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Rank-Based Analysis of Linear Models and Beyond: A Review

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Robust Rank-Based and Nonparametric Methods

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 168))

Abstract

In the 1940s Wilcoxon, Mann and Whitney, and others began the development of rank based methods for basic one and two sample models. Over the years a multitude of papers have been written extending the use of ranks to more and more complex models. In the late 60s and early 70s Jurečková and Jaeckel along with others provided the necessary asymptotic machinery to develop rank based estimates in the linear model. Geometrically Jaeckel’s fit of linear model is the minimization of the distance between the vector of responses and the column space of the design matrix where the norm is not the squared-Euclidean norm but a norm that leads to robust fitting. Beginning with his 1975 thesis, Joe McKean has worked with many students and coauthors to develop a unified approach to data analysis (model fitting, inference, diagnostics, and computing) based on ranks. This approach includes the linear model and various extensions, for example multivariate models and models with dependent error structure such as mixed models, time series models, and longitudinal data models. Moreover, McKean and Kloke have developed R libraries to implement this methodology. This paper reviews the development of this methodology. Along the way we will illustrate the surprising ubiquity of ranks throughout statistics.

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References

  • Abebe, A., & McKean, J. W. (2007). Highly efficient nonlinear regression based on the Wilcoxon norm. In D. Umbach (Ed.), Festschrift in Honor of Mir Masoom Ali on the Occasion of his Retirement (pp. 340–357).

    Google Scholar 

  • Abebe, A., & McKean, J. W. (2013). Weighted Wilcoxon estimators in nonlinear regression. Australian & New Zealand Journal of Statistics, 55, 401–420.

    Article  MathSciNet  MATH  Google Scholar 

  • Abebe, A., McKean, J. W., Kloke, J. D., & Bilgic, Y. (2016). Iterated reweighted rank-based estimates for GEE models. In R. Y. Liu & J. W. McKean (Eds.), Robust rank-based and nonparametric methods. New York: Springer

    Google Scholar 

  • Arbuthnott, J. (1710). An argument for divine providence taken from the constant regularity observed in the birth of both sexes. Philosophical Transactions, 27, 186–190.

    Article  Google Scholar 

  • Chang, W., McKean, J. W., Naranjo, J. D., & Sheather, S. J. (1999). High breakdown rank-based regression. Journal of the American Statistical Association, 94, 205–219.

    Article  MathSciNet  MATH  Google Scholar 

  • Davis, J., & McKean, J. W. (1993). Rank based methods for multivariate linear Models. The Journal of the American Statistical Association, 88, 241–251

    MathSciNet  MATH  Google Scholar 

  • Hájek, J., & Šidák, Z. (1967). Theory of rank tests. New York: Academic.

    MATH  Google Scholar 

  • Hájek, J., Šidák, Z., & Sen, P. K. (1999). Theory of rank tests (2nd ed.). New York: Academic.

    MATH  Google Scholar 

  • Hampel, F. R. (1974). The influence curve and its role in robust estimation. Journal of the American Statistical Association, 69, 383–393.

    Article  MathSciNet  MATH  Google Scholar 

  • Hettmansperger, T. P., & McKean, J. W. (2011). Robust nonparametric statistical methods (2nd ed.). Boca Raton, FL: Chapman-Hall.

    MATH  Google Scholar 

  • Hodges, J. L., Jr., & Lehmann, E. L. (1956). The efficiency of some nonparametric competitors of the t-test. Annals of Mathematical Statistics, 27, 324–335.

    Article  MathSciNet  MATH  Google Scholar 

  • Hodges, J. L., Jr., & Lehmann, E. L. (1960). Comparison of the normal scores and Wilcoxon tests. In Proceedings 4th Berkeley Symposium (Vol. 1, pp. 307–317).

    Google Scholar 

  • Hodges, J. L., Jr., & Lehmann, E. L. (1963). Estimates of location based on rank tests. Annals of Mathematical Statistics, 34, 598–611.

    Article  MathSciNet  MATH  Google Scholar 

  • Hollander, M., & Wolfe, D. (1999). Nonparametric statistical methods (2nd ed.). New York: Wiley.

    MATH  Google Scholar 

  • Huber, P. J. (1964). Robust estimation of a location parameter. Annals of Mathematical Statistics, 35, 73–101.

    Article  MathSciNet  MATH  Google Scholar 

  • Jaeckel, L. A. (1972). Estimating regression coefficients by minimizing the dispersion of the residuals. Annals of Mathematical Statistics, 43, 1449–1458.

    Article  MathSciNet  MATH  Google Scholar 

  • Jurečková, J. (1969). Asymptotic linearity of rank statistics in regression parameters. Annals of Mathematical Statistics, 40, 1449–1458.

    Article  MathSciNet  Google Scholar 

  • Jurečková, J. (1971). Nonparametric estimate of regression coefficients. Annals of Mathematical Statistics, 42, 1328–1338.

    Article  MathSciNet  MATH  Google Scholar 

  • Kloke, J. D., & McKean, J. W. (2012). Rfit: Rank-based estimation for linear models. The R Journal, 4, 57–64.

    Google Scholar 

  • Kloke, J. D., & McKean, J. W. (2014). Nonparametric statistical methods using R. Boca Raton, FL: Chapman-Hall.

    Book  Google Scholar 

  • Kloke, J., McKean, J. W., & Rashid, M. (2009). Rank-based estimation and associated inferences for linear models with cluster correlated errors. Journal of the American Statistical Association, 104, 384–390.

    Article  MathSciNet  MATH  Google Scholar 

  • Koul, H. L., & Saleh, A. K. M. E. (1993). R-estimation of the parameters of autoregressive [AR(p)] models. The Annals of Statistics, 21, 534–551.

    Article  MathSciNet  MATH  Google Scholar 

  • Koul, H. L., Sievers, G. L., & McKean, J. W. (1987). An estimator of the scale parameter for the rank analysis of linear models under general score functions. Scandinavian Journal of Statistics, 14, 131–141.

    MathSciNet  MATH  Google Scholar 

  • Mann, H. B., & Whitney, D. R. (1947). On a test of whether one or two random variables is stochastically larger than the other. Annals of Mathematical Statistics, 18, 50–60.

    Article  MathSciNet  MATH  Google Scholar 

  • McKean, J. W., Jr. (1975). Tests of Hypotheses Based on Ranks in the General Linear Model. Ph.D. dissertation, University Park, Penn State University.

    Google Scholar 

  • McKean, J. W., & Hettmansperger, T. P. (1976). Tests of hypotheses of the general linear model based on ranks. Communications in Statistics, Part A-Theory and Methods, 5, 693–709.

    Article  MathSciNet  MATH  Google Scholar 

  • McKean, J. W., & Kloke, J. D. (2014). Efficient and adaptive rank-based fits for linear models with skewed-normal errors. Journal of Statistical Distributions and Applications, 1, 18. http://www.jsdajournal.com/content/1/1/18.

  • McKean, J. W., Naranjo, J. D., & Sheather, S. J. (1996). Diagnostics to detect differences in robust fits of linear models. Computational Statistics, 11, 223–243.

    MathSciNet  MATH  Google Scholar 

  • McKean, J. W., Naranjo, J. D., & Sheather, S. J. (1999). Diagnostics for comparing robust and least squares fits. Journal of Nonparametric Statistics, 11, 161–188.

    Article  MathSciNet  MATH  Google Scholar 

  • McKean, J. W., & Schrader, R. (1980). The geometry of robust procedures in linear models. Journal of the Royal Statistical Society, Series B, Methodological, 42, 366–371.

    MathSciNet  MATH  Google Scholar 

  • McKean, J. W., & Sheather, S. J. (1991). Small sample properties of robust analyses of linear models based on r-estimates. In W. Stahel & S. Weisberg (Eds.), Directions in robust statistics and diagnostics, part II (Vols. 1–20). New York: Springer.

    Chapter  Google Scholar 

  • McKean, J. W., Sheather, S. J., & Hettmansperger, T. P. (1990). Regression diagnostics for rank-based methods, Journal of the American Statistical Association, 85, 1018–1028.

    Article  Google Scholar 

  • Noether, G. E. (1955). On a theorem of Pitman. Annals of Mathematical Statistics, 26, 64–68.

    Article  MathSciNet  MATH  Google Scholar 

  • Nordhausen, K., & Oja, H. (2011). Multivariate L1 methods: The package MNM. Journal of Statistical Software, 43(5), 1–28. http://www.jstatsoft.org/v43/i05/.

  • Oja, H. (2010). Multivariate nonparametric methods with R. New York: Springer.

    Book  MATH  Google Scholar 

  • Pitman, E. J. G. (1948). Notes on nonparametric statistical inference (Unpublished notes).

    Google Scholar 

  • Rousseeuw, P., & Van Driessen, K. (1999). A fast algorithm for the minimum covariance determinant estimator. Technometrics, 41, 212–223.

    Article  Google Scholar 

  • Terpstra, J., McKean, J. W., & Naranjo, J. D. (2000). Highly efficient weighed Wilcoxon estimates for autoregression. Statistics, 35, 45–80.

    Article  MathSciNet  MATH  Google Scholar 

  • Terpstra, J., McKean, J. W., & Naranjo, J. D. (2001). GR-estimates for an autoregressive time series. Statistics and Probability Letters, 51, 172–180.

    Article  MathSciNet  MATH  Google Scholar 

  • Tukey, J. W. (1949). The simplest signed rank tests. Princeton University Stat. Res. Group, Memo Report no. 17.

    Google Scholar 

  • Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics, 1, 80–83.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Statistics, Western Michigan University, Kalamazoo, MI, 49008, USA

    Joseph W. McKean

  2. Department of Statistics, Penn State University, University Park, PA, 16002, USA

    Thomas P. Hettmansperger

Authors
  1. Joseph W. McKean
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  2. Thomas P. Hettmansperger
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Corresponding author

Correspondence to Joseph W. McKean .

Editor information

Editors and Affiliations

  1. Department of Statistics, Rutgers University, New Brunswick, New Jersey, USA

    Regina Y. Liu

  2. Department Statistics, Western Michigan University, Kalamazoo, Michigan, USA

    Joseph W. McKean

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McKean, J.W., Hettmansperger, T.P. (2016). Rank-Based Analysis of Linear Models and Beyond: A Review. In: Liu, R., McKean, J. (eds) Robust Rank-Based and Nonparametric Methods. Springer Proceedings in Mathematics & Statistics, vol 168. Springer, Cham. https://doi.org/10.1007/978-3-319-39065-9_1

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