RRTCS (version 0.0.4)

Horvitz: Horvitz model

Description

Computes the randomized response estimation, its variance estimation and its confidence interval through the Horvitz model. The function can also return the transformed variable. The Horvitz model was proposed by Horvitz et al. (1967) and by Greenberg et al. (1969).

Usage

Horvitz(z,p,alpha,pi,type=c("total","mean"),cl,N=NULL,pij=NULL)

Arguments

z

vector of the observed variable; its length is equal to \(n\) (the sample size)

p

proportion of marked cards with the sensitive question

alpha

proportion of people with the innocuous attribute

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

Value

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

Details

In the Horvitz model, the randomized response device presents to the sampled person labelled \(i\) a box containing a large number of identical cards, with a proportion \(p,(0 <p < 1)\) bearing the mark \(A\) and the rest marked \(B\) (an innocuous attribute whose population proportion \(\alpha\) is known). The response solicited denoted by \(z_i\) takes the value \(y_i\) if \(i\) bears \(A\) and the card drawn is marked \(A\) or if \(i\) bears \(B\) and the card drawn is marked \(B\). Otherwise \(z_i\) takes the value 0.

The transformed variable is \(r_i=\frac{z_i-(1-p)\alpha}{p}\) and the estimated variance is \(\widehat{V}_R(r_i)=r_i(r_i-1)\).

References

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

HorvitzData

HorvitzDataStCl

HorvitzDataRealSurvey

HorvitzUB

SoberanisCruz

ResamplingVariance

Examples

# NOT RUN {
N=10777
data(HorvitzData)
dat=with(HorvitzData,data.frame(z,Pi))
p=0.5
alpha=0.6666667
cl=0.95
Horvitz(dat$z,p,alpha,dat$Pi,"mean",cl,N)

#Horvitz real survey
N=10777
n=710
data(HorvitzDataRealSurvey)
p=0.5
alpha=1/12
pi=rep(n/N,n)
cl=0.95
Horvitz(HorvitzDataRealSurvey$sex,p,alpha,pi,"mean",cl,N)
# }