RRTCS (version 0.0.4)

Saha: Saha model

Description

Computes the randomized response estimation, its variance estimation and its confidence interval through the Saha model. The function can also return the transformed variable. The Saha model was proposed by Saha in 2007.

Usage

Saha(z,mu,sigma,pi,type=c("total","mean"),cl,N=NULL,method="srswr")

Arguments

z

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

mu

vector with the means of the scramble variables \(W\) and \(U\)

sigma

vector with the standard deviations of the scramble variables \(W\) and \(U\)

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

method

method used to draw the sample: srswr or srswor. By default it is srswr

Value

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

Details

In the Saha model, each respondent selected is asked to report the randomized response \(z_i=W(y_i+U)\) where \(W,U\) are scramble variables whose distribution is assumed to be known.

To estimate \(\bar{Y}\) a sample of respondents is selected according to simple random sampling with replacement. The transformed variable is $$r_i=\frac{z_i-\mu_W\mu_U}{\mu_W}$$ where \(\mu_W,\mu_U\) are the means of \(W,U\) scramble variables respectively

The estimated variance in this model is $$\widehat{V}(\widehat{\bar{Y}}_R)=\frac{s_z^2}{n\mu_W^2}$$ where \(s_z^2=\sum_{i=1}^n\frac{(z_i-\bar{z})^2}{n-1}\).

If the sample is selected by simple random sampling without replacement, the estimated variance is $$\widehat{V}(\widehat{\bar{Y}}_R)=\frac{s_z^2}{n\mu_W^2}\left(1-\frac{n}{N}\right)$$

References

Saha, A. (2007). A simple randomized response technique in complex surveys. Metron LXV, 59-66.

See Also

SahaData

ResamplingVariance

Examples

# NOT RUN {
N=228
data(SahaData)
dat=with(SahaData,data.frame(z,Pi))
mu=c(1.5,5.5)
sigma=sqrt(c(1/12,81/12))
cl=0.95
Saha(dat$z,mu,sigma,dat$Pi,"mean",cl,N)
# }