Computes the randomized response estimation, its variance estimation and its confidence interval through the Diana-Perri-1 model. The function can also return the transformed variable. The Diana-Perri-1 model was proposed by Diana and Perri (2010, page 1877).
DianaPerri1(z,p,mu,pi,type=c("total","mean"),cl,N=NULL,method="srswr")vector of the observed variable; its length is equal to \(n\) (the sample size)
probability of direct response
vector with the means of the scramble variables \(W\) and \(U\)
vector of the first-order inclusion probabilities
the estimator type: total or mean
confidence level
size of the population. By default it is NULL
method used to draw the sample: srswr or srswor. By default it is srswr
Point and confidence estimates of the sensitive characteristics using the Diana-Perri-1 model. The transformed variable is also reported, if required.
In the Diana-Perri-1 model let \(p\in [0,1]\) be a design parameter, controlled by the experimenter, which is used to randomize the response as follows: with probability \(p\) the interviewer responds with the true value of the sensitive variable, whereas with probability \(1-p\) the respondent gives a coded value, \(z_i=W(y_i+U)\) 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 $$r_i=\frac{z_i-(1-p)\mu_W\mu_U}{p+(1-p)\mu_W}$$ where \(\mu_W,\mu_U\) are the means of \(W,U\) scramble variables, respectively.
The estimated variance in this model is $$\widehat{V}(\widehat{\bar{Y}}_R)=\frac{s_z^2}{n(p+(1-p)\mu_W)^2}$$ 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 $$\widehat{V}(\widehat{\bar{Y}}_R)=\frac{s_z^2}{n(p+(1-p)\mu_W)^2}\left(1-\frac{n}{N}\right)$$
Diana, G., Perri, P.F. (2010). New scrambled response models for estimating the mean of a sensitive quantitative character. Journal of Applied Statistics 37 (11), 1875-1890.
# NOT RUN {
N=417
data(DianaPerri1Data)
dat=with(DianaPerri1Data,data.frame(z,Pi))
p=0.6
mu=c(5/3,5/3)
cl=0.95
DianaPerri1(dat$z,p,mu,dat$Pi,"mean",cl,N,"srswor")
# }
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