The Optimal Sample Size in Crosswise Model for Sensitive Questions Stanis law Jaworski

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The Optimal Sample Size

in Crosswise Model for Sensitive Questions

Stanis law Jaworski∗

Wojciech Zieli´nski

Warsaw University of Life Sciences (Poland)

e-mail: stanislaw jaworski@sggw.edu.pl

e-mail: wojciech zielinski@sggw.edu.pl

Abstract

The problem is in the estimation of the fraction of population with a stigmatizing characteristic. In the

paper the nonrandomized response model proposed by Tian, Yu, Tang, and Geng (2007) is considered. The

exact conﬁdence interval for this fraction is constructed. Also the optimal sample size for obtaining the

conﬁdence interval of a given length is derived.

Keywords: sensitive questions, NNR model, exact conﬁdence interval

2010 Mathematics Subject Classification: 62F25, 62P20

1 Introduction

The problem is in the estimation of the percentage of population who “committed” socially stigmatizing “crimes”

such as corruption, tax frauds, illegal work (black market), drug uses, violence against children and other.

Mathematicaly, let Ybe a random variable such that

P{Y= 1}=π= 1 −P{Y= 0}.

The r.v. takes on the value 1 when the answer to the sensitive question is YES and the value 0 otherwise.

The number π∈(0,1) is the probability of the positive answer to the sensitive question, i.e. π·100% is the

percentage of interest. We want to estimate the probability π, i.e. we are going to construct a conﬁdence

interval for π.

Let Y1, . . . , Ynbe a sample. The statistical model for the sample is

({0,1, . . . , n},{Bin(n, π), π ∈(0,1)}),

where Bin(·,·) denotes a Binomial distribution.

The diﬃculty which arises is such that random variables Y1, . . . , Ynare not observable. Answers to the sensitive

question are “hidden” through asking a “neutral” question, which is answered YES or NO. It is assumed that

the “neutral” question is independent from the sensitive question. In a questionnaire two questions are asked:

sensitive and neutral. But only one answer is registered and the interviewer does not know which of the two

questions the interviewee answered.

The ﬁrst method of obscuring the answer to a sensitive question was proposed by (Warner, 1965). His method

consists in the randomization of answers. This randomization is done by the respondent and the interviewer does

not know what the answer to the sensitive question is. This model was extended in diﬀerent ways (Horvitz,

Shah, & Simmons, 1967; Greenberg, Abul-Ela, & Horvitz, 1969; Raghavarao, 1978; Franklin, 1989; Arnab,

Shangodoyin, & Arcos, 2019; Arnab, 1990, 1996; Kuk, 1990; Rueda, Cobo, & Arcos, 2015).

∗Corresponding author: Stanis law Jaworski, Department of Econometrics and Statistics, Warsaw University of Life

Sciences, Nowoursynowska 159, PL-02-767 Warsaw

arXiv:2210.15245v1 [stat.ME] 27 Oct 2022

Jaworski, Zieli´nski: Optimal Sample Size... 2

(Tian et al., 2007) proposed a nonrandomized response model (NRR). Their idea consists in asking two questions

simultaneously: one sensitive and one neutral. This model was extended to other, similar approaches: (Yu,

Tian, & Tang, 2008; Tan, Tian, & Tang, 2009; Tian, 2014).

Unfortunately, the problem of constructing conﬁdence intervals for πwas considered rather rarely. Moreover,

proposed conﬁdence intervals are asymptotic. These conﬁdence intervals are not c.i. in (Neyman, 1934, p. 562)

sense: they do not keep prescribed conﬁdence level. In what follows the ﬁnite sample size conﬁdence interval is

proposed. Its construction is based on the distribution of the Maximum Likelihood estimator of π. We consider

only the crosswise model proposed by (Yu et al., 2008).

In section 2. we give the method of the construction of a new conﬁdence interval for π. In section 3. we recall

the construction of asymptotic conﬁdence intervals. We also discuss the probability of the coverage of presented

conﬁdence intervals. In the next section we present the methods of sample size selection. This section plays the

main role in our paper. In section 5. some concluding remarks are given.

2 Conﬁdence interval in Crosswise Model

In the crosswise model (CM) respondents are presented with two questions simultaneously, one neutral and one

sensitive. They are instructed to report 1 only if answers to both questions are the same, i.e. the observable

variable in this model is Z, where

Z=(1,if both answers are YES or NO,

0,otherwise.

The answers of nrespondents may be treated as the realizations of a Binomial distribution with parameters

(n, %), where %is the probability of receiving an outcome 1 of the Zvariable. Assume that the asked questions

are independent and the probability of the answer YES to the sensitive question is π(qfor the neutral question).

It is assumed that qis known. Hence, in CM model

%=qπ + (1 −q)(1 −π) = (2q−1)π+ (1 −q).

In this model

π=%−(1 −q)

2q−1.

Without loss of generality we assume that q < 0.5.

Let Z1, . . . , Znbe a sample. MLE of %is ˆ%=1

nPn

i=1 Zi. The distribution of nˆ%is Bin(n;%).

The MLE of πhas the form

ˆπCM = max min ˆ%−(1 −q)

2q−1,1,0.

Let Bin (·, n;%) denote the CDF of the binomial distribution with the probability of success equal to %and let

B(a, b;·) denote the CDF of the Beta distribution with parameters (a, b).

In the derivation of the pdf of ˆπCM the following known relationship will be applied: if ξis a random variable

distributed as binomial with parameters (n, ρ) then

Pρ{ξ≤x}=

i=0 n

iρi(1 −ρ)n−i=B(n−x, x + 1; 1 −ρ).

The pdf of the distribution of ˆπCM is

Pπ{ˆπCM =x}=









Pπ{nˆ%≥ d(1 −q)ne},for x= 0,

n

due((2q−1)π+ (1 −q))due(1 −(2q−1)π−(1 −q))n−due,for 0 <x<1

Pπ{nˆ%≤ bqnc},for x= 1

=









B(dn(1 −q)e, n − dn(1 −q)e+ 1; (2q−1)π+ (1 −q)) ,for x= 0

n

due((2q−1)π+ (1 −q))due(1 −(2q−1)π−(1 −q))n−due,for 0 <x<1

1−B(bnqc+ 1, n − bnqc; (2q−1)π+ (1 −q)) for x= 1

where u=n(x(2q−1) + (1 −q)). Here dxedenotes the smallest integer not smaller than xand bxcdenotes the

greatest integer not greater than x.

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TheOptimalSampleSizeinCrosswiseModelforSensitiveQuestionsStanislawJaworski*WojciechZielinskiWarsawUniversityofLifeSciences(Poland)e-mail:stanislawjaworski@sggw.edu.ple-mail:wojciechzielinski@sggw.edu.plAbstractTheproblemisintheestimationofthefractionofpopulationwithastigmatizingcharacteristic.Inthe...

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The Optimal Sample Size in Crosswise Model for Sensitive Questions Stanis law Jaworski

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