HorvitzUB {RRTCS}R Documentation

Horvitz-UB model

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 nn (the sample size)

J

second vector of the observed variable; its length is equal to nn (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\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 p1(0<p1<1)p_1(0<p_1<1) of them is marked AA and the complementary proportion (1p1)(1-p_1) each bearing the mark BB, while in the second box these proportions are p2p_2 and 1p21-p_2, maintaining p2p_2 different from p1p_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

Ii={1if card type draws "matches" the sensitive trait A or the innocuous trait B0if there is "no match" with the first box 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

Ji={1if there is "match" for the second box0if there is "no match" for the second boxJ_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 ri=(1p2)Ii(1p1)Jip1p2r_i=\frac{(1-p_2)I_i-(1-p_1)J_i}{p_1-p_2} and the estimated variance is V^R(ri)=ri(ri1)\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)

[Package RRTCS version 0.0.4 Index]