Circular interpretation of regression coefficients. (English) Zbl 1460.62104
Summary: The interpretation of the effect of predictors in projected normal regression models is not straight-forward. The main aim of this paper is to make this interpretation easier such that these models can be employed more readily by social scientific researchers. We introduce three new measures: the slope at the inflection point (\(b_c\)), average slope (AS) and slope at mean (SAM) that help us assess the marginal effect of a predictor in a Bayesian projected normal regression model. The SAM or AS are preferably used in situations where the data for a specific predictor do not lie close to the inflection point of a circular regression curve. In this case \(b_c\) is an unstable and extrapolated effect. In addition, we outline how the projected normal regression model allows us to distinguish between an effect on the mean and spread of a circular outcome variable. We call these types of effects location and accuracy effects, respectively. The performance of the three new measures and of the methods to distinguish between location and accuracy effects is investigated in a simulation study. We conclude that the new measures and methods to distinguish between accuracy and location effects work well in situations with a clear location effect. In situations where the location effect is not clearly distinguishable from an accuracy effect not all measures work equally well and we recommend the use of the SAM.
Keywords:
Bayesian analysis; circular regression; projected normal distribution; normal regression; regression coefficients; average slope (AS); slope at mean (SAM)References:
| [1] | Brunyé, T. T., Burte, H., Houck, L. A., & Taylor, H. A. (2015). The map in our head is not oriented north: Evidence from a real‐world environment. PLoS One, 10(9), 1-12. https://doi.org/10.1371/journal.pone.0135803 |
| [2] | Fisher, N. I., & Lee, A. J. (1992). Regression models for an angular response. Biometrics, 48, 665-677. https:doi.org/10.2307/2532334 |
| [3] | Hernandez‐Stumpfhauser, D., Breidt, F. J., & van der Woerd, M. J. (2017). The general projected normal distribution of arbitrary dimension: Modeling and Bayesian inference. Bayesian Analysis, 12(1), 112-133. https://doi.org/10.1214/15-BA989 · Zbl 1384.62176 |
| [4] | Heyes, S. B., Zokaei, N., & Husain, M. (2016). Longitudinal development of visual working memory precision in childhood and early adolescence. Cognitive Development, 39, 36-44. https://doi.org/10.1016/j.cogdev.2016.03.004 |
| [5] | Kendall, D. G. (1974). Pole‐seeking Brownian motion and bird navigation. Journal of the Royal Statistical Society, Series B, 37, 365-417. · Zbl 0291.92005 |
| [6] | Kirschner, S., & Tomasello, M. (2009). Joint drumming: Social context facilitates synchronization in preschool children. Journal of Experimental Child Psychology, 102, 299-314. https://doi.org/10.1016/j.jecp.2008.07.005 |
| [7] | König, J., Onnen, M., Karl, R., Rosner, R., & Butollo, W. (2016). Interpersonal subtypes and therapy response in patients treated for posttraumatic stress disorder. Clinical Psychology & Psychotherapy, 23, 97-106. https://doi.org/10.1002/cpp.1946 |
| [8] | Locke, K. D., Sayegh, L., Weber, C., & Turecki, G. (2017). Interpersonal self‐efficacy, goals, and problems of persistently depressed outpatients: Prototypical circumplex profiles and distinctive subgroups. Assessment, Advance online publication. https://doi.org/10.1177/1073191116672330 |
| [9] | Mardia, K. V., & Jupp, P. E. (2000). Directional statistics. Chichester, UK: John Wiley & Sons. · Zbl 0935.62065 |
| [10] | Maruotti, A. (2016). Analyzing longitudinal circular data by projected normal models: A semi‐parametric approach based on finite mixture models. Environmental and Ecological Statistics, 23, 257-277. https://doi.org/10.1007/s10651-015-0338-3 |
| [11] | Mastrantonio, G., Lasinio, G. J., & Gelfand, A. E. (2016). Spatio‐temporal circular models with nonseparable covariance structure. Test, 25, 331-350. https://doi.org/10.1007/s11749-015-0458-y · Zbl 1402.62102 |
| [12] | Matsushima, E. H., Vaz, A. M., Cazuza, R. A., & Ribeiro Filho, N. P. (2014). Independence of egocentric and exocentric direction processing in visual space. Psychology & Neuroscience, 7, 277-284. https://doi.org/10.3922/j.psns.2014.050 |
| [13] | Neal, R. M. (2003). Slice sampling. Annals of Statistics, 31(3), 705-741. https://doi.org/10.1214/aos/1056562461 · Zbl 1051.65007 |
| [14] | Nuñez‐Antonio, G., & Gutiérrez‐Peña, E. (2005). A Bayesian analysis of directional data using the von mises-fisher distribution. Communications in Statistics—Simulation and Computation, 34, 989-999. https://doi.org/10.1080/03610910500308495 · Zbl 1078.62025 |
| [15] | Nuñez‐Antonio, G., & Gutiérrez‐Peña, E. (2014). A Bayesian model for longitudinal circular data based on the projected normal distribution. Computational Statistics & Data Analysis, 71, 506-519. https://doi.org/10.1016/j.csda.2012.07.025 · Zbl 1471.62153 |
| [16] | Nuñez‐Antonio, G., Gutiérrez‐Peña, E., & Escarela, G. (2011). A Bayesian regression model for circular data based on the projected normal distribution. Statistical Modelling, 11, 185-201. https://doi.org/10.1177/1471082X1001100301 |
| [17] | Ouwehand, P. E. W., & Peper, C. L. E. (2015). Does interpersonal movement synchronization differ from synchronization with a moving object?Neuroscience Letters, 606, 177-181. https://doi.org/10.1016/j.neulet.2015.08.052 |
| [18] | Presnell, B., Morrison, S. P., & Littell, R. C. (1998). Projected multivariate linear models for directional data. Journal of the American Statistical Association, 93, 1068-1077. https://doi.org/10.1080/01621459.1998.10473768 · Zbl 1063.62546 |
| [19] | Rivest, L.‐P., Duchesne, T., Nicosia, A., & Fortin, D. (2016). A general angular regression model for the analysis of data on animal movement in ecology. Applied Statistics, 63, 445-463. https://doi.org/10.1111/rssc.12124 |
| [20] | Santos, G. C., Vandenberghe, L., & Tavares, W. M. (2015). Interpersonal interactions in the marital pair and mental health: A comparative and correlational study. Paidéia (Ribeirão Preto), 25, 373-382. https://doi.org/10.1590/1982-43272562201511 |
| [21] | Stoffregen, T. A., Bardy, B. G., Merhi, O. A., & Oullier, O. (2004). Postural responses to two technologies for generating optical flow. Presence: Teleoperators and Virtual Environments, 13, 601-615. https://doi.org/10.1162/1054746042545274 |
| [22] | Wang, F., & Gelfand, A. E. (2013). Directional data analysis under the general projected normal distribution. Statistical Methodology, 10(1), 113-127. https://doi.org/10.1016/j.stamet.2012.07.005 · Zbl 1365.62195 |
| [23] | Wang, F., & Gelfand, A. E. (2014). Modeling space and space‐time directional data using projected Gaussian processes. Journal of the American Statistical Association, 109, 1565-1580. https://doi.org/10.1080/01621459.2014.934454 · Zbl 1368.62304 |
| [24] | Wright, A. G., Pincus, A. L., Conroy, D. E., & Hilsenroth, M. J. (2009). Integrating methods to optimize circumplex description and comparison of groups. Journal of Personality Assessment, 91, 311-322. https://doi.org/10.1080/00223890902935696 |
| [25] | Zilcha‐Mano, S., McCarthy, K. S., Dinger, U., Chambless, D. L., Milrod, B. L., Kunik, L., & Barber, J. P. (2015 |
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