隨機作答模式之邏輯斯迴歸參數估計

Logistic Regression Analysis of Randomized Response Data

謝淑惠(Shu-Hui Hsieh) ； 彭德昭(Ter-Chao Peng) ； 李燊銘(Shen-Ming Lee)

《智慧科技與應用統計學報》 8卷2期 (2010/12) Pp. 69-80

https://doi.org/10.29807/JTITAS.201012.0005

摘要

調查研究中牽涉到『敏感性問題』時，一般建議採用『隨機作答模式』來降低不誠實回答的可能性，並得到較為精確的推論。本文藉由Greenberg et al.(1969)所提出不相關問題之隨機作答設計並結合Scheer and Dayton (1988)之敏感性問題邏輯斯迴歸模型來估計迴歸的參數。在此，我們將藉由潛在隨機選題之變數來引出邏輯斯迴歸的估計方程式，並藉由EM演算法來估計迴歸參數。我們也可以直接引出最大概似估計法之估計方程式，同時證明利用潛在隨機選題之變數的觀念，以條件期望值替代此變數所得到之迴歸參數估計與最大概似估計法直接估計結果是相同的。另外，我們也將比較Warner (1965)所提出相關問題之隨機作答模式與Greenberg et al.(1969)所提出不相關問題之隨機作答模式於參數間的關係，並證明Warner (1965)可視為Greenberg et al.(1969)之特殊情況。最後，將所提出之方法應用於有線電視收視戶的調查資料，以分析與評估台北縣、台中市及雲林縣的私接率。實證結果證明不同縣市存在不同的私接率，且有線電視的系統商提供收視費率的優惠會顯著降低收視戶的私接動機。

關鍵字

隨機作答模式 ； 敏感性問題 ； 邏輯斯迴歸 ； 最大概似估計 ； EM演算法

並列摘要

The randomized response technique (RRT) is survey procedure that requires respondents to decide randomly whether to respond to a sensitive question in its positive or negative form (Warner 1965) or respond to a sensitive question and an unrelated (Greenberg et al. 1965). Under RRT, a respondent privacy can be protected, the tendency to refuse co-operation or to give non-incriminating or socially acceptable answers will decrease and thus the validity of the data will increase. Scheers and Dayton (1988) presented theory for a covariate randomized response model that is an extension of the Warner (1965) procedure and for a covariate extension of the unrelated-question RRT (Greenberg et al. 1969). In this article, we present logistic regression analysis of data from unrelated-question RRT (Greenberg et al. 1965) and theory for estimate parameters in latent variable. We compare the relationship of related-question RRT (Warner 1965) and unrelated-question (Greenberg et al. 1965), and present the unrelated-question RRT (Greenberg et al. 1965) can be extended the related-question RRT proposed by Warner (1965). We also apply the method to estimate the rate of illegal cable TV viewers of Taipei county, Taichung city and Yunlin county. Factors that can have impact on decreasing the rate also be examined. The results show subscription fare discount can significantly reduce the rate of illegal cable TV viewers.

並列關鍵字

randomized response technique ； sensitive question ； logistic regression ； maximum likelihood method ； EM algorithm