessment annual meeting, New Orleans, Louisiana, March 28, 2008. A. G. C. Wright thanks Burt Monroe and the Quantitative Social Science Initiative at the Pennsylvania State University for funding that made much of the work behind this manuscript possible. We thank Andrew J. Elliot for his generosity in sharing of some of the included data. We also thank Wendy Eichler, Christopher Hopwood, and Mark Lukowitsky for comments on an earlier draft of this article.

Notes

¹Example computations of these methods are presented in the Appendix, and a user-friendly, spreadsheet-based calculator is available on the Journal of Personality Assessment Web site.

²The basic trigonometric functions of the sine (sin), cosine (cos), and tangent (tan) allow for alternating between angular measurement (radians or degrees) and Cartesian distances on the two dimensions of a circumplex. Thus, this enables the two resulting scores on the dimensions to be translated into an angular location and back with ease. One or more of these basic functions are used in each of the equations provided in this primer and therefore we define them here for conceptual clarity. The trigonometric functions are perhaps easiest understood when thinking of a triangle. All angles in a circumplex-based measure can be understood in terms of a right triangle, with the scores on the x- and y-axes of the measure forming the legs of the triangle. In Figure 1, notice the outline of a right triangle with sides of X, Y, and hypotenuse Z, with the angle of interest defined as θ. The sine of angle θ is equivalent to the ratio of Y to Z (i.e., sin θ = Y/Z) or the ratio of the score on the vertical dimension divided by the length of the hypotenuse. The cosine of angle θ in is equivalent to the ratio of X to Z (i.e., cos θ = X/Z) or the ratio of the score on the horizontal dimension divided by the length of the hypotenuse. The tangent of angle θ is equal to the ratio of Y to X (i.e., tan θ = X/Y) or the ratio of the score on the vertical axis to the score on the horizontal axis. More important for the applications in this primer is the arctangent or the inverse of the tangent (i.e., tan^–1). A given angle θ is equivalent to the arctangent of the ratio of Y to X (i.e., θ = tan ⁻¹ X/Y) or the inverse of the ratio of the score on the vertical dimension to the score of the horizontal dimension. Therefore, for each individual or group, it is possible to get angular locations using the arctangent and the ratio of dimension (i.e., axis) scores or decompose angles into sines and cosines, which can be readily subjected to standard arithmetic.

³In calculating the sum of the cosine deviation, the same result will be obtained if the individual case is subtracted from the mean or the other way around. The cosine of an angle, positive or negative, is equivalent. For consistency with the literature, we report this formula with the individual case subtracted from the mean.

⁴Alternatively, significance testing for group means can be accomplished using Stephens' A statistic in the case of two groups (Stephens, 1972), or Stephens' C test in the case of more than two groups (see also Upton & Fingleton, 1989). Consistent with the American Psychological Association's Task Force on Statistical Inference (Wilkinson & Task Force on Statistical Inference, 1999) and Publication Manual (American Psychological Association, 2001), we suggest the use of CIs for the comparison and reporting of differences between groups.

⁵The goodness-of-fit statistic (R ²) correlates highly with amplitude (≈ 0.7), as differentiated profiles are often more defined and tend to be sinusoidal. However, they need not be so, and it is possible for a high R ² curve to be relatively undifferentiated (i.e., a shallow wave) and for a high amplitude profile to not be sinusoidal (see Pincus and Gurtman, 2003, for an example with an individual profile).