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Latest revision as of 14:49, 6 October 2016
Abstract
This paper focuses on presenting a generalization of the scrambled response models of Hussain and Shabbir [Hussain, Z. and Shabbir, J. “On estimation of mean of a sensitive quantitative variable”, InterStat , (#006), (2007)] and Gjestvang and Singh [Gjestvang, C.R. and Singh, S. “An improved randomized response model: estimation of mean”, Journal of Applied Statistics , 36(12), pp. 1361–1367 (2009)]. The suggested generalization is helpful in procuring honest data on socially undesirable characteristics. The suggested estimator is found to be unconditionally more efficient in terms of variablity. From a privacy point of view, comparison of the proposed class of models is made using the privacy protection measure by Zaizai et al. [Zaizai, Y., Jingu, W. and Junfeng, L. “An efficiency and protection based comparison among the quantitative randomized response strategies”, Communications in Statistics-Theory and Methods , 38, pp. 400–408 (2009)]. Unlike many scrambled response models, the proposed class of models is free from the need of known parameters of scrambling variables. The relative numerical efficiency of the proposed model is simulated for some fixed values of the parameters. The practical application of the proposed model is also studied through a small scale survey.
Keywords
Randomized response technique ; Sensitive character, Estimation of mean, anonymity ; Social desirability bias ; Scrambled response
1. Introduction
One of the leading paraphernalia for obtaining data pertaining to human populations is the social survey. To measure opinions, attitudes, and behaviors that cover a wide band of interests, the social survey has been established as being tremendously practical. The surveys are conducted due to many reasons, non-availability of certain facts/information in the archives being the most understandable and apparent. For instance, if one is interested in knowing crime rates, information about unseen crimes or unreported victimization experience is not available in formal records on crime. Sometimes the facts about the individuals (in a population) are inaccessible to the investigators for legal reasons. For example, in many countries, certain information about criminals is kept confidential, due to security and privacy concerns. In most studies, the study population may be so geographically dispersed that studying a whole population is simply infeasible.
Questionnaires, in particular social surveys, generally consist of many items. Some of the items may be about sensitive/high risk behavior, due to the social stigma carried by them. One problem with research on high-risk behavior is that respondents may consciously or unconsciously provide incorrect information. In psychological surveys, a social desirability bias has been observed as a major cause of distortion in standardized personality measures. Survey researchers have similar concerns about the truth of survey results/findings about such topics as drunk driving, use of marijuana, tax evasion, illicit drug use, induced abortion, shop lifting, cheating in exams, and sexual behavior.
The most serious problem in studying certain social problems that are sensitive in nature (e.g. induced abortion, drug usage, tax evasion, etc.) is lack of a reliable measure of their incidence or prevalence. Social stigma and fear of reprisal usually result in lying by the respondents when approached with the conventional or direct-response survey method. An obvious consequence of false reporting is unavoidable estimation bias. Warner [1] showed this evasive answer bias to prevail in the estimate obtained by direct questioning, and proposed a Randomized Response Model (RRM) to estimate the proportion of prevalence of sensitive characteristics in a population. Greenberg et al. [2] extended the RRM to the estimation of mean of a sensitive quantitative variable. The recent articles on the estimation of mean of a sensitive variable include: Eichhorn and Hayre [3] , Singh et al. [4] , Gupta et al. [5] , Bar-Lev et al. [6] , Ryu et al. [7] , Hussain et al. [8] , Hussain and Shabbir [9] and [10] , Huang [11] and [12] , Gjestvang and Singh [13] and [14] , Gupta et al. [15] and the references cited therein. For a detailed understanding of RRM, interested readers may be referred to Chaudhuri and Mukerjee [16] .
In the literature on estimation of scrambled randomized response models, we can find two types of scrambling, namely, the additive and multiplicative. The additive scrambled response model is due to Himmelfarb and Edgell [17] , and has been advocated by many authors due to its simplicity of application (cf. [11] , [12] , [13] and [15] ). Keeping in mind this advocacy, it is obvious that we need to search for an improvement in additive randomized response models. In this study, we present an unbiased estimator of the mean, assuming Simple Random Sampling with Replacement (SRSWR) and Stratified Random Sampling (STRS) protocols. The paper is organized as follows. In Section 2 , we briefly present Hussain and Shabbir [10] and Gjestvang and Singh [14] models. The proposed generalization is showcased in Section 3 under SRSWR and STRS schemes. Section 4 of the paper consists of efficiency comparisons and Section 5 is about a practical study, followed by conclusions of this study in Section 6 .
2. Gjestvang and Singh [14] , Hussain and Shabbir [10] RRMs and motivation
Following Gjestvang and Singh [13] , Gjestvang and Singh [14] proposed an additive Randomized Response Model. For the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
th individual, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S_i}
be the values of the sensitive and scrambling variables, respectively. The distribution of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S}
is completely known, with mean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu _S(-\infty <\mu _S<\infty )}
and variance, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _S^2}
. Assuming Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b}
to be positive real constants, the Gjestvang and Singh [14] model provides two options for the respondents; (i) “Report the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i+aS_i}
”, and (ii) “Report the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i-bS_i}
”, with pre-assigned probabilities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_1=\frac{b}{a+b}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (1-P_1)=\frac{a}{a+b}}
, respectively, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S}
is a scrambling variable with mean, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu _S}
, and variance, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _S^2}
. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Y_i}
be the reported response of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
th respondent, then, it can be written as:
|
( 1) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta _i}
is a Bernoulli random variable having value ‘1’, if statement (i) is randomly chosen by the respondent and ‘0’, otherwise. They proposed an unbiased estimator of the population mean, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu _X}
, of the sensitive variable, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X}
, as:
|
( 2) |
with variance:
|
( 3) |
It is to be noted that borrowing the idea from Gjestvang and Singh [13] , Hussain and Shabbir [10] proposed an improved version of Gjestvang and Singh [14] RRM. The RRM of Hussain and Shabbir [10] is actually a two stage RRM. In their proposed RRM, a sample of size Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
is drawn from the population with a SRSWR sampling scheme. Each individual in the sample is requested to use a randomization device, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_1}
, which consists of the two statements:
- “report your true response, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i}
, of the sensitive question” and
- “go to the randomization device, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_2}
”, represented with the probabilities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (1-P_1)}
, respectively. The randomization device, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_2}
, consists of the two statements:
- “report the scrambled response, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i+aS_i}
”, and
- “report your scrambled response, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i-bS_i}
”, represented with probabilities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_2=\frac{b}{a+b}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1-P_2=\frac{a}{a+b}}
, respectively. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Z_i}
be the response of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
th respondent, then, it can be written as:
|
( 4) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha _i=1}
, if statement (i) is randomly chosen in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_1}
, and ‘0’, otherwise. Similarly, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta _i=1}
, if statement (i) is randomly chosen in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_2}
and ‘0’, otherwise. An unbiased estimator of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu _X}
is given by:
|
( 5) |
The variance of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\mbox{ˆ}}{\mu }_{A(HS)}}
is given by:
|
( 6) |
Hussain and Shabbir [9] used the idea of distributing the probability of reporting on the true value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X}
into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k(>2)}
stages using the multiplicative randomized response models and reported the following advantages: (i) the inability of a clever respondent to correctly guess the total probability on reporting ‘Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X}
’. (ii) Provision of more protection against the privacy of the respondents, and, therefore, making the interviewer unable to know at which stage respondents actually reported his response, and (iii) the increased degrees of freedom to set the values for design probabilities, in order to keep the total probability of reporting on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X}
at some desired level. As the use of additive randomized response models has been advocated by many authors, like Gjestvang and Singh [14] , Gupta et al. [15] , and Huang [11] and [12] , we plan to study the additive RRM of Gjestvang and Singh [14] in increased numbers of randomization stages. In the next section, we present the proposed RRM.
3. Proposed class of RRMs
We present the proposed model under two sampling schemes, namely, SRSWR and STRS, in following Sections 3.1 and 3.2 , respectively.
3.1. Case of SRSWR
In the proposed RRM, a sample of size Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
is drawn from the population with the SRSWR sampling scheme. Each individual in the sample is provided Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k(>2)}
randomization devices, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_1,R_2,\ldots ,R_k}
, and requested to use these randomization devices in the following order:
Use the randomization device, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_1}
, which consists of the two statements:
- “report your true response, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i}
, of the sensitive question” and
- “go to the randomization device, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_2}
”, represented with the probabilities, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (1-P_1)}
, respectively.
The randomization device, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_2}
, consists of the two statements:
- “report your true response, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i}
, of the sensitive question” and
- “go to the randomization device, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_3}
”, represented with the probabilities, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_2}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (1-P_2)}
, respectively. Continuing in this way, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (k-1)}
th randomization device, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_{k-1}}
, consists of the two statements:
- “report your true response, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i}
, of the sensitive question” and
- “go to the randomization device,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_k}
”, represented with probabilities, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{k-1}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (1-P_{k-1})}
, respectively.
The randomization device, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_k}
, consists of the two statements:
- “report the scrambled response, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i+aS_i}
”, and
- “report your scrambled response, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i-bS_i}
”, represented with probabilities, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_k=\frac{b}{a+b}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (1-P_k)=\frac{a}{a+b}}
, respectively. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V_i}
be the response of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
threspondent, then, it can be written as:
|
( 7) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha _{ji}=1}
, if the statement (i) is chosen randomly by the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i^{th}(i=1,2,\ldots ,n)}
respondents using the randomization device, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_j(j=1,2,\ldots ,k)}
.
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E_1}
be the expectation operator over all possible samples and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E_2}
be the expectation operator over the randomization device, then:
|
where:
|
Thus:
|
( 8) |
We propose an unbiased estimator of population mean, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu _X}
, as:
|
( 9) |
The variance of the proposed estimator is given by:
|
( 10) |
It is to be noted that for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=1}
, the proposed model is essentially the Gjestvang and Singh [14] RRM, and, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=2} , it reduces to Hussain and Shabbir [10] RRM. For all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k\geq 3} , the responses of the respondents can be expressed as:
|
The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
-stage randomization device can be viewed as a two stage randomization procedure withFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lbrace 1-\prod _{h=1}^{k-1}(1-P_h)\rbrace =P_1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_k=P_2}
. Thus, the Hussain and Shabbir [10] procedure is a special case of the proposed procedure. In addition, the proposed procedure has the advantage of distributing the total probability of reporting on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X_i}
into an increased number of stages. In the lines to follow, we illustrate the working of the proposed RRM for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=3}
. Following these lines, we can easily derive the generalized results given by Eqs. (3) , (6) and (10) .
For the purpose of illustrating the idea, suppose we have three different urns Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (U_1,U_2,U_3)}
containing black and white cards, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_1,P_2}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_3}
being the proportions of white cards, respectively. A selected respondent is asked to pick a card randomly from the urn ‘Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle U_1}
’. If a white card is picked, he/she is asked to report the true value of ‘Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X}
’, otherwise, he/she is directed to go to the second urn, ‘Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle U_2}
’. At this stage, again, he/she is requested to randomly draw a card from the urn, ‘Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle U_2}
’, and report the value of ‘Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X}
’, if the white card is drawn, otherwise, directed to go to third urn, ‘Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle U_3}
’, and randomly draw a card from the third urn. Then, report ‘Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X+aS}
’ if a white card is drawn, otherwise, report ‘Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle X-bS}
’.
The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
th respondent selected in the sample of size Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
, drawn by using simple random sampling with replacement (SRSWR), is requested to report the value:
|
( 11) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha _{ji}(j=1,2,3,i=1,2,\ldots ,n)}
is defined as earlier. The expected value of the observed response is:
|
( 12) |
The unbiased estimator of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu _X}
is then given, as in Eq. (9) , with variance:
|
( 14) |
As pointed out by one of the referees, one of the two key issues in the scrambling model is the degree of privacy protection provided and competing models should also be compared at equal levels of privacy protection. For this purpose, we take the privacy measure proposed by Zaizai et al. [18] . The privacy measure proposed by Zaizai et al. [18] is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E(X_i-T_i)^2}
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T_i}
is the scrambled response obtained through a given scrambling model. The model with the larger value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E(X_i-T_i)^2}
is taken as a more protective model. The privacy measure by Zaizai et al. [18] is not a normalized privacy measure. We normalize it as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E(\frac{X_i-T_i}{\mu _S^2})^2}
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu _S}
is defined as earlier. The normalized privacy measures for the proposed, Hussain and Shabbir [10] and Gjestvang and Singh [14] models are given, respectively, as below:
|
( 15) |
From Eqs. , and , it is observed that the model of Gjestvang and Singh [14] is more protective than the other two models and the model of Hussain and Shabbir [10] is better only to the proposed model. It is also observed that privacy protection (efficiency) is the decreasing (increasing) function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
. Thus, it is a tradeoff between privacy and efficiency. The value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
should be fixed, depending upon the required privacy protection and efficiency.
Acceptance of the unrelated variable, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S}
, by the respondents, as pointed out by one of the referee, is another key issue of concern. Explaining the working of the whole procedure to respondents may be needed in some situations, but not always. It depends upon the nature of the study variable and the sampled population. If the study variable is sensitive enough, the procedure should be explained to the respondents, assuring them that their individual answers cannot be traced back to their true values on the study variable and that only the population mean is estimable. The explanation of the procedure would help decrease suspicion among the respondents. Though any unrelated variable with a known population mean and variance may be fairly used, we recommend using generating random numbers from a known distribution through the computer, writing them on cards and putting them into a box. Otherwise, the number of siblings, family size, last digit of the social security number, etc. may be used as an unrelated variable.
3.2. Case of STRS
Suppose the population is partitioned into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H}
strata, and a sample is selected by simple random sampling with replacement from each stratum. Using the results in Section 3.1 , we can show that for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} th stratum, the estimator of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu _{Xh}} is given by:
|
( 18) |
Its variance is given by:
|
( 19) |
The mean estimators for individual strata can be added together to obtain a mean estimator for the whole population. The mean estimator of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu _X}
is:
|
( 20) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
is the number of units in the whole population, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_h}
is total number of units in stratum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h}
, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_h=\frac{N_h}{N}}
for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=1,2,\ldots ,k}
, so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sum _{h=1}^kW_h=1}
.
It is obvious that the proposed mean estimator, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mu }_{X(p)}}
, is an unbiased estimate for the population mean, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu _X}
. Since the selections in different strata are made independently, each unbiased mean estimator, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\mbox{ˆ}}{\mu }_{X_h(p)}}
, has its own variance. The variance of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mu }_{X(p)}}
is given by:
|
( 21) |
3.2.1. Optimum sample sizes
The optimal allocation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n\mbox{ to }n_1,n_2,\ldots ,n_{k-1}\mbox{ and }n_k}
to derive the minimum variance of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mu }_{X(p)}}
, subject to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=\sum _{h=1}^Hn_h}
, is approximately given by:
|
( 22) |
The minimal variance of the estimator, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mu }_{X(p)}}
, is given by:
|
( 23) |
Application of the Gjestvang and Singh [14] model in the stratified sampling with fixed total sample size and optimum allocation of sample sizes in different strata yields the following mean estimator:
|
( 24) |
with minimal variance:
|
( 25) |
4. Efficiency comparisons
The proposed estimator, based on SRSWR, will be more efficient than that of Gjestvang and Singh [14] , if:
|
or if:
|
or if:
|
or if:
|
( 26) |
which is always true.
Similarly the efficiency condition, with respect to Hussain and Shabbir [10] , is given by:
|
which is always true.
Our proposed stratified mean estimator is more efficient than the Gjestvang and Singh [14] stratified mean estimator, iff:
|
That is:
|
Or:
|
( 27) |
If, for each stratum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Var(\overset{\mbox{ˆ}}{\mu }_{X_h(p)})\leq Var(\overset{\mbox{ˆ}}{\mu }_{X_h(GS)})}
, then, above inequality is always true. Using Eq. (26) for each stratum, we can see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Var(\overset{\mbox{ˆ}}{\mu }_{X_h(p)})\leq Var(\overset{\mbox{ˆ}}{\mu }_{X_h(GS)})} . Thus, Eq. (27) is always true. Hence, the proposed stratified mean estimator is more efficient than that of Gjestvang and Singh [14] . To know the extent of Relative Efficiency (RE ) we have done a simulation study, assuming Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_1=0.5,P_2=0.8} and different values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b} . The REs of the proposed model relative to Hussain and Shabbir [10] and Gjestvang and Singh [14] models are defined, respectively, as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle RE_1=\frac{Var(\overset{\mbox{ˆ}}{\mu }_{X(HS)})}{Var(\overset{\mbox{ˆ}}{\mu }_{X(p)})}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle RE_2=\frac{Var(\overset{\mbox{ˆ}}{\mu }_{X(GS)})}{Var(\overset{\mbox{ˆ}}{\mu }_{X(p)})}} . The RE results are shown in Table A.2 in the Appendix .
| Model | Estimated mean | Estimated variance | Large sample 95% confidence interval |
|---|---|---|---|
| Proposed | 2.718 | 1.830 | (2.359, 3.077) |
| Hussain and Shabbir [10] | 3.048 | 1.940 | (2.344, 3.105) |
| Gjestvang and Singh [14] | 2.711 | 2.023 | (2.314, 3.107) |
| Direct questioning | 3.048 | 0.269 | (2.936, 3.159) |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu _S} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b} | |||||||
|---|---|---|---|---|---|---|---|---|---|
| 0.1 | 0.5 | 1.0 | 1.5 | 2.0 | 3.0 | 4.0 | |||
| 5 | 0.01 | RE1 | 1.006 | 1.033 | 1.062 | 1.111 | 1.122 | 1.226 | 1.230 |
| RE2 | 1.063 | 1.325 | 1.598 | 1.924 | 2.193 | 2.644 | 3.180 | ||
| 0.05 | RE1 | 1.034 | 1.157 | 1.281 | 1.366 | 1.453 | 1.479 | 1.482 | |
| RE2 | 1.332 | 2.421 | 3.452 | 4.151 | 4.872 | 5.664 | 6.296 | ||
| 0.1 | RE1 | 1.071 | 1.273 | 1.459 | 1.517 | 1.612 | 1.720 | 1.730 | |
| RE2 | 1.628 | 3.497 | 4.957 | 5.762 | 6.102 | 7.168 | 7.673 | ||
| 0.15 | RE1 | 1.101 | 1.341 | 1.537 | 1.658 | 1.727 | 1.809 | 1.824 | |
| RE2 | 1.918 | 4.237 | 5.866 | 6.765 | 7.296 | 8.322 | 7.925 | ||
5. Practical application
We considered application of the proposed model withFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=3}
, in estimating the average GPA of the students at Quaid-i-Azam University, Islamabad. To estimate the average GPA of the students, we took a sample of 100 s semester students. Each student was requested to report the responses using the proposed Hussain and Shabbir [10] and Gjestvang and Singh [14] RR models. The responses obtained from the students are reported in the Table A.3 , Table A.4 and Table A.5 (see Appendix ). We generated 100 random numbers from a normal distribution, with mean 5 and standard deviation 0.5, wrote on the cards (white and black) and placed them in a transparent box. Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S\sim N(5,0.5)} . We decided to choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_1=0.2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_2=0.12,a=0.2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b=0.6} , that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_3=\frac{0.6}{0.2+0.6}=0.75} .
| 2.23316076 | 0.11682901 | 3.69261291 | 2.79361833 | 3.31754616 | 4.48523422 |
| 0.07177086 | 2.54567059 | 2.00454378 | −0.18521662 | 4.02623078 | 4.78814901 |
| 4.34181631 | −1.20497925 | 3.03632159 | 0.92956271 | 3.18827191 | −0.39164659 |
| 5.05778842 | −0.04244069 | 4.82164560 | 4.70977953 | 3.16458601 | 3.44016293 |
| 3.33889340 | 3.98265516 | 4.47154482 | 5.05657143 | −1.03031096 | 3.75428966 |
| 4.71087638 | 3.04748875 | −0.12171074 | 3.93705588 | 3.19369396 | 2.93455250 |
| 3.31228968 | 4.81156619 | 3.99121545 | 3.47980705 | 3.41929935 | 4.30847776 |
| 0.61209063 | 4.45097525 | −0.59982820 | −0.38546359 | 4.87526643 | 3.34191693 |
| 2.16227070 | −0.61183118 | 4.46274876 | 3.83178736 | 2.93766344 | 4.65216277 |
| 4.44753724 | 3.75060349 | 3.68410234 | 4.66868069 | 0.07913056 | −0.12381204 |
| 0.25315506 | 2.44209882 | 4.31549906 | 3.31681455 | 2.96894543 | 3.35360459 |
| 0.07256166 | 1.02781537 | −0.45570842 | −0.87029593 | 4.11836979 | 2.64111698 |
| −0.27625023 | 3.16656813 | 4.63397434 | 3.29360005 | 4.48240265 | 4.23193867 |
| 0.01389507 | 3.57583500 | 4.81195764 | 2.38847305 | 3.26900358 | 3.35566998 |
| −0.23058781 | 3.52512426 | 3.02007434 | −0.87307810 | 4.56112139 | 3.93763373 |
| 0.80805500 | 3.26219485 | 3.74346405 | 3.16233729 | 3.34006163 | 3.15538427 |
| 3.47715985 | 3.88315901 | 0.62564945 | 4.55753575 |
| 3.11109635 | 0.11682901 | 4.78965708 | 2.79361833 | 3.31754616 | 4.48523422 |
| 0.07177086 | 3.77153499 | 2.00454378 | −0.18521662 | 4.02623078 | 4.78814901 |
| 4.34181631 | −1.20497925 | 3.03632159 | 0.92956271 | 3.18827191 | −0.39164659 |
| 5.05778842 | −0.04244069 | 4.82164560 | 4.70977953 | 3.16458601 | 4.34426764 |
| 3.33889340 | 3.98265516 | 4.47154482 | 5.05657143 | −1.03031096 | 4.74066931 |
| 4.71087638 | 3.04748875 | −0.12171074 | 3.93705588 | 0.48483138 | 2.93455250 |
| 3.31228968 | 4.81156619 | 3.99121545 | 0.62013548 | 3.41929935 | 4.30847776 |
| 0.61209063 | 4.45097525 | −0.59982820 | −0.38546359 | 4.87526643 | 3.34191693 |
| 3.11559258 | −0.61183118 | 4.46274876 | 3.83178736 | 4.05210038 | 4.65216277 |
| 4.44753724 | 3.75060349 | 3.68410234 | 4.66868069 | 0.07913056 | −0.12381204 |
| 0.25315506 | 2.44209882 | 4.31549906 | 4.44771291 | 3.97064382 | 3.35360459 |
| 0.07256166 | 1.02781537 | −0.45570842 | −0.87029593 | 4.11836979 | −0.17281558 |
| −0.27625023 | 3.16656813 | 4.63397434 | 4.25109065 | 4.48240265 | 4.23193867 |
| 0.01389507 | 3.57583500 | 4.81195764 | 2.38847305 | 3.26900358 | 3.35566998 |
| −0.23058781 | 3.52512426 | 3.02007434 | −0.87307810 | 4.56112139 | 3.93763373 |
| 0.80805500 | 3.26219485 | 4.66407697 | 3.16233729 | 4.28249550 | 3.15538427 |
| 3.47715985 | 0.84556881 | 0.62564945 | 4.55753575 |
| 3.11109635 | 0.11682901 | 4.78965708 | 0.07148237 | 3.31754616 | 4.48523422 |
| 0.07177086 | 3.77153499 | 2.98748964 | −0.18521662 | 4.02623078 | 4.78814901 |
| 4.34181631 | −1.20497925 | 0.20104257 | 0.92956271 | 4.16780198 | −0.39164659 |
| 5.05778842 | −0.04244069 | 4.82164560 | 4.70977953 | 4.31361666 | 4.34426764 |
| 3.33889340 | 3.98265516 | 4.47154482 | 5.05657143 | −1.03031096 | 4.74066931 |
| 4.71087638 | 3.04748875 | −0.12171074 | 3.93705588 | 0.48483138 | 2.93455250 |
| 3.31228968 | 4.81156619 | 3.99121545 | 0.62013548 | 3.41929935 | 4.30847776 |
| 0.61209063 | 4.45097525 | −0.59982820 | −0.38546359 | 4.87526643 | 4.41765019 |
| 3.11559258 | −0.61183118 | 4.46274876 | 3.83178736 | 4.05210038 | 4.65216277 |
| 4.44753724 | 3.75060349 | 3.68410234 | 4.66868069 | 0.07913056 | −0.12381204 |
| 0.25315506 | −0.42428630 | 4.31549906 | 4.44771291 | 3.97064382 | 3.35360459 |
| 0.07256166 | 1.02781537 | −0.45570842 | −0.87029593 | 4.11836979 | −0.17281558 |
| −0.27625023 | 3.16656813 | 4.63397434 | 4.25109065 | 4.48240265 | 4.23193867 |
| 0.01389507 | 3.57583500 | 4.81195764 | 3.26489306 | 3.26900358 | 3.35566998 |
| −0.23058781 | 3.52512426 | 4.02475847 | −0.87307810 | 4.56112139 | 3.93763373 |
| 0.80805500 | 3.26219485 | 4.66407697 | 4.10066525 | 4.28249550 | 3.15538427 |
| 3.47715985 | 0.84556881 | 0.62564945 | 4.55753575 |
In the first deck of 100 cards, on 20 we wrote the statement: Please “Report GPA” and on the remaining 80, we wrote the statement “go to second box”. In the second deck of 100 cards, on 12 cards, we wrote the statement: Please “Report GPA” and on the remaining 88 cards, we wrote the statement “go to second box”. Similarly, in the third deck of cards, on 75, we wrote the statement: Please “Report: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle GPA+0.2}
(Random number)”, and on the remaining 25 cards, we wrote the statement “Report: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle GPA-0.6}
(Random number)”. Assuming that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_1=0}
, the data were obtained from the same respondents, which were, essentially, the data obtained through Hussain and Shabbir [10] RRM. Similarly, assuming Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_1=P_2=0}
the data were obtained, again, from the same respondents. Obviously, those were the data obtained by Gjestvang and Singh [14] RRM. At the end, we requested them to write their true GPA on a paper chit and drop it into a box without disclosing their identity. The true data are given in Table A.6 (see Appendix ).
| 2.233161 | 3.136780 | 3.692613 | 2.793618 | 2.205516 | 3.431046 | 2.707550 | 2.545671 |
| 2.004544 | 3.460874 | 3.110159 | 3.619516 | 3.217301 | 2.058742 | 3.036322 | 3.914756 |
| 3.188272 | 3.005791 | 3.982075 | 3.275306 | 3.930744 | 3.717378 | 3.164586 | 3.440163 |
| 2.099116 | 2.988803 | 3.171050 | 3.958999 | 2.080994 | 3.754290 | 3.707987 | 2.036558 |
| 2.655664 | 2.910881 | 3.193694 | 2.046447 | 2.261989 | 3.656463 | 2.845359 | 3.479807 |
| 2.492049 | 3.277419 | 3.719893 | 3.494468 | 2.591614 | 2.792680 | 3.810854 | 3.341917 |
| 2.162271 | 2.996835 | 3.462074 | 2.922333 | 2.937663 | 3.800106 | 3.252035 | 2.894699 |
| 2.741574 | 3.590808 | 2.611402 | 3.730712 | 3.321683 | 2.442099 | 3.223477 | 3.316815 |
| 2.968945 | 2.312281 | 3.336032 | 3.967372 | 2.366828 | 2.500851 | 3.303393 | 2.641117 |
| 3.461387 | 2.230421 | 3.716620 | 3.293600 | 3.488766 | 3.251924 | 2.901757 | 2.566012 |
| 3.843604 | 2.388473 | 2.085841 | 2.310979 | 2.968765 | 2.624486 | 3.020074 | 2.626089 |
| 3.553566 | 2.784691 | 3.598225 | 2.165920 | 3.743464 | 3.162337 | 3.340062 | 2.142230 |
| 2.391306 | 3.883159 | 3.739664 | 3.490303 |
The summary of the survey results is given in Table A.1 (see Appendix ). From Table A.1 , it is observed that the estimates based on the responses through the proposed model are closer to the estimates based on true responses than those of the other two methods.
6. Conclusion
Using the idea of distributing the probability of reporting the true value of sensitive variables into an increased number of stages, we proposed a general class of the scrambling model. The models by Hussain and Shabbir [10] and Gjestvang and Singh [14] have been shown as special cases of the proposed class of models. The efficiency and privacy protection of the models in the proposed class are functions of the number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (k)}
of randomization stages. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
increases, the efficiency (privacy protection) of the proposed class of models increases (decreases). Thus, a suitable value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
is the value which satisfies the objectives (greater efficiency and privacy protection) of the study. It is also established that the proposed class of models is actually a class of two stage models, having the additional advantage of distributing the probability of reporting on sensitive variables into an increased number of stages. Although the estimator given in Eq. (9) is unbiased and has smaller variance, its application in field surveys may be problematic because the individuals in the samples may get annoyed/irritated at reporting again and again. Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
must be chosen, at most, 3 or 4, in order to have the proposed model practically feasible.
A small scale practical application of the proposed class of model for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=3}
is also given. In this application, we compared two types of estimate, one based on direct responses and the others based on scrambled responses. These estimates may not represent the true average GPA of the whole campus, as we have taken only the second semester students. This study could have been extended to a large scale by including all the students in the university and getting their actual average GPA from the controller of the examination office. Then, comparing the true average GPA with the estimates would shed more light on the performance of the proposed estimators. Nevertheless, it is established that the proposed estimator performs well compared to estimators considered in this paper. In conclusion, we must say that the proposed method of obtaining scrambled responses can be used safely and securely in field surveys on sensitive variables.
Acknowledgments
The authors are most grateful to the two learned referees for their guidance in improving the earlier draft of this article. The first author greatly appreciates the research facilities provided by King Abdulaziz University.
Appendix.
See Table A.1 , Table A.2 , Table A.3 , Table A.4 , Table A.5 and Table A.6 .
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