RRTCS package. Application of Horvitz model to a real survey

##Horvitz model 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 α 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. Sin titulo

##Horvitz model example The file data “HorvitzDataRealSurvey.rda” stores a sample, extracted by simple random sampling without replacement, of size 710. With these data we estimate the prevalence of students who have sensitive characteristics set out below.
The randomized response technique used is Horvitz model (Horvitz et al, 1967 and Greenberg et al., 1969) with parameter p = 0.5

###Horvitz model survey design In this model an additional questionnaire with the following characteristics is required:

The questions should not be sensitive.
The questions should not be related to the theme of “sensitive” questions.
For each of the questions that measure qualitative variables, the answers should have a known probability.

The survey is

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To estimate results, you must have the response probabilities of non-sensitive questions:

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###Horvitz model technique description In order to give more confidence to the respondent, he was given a booklet of instructions:

Step 1: Flip a coin and did not show anyone the side obtained.
Step 2: The question you must answer in each game depends on the side obtained on the coin:
- If you get head, answer the question preceded by the word HEAD
- If you get tail, answer the question preceded by the word TAIL

This procedure must be repeated for each question.

The respondent only had to put on the question sheet a cross in the chosen answers, and then he deposited the survey sheet in an urn.

We store the answers to senstive questions in data.frame (HorvitzDataRealSurvey.rda).

###Horvitz function We use the Horvitz function to obtain the estimates:

library("RRTCS")
N=10777
n=710
data(HorvitzDataRealSurvey)
p=0.5
alpha=c(1/12,1/10,20/30,1/10,10/30,1/12)
pi=rep(n/N,n)
cl=0.95
out1=Horvitz(HorvitzDataRealSurvey$copied,p,alpha[1],pi,"mean",cl,N)
out1

## 
## Call:
## Horvitz(z = HorvitzDataRealSurvey$copied, p = p, alpha = alpha[1], 
##     pi = pi, type = "mean", cl = cl, N = N)
## 
## Qualitative model
## Horvitz model for the mean estimator
## Parameters: p=0.5; alpha=0.083
## 
## Estimation: 0.8406103
## Variance: 0.001389716
## Confidence interval (95%)
##     Lower bound: 0.767545 
##     Upper bound: 0.9136756

out2=Horvitz(HorvitzDataRealSurvey$fought,p,alpha[2],pi,"mean",cl,N)
out2

## 
## Call:
## Horvitz(z = HorvitzDataRealSurvey$fought, p = p, alpha = alpha[2], 
##     pi = pi, type = "mean", cl = cl, N = N)
## 
## Qualitative model
## Horvitz model for the mean estimator
## Parameters: p=0.5; alpha=0.1
## 
## Estimation: 0.4070423
## Variance: 0.001045196
## Confidence interval (95%)
##     Lower bound: 0.3436776 
##     Upper bound: 0.4704069

out3=Horvitz(HorvitzDataRealSurvey$bullied,p,alpha[3],pi,"mean",cl,N)
out3

## 
## Call:
## Horvitz(z = HorvitzDataRealSurvey$bullied, p = p, alpha = alpha[3], 
##     pi = pi, type = "mean", cl = cl, N = N)
## 
## Qualitative model
## Horvitz model for the mean estimator
## Parameters: p=0.5; alpha=0.67
## 
## Estimation: 0.1220657
## Variance: 0.001337415
## Confidence interval (95%)
##     Lower bound: 0.05038851 
##     Upper bound: 0.1937429

out4=Horvitz(HorvitzDataRealSurvey$bullying,p,alpha[4],pi,"mean",cl,N)
out4

## 
## Call:
## Horvitz(z = HorvitzDataRealSurvey$bullying, p = p, alpha = alpha[4], 
##     pi = pi, type = "mean", cl = cl, N = N)
## 
## Qualitative model
## Horvitz model for the mean estimator
## Parameters: p=0.5; alpha=0.1
## 
## Estimation: 0.128169
## Variance: 0.0005597858
## Confidence interval (95%)
##     Lower bound: 0.08179667 
##     Upper bound: 0.1745414

out5=Horvitz(HorvitzDataRealSurvey$drug,p,alpha[5],pi,"mean",cl,N)
out5

## 
## Call:
## Horvitz(z = HorvitzDataRealSurvey$drug, p = p, alpha = alpha[5], 
##     pi = pi, type = "mean", cl = cl, N = N)
## 
## Qualitative model
## Horvitz model for the mean estimator
## Parameters: p=0.5; alpha=0.33
## 
## Estimation: 0.1286385
## Variance: 0.000991658
## Confidence interval (95%)
##     Lower bound: 0.06691805 
##     Upper bound: 0.1903589

out6=Horvitz(HorvitzDataRealSurvey$sex,p,alpha[6],pi,"mean",cl,N)
out6

## 
## Call:
## Horvitz(z = HorvitzDataRealSurvey$sex, p = p, alpha = alpha[6], 
##     pi = pi, type = "mean", cl = cl, N = N)
## 
## Qualitative model
## Horvitz model for the mean estimator
## Parameters: p=0.5; alpha=0.083
## 
## Estimation: 0.06596244
## Variance: 0.000383954
## Confidence interval (95%)
##     Lower bound: 0.02755745 
##     Upper bound: 0.1043674

##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.