HorvitzUB: Horvitz-UB model

In RRTCS: Randomized Response Techniques for Complex Surveys

Description

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).

Usage

HorvitzUB(I,J,p1,p2,pi,type=c("total","mean"),cl,N=NULL,pij=NULL)

Arguments

I	first vector of the observed variable; its length is equal to n (the sample size)
J	second vector of the observed variable; its length is equal to n (the sample size)
p1	proportion of marked cards with the sensitive attribute in the first box
p2	proportion of marked cards with the sensitive attribute in the second box
pi	vector of the first-order inclusion probabilities
type	the estimator type: total or mean
cl	confidence level
N	size of the population. By default it is NULL
pij	matrix of the second-order inclusion probabilities. By default it is NULL

Details

In the Horvitz model, when the population proportion α 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=≤ft\{\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=≤ft\{\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).

Value

Point and confidence estimates of the sensitive characteristics using the Horvitz-UB model. The transformed variable is also reported, if required.

References

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.

See Also

HorvitzUBData

Horvitz

ResamplingVariance

Examples

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)

RRTCS documentation built on April 21, 2021, 9:06 a.m.