Metric Methods for Analyzing Partially Ranked Data

I. Introduction and Outline.- II. Metrics on Fully Ranked Data.- A. Permutations: Some Important Conventions.- B. Metrics on Permutations: Discussion and Exampl es.- C. The Requirement of Right—Invariance.- III. Metrics on Partially Ranked Data: The Case where Each Judge Lists His k Favorite Items Out of n.- A. The Coset Space Sn/Sn-k.- B. The Hausdorff Metrics on Sn/Sn-k.- C. The Fixed Vector Metrics on Sn/Sn-k.- IV. Metrics on Other Types of Partially Ranked Data.- A. The Coset Space Sn/S, Where S = Sn1 ×Sn2 × …×Snr.- B. The Hausdorff Metrics on Sn/S.- C. The Fixed Vector Metrics on Sn/S.- D. Hausdorff Distances between Different Types of Partially Ranked Data: A Complete Proof of the Main Theorem.- E. The Tied Ranks Approach to Metrizing Partially Ranked Data.- 1. A Description of the Tied Ranks Approach.- 2. Relations among the Tied Ranks, Hausdorff, and Fixed Vector Metrics.- 3. Limitations of the Tied Ranks Approach.- V. Distributional Properties of the Metrics.- A. Exact Distributions.- B. Asymptotic Distributions.- VI. Data Analysis, Using the Metrics.- A. Fitting Probability Models to Partially Ranked Data.- 1. Mallows’ Model for Fully Ranked Data.- 2. The Extension of Mallows’ Model to Partially Ranked Data.- 3. A Likelihood Ratio Interpretation of the Triangle Inequality.- 4. Maximum Likelihood Estimation for the Model.- 5. A Goodness-of-Fit Result.- 6. An Example: The Educational Testing Service Word Association Data.- B. Multidimensional Scaling for Partially Ranked Data.- 1. An Example, Using Leann Lipps Birch’s Cracker Preference Data.- C. Two Sample Problems for Partially Ranked Data.- 1. A Two-Sample Test Based on the Minimal Spanni ng Tree.- 2. A Two-Sample Test Based on the Nearest Neighbors Graph.- Appendix A — The Existence Of Fixed Vectors.- Appendix C — Fortran Subroutines For Fitting Mallows’ Model To Partially Ranked Data.- Appendix E — Comparison Of Exact And Asymptotic Distributions.- Index Of Notation.