Computes the randomized response estimation, its variance estimation and its confidence interval through the Horvitz model (Horvitz et al., 1967, and Greenberg et al., 1969) when the proportion of people bearing the innocuous attribute is unknown. The function can also return the transformed variable. The Horvitz-UB model can be seen in Chaudhuri (2011, page 42).
HorvitzUB(I,J,p1,p2,pi,type=c("total","mean"),cl,N=NULL,pij=NULL)first vector of the observed variable; its length is equal to \(n\) (the sample size)
second vector of the observed variable; its length is equal to \(n\) (the sample size)
proportion of marked cards with the sensitive attribute in the first box
proportion of marked cards with the sensitive attribute in the second box
vector of the first-order inclusion probabilities
the estimator type: total or mean
confidence level
size of the population. By default it is NULL
matrix of the second-order inclusion probabilities. By default it is NULL
Point and confidence estimates of the sensitive characteristics using the Horvitz-UB model. The transformed variable is also reported, if required.
In the Horvitz model, when the population proportion \(\alpha\) is not known, two independent samples are taken. Two boxes are filled with a large number of similar cards except that in the first box a proportion \(p_1(0<p_1<1)\) of them is marked \(A\) and the complementary proportion \((1-p_1)\) each bearing the mark \(B\), while in the second box these proportions are \(p_2\) and \(1-p_2\), maintaining \(p_2\) different from \(p_1\). A sample is chosen and every person sampled is requested to draw one card randomly from the first box and to repeat this independently with the second box. In the first case, a randomized response should be given, as $$I_i=\left\{\begin{array}{lcc} 1 & \textrm{if card type draws "matches" the sensitive trait } A \textrm{ or the innocuous trait } B \\ 0 & \textrm{if there is "no match" with the first box } \end{array} \right.$$ and the second case given a randomized response as $$J_i=\left\{\begin{array}{lcc} 1 & \textrm{if there is "match" for the second box} \\ 0 & \textrm{if there is "no match" for the second box} \end{array} \right.$$ The transformed variable is \(r_i=\frac{(1-p_2)I_i-(1-p_1)J_i}{p_1-p_2}\) and the estimated variance is \(\widehat{V}_R(r_i)=r_i(r_i-1)\).
Chaudhuri, A. (2011). Randomized response and indirect questioning techniques in surveys. Boca Raton: Chapman and Hall, CRC Press.
Greenberg, B.G., Abul-Ela, A.L., Simmons, W.R., Horvitz, D.G. (1969). The unrelated question RR model: Theoretical framework. Journal of the American Statistical Association, 64, 520-539.
Horvitz, D.G., Shah, B.V., Simmons, W.R. (1967). The unrelated question RR model. Proceedings of the Social Statistics Section of the American Statistical Association. 65-72. Alexandria, VA: ASA.
# NOT RUN {
N=802
data(HorvitzUBData)
dat=with(HorvitzUBData,data.frame(I,J,Pi))
p1=0.6
p2=0.7
cl=0.95
HorvitzUB(dat$I,dat$J,p1,p2,dat$Pi,"mean",cl,N)
# }
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