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FinchModern methods for the detection of multivariate outliersmight well yield statistical results that continue to be influencedby the presence of outliers. Thus, other methods described hereshould be considered as viable options when multivariate outliersare present. In the final analysis, such an approach mustbe based on the goals of the data analysis and the study as awhole. The removal of outliers, when done, must be carriedout thoughtfully and with purpose so that the resulting datasetis both representative of the population of interest and usefulwith the appropriate statistical tools to address the researchquestions.REFERENCESBrown, T. A. (2006). Confirmatory FactorAnalysis for Applied Research.New York: The Guilford Press.Donoho, D. L., and Gasko, M. (1992).Breakdown properties of the locationestimates based on halfspacedepth and projected outlyingness.Ann. Stat. 20, 1803–1827.Evans, V. P. 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(1988). Miracles and statistics:the causal assumption of independence.J. Am. Stat. Assoc. 83,929–940.Mahalanobis, P. C. (1936). On the generalizeddistance in statistics. Proc.Indian Natl. Sci. Acad. B Biol. Sci. 2,49–55.Marascuilo, L. A., and Serlin, R. C.(1988). Statistical Methods for theSocial and Behavioral Sciences. NewYork: W. H. Freeman.Maronna, R., Martin, D., and Yohai, V.(2006). Robust Statistics: Theory andMethods. Hoboken, NJ: John Wiley& Sons, Inc.Mourão-Miranda, J., Hardoon, D. R.,Hahn, T., Marquand, A. F., Williams,S. C. R., Shawe-Taylor, J., and Brammer,M. (2011). Patient classificationas an outlier detection problem: anapplication of the one-class supportvector machine. Neuroimage 58,793–804.Osborne, J. W., and Overbay, A. (2004).The power of outliers (and whyresearchers should always check forthem). Pract. Assess. Res. Eval. 9.Available at: http://PAREonline.net/getvn.asp?v=9&n=6Pedhazur, E. J. (1997). 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Fundamentals ofModern Statistical Methods: SubstantiallyImproving Power and Accuracy.New York: Springer.Zijlstra, W. P., van der Ark, L. A., andSijtsmal, K. (2007). Robust Mokkenscale analysis by means of a forwardsearch algorithm for outlier detection.Multivariate Behav. Res. 46,58–89.Zijlstra, W. P., van der Ark, L. A., andSijtsmal, K. (2011). Outliers in questionnairedata: can they be detectedand should they be removed? J. Educ.Behav. Stat. 36, 186–212.Conflict of Interest Statement: Theauthor declares that the research wasconducted in the absence of any commercialor financial relationships thatcould be construed as a potential conflictof interest.Received: 20 December 2011; paper pendingpublished: 17 January 2012; accepted:06 June 2012; published online: 05 July2012.Citation: Finch WH (2012) Distributionof variables by method of outlierdetection. Front. Psychology 3:211. doi:10.3389/fpsyg.2012.00211This article was submitted to Frontiersin Quantitative Psychology and Measurement,a specialty of Frontiers in Psychology.Copyright © 2012 Finch. This is an openaccessarticle distributed under the termsof the Creative Commons AttributionLicense, which permits use, distributionand reproduction in other forums, providedthe original authors and sourceare credited and subject to any copyrightnotices concerning any third-partygraphics etc.www.frontiersin.org July 2012 | Volume 3 | Article 211 | 69
FinchModern methods for the detection of multivariate outliersAPPENDIXlibrary(MASS)mahalanobis.out < -mahalanobis(full.data,colMeans(full.data),cov(full.data))mcd.output < -cov.rob(full.data,method =“mcd,” nsamp =“best”)mcd.keep < -full.data[mcd.output$best,]mve.output < -cov.rob(full.data,method =“mve,” nsamp =“best”)mve.keep < -full.data[mve.output$best,]mgv.output < -outmgv(full.data,y = NA,outfun = outbox)mgv.keep < -full.data[mgv.output$keep,]projection1.output < -outpro(full.data,cop = 2)projection1.keep < -full.data[projection1.output$keep,]projection2.output < -outpro(full.data.matrix,cop = 3)projection2.keep < -full.data[projection2.output$keep,]projection3.output < -outpro(full.data,cop = 4)projection3.keep < -full.data[projection3.output$keep,]∗ Note that for the projection methods, the center of the distributionmay be determined using one of four possible approaches.The choice of method is specified in the cop command. For thisstudy, three of these were used, including MCD (cop = 2), medianof the marginal distributions (cop = 3), and MVE (cop = 4).∗∗ The functions for obtaining the mahalanobis distance(mahalanobis) and MCD/MVE (cov.rob) are part of the MASSlibrary in R, and will be loaded when it is called. The outmgvand outpro functions are part of a suite of functionswritten by Rand Wilcox with information for obtaining themavailable at his website (http://dornsife.usc.edu/cf/faculty-andstaff/faculty.cfm?pid= 1003819&CFID = 259154&CFTOKEN =86756889) and his book, Introduction to Robust Estimation andHypothesis Testing.Frontiers in Psychology | Quantitative Psychology and Measurement July 2012 | Volume 3 | Article 211 | 70
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