Tests for Trend and Association

  • Mayer Alvo
  • Philip L. H. Yu
Part of the Springer Series in the Data Sciences book series (SSDS)


In this chapter, we consider additional applications of the smooth model paradigm described earlier in Chapter  4 We begin by considering tests for trend. We then proceed with the study of the one-sample test for a randomized block design. We obtain a different proof of the asymptotic distribution of Friedman’s statistic based on Alvo (2016) who developed a likelihood function approach for the analysis of ranking data. Further, we derive a test statistic for the two-sample problem as well as for problems involving various two-way experimental designs. We exploit the parametric paradigm further by introducing the use of penalized likelihood in order to gain further insight into the data. Specifically, if judges provide rankings of t objects, penalized likelihood enables us to focus on those objects which exhibit the greatest differences.


  1. Alvo, M. (2016). Bridging the gap: a likelihood: function approach for the analysis of ranking data. Communications in Statistics - Theory and Methods, Series A, 45:5835–5847.MathSciNetCrossRefGoogle Scholar
  2. Alvo, M. and Cabilio, P. (1991). On the balanced incomplete block design for rankings. The Annals of Statistics, 19:1597–1613.MathSciNetCrossRefGoogle Scholar
  3. Alvo, M. and Cabilio, P. (1994). Rank test of trend when data are incomplete. Environmetrics, 5:21–27.CrossRefGoogle Scholar
  4. Alvo, M. and Cabilio, P. (1998). Applications of Hamming distance to the analysis of block data. In Szyszkowicz, B., editor, Asymptotic Methods in Probability and Statistics: A Volume in Honour of Miklós Csörgõ, pages 787–799. Elsevier Science, Amsterdam.CrossRefGoogle Scholar
  5. Alvo, M. and Cabilio, P. (1999). A general rank based approach to the analysis of block data. Communications in Statistics: Theory and Methods, 28:197–215.MathSciNetCrossRefGoogle Scholar
  6. Alvo, M., Cabilio, P., and Feigin, P. (1982). Asymptotic theory for measures of concordance with special reference to Kendall’s tau. The Annals of Statistics, 10:1269–1276.MathSciNetCrossRefGoogle Scholar
  7. Alvo, M. and Xu, H. (2017). The analysis of ranking data using score functions and penalized likelihood. Austrian Journal of Statistics, 46:15–32.CrossRefGoogle Scholar
  8. Alvo, M. and Yu, P. L. H. (2014). Statistical Methods for Ranking Data. Springer.Google Scholar
  9. Anderson, R. (1959). Use of contingency tables in the analysis of consumer preference studies. Biometrics, 15:582–590.CrossRefGoogle Scholar
  10. Casella, G. and George, E. I. (1992). Explaining the Gibbs sampler. The American Statistician, 46:167–174.MathSciNetGoogle Scholar
  11. Feigin, P. D. and Alvo, M. (1986). Intergroup diversity and concordance for ranking data: an approach via metrics for permutations. The Annals of Statistics, 14:691–707.MathSciNetCrossRefGoogle Scholar
  12. Friedman, M. (1937). The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American Statistical Association, 32:675–701.CrossRefGoogle Scholar
  13. John, J. and Williams, E. (1995). Cyclic Designs. Chapman Hall, New York.CrossRefGoogle Scholar
  14. Kannemann, K. (1976). An incidence test for k related samples. Biometrische Zeitschrift, 18:3–11.MathSciNetzbMATHGoogle Scholar
  15. McCullagh, P. (1993). Models on spheres and models for permutations. In Fligner, M. A. and Verducci, J. S., editors, Probability Models and Statistical Analyses for Ranking Data, pages 278–283. Springer-Verlag.Google Scholar
  16. Schach, S. (1979). An alternative to the Friedman test with certain optimality properties. Ann. Statist., 7(3):537–550.MathSciNetCrossRefGoogle Scholar
  17. Yu, P. L. H., Lam, K. F., and Alvo, M. (2002). Nonparametric rank test for independence in opinion surveys. Austrian Journal of Statistics, 31:279–290.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Mayer Alvo
    • 1
  • Philip L. H. Yu
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  2. 2.Department of Statistics and Actuarial ScienceUniversity of Hong KongHong KongChina

Personalised recommendations