The Analysis of Ranking Data Using Score Functions and Penalized Likelihood

Authors

  • Mayer Alvo Department of Mathematics and Statistics University of Ottawa
  • Hang Xu University of Hong Kong

DOI:

https://doi.org/10.17713/ajs.v46i1.133

Abstract

In this paper, we consider different score functions in order summarize certain characteristics for one and two sample ranking data sets. Our approach is flexible and is based on embedding the nonparametric problem in a parametric framework. We make use of the von Mises-Fisher distribution to approximate the normalizing constant in our model. In order to gain further insight in the data, we make use of penalized likelihood to narrow down the number of items where the rankers differ. We applied our method on various real life data sets and we conclude that our methodology is consistent with the data.

Author Biography

Hang Xu, University of Hong Kong

Department of Statistics and Actuarial Science

References

Aslam JA, Montague M (2001). Models for metasearch. Proceedings of the 24th annual international ACM SIGIR conference on Research and development in information retrieval, pp. 276--284. ACM.

Baba Y (1986). Graphical analysis of rank data.

Behaviormetrika, 19, 1--15.

Broyden CG (1970). The convergence of a class of double-rank minimization algorithms 1. general considerations.

IMA Journal of Applied Mathematics, 6 (1), 76--90.

Byrd RH, Gilbert JC, Nocedal J (2000). A trust region method based on interior point techniques for nonlinear programming.

Mathematical Programming, 89 (1), 149--185.

Byrd RH, Hribar ME, Nocedal J (1999). An interior point algorithm for large-scale nonlinear programming.

SIAM Journal on Optimization, 9 (4), 877--900.

Cohen A, Mallows C (1980). Analysis of ranking data.

Technical memorandum, AT&T Bell Laboratories, Murray Hill, N.J.

Coleman T, Li Y (1994). On the Convergence of Reflective Newton Methods for Large-scale Nonlinear Minimization Subject to Bounds vol. 67.

Ithaca, NY, USA: Cornell University.

Coleman TF, Li Y (1996). An interior trust region approach for nonlinear minimization subject to bounds.

SIAM Journal on optimization, 6 (2), 418--445.

Critchlow DE, Fligner MA, Verducci JS (1991). Probability models on rankings.

Journal of Mathematical Psychology, 35 (3), 294--318.

Croon MA (1989). Latent class models for the analysis of rankings.

Advances in psychology, 60, 99--121.

DeConde RP, Hawley S, Falcon S, Clegg N, Knudsen B, Etzioni R (2006). Combining results of microarray experiments: a rank aggregation approach.

Statistical Applications in Genetics and Molecular Biology, 5 (1).

Deng K, Han S, Li KJ, Liu JS (2014). Bayesian aggregation of order-based rank data.

Journal of the American Statistical Association, 109 (507), 1023--1039.

Dwork C, Kumar R, Naor M, Sivakumar D (2001). Rank aggregation methods for the web.

In Proceedings of the 10th international conference on World Wide Web, pp. 613--622. ACM.

Feigin PD, Cohen A (1978). On a model for concordance between judges.

Journal of the Royal Statistical Society Series B, 40, 203--213.

Fletcher R (1970). A new approach to variable metric algorithms.

The computer journal, 13 (3), 317--322.

Fok D, Paap R, Van Dijk B (2012). A Rank-Ordered Logit Model With Unobserved Heterogeneity In Ranking Capabilities.

Journal of applied econometrics, 27 (5), 831--846.

Goldfarb D (1970). A family of variable-metric methods derived by variational means.

Mathematics of computation, 24 (109), 23--26.

Kemeny JG, Snell JL (1962). Mathematical models in the social sciences, volume 9.

Ginn New York.

Lee PH, Philip L (2012). Mixtures of weighted distance-based models for ranking data with applications in political studies.

Computational Statistics & Data Analysis, 56 (8), 2486--2500.

Mallows CL (1957). Non-null ranking models. I.

Biometrika, 44, 114--130.

McCullagh P (1993). Models on spheres and models for permutations.

In MA Fligner, JS Verducci (eds.), Probability Models and Statistical Analyses for Ranking Data, pp. 278--283. Springer-Verlag.

Neyman J (1937). Smooth test for goodness of fit.

Skandinavisk Aktuarietidskrift, 20, 149--199.

Shanno DF (1970). Conditioning of quasi-Newton methods for function minimization.

Mathematics of computation, 24 (111), 647--656.

Waltz RA, Morales JL, Nocedal J, Orban D (2006). An interior algorithm for nonlinear optimization that combines line search and trust region steps.

Mathematical Programming, 107 (3), 391--408.

Downloads

Published

2017-01-04

How to Cite

Alvo, M., & Xu, H. (2017). The Analysis of Ranking Data Using Score Functions and Penalized Likelihood. Austrian Journal of Statistics, 46(1), 15–32. https://doi.org/10.17713/ajs.v46i1.133

Issue

Section

Articles