A Bayesian Estimation Study of the Sensitive Character Proportion in Stratified Randomized Randomized Response Model
許多研究者在估計母體具敏感性特質A 的比例π時，受訪者常因不想承認具敏感性特質A，而拒絕合作或給予不真實的回答，使調查結果有所偏差。Stanley Warner提出隨機作答模式（randomized response model, RRM），這種模式能保護受訪者的隱私，亦可增加受訪者的合作性，故較願意提供誠實的答案。但Warner和之後的相關文獻都專注於最大概似法的應用，而忽視受訪者針對RRM所回答「是」的比例θ應限制於參數空間［1－P,P］之內（P為隨機器之設定機率），因此造成π的估計值可能為負值或大於1。Robert Winkler與Leroy Franklin提出針對RRM使用範圍為［0, 1］的非共軛事前Beta分配的貝氏估計法。Shaul Bar-Lev等學者針對一些RRMs使用共軛事前Beta分配的貝氏估計法。本文將貝氏法應用於王智立與蔡宛容之RRM，和本文所提之兩階段RRM，兩個模式並採用Jong-Min Kim與William Warde分層RRM的觀念，再以適當的截切Beta分配當作共軛事前分配獲得比例π的貝氏估計量。結果顯示本文所提的貝氏估計量可改善最大概似法之估計結果不在參數空間內的缺點。此外，本文所提之兩階段RRM與其他模式進行比較有更好的估計效率。
Many researchers are faced with the problem of estimating the sensitive characteristic proportion π of a human population that has a particular characteristic A. Often reluctant to admit to having this particular characteristic A, respondents may refuse to participate or provide untruthful answers, which causes bias in study results. Stanley Warner first proposed the Randomized Response Model (RRM), which protects the respondent's privacy while simultaneously increasing both their cooperation in the survey and their willingness to provide an honest answer. However, Warner's study and related subsequent literature all focus on the application of the maximum likelihood method and ignore the fact that the ratio θ of the respondents' "yes" response in the model should be restricted within the parameter space [1－P, P], where P is the probability of the randomized device, resulting in the possibility of the estimated value of π being negative or greater than one. Robert Winkler and Leroy Franklin proposed a Bayesian approach for Warner's model using a non-conjugate prior Beta distribution for π over [0, 1]. Shaul Bar-Lev et al. (2003) presented a conjugate prior Beta distribution to some RRMs. The aim of this study is to apply the Bayesian approach to the RRM provided by Chih-li Wang and Wang-jung Tsai, coupled with a two-stage randomized response model, which uses the concept of stratified randomized response model proposed by Jong-Min Kim and William Warde and suitable truncated Beta distributions in a common conjugate prior structure to obtain the Bayes estimates for the proportion of a "sensitive character A" in the population of interest. Final results reveal that the Bayesian estimator proposed by this study can improve the drawback of the absence of the maximum likelihood method results in the parameter space. In addition, results show that the RRM proposed by this study will have better estimation efficiency when compared to other models.