Computes the randomized response estimation, its variance estimation and its confidence interval through the Christofides model.
The function can also return the transformed variable.
The Christofides model was proposed by Christofides in 2003.
vector of the observed variable; its length is equal to \(n\) (the sample size)
mm
vector with the marks of the cards
pm
vector with the probabilities of previous marks
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 Christofides model. The transformed variable is also reported, if required.
Details
In the Christofides randomized response technique, a sampled person \(i\) is given a box with identical cards, each bearing a separate mark as
\(1,\dots,k,\dots m\) with \(m\geq 2\) but in known proportions \(p_1,\dots,p_k,\dots p_m\) with \(0<p_k< 1\) for \(k=1,\dots,m\) and
\(\sum_{k=1}^{m}p_k=1\). The person sampled is requested to draw one of the cards and respond
$$z_i=\left \{\begin{array}{lcc}
k & \textrm{if a card marked } k \textrm{ is drawn and the person bears } A^c\\
m-k+1 & \textrm{if a card marked } k \textrm{ is drawn but the person bears } A
\end{array}
\right .$$
The transformed variable is \(r_i=\frac{z_i-\mu}{m+1-2\mu}\) where \(\mu=\sum_{k=1}^{m}kp_k\) and the estimated variance is
\(\widehat{V}_R(r_i)=\frac{V_R(k)}{(m+1-2\mu)^2}\), where \(V_R(k)=\sum_{k=1}^{m}k^2p_k-\mu^2\).
# NOT RUN {N=802data(ChristofidesData)
dat=with(ChristofidesData,data.frame(z,Pi))
mm=c(1,2,3,4,5)
pm=c(0.1,0.2,0.3,0.2,0.2)
cl=0.95Christofides(dat$z,mm,pm,dat$Pi,"mean",cl,N)
# }