Some results on a Winsorized correlation coefficient
Abstract
Recent investigations indicate that psychometric measures often have distributions with very heavy tails, and that outliers are quite common. It has long been known that in terms of both power and Type I errors, even slight departures from normality can have serious consequences, and it is fairly evident that there are problems when using conventional measures of effect size, as is briefly illustrated in this paper. Similar problems arise when dealing with the usual correlation coefficient. Robust correlation coefficients have already been proposed that reflect the linear relationship between two random variables, but many have the unfortunate property of not always being equal to 0 under independence. One exception is the Winsorized correlation. The primary goal in this paper is to suggest a simple method for testing the hypothesis that the Winsorized correlation is equal to zero. A minor goal is to describe a formal definition of the Winsorized correlation coefficient in terms of Winsorized expected values.
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We analyzed MAX-PC for normality and Winsorized outliers of >3 SD from the mean.
“We analyzed MAX-PC for normality and Winsorized outliers of >3 SD from the mean [21].”
From Brief report of a tablet-delivered psychosocial intervention for men with advanced prostate cancer: Acceptability and efficacy by race (2019) · cites this paper as [21]
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de Winter, J. C. F., Gosling, S. D., & Potter, J. (2016) · Psychological Methods · doi:10.1037/met0000079 · Cited by 1,201 · see citing works
“Alternative measures of association, such as the percentage bend correlation (Wilcox, 1994), the Winsorized correlation (Wilcox, 1993), and the Kendall’s tau rank correlation coefficient (rt), may be even more robust and efficient than rs (see Croux & Dehon, 2010).”
MethodStagl, J. M., Bouchard, L. C., Lechner, S. C., Blomberg, B. B., Gudenkauf, L. M., Jutagir, D. R., Glück, S., Derhagopian, R. P., Carver, C. S., & Antoni, M. H. (2015) · Cancer · doi:10.1002/cncr.29076 · Cited by 174 · see citing works
“were Winsorized to fall within 3 standard deviations of the mean.(20) All analyses controlled for baselines of the comparable measure of well-being: analyses of QOL controlled for initial QOL as assessed by the FACT-B, and analyses of depressive symptoms controlled for initial scores on the Hamilton Depression Rating…”
MethodGudenkauf, L. M., Antoni, M. H., Stagl, J. M., Lechner, S. C., Jutagir, D. R., Bouchard, L. C., Blomberg, B. B., Glück, S., Derhagopian, R. P., Giron, G. L., et al. (2015) · Journal of Consulting and Clinical Psychology · doi:10.1037/ccp0000020 · Cited by 108 · see citing works
“Variables with outliers greater than 3 standard deviations from the mean were winsorized (Wilcox, 1993).”
BackgroundThe Percentage Bend Correlation Coefficient
Wilcox, R. R. (1994) · Psychometrika · doi:10.1007/bf02294395 · Cited by 105 · see citing works
Self-citationMilrad, S. F., Hall, D. L., Jutagir, D. R., Lattie, E. G., Ironson, G. H., Wohlgemuth, W. K., Vera-Nunez, M. A., Garcia, L., Czaja, S. J., Perdomo, D. M., et al. (2016) · Journal of Neuroimmunology · doi:10.1016/j.jneuroim.2016.12.008 · Cited by 89 · see citing works
A One-Way Random Effects Model for Trimmed Means
Wilcox, R. R. (1994) · Psychometrika · doi:10.1007/bf02296126 · Cited by 69 · see citing works
Self-citationYanez, B. R., Oswald, L. B., Baik, S. H., Buitrago, D., Iacobelli, F., Perez-Tamayo, A. M., Guitelman, J., Penedo, F. J., & Buscemi, J. (2019) · Psycho-Oncology · doi:10.1002/pon.5281 · Cited by 48 · see citing works
“We screened our primary outcomemeasures for normality and Winsorized outliers greater than 3 standard deviations from the mean.(38) We calculated means, standard deviations, frequencies, t tests, and chi‐squared analyses to characterize the sample.”
MethodMarital interaction prior to parenthood predicts parent–child interaction 9 years later.
Tanner, L. R., & Bradbury, T. N. (2012) · Journal of Family Psychology · doi:10.1037/a0029051 · Cited by 28 · see citing works
Jutagir, D. R., Blomberg, B. B., Carver, C. S., Lechner, S. C., Timpano, K. R., Bouchard, L. C., Gudenkauf, L. M., Jacobs, J. M., Diaz, A., Lutgendorf, S. K., et al. (2017) · Breast Cancer Research and Treatment · doi:10.1007/s10549-017-4316-3 · Cited by 29 · see citing works
“Outliers C3 standard deviations outside the mean were winsorized [26] and variables were then analyzed for normal distribution (skewness \3.”
MethodA Simulation Based Comparison of Correlation Coefficients with Regard to Type I Error Rate and Power
Tuğran, E., Koçak, M. Ç., Mirtagioğlu, H., Yiğit, S., & Mendeş, M. (2015) · Journal of Data Analysis and Information Processing · doi:10.4236/jdaip.2015.33010 · Cited by 27 · see citing works
“Therefore, the observations are winsorizedat each end of both X and Y [5] [18].”
BackgroundBouchard, L. C., Yanez, B., Dahn, J. R., Flury, S. C., Perry, K. T., Mohr, D. C., & Penedo, F. J. (2019) · Translational Behavioral Medicine · doi:10.1093/tbm/iby089 · Cited by 24 · see citing works
“We analyzed MAX-PC for normality and Winsorized outliers of >3 SD from the mean [21].”
MethodOstojski, M. S., Gębala, J., Orlińska‐Woźniak, P., et al. (2016) · Ecological Modelling · doi:10.1016/j.ecolmodel.2016.04.004 · Cited by 21 · see citing works
My health smartphone intervention decreases daily fat sources among Latina breast cancer survivors
Buscemi, J., Oswald, L. B., Baik, S. H., Buitrago, D., Iacobelli, F., Phillips, S. M., Perez-Tamayo, A. M., Guitelman, J., Penedo, F. J., & Yanez, B. R. (2020) · Journal of Behavioral Medicine · doi:10.1007/s10865-020-00136-3 · Cited by 20 · see citing works
“We Winsorized outlier values > 3 standard deviations from the mean and screened the outcome measures for normality (Wilcox, 1993).”
MethodCesare, F. D., Luchinat, C., Tenori, L., & Saccenti, E. (2021) · The Journals of Gerontology Series A · doi:10.1093/gerona/glab335 · Cited by 19 · see citing works
“The Pearson’s correlation coefficient is then calculated on the Winsorized variables (20).”
MethodJacob, J., & Varadharajan, R. (2024) · Sankhya B · doi:10.1007/s13571-024-00342-y · Cited by 15 · see citing works
“Some of the methods included in the first category are Percent-age Bent Correlation (PBC) Wilcox (1994), Winsorized correlation (WINZ) Wilcox (1993) and Bi-weight Mid-correlation (BIW) Wilcox (2005).”
Stagl, J. M., Antoni, M. H., Lechner, S. C., Carver, C. S., & Lewis, J. E. (2013) · Psychology and Health · doi:10.1080/08870446.2013.843682 · Cited by 14 · see citing works
“One outlier for MVPA fell outside three standard deviations from the mean and was Winsorised (Wilcox, 1993) to fall within three standard deviations of the mean.”
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Jutagir, D. R., Gudenkauf, L. M., Stagl, J. M., Carver, C. S., Bouchard, L. C., Lechner, S. C., Glück, S., Blomberg, B. B., & Antoni, M. H. (2015) · Ethnicity and Health · doi:10.1080/13557858.2015.1066494 · Cited by 13 · see citing works
“5 or more standard deviations from the mean were winsorized (Wilcox 1993).”
BackgroundMethodInfluentialA Foundation for Robust Methods
Wilcox, R. R. (2012) · Elsevier eBooks · doi:10.1016/b978-0-12-386983-8.00002-0 · Cited by 11 · see citing works
Self-citationCorrelation and Tests of Independence
Wilcox, R. R. (2012) · Elsevier eBooks · doi:10.1016/b978-0-12-386983-8.00009-3 · Cited by 8 · see citing works
Self-citationRobust two-sample test of high-dimensional mean vectors under dependence
Wang, W., Lin, N., & Tang, X. (2018) · Journal of Multivariate Analysis · doi:10.1016/j.jmva.2018.09.013 · Cited by 6 · see citing works
Çek, D., Reese, H. E., Broos, H. C., & Timpano, K. R. (2022) · Mindfulness · doi:10.1007/s12671-022-01934-3 · Cited by 5 · see citing works
“This participant’s YBOCS and anxiety scores were replaced with the next highest score that is not an outlier, using a standard winsorizing approach (Field, 2015; Wilcox, 1993).”
MethodCorrelation and Tests of Independence
Wilcox, R. R. (2017) · Elsevier eBooks · doi:10.1016/b978-0-12-804733-0.00009-3 · Cited by 4 · see citing works
Self-citationAnalysis and Interpretation of Metabolite Associations Using Correlations
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Goodwin, G. J., Briley, D. A., Singsank, K. T., Tanner, D., Maloy-Robertson, M., & John, S. E. (2025) · Journal of the International Neuropsychological Society · doi:10.1017/s1355617724000730 · Cited by 2 · see citing works
“We Winsorized NPI-Q total symptom severity scores that were greater than three standard deviations from the mean to the next highest value (Wilcox, 1993).”
Robustness and monotonicity properties of generalized correlation coefficients
Chen, V. Y.-J., Chinchilli, V. M., & Richards, D. S. P. (2010) · Journal of Statistical Planning and Inference · doi:10.1016/j.jspi.2010.08.016 · Cited by 2 · see citing works
“3 Mixtures of bivariate normal distributions Departures from multivariate normality are well-known to arise naturally in applied work (e.g. Kowalski, 1972; Wilcox, 1993).”
BackgroundCorrelation and Tests of Independence
Wilcox, R. R. (2021) · Elsevier eBooks · doi:10.1016/b978-0-12-820098-8.00015-4 · Cited by 2 · see citing works
Self-citationBootstrapping Confidence Intervals For Robust Measures Of Association
King, J. E. (2003) · Journal of Modern Applied Statistical Methods · doi:10.22237/jmasm/1067646240 · Cited by 1 · see citing works
“Along with possessing properties that curb the influence of distributional anomalies, both the Winsorized correlation (Devlin et al., 1975; Gnanadesikan & Kettenring, 1972; Wilcox, 1993) and the percentage bend correlation (Wilcox, 1994) yield interpretations analogous to the popular Pearson r.”
BackgroundInfluentialA Foundation for Robust Methods
Wilcox, R. R. (2021) · Elsevier eBooks · doi:10.1016/b978-0-12-820098-8.00008-7 · Cited by 1 · see citing works
Self-citationA Foundation for Robust Methods
Wilcox, R. R. (2017) · Elsevier eBooks · doi:10.1016/b978-0-12-804733-0.00002-0 · Cited by 1 · see citing works
Self-citation(2012) · Elsevier eBooks · doi:10.1016/b978-0-12-386983-8.00012-3 · Cited by 1 · see citing works
Postsurgical physical activity and fatigue-related daily interference in women with breast cancer
Stagl, J. M., Antoni, M. H., Lechner, S. C., Carver, C. S., Lewis, J. E., & Gables, C. (2014)
(2021) · Elsevier eBooks · doi:10.1016/b978-0-12-820098-8.00019-1
Editors, J. (2003) · Journal of Modern Applied Statistical Methods · doi:10.22237/jmasm/1067644800
(2017) · Elsevier eBooks · doi:10.1016/b978-0-12-804733-0.00018-4
Influence of truthful and misleading instructions on statistical learning across the autism spectrumPreprint
Chrysaitis, N. A., & Seriès, P. (2024) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2024.07.12.603256
“Then we winsorised the data, meaning that we equated the outliers with the most extreme non-outlier values [25].”
MethodA meta-possibilistic fuzzy framework for robust bivariate correlation estimation
Karakullukçu, E. (2026) · Applied Soft Computing · doi:10.1016/j.asoc.2026.115411
- Authors
- Wilcox, R. R.
- Affiliations
- University of Southern California (US)
- Corresponding author
- Wilcox, Rand R.
- ORCID
- Wilcox, Rand R.
- Venue
- British Journal of Mathematical and Statistical Psychology
- Publisher
- Wiley
- Published
- 1993-11
- Volume
- 46
- Issue
- 2
- Pages
- 339-349
- ISSN
- 0007-1102 (ISSN-L)
- 2044-8317
- OpenAlex
- W2042507563
- Cited by
- 36
- References
- 0
- Citations by year
- 2026: 1 · 2025: 1 · 2024: 2 · 2023: 1 · 2022: 1 · 2021: 4 · 2020: 1 · 2019: 1 · 2018: 2 · 2017: 4 · 2016: 3 · 2015: 4 · 2014: 1 · 2013: 1 · 2012: 4
- FWCI
- 1.01
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- British Journal of Mathematical and Statistical Psychology (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Social Sciences › Decision Sciences › Statistics, Probability and Uncertainty › Multidisciplinary Science and Engineering Research
- Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
- Keywords
- robust correlation, heavy-tailed distributions, outliers, effect size measures, non-normality, psychometric measures
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