1. Introduction
Many issues that are of interest to social scientists and policymakers are of a sensitive nature. A topic can be said to be ‘sensitive’ when the disclosure of information regarding this topic poses to be threatening for the respondent in the form of being potentially stigmatizing, incriminating or severely intrusive [17, 18]. Due to the potential threats, people have selfrepresentational concerns, leading to refusals to cooperate or evasive cooperation [see, e.g., 13]. The research problems associated with these behaviors tend to inhibit their adequate measurement, and lead one to dismiss the general assumption of sampling theory that data collected on units in the sample are accurate representations of the values associated with the units sampled.
In a seminal article Stanley Warner [30]
proposed to use the element of chance stemming from a randomization device to inoculate individual responses to sensitive inquiries in order to reduce nonsampling bias. Subsequently the population estimates of
, the population proportion indulging in sensitive trait X, will become more accurate. This initial idea grew out to a family of techniques commonly referred to as randomized response (RR), which as a core characteristic uses the insertion of random error by an element of chance to provide a respondent optimal privacy protection. For an indepth review of developments in RR methods we confine by referring to Deffaa [7], Fox & Tracy [9], Chaudhuri & Mukerjee [5], and Tracy & Mangat [27].The psychological premises of the RR method should be clearly understood: If the respondent understands that RR objectively guarantees privacy or trusts the method to provide full anonymity in the disclosure of information (whether fully understanding it or not), he or she is relieved from selfrepresentational concerns and will be more inclined to cooperate and will do so in a nonevasive manner. While generally RR is thus thought able to eliminate or relieve both cooperation refusal and evasive cooperation, we view the RR method mainly as a technique for relieving the latter. As one has to actually endeavor on responding to an RR survey before fully grasping its relative merit and assurances with regard to anonymity and data confidentiality, it can be expected that RR will actually have a stronger stance in reducing evasiveness than in reducing response refusal. It is in this light that the developments in the remainder should be read.
Methods for quantitative RR have been given less attention in the past than its binary family members and existing models and proposed randomization efforts often prove impractical. Many new developments in RR seem absorbed by the technical fix, neglecting its purpose as a technique for improving the validity of observational data. This short research note has two aims with regard to communicating our current work on the RR method: (1) to propose a simple unified framework for eliciting both binary and quantitative information with RR (Section 2) and (2) to describe features of the incorporation of the proposed RR procedures in a computerassisted environment (Section 3). Our framework stresses RR as a survey technique that should have high practicability as we will argue in the concluding discussion.
2. Proposed Framework
An efficient procedure for RR is to let the randomization device provide for the inoculating response. Richard Morton [as cited in 10] and Robert Boruch [2, 3] almost simultaneously proposed an equivalent dichotomous procedure that has this specific design characteristic, now commonly referred to as forced randomized response (FRR). Letting the randomization device provide for the inoculating response combines efficiency, simplicity, and practicability when developing inquiries into large numbers of sensitive behaviors. Here we will give an account of the dichotomous FRR procedure and subsequently extend this model to a polychotomous appreciation of quantitative RR, so as to come to certain unity in the RR framework.
2.1. Binary Forced Randomized Response Model
Consider a sensitive trait X for which the population is dichotomous. A random sample of persons is drawn. The goal is the estimation of the population prevalence of X (). To this purpose each person in the sample is asked the question: “Do you possess trait X?” Each respondent receives a randomization device. In previous studies a pair of dice was used accompanied with the instructions to answer truthfully if the outcome is 5, 6, 7, 8, 9 or 10; to answer with ‘yes’ if the outcome is 2, 3 or 4; or to answer with ‘no’ if the outcome is 11 or 12 [29]. We propose to use a (digital) binary spinner along with the following instructions: Turn the spinner; if it stops on an empty area, respond with ‘yes’ or ‘no’ truthfully; if it stops on an area imprinted with ‘yes’, answer ‘yes’; if it stops on an area imprinted with ‘no’, answer ‘no’.
Taking into account the assumptions regarding RR, events can be expressed in probabilities, so that
(1) 
As the probability of a ‘yes’ response can be estimated by the sample proportion answering ‘yes’ and as and
are fixed by design, an unbiased moment estimator of the population prevalence of X can be obtained by
(2) 
with sampling variance
(3) 
and unbiased sampling estimate
(4) 
2.2. Discrete Quantitative Forced Randomized Response Model
In inquiries into sensitive behaviors not only questions of prevalence are of interest, but also questions of frequency of occurrence. To evaluate frequency of occurrence, RR models must be used that deal with quantitative data. Here, a discrete quantitative FRR model is developed for which previous work by Greenberg et al. [11], Eriksson [8], Liu & Chow [21], and Stem & Steinhorst [26] is acknowledged. Consider a sensitive trait X, for which the population is supposed to be continuous or discretely quantitative. A random sample of persons is drawn. Instead of focussing on a mean we can device a model which redirects X to be discrete, assuming values with respective unknown true proportions , where and . Each of those values can be assigned a category of numbers, for example in a setup where k = 6: The responses 1, 2, 3, 4, 5 and 6 could correspond with the respective categories ‘0’, ‘1 time’, ‘2 to 3 times’, ‘4 to 5 times’, ‘6 to 10 times’, and ‘more than 10 times’. We are then interested in estimating the proportion in each of discrete quantitative categories, providing for a multiproportional or polychotomous appreciation of quantitative RR [25].
For such a setup we could again use two dice as randomizing device. But in this case 2 throws with 3 dice are needed, where the selection probabilities of and (as in the previous section) require for the third die to be thrown. The number that turns up in this throw gives the forced response. There are two problems associated with the use of three dice as randomization device. First, twostep procedures introduce an extra source of error due to increased respondent burden. Second, using dice provides a setup where any piece of quantitative information can be misclassified into exactly 6 discrete categories. To move beyond these 6 categories and to decrease the respondent burden one could also supply each respondent with a (digital) discrete quantitative spinner such as given in Section 3 along with the following instructions: Turn the spinner; if it stops on an empty area, respond with ‘1’, ‘2’, …, or ‘k’ truthfully; if it stops on an area imprinted with ‘1’, ‘2’, …, or ‘k’, respond accordingly. The selection probabilities for the truthful answer and the forced responses are and respectively, with and .
In this setup we define
and
with .
If we let denote the probability of an affirmative quantitative response, again with denoting the response pointing to one of our discrete quantitative categories and taking into account the assumptions regarding RR, it can then be shown analogous to (2.1) that
(5) 
As the probability of a certain quantitative response within our category parameter space can be estimated by the sample proportion giving an affirmative response to that certain quantitative category , where
denotes the characteristic function
, and as and are fixed by design, each of ourpopulation proportions can be unbiasedly estimated by solving (
2.2) for :(6) 
The property of unbiasedness follows from
Utilizing the quality of a multinomial distribution as a joint structure of binomials, the sampling variance for is
(7) 
with unbiased sampling estimate
(8) 
2.3. General Form
Let denote the probabilities of the observed responses with categories, and let denote the probabilities of true answers/status with categories. The models as developed here can then be generally captured in the function
(9) 
so that the moment estimator is given by [5]
(10) 
In the aforementioned
(11) 
the matrix of conditional misclassification probabilities where denotes observed answers and denotes true status. Thus, when having a nonsingular and an unbiased estimate of , then can be estimated by (10). As van den Hout and van der Heijden [28] point out, the assumption that is nonsingular does not impose much restriction on the misclassification design matrix as exists when the diagonal of dominates, meaning: , for . This is reasonable as the diagonal elements represent correct classification and these probabilities should be taken relatively high for the design to be efficient.
3. ComputerAssisted Randomized Response
The incorporation of computerassisted self interviewing (CASI) and RR in computerassisted randomized response (CARR) interviewing has been proposed and successfully tested by Musch, Bröder & Klauer [23], LensveltMulders et al. [20] and LensveltMulders & Boeije [19]. CARR could prove to be very important for future inquiries into socially sensitive issues as other computerbased techniques lose their magic due to penetration of mainstream culture [1, 20], as respondents’ awareness of the negative possibilities of computers regarding his or her privacy issues increases with increasing ‘computerliteracy’, and as the use of internet makes possible the dissemination of data on a large scale which may possibly relieve the efficiency problems inherent in RR designs [23]. Moreover, Boeije & LensveltMulders [1] and LensveltMulders & Boeije [19] have shown that respondents prefer computerassisted RR above paperandpencil and facetoface RR.
The use of CARR makes possible the use of accompanying digital randomizers. Here we propose digital spinners for binary and multiproportional FRR. Spinners have the advantage of being widely known at least across western populations and it may be argued that, when unfamiliar with spinners, their concept may be easily grasped. Also, two important problems are solved concerning the randomizers when a digital makeup is used. The first of which is randomness. Randomness has proven hard to achieve with physical spinners. For example, Stem and Steinhorst [26] used a physical spinner in a mail questionnaire. This spinner could get bend in the mail, and subsequently affect the randomness of the device. With sufficient programming the digital randomizers give very acceptable and consistent levels of pseudorandomness. The second problem alleviated with a digital makeup of the randomizing device is the problem of inconclusive outcomes. In the same study by Stem and Steinhorst, an answer could land on a line on their physical spinner, proving the need for additional instructions, which increases the cognitive load for the respondent. Sufficient programming curbs this problem.
Figure 1^{1}^{1}1The spinners were developed by Peeters [25] and programmed by Rens Lensvelt of WetenschapWerkt. Webpage questionnaires for CASI and CARR as used in Peeters [25] are properties of C.F.W. Peeters, WetenschapWerkt, Utrecht University, and VU University Amsterdam. gives an impression of our incorporation of a (discrete quantitative) FRR spinner into CASI to make for a CARR questionnaire. The upper left corner contains instructions as in Section 2, the upper right corner contains the randomizer, and the bottom part contains the sensitive question with accompanying answering categories. The discrete quantitative FRR spinner of Figure 1 is constructed with , a of threefourths and a of onefourth. These selection probabilities are translated into the randomization device by converting probabilities into degrees. Moriarty and Wiseman [22] give evidence that respondents, due to the involved permutations leading to a certain outcome, misperceive selection probabilities when dice are used as a randomization device. If the misperceiving of probabilities can be incorporated in the randomization device, it is possible to provide for sufficient protection, while pertaining relatively high efficiency. To enhance the misperceiving of selection probabilities the empty and imprinted areas are evenly divided over the spinner, which is divided in 24 subareas. This gives every empty area a selection probability of 3/24 () and the analogous selection probability of each of the 6 forced numerical responses is 1/24. These probabilities and number of categories are by no way necessary. The selection probabilities and number of discrete quantitative categories can be chosen to convenience.
Note that in the aforementioned setup of the discrete quantitative FRR spinner a onestep solution is given for the multiplestep misclassification of sensitive data in 6 categories using 3 dice as described in Section 2.2. Also note that by replacing the numbers in the imprinted areas with ‘yes’ and ‘no’, a binary computerassisted FRR questionnaire is obtained.
4. Discussion
The unified framework for binary and quantitative RR provides several advantages. The design can be adjusted to any efficiency level by selecting certain parameters of the randomization device. Additionally, the spinners are easily incorporated in computerized settings and provide the opportunity of utilizing the psychological advantage of the misperceiving of actual misclassification probabilities [9, 25].
Furthermore, sensitive quantitative information can be misclassified in discrete categories, where the scope of the frequencies denoted with each category can be adapted to the researcher’s need. In previous quantitative designs the desired population means regarding certain sensitive behaviors had to be estimated before any field research efforts, so as to adapt the setup of the randomizer to the expected range of frequencies in the population. After all, the range of responses to the innocuous question has to be similar to the range of possible responses to the sensitive question if one is to unbiasedly estimate frequency of occurrence while still providing sufficient respondent protection [see, e.g., 21, 26]. In the proposed unified approach, the categories can be very easily adopted to expected frequencies while retaining setup and efficiency, as the forced responses have a range that is equal to the range of possible responses on the sensitive question, due to the multiproportional construction. Respondent protection is thus also statistically guaranteed in the quantitative RR setup.
Notwithstanding the relieve RR provides with regard to selfrepresentational concerns, respondents may still cheat, that is, they may not comply with the RR instructions. Assuming clear instructions and respondent understanding, two constructs are theorized to be responsible for noncompliance: respondent jeopardy and risk of suspicion [12]. The former refers to guilty respondents’ risk of being identified as such when responding affirmatively. The latter refers to innocent respondents’ risk of being identified as guilty when responding affirmatively. These risks may advocate selfprotective response behavior: evasive answering irrespective of the outcome of the randomization device. Provisions exist in the form of parameter selection and design symmetry.
The optimization of parameter selection for both the binary and quantitative FRR models in terms of reconciling respondent hazards and efficiency can be captured in the Bayesian framework for conditional RR probabilities proposed by Lanke [15, 16] and Greenberg et al. [12]. Moreover, the proposed FRR design is symmetric [4], meaning that no possible response option, in itself, conveys information on one’s true status. While less efficient than asymmetric designs, there is evidence that symmetric designs spur less cheating due to reduced risk of suspicion relative to their asymmetric counterparts [24].
These provisions may not completely eliminate selfprotective response behavior. It may subsequently be possible that the percentage of affirmative responses (in certain response categories) falls below chance level. We would then have an estimator that lies outside the interior of the parameter space, implying that the moment estimator (10) is no longer equivalent to the maximum likelihood estimator. It is here that we may also see the analytical advantage of the unified RR framework in that it allows for a certain unity in the analysis of dichotomous and quantitative RR data. van Den Hout and Van Der Heijden [28] give an elegant framework for analyzing RR data based on a loglinear latent class model analogy. Their framework unifies the chisquare test of independence and numerical maximum likelihood estimation for models of the general form (9). Their loglinear RR approach may be extended to provide prevalence estimates corrected for selfprotective response behavior [6].
Adding to the attractiveness of practicality, one may note that it is straightforward to incorporate Likertscale type questions (e.g., to study sensitive attitudes) into the quantitative FRR setup, as the distinct categories 16 could be labeled as a Likert scale . The RR framework given earlier thus provides for simple binary and quantitative RR models whose incorporation in a computerized setting may prove to be more practically feasible in real research settings. The models and CARR questionnaires as described above have been tested in our lab and fieldresearch efforts are currently being undertaken.
Acknowledgements
At the time of writing C.F.W.P. was affiliated with the Department of Methodology & Statistics at Utrecht University and K.L. was affiliated with the Department of Governance Studies, VU University Amsterdam. This version is a postprint of: Peeters, C.F.W., LensveltMulders, G.J.L.M., & Lasthuizen, K. (2010). A Note on a Simple and Practical Randomized Response Framework for Eliciting Sensitive Dichotomous and Quantitative Information. Sociological Methods & Research, 39: 283–296.
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