部分排名資料的無母數檢定方法

A Nonparametric Procedure for Testing Partially Ranked Data

謝旻宜(Min-Yi Hsieh)

指導教授 ： 伍志祥

淡江大學/理學院/數學學系碩士在職專班/碩士(2018年)

https://doi.org/10.6846/TKU.2018.00690

摘要

進行消費者喜好度調查時，受訪者往往受限於時間、注意力等因素，難以完整評比出所有選項。而發起調查的機構，比起「所有選項皆有分數」這件事，他們可能更在意哪些選項「最受人喜愛(排名在前段)」或「最不受人喜愛(排名在後段)」。這指出了一個方便且可行的評比方法-部分排名。在本文中，我們修改喜好度調查方式，請每位受訪者評比N個不同選項時，只須選出他偏好的前k名，收集n名受訪者的調查結果，此類型的資料被我們稱為「部分排名的資料」。我們將其餘未列入前k名的產品併列為第k+1名，則部分排名的資料可按照選項編號（共N個）及排名次序（共k+1名）整理成一個N×(k+1)的列聯表。我們擴展Anderson (1959)及Wu & Deng (2017)的程序，根據上述的N×(k+1)列聯表，提出一個建議統計量，與Anderson檢定統計量相比，在顯著水準α=0.05時，我們建議的統計量檢定力較佳。

關鍵字

Anderson檢定 ； 卡方檢定 ； 列聯表 ； 部分排名資料 ； 喜好度調查

並列摘要

In consumer preference survey, it is difficult to rate all alternatives, because respondents are often limited by factors such as time, attention, etc. The institution that initiated the survey may be more concern about which alternatives are “popular (ranked top)” or “unpopular (ranked bottom)”, compare to “all alternatives have ranked”. This points to a convenient and feasible method --- partially ranked survey. In this paper, we propose a modified preference survey that each respondent ranks only her/his most preferred k out of N alternatives, the survey result of n respondents is referred to as “partially ranked data”. In our procedure, the alternatives which are not ranked as top k receive the (k+1)-th rank imposed by the investigator. The partially ranked data is constructed as an N×(k+1) contingency table, according to N alternatives and (k+1) ranks. We extend the procedures of Anderson (1959) and Wu & Deng (2017), and propose a test statistic based on the N×(k+1) contingency table. Compare with the Anderson's test, we prove the power of our proposed test is better at significance level α=0.05.

並列關鍵字

Anderson’ test ； chi-square test ； contingency table ； partially ranked data ； preference survey

參考文獻

[1] Anderson, R.L. (1959), Use of contingency tables in the analysis of consumer preference studies, Biometrics, 15, 582-590.

[2] Best, D.J. (1993), Extended analysis for ranked data, Australian Journal of Statistics, 35, 257-262. (doi: 10.1111/j.1467-842X.1993.tb01333.x)

[3] Best, D.J.; Rayner, J.C.W. (1997), Product maps for ranked preference data, The Statistician, 46, 347-354. (doi: 10.1111/1467-9884.00088)

[4] Best, D.J. (1999), Test of fit and other nonparametric data analysis, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong. Retrieved from http://ro.uow.edu.au/theses/2061.

[5] Brockhoff, P.B.; Best, D.J. and Rayner, J.C.W. (2004a), Partitioning Anderson's statistic for tied data, Journal of Statistical Planning and Inference, 121, 93-111. (doi: 10.1016/ S0378-3758(02)00508-6)