A Mixture Quantitative Randomized Response Model That Improves Trust in RRT Methodology

: The Quantitative Randomized Response Technique (RRT) can be used by researchers to obtain honest answers to questions that, due to their sensitive (socially undesirable, dangerous, or even illegal) nature, might otherwise invoke partially or completely falsified responses. Over the years, Quantitative RRT models, sometimes called Scrambling models, have been developed to incorporate such advancements as mixture, optionality and enhanced trust, each of which has important benefits. However, no single model incorporates all of these features. In this study, we propose just such a unified model, which we call the Mixture Optional Enhanced Trust (MOET) model. After developing methodologies to assess MOET based on standard approaches and using them to explore the key characteristics of the new model, we show that MOET has superior efficiency compared to the Quantitative Optional Enhanced Trust (OET) model. We also show that use of the model’s mixture capability allows practitioners to optimally balance the model’s efficiency with its privacy, making the model adaptable to a wide variety of research scenarios.


Introduction
When faced with uncomfortable or sensitive quantitative questions (for example, "What is your IQ?" or "What is your personal income level?"),respondents may modify or outright falsify their answers.This untruthfulness creates a significant problem for researchers; consequently, statisticians have developed a variety of clever techniques designed to encourage truthfulness in scenarios like these.Some of these techniques, for example, the Unmatched Count Technique (Raghavarao and Federer, 1979) and Social Desirability Scale based techniques (Reynolds, 1982), are best suited for binary applications [1,2].Others, for example, the Bogus Pipeline technique (Jones and Sigall, 1971) and some Randomized Response Techniques (Warner 1971, Greenberg et al., 1971, Gupta et al., 2022, etc.), apply well in quantitative scenarios [3][4][5][6].
This study focuses on certain Quantitative Randomized Response Techniques.The RRT was first pioneered by Warner when he proposed a binary-question RRT in 1965.Six years later, Warner proposed a new technique, this one applicable to quantitative questions.According to this technique, researchers would instruct respondents to apply random noise to their quantitative responses via additive or multiplicative scrambling variables.As the researcher would only see scrambled responses, the respondents' true answers would remain "hidden" or "confidential."The idea was to make the respondent feel comfortable answering a sensitive question truthfully, knowing that their response would be obfuscated by the scrambling.Statistical techniques, taking into account the known distribution of the Axioms 2024, 13, 11 2 of 13 scrambling variable, could then be used across the group of responses to backsolve for the group level mean response to the sensitive question.Greenberg (1971) developed another Quantitative RRT model based on an entirely different mechanism [5].Rather than asking each respondent the sensitive question and then instructing them to scramble their response, Greenberg suggested that each respondent should answer one of two questions-either the sensitive quantitative question or some unrelated and nonsensitive quantitative question with a known probability distribution-based on a random assignment unknown to the researcher.While the researcher would see all of the respondents' responses, the respondents' confidentiality would none the less be maintained because the researcher would not know which question each individual respondent was answering.
Several advancements to Warner's and Greenberg's models were made over the years.Metha, and Aggarwal (2018) proposed a means of estimating the sensitivity level of a sensitive binary question [7].Different kinds of scrambling techniques were explored by Diana and Perri (2011), Singh et al. (2018), and Priyanka and Trisandhya (2019) [8][9][10].Gupta et al. (2002) showed that adding optionality to RRT models (respondents may opt in or opt out of the RRT according to whether they personally find the "sensitive" question to be sensitive) significantly improved model efficiency [11].This same concept of optionality enabled measurement of the level of a quantitative question's sensitivity.Additionally, Gupta et al. (2022) introduced an enhanced trust feature that enabled respondents to opt for greater levels of scrambling if they felt their responses were not being sufficiently obscured by additive scrambling alone [6].
The importance of respondent privacy was formally recognized during this timeframe as a key RRT model attribute, necessary for motivating respondent truthfulness, and Gupta et al. (2018) developed a "unified measure" designed to evaluate quantitative RRT models based on a single statistic that incorporated the competing elements of privacy and efficiency [12].Vishwakarma et al. (2023) developed a two-stage unrelated randomized response model [13].Narjis and Shabir (2021) proposed an RRT model for estimating rare sensitive attributes using a Poisson distribution [14].The concept of "mixture"-combining Warner-based and Greenberg-based constructs into a single model where respondents are randomly assigned to one technique or the other-was incorporated into binary RRT models by Lovig (2021), but "mixture" has not been incorporated into a quantitative model prior to this study [15].
The model we propose in this study, which we call the Mixture Optional Enhanced Trust (MOET) model incorporates optionality, enhanced trust, and mixture into a single model, thereby consolidating many of the advantages of predecessor models into a single model.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy ( Axioms 2024, 12, x FOR PEER REVIEW 2 of 13 known distribution of the scrambling variable, could then be used across the group of responses to backsolve for the group level mean response to the sensitive question.Greenberg (1971) developed another Quantitative RRT model based on an entirely different mechanism [5].Rather than asking each respondent the sensitive question and then instructing them to scramble their response, Greenberg suggested that each respondent should answer one of two questions-either the sensitive quantitative question or some unrelated and nonsensitive quantitative question with a known probability distribution-based on a random assignment unknown to the researcher.While the researcher would see all of the respondents' responses, the respondents' confidentiality would none the less be maintained because the researcher would not know which question each individual respondent was answering.
Several advancements to Warner's and Greenberg's models were made over the years.Metha, and Aggarwal (2018) proposed a means of estimating the sensitivity level of a sensitive binary question [7].Different kinds of scrambling techniques were explored by Diana and Perri (2011), Singh et al. (2018), and Priyanka and Trisandhya (2019) [8][9][10].Gupta et al. (2002) showed that adding optionality to RRT models (respondents may opt in or opt out of the RRT according to whether they personally find the "sensitive" question to be sensitive) significantly improved model efficiency [11].This same concept of optionality enabled measurement of the level of a quantitative question's sensitivity.Additionally, Gupta et al. (2022) introduced an enhanced trust feature that enabled respondents to opt for greater levels of scrambling if they felt their responses were not being sufficiently obscured by additive scrambling alone [6].
The importance of respondent privacy was formally recognized during this timeframe as a key RRT model attribute, necessary for motivating respondent truthfulness, and Gupta et al. (2018) developed a "unified measure" designed to evaluate quantitative RRT models based on a single statistic that incorporated the competing elements of privacy and efficiency [12].Vishwakarma et al. (2023) developed a two-stage unrelated randomized response model [13].Narjis and Shabir (2021) proposed an RRT model for estimating rare sensitive attributes using a Poisson distribution [14].The concept of "mixture"-combining Warner-based and Greenberg-based constructs into a single model where respondents are randomly assigned to one technique or the other-was incorporated into binary RRT models by Lovig (2021), but "mixture" has not been incorporated into a quantitative model prior to this study [15].
The model we propose in this study, which we call the Mixture Optional Enhanced Trust (MOET) model incorporates optionality, enhanced trust, and mixture into a single model, thereby consolidating many of the advantages of predecessor models into a single model.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.
), and a unified measure that incorporates both efficiency and privacy (δ ) are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.

Efficiency Metric
The efficiency of any estimator µ Y can be quantified by its MSE, denoted by MSE( µ Y ).

MSE( µ
Smaller MSE values are preferred, as high levels of efficiency are achieved when MSE is small.

Privacy Metric
A measure of privacy proposed by Yan et al. (2008) [16] commonly used in quantitative models is given by timating rare sensitive attributes using a Poisson distribution [14].The concept of "mixre"-combining Warner-based and Greenberg-based constructs into a single model here respondents are randomly assigned to one technique or the other-was incorpoted into binary RRT models by Lovig (2021), but "mixture" has not been incorporated to a quantitative model prior to this study [15].
The model we propose in this study, which we call the Mixture Optional Enhanced ust (MOET) model incorporates optionality, enhanced trust, and mixture into a single odel, thereby consolidating many of the advantages of predecessor models into a single odel.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate d compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced ust (MOET) model and derive estimators for all of its key attributes.Section 4 will be voted to computer simulations.We will observe the simulations' output both as a way understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics RRT models, which enable the comparisons between models shown later in the study.measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unid measure that incorporates both efficiency and privacy () are provided.All three of ese metrics have been commonly used by other researchers to quantify and compare the cacy of RRT models.These are the metrics we will use to quantify key attributes of the OET model.We will use the same metrics to compare the characteristics of the MOET odel to those of the OET model.
where Y represents a respondent's true response to a sensitive question, while Z represents the respondent's reported response (which may be scrambled) [16].One can think of privacy and efficiency [12].Vishwakarma et al. (2023) developed a two-stage unrela randomized response model [13].Narjis and Shabir (2021) proposed an RRT model estimating rare sensitive attributes using a Poisson distribution [14].The concept of "m ture"-combining Warner-based and Greenberg-based constructs into a single mo where respondents are randomly assigned to one technique or the other-was incor rated into binary RRT models by Lovig (2021), but "mixture" has not been incorpora into a quantitative model prior to this study [15].The model we propose in this study, which we call the Mixture Optional Enhan Trust (MOET) model incorporates optionality, enhanced trust, and mixture into a sin model, thereby consolidating many of the advantages of predecessor models into a sin model.
In Section 2 of this study, we will explore the key metrics that we will use to evalu and compare RRT models.In Section 3, we will propose the Mixture Optional Enhan Trust (MOET) model and derive estimators for all of its key attributes.Section 4 wil devoted to computer simulations.We will observe the simulations' output both as a w of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteris of RRT models, which enable the comparisons between models shown later in the stu A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a u fied measure that incorporates both efficiency and privacy () are provided.All thre these metrics have been commonly used by other researchers to quantify and compare efficacy of RRT models.These are the metrics we will use to quantify key attributes of MOET model.We will use the same metrics to compare the characteristics of the MO model to those of the OET model.as a measurement of the "hiddenness" of a respondent's true response.Clearly, when respondents' reported responses lie far from respondents' true responses, this metric will be large.Higher values of key RRT model attribute, necessary for motivating respondent truthfula et al. (2018) developed a "unified measure" designed to evaluate quantiels based on a single statistic that incorporated the competing elements of ciency [12].Vishwakarma et al. (2023) developed a two-stage unrelated sponse model [13].Narjis and Shabir (2021) proposed an RRT model for sensitive attributes using a Poisson distribution [14].The concept of "mixing Warner-based and Greenberg-based constructs into a single model ents are randomly assigned to one technique or the other-was incorpory RRT models by Lovig (2021), but "mixture" has not been incorporated tive model prior to this study [15].l we propose in this study, which we call the Mixture Optional Enhanced model incorporates optionality, enhanced trust, and mixture into a single consolidating many of the advantages of predecessor models into a single 2 of this study, we will explore the key metrics that we will use to evaluate RT models.In Section 3, we will propose the Mixture Optional Enhanced model and derive estimators for all of its key attributes.Section 4 will be puter simulations.We will observe the simulations' output both as a way ng the model's behavior and in contrast to the OET model.d Methods tion, we will discuss the methods used to measure the key characteristics , which enable the comparisons between models shown later in the study.fficiency (Mean Squared Error, MSE), a measure of privacy (), and a uniat incorporates both efficiency and privacy () are provided.All three of ave been commonly used by other researchers to quantify and compare the models.These are the metrics we will use to quantify key attributes of the We will use the same metrics to compare the characteristics of the MOET of the OET model.
when Z is defined in various ways (we will call Z "Z i " when Z is defined in the ith unique way).These calculations will facilitate the calculation of an expression for the privacy of the MOET model in Section 3 of this study.
First, consider the trivial case of no scrambling.In this case we have In a case where scrambling is introduced into Z via an additive scrambling variable S (where the distribution of S is known and E(S) = 0), we have that Z = Z 2 = Y + S, and When multiplicative scrambling is implemented via a scrambling variable T (where the distribution of T is known and E(T) = 1) and additive scrambling is also included as above, we have that Z = Z 3 = TY + S, and Finally, when an unrelated question is implemented, we have that Z = Z 4 = R, and We now recall that Gupta et al. (2018) showed that optionality does not compromise privacy because respondents who do not consider a question to be sensitive do not value privacy [12].Hence, the privacy of an optional model is the same as that of a model where W, which we define as the sensitivity level of the sensitive question, is equal to 1. Finally, we note that the privacy of a composite mixture model (MOET mixes Greenberg and Warner components) can be represented as the privacy associated with Z being defined in each of m unique ways within the model (denoted with probability q 1 Z 2 with probability q 2 . . . .

Z m with probability q m
From this we can write: Section 2 of this study, we will explore the key metrics that we will use to evaluate pare RRT models.In Section 3, we will propose the Mixture Optional Enhanced OET) model and derive estimators for all of its key attributes.Section 4 will be to computer simulations.We will observe the simulations' output both as a way rstanding the model's behavior and in contrast to the OET model.

rials and Methods
this section, we will discuss the methods used to measure the key characteristics models, which enable the comparisons between models shown later in the study.ure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a uniasure that incorporates both efficiency and privacy () are provided.All three of etrics have been commonly used by other researchers to quantify and compare the of RRT models.These are the metrics we will use to quantify key attributes of the model.We will use the same metrics to compare the characteristics of the MOET o those of the OET model.
• m is the number of ways Z is uniquely defined within the model.• j is a particular categorical way that Z is defined in the model.

•
q j is the probability that category 'j' captures the respondent's response.• The superscript a indicates that privacy is adjusted according to Gupta et al.'s (2018) [12] optionality adjustment (W = 1).
Section 3.3 of this study shows how this formula for where respondents are randomly assigned to one technique or the other-was incorporated into binary RRT models by Lovig (2021), but "mixture" has not been incorporated into a quantitative model prior to this study [15].The model we propose in this study, which we call the Mixture Optional Enhanced Trust (MOET) model incorporates optionality, enhanced trust, and mixture into a single model, thereby consolidating many of the advantages of predecessor models into a single model.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.
a applies to the MOET model.

Unified Measure of Efficiency and Privacy
The following unified measure from Gupta et al. (2018) simultaneously evaluates a quantitative RRT model for its efficiency and for its privacy [12]: timeframe as a key RRT model attribute, necessary for motivating respondent truthfulness, and Gupta et al. (2018) developed a "unified measure" designed to evaluate quantitative RRT models based on a single statistic that incorporated the competing elements of privacy and efficiency [12].Vishwakarma et al. (2023) developed a two-stage unrelated randomized response model [13].Narjis and Shabir (2021) proposed an RRT model for estimating rare sensitive attributes using a Poisson distribution [14].The concept of "mixture"-combining Warner-based and Greenberg-based constructs into a single model where respondents are randomly assigned to one technique or the other-was incorporated into binary RRT models by Lovig (2021), but "mixture" has not been incorporated into a quantitative model prior to this study [15].The model we propose in this study, which we call the Mixture Optional Enhanced Trust (MOET) model incorporates optionality, enhanced trust, and mixture into a single model, thereby consolidating many of the advantages of predecessor models into a single model.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.
Low values of δ a are preferred, as both high privacy and high efficiency (low MSE) lead to a smaller δ a .

Proposed Mixture Optional Enhanced Trust Model (MOET)
We now propose the Mixture Optional Enhanced Trust model.This model features strong efficiency and unified measure performance.Additionally, the model's many features-optionality, mixture, and enhanced trust scrambling-make it highly flexible and therefore useful in a wide variety of research settings characterized by different kinds of sensitive questions and demanding different levels of efficiency and privacy.

MOET Model Introduction
Let Y be the respondent's true response to the sensitive question, while Z is their reported response.S and T are additive and multiplicative scrambling random variables, and R is a random variable representing a respondent's response to the Greenberg unrelated question.Here Y, S, T, and R are mutually independent.W represents the respondent's choice to take part in some form of an RRT (rather than simply giving a straight answer to the sensitive question without an RRT) and therefore can be considered a measure of the sensitivity of the sensitive question.The parameter α represents the proportion of respondents randomly assigned to the Warner-based model.Note that when α = 1, the MOET model becomes the OET model (See Appendix A).A represents a respondent's trust in the RRT model in absence of additional scrambling, while (1 − A) represents the proportion of respondents who require additional scrambling in order to trust the RRT.
Below, Figure 1 presents a diagram of the MOET model: portion of respondents randomly assigned to the Warner-based model.Note that when = 1, the MOET model becomes the OET model (See Appendix A).  represents a respond ent's trust in the RRT model in absence of additional scrambling, while (1 − ) repre sents the proportion of respondents who require additional scrambling in order to trus the RRT.Below, Figure 1 presents a diagram of the MOET model:

MOET: Mean Estimator
Choice of random variables S and T should be made such that E(S) = 0 and E(T) = 1.It follows from the MOET model ( 9) that Using a Split Sample approach with p 1 , p 2 , where p 1 ̸ = p 2 and E(Z i ) is estimated by Z i , we have Therefore, the mean of the sensitive trait µ Y can be estimated by The variance/MSE of our unbiased estimator, using an equally-split, split sample approach for convenience, is given by Neither A nor W values are needed to estimate µ Y .Var( µ Y ) may be estimated by the sample variance of µ Y values.

MOET: Privacy Measure
Per Equation (7), the privacy of the MOET model is given by 2 of 13 stribution of the scrambling variable, could then be used across the group of to backsolve for the group level mean response to the sensitive question.nberg (1971) developed another Quantitative RRT model based on an entirely mechanism [5].Rather than asking each respondent the sensitive question and ucting them to scramble their response, Greenberg suggested that each respondd answer one of two questions-either the sensitive quantitative question or elated and nonsensitive quantitative question with a known probability distriased on a random assignment unknown to the researcher.While the researcher all of the respondents' responses, the respondents' confidentiality would none maintained because the researcher would not know which question each indipondent was answering.ral advancements to Warner's and Greenberg's models were made over the tha, and Aggarwal (2018) proposed a means of estimating the sensitivity level of binary question [7].Different kinds of scrambling techniques were explored by Perri ( 2011 2002) showed that adding optionality to RRT models (respondents may opt ut of the RRT according to whether they personally find the "sensitive" question itive) significantly improved model efficiency [11].This same concept of optionled measurement of the level of a quantitative question's sensitivity.Additiona et al. ( 2022) introduced an enhanced trust feature that enabled respondents to eater levels of scrambling if they felt their responses were not being sufficiently by additive scrambling alone [6].importance of respondent privacy was formally recognized during this as a key RRT model attribute, necessary for motivating respondent truthful-Gupta et al. ( 2018) developed a "unified measure" designed to evaluate quantimodels based on a single statistic that incorporated the competing elements of nd efficiency [12].Vishwakarma et al. ( 2023) developed a two-stage unrelated ed response model [13].Narjis and Shabir (2021) proposed an RRT model for rare sensitive attributes using a Poisson distribution [14].The concept of "mixmbining Warner-based and Greenberg-based constructs into a single model pondents are randomly assigned to one technique or the other-was incorpobinary RRT models by Lovig (2021), but "mixture" has not been incorporated ntitative model prior to this study [15].
odel we propose in this study, which we call the Mixture Optional Enhanced ET) model incorporates optionality, enhanced trust, and mixture into a single ereby consolidating many of the advantages of predecessor models into a single ction 2 of this study, we will explore the key metrics that we will use to evaluate are RRT models.In Section 3, we will propose the Mixture Optional Enhanced ET) model and derive estimators for all of its key attributes.Section 4 will be o computer simulations.We will observe the simulations' output both as a way tanding the model's behavior and in contrast to the OET model.

ls and Methods
is section, we will discuss the methods used to measure the key characteristics odels, which enable the comparisons between models shown later in the study.e of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a uniure that incorporates both efficiency and privacy () are provided.All three of rics have been commonly used by other researchers to quantify and compare the f RRT models.These are the metrics we will use to quantify key attributes of the odel.We will use the same metrics to compare the characteristics of the MOET those of the OET model.
own distribution of the scrambling variable, could then be used across the group of sponses to backsolve for the group level mean response to the sensitive question.Greenberg (1971) developed another Quantitative RRT model based on an entirely fferent mechanism [5].Rather than asking each respondent the sensitive question and en instructing them to scramble their response, Greenberg suggested that each respondt should answer one of two questions-either the sensitive quantitative question or me unrelated and nonsensitive quantitative question with a known probability distrition-based on a random assignment unknown to the researcher.While the researcher ould see all of the respondents' responses, the respondents' confidentiality would none e less be maintained because the researcher would not know which question each indidual respondent was answering.
pta et al. (2002) showed that adding optionality to RRT models (respondents may opt or opt out of the RRT according to whether they personally find the "sensitive" question be sensitive) significantly improved model efficiency [11].This same concept of optionity enabled measurement of the level of a quantitative question's sensitivity.Additionly, Gupta et al. ( 2022) introduced an enhanced trust feature that enabled respondents to t for greater levels of scrambling if they felt their responses were not being sufficiently scured by additive scrambling alone [6].
The importance of respondent privacy was formally recognized during this eframe as a key RRT model attribute, necessary for motivating respondent truthfulss, and Gupta et al. ( 2018) developed a "unified measure" designed to evaluate quantitive RRT models based on a single statistic that incorporated the competing elements of ivacy and efficiency [12].Vishwakarma et al. ( 2023) developed a two-stage unrelated ndomized response model [13].Narjis and Shabir (2021) proposed an RRT model for timating rare sensitive attributes using a Poisson distribution [14].The concept of "mixre"-combining Warner-based and Greenberg-based constructs into a single model here respondents are randomly assigned to one technique or the other-was incorpoted into binary RRT models by Lovig (2021), but "mixture" has not been incorporated to a quantitative model prior to this study [15].
The model we propose in this study, which we call the Mixture Optional Enhanced ust (MOET) model incorporates optionality, enhanced trust, and mixture into a single odel, thereby consolidating many of the advantages of predecessor models into a single odel.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate d compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced ust (MOET) model and derive estimators for all of its key attributes.Section 4 will be voted to computer simulations.We will observe the simulations' output both as a way understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics RRT models, which enable the comparisons between models shown later in the study.measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unid measure that incorporates both efficiency and privacy () are provided.All three of ese metrics have been commonly used by other researchers to quantify and compare the cacy of RRT models.These are the metrics we will use to quantify key attributes of the OET model.We will use the same metrics to compare the characteristics of the MOET odel to those of the OET model.
known distribution of the scrambling variable, could then be used across the group of responses to backsolve for the group level mean response to the sensitive question.Greenberg (1971) developed another Quantitative RRT model based on an entirely different mechanism [5].Rather than asking each respondent the sensitive question and then instructing them to scramble their response, Greenberg suggested that each respondent should answer one of two questions-either the sensitive quantitative question or some unrelated and nonsensitive quantitative question with a known probability distribution-based on a random assignment unknown to the researcher.While the researcher would see all of the respondents' responses, the respondents' confidentiality would none the less be maintained because the researcher would not know which question each individual respondent was answering.
Several advancements to Warner's and Greenberg's models were made over the years.Metha, and Aggarwal (2018) proposed a means of estimating the sensitivity level of a sensitive binary question [7].Different kinds of scrambling techniques were explored by Diana and Perri (2011), Singh et al. (2018), and Priyanka and Trisandhya (2019) [8][9][10].Gupta et al. (2002) showed that adding optionality to RRT models (respondents may opt in or opt out of the RRT according to whether they personally find the "sensitive" question to be sensitive) significantly improved model efficiency [11].This same concept of optionality enabled measurement of the level of a quantitative question's sensitivity.Additionally, Gupta et al. ( 2022) introduced an enhanced trust feature that enabled respondents to opt for greater levels of scrambling if they felt their responses were not being sufficiently obscured by additive scrambling alone [6].
The importance of respondent privacy was formally recognized during this timeframe as a key RRT model attribute, necessary for motivating respondent truthfulness, and Gupta et al. (2018) developed a "unified measure" designed to evaluate quantitative RRT models based on a single statistic that incorporated the competing elements of privacy and efficiency [12].Vishwakarma et al. ( 2023) developed a two-stage unrelated randomized response model [13].Narjis and Shabir (2021) proposed an RRT model for estimating rare sensitive attributes using a Poisson distribution [14].The concept of "mixture"-combining Warner-based and Greenberg-based constructs into a single model where respondents are randomly assigned to one technique or the other-was incorporated into binary RRT models by Lovig (2021), but "mixture" has not been incorporated into a quantitative model prior to this study [15].
The model we propose in this study, which we call the Mixture Optional Enhanced Trust (MOET) model incorporates optionality, enhanced trust, and mixture into a single model, thereby consolidating many of the advantages of predecessor models into a single model.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.
known distribution of the scrambling variable, could then be used across the group of responses to backsolve for the group level mean response to the sensitive question.Greenberg (1971) developed another Quantitative RRT model based on an entirely different mechanism [5].Rather than asking each respondent the sensitive question and then instructing them to scramble their response, Greenberg suggested that each respondent should answer one of two questions-either the sensitive quantitative question or some unrelated and nonsensitive quantitative question with a known probability distribution-based on a random assignment unknown to the researcher.While the researcher would see all of the respondents' responses, the respondents' confidentiality would none the less be maintained because the researcher would not know which question each individual respondent was answering.
Several advancements to Warner's and Greenberg's models were made over the years.Metha, and Aggarwal (2018) proposed a means of estimating the sensitivity level of a sensitive binary question [7].Different kinds of scrambling techniques were explored by Diana and Perri (2011), Singh et al. (2018), and Priyanka and Trisandhya (2019) [8][9][10].Gupta et al. (2002) showed that adding optionality to RRT models (respondents may opt in or opt out of the RRT according to whether they personally find the "sensitive" question to be sensitive) significantly improved model efficiency [11].This same concept of optionality enabled measurement of the level of a quantitative question's sensitivity.Additionally, Gupta et al. ( 2022) introduced an enhanced trust feature that enabled respondents to opt for greater levels of scrambling if they felt their responses were not being sufficiently obscured by additive scrambling alone [6].
The importance of respondent privacy was formally recognized during this timeframe as a key RRT model attribute, necessary for motivating respondent truthfulness, and Gupta et al. (2018) developed a "unified measure" designed to evaluate quantitative RRT models based on a single statistic that incorporated the competing elements of privacy and efficiency [12].Vishwakarma et al. ( 2023) developed a two-stage unrelated randomized response model [13].Narjis and Shabir (2021) proposed an RRT model for estimating rare sensitive attributes using a Poisson distribution [14].The concept of "mixture"-combining Warner-based and Greenberg-based constructs into a single model where respondents are randomly assigned to one technique or the other-was incorporated into binary RRT models by Lovig (2021), but "mixture" has not been incorporated into a quantitative model prior to this study [15].
The model we propose in this study, which we call the Mixture Optional Enhanced Trust (MOET) model incorporates optionality, enhanced trust, and mixture into a single model, thereby consolidating many of the advantages of predecessor models into a single model.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.
known distribution of the scrambling variable, could then be used across the group of responses to backsolve for the group level mean response to the sensitive question.Greenberg (1971) developed another Quantitative RRT model based on an entirely different mechanism [5].Rather than asking each respondent the sensitive question and then instructing them to scramble their response, Greenberg suggested that each respondent should answer one of two questions-either the sensitive quantitative question or some unrelated and nonsensitive quantitative question with a known probability distribution-based on a random assignment unknown to the researcher.While the researcher would see all of the respondents' responses, the respondents' confidentiality would none the less be maintained because the researcher would not know which question each individual respondent was answering.
Several advancements to Warner's and Greenberg's models were made over the years.Metha, and Aggarwal (2018) proposed a means of estimating the sensitivity level of a sensitive binary question [7].Different kinds of scrambling techniques were explored by Diana and Perri (2011), Singh et al. (2018), and Priyanka and Trisandhya (2019) [8][9][10].Gupta et al. (2002) showed that adding optionality to RRT models (respondents may opt in or opt out of the RRT according to whether they personally find the "sensitive" question to be sensitive) significantly improved model efficiency [11].This same concept of optionality enabled measurement of the level of a quantitative question's sensitivity.Additionally, Gupta et al. ( 2022) introduced an enhanced trust feature that enabled respondents to opt for greater levels of scrambling if they felt their responses were not being sufficiently obscured by additive scrambling alone [6].
The importance of respondent privacy was formally recognized during this timeframe as a key RRT model attribute, necessary for motivating respondent truthfulness, and Gupta et al. (2018) developed a "unified measure" designed to evaluate quantitative RRT models based on a single statistic that incorporated the competing elements of privacy and efficiency [12].Vishwakarma et al. (2023) developed a two-stage unrelated randomized response model [13].Narjis and Shabir (2021) proposed an RRT model for estimating rare sensitive attributes using a Poisson distribution [14].The concept of "mixture"-combining Warner-based and Greenberg-based constructs into a single model where respondents are randomly assigned to one technique or the other-was incorporated into binary RRT models by Lovig (2021), but "mixture" has not been incorporated into a quantitative model prior to this study [15].
The model we propose in this study, which we call the Mixture Optional Enhanced Trust (MOET) model incorporates optionality, enhanced trust, and mixture into a single model, thereby consolidating many of the advantages of predecessor models into a single model.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.

.
( Analysis of the diagram of the MOET model in Section 3.1 above shows that dding optionality to RRT models (respondents may opt to whether they personally find the "sensitive" question oved model efficiency [11].This same concept of optionlevel of a quantitative question's sensitivity.Additiond an enhanced trust feature that enabled respondents to g if they felt their responses were not being sufficiently alone [6].dent privacy was formally recognized during this ttribute, necessary for motivating respondent truthfulloped a "unified measure" designed to evaluate quantigle statistic that incorporated the competing elements of akarma et al. ( 2023) developed a two-stage unrelated .Narjis and Shabir (2021) proposed an RRT model for s using a Poisson distribution [14].The concept of "mixand Greenberg-based constructs into a single model assigned to one technique or the other-was incorpo-Lovig (2021), but "mixture" has not been incorporated this study [15].is study, which we call the Mixture Optional Enhanced s optionality, enhanced trust, and mixture into a single y of the advantages of predecessor models into a single will explore the key metrics that we will use to evaluate tion 3, we will propose the Mixture Optional Enhanced estimators for all of its key attributes.Section 4 will be .We will observe the simulations' output both as a way avior and in contrast to the OET model.ss the methods used to measure the key characteristics comparisons between models shown later in the study.uared Error, MSE), a measure of privacy (), and a unith efficiency and privacy () are provided.All three of y used by other researchers to quantify and compare the the metrics we will use to quantify key attributes of the me metrics to compare the characteristics of the MOET .
e scrambling variable, could then be used across the group of r the group level mean response to the sensitive question.veloped another Quantitative RRT model based on an entirely Rather than asking each respondent the sensitive question and cramble their response, Greenberg suggested that each respondf two questions-either the sensitive quantitative question or ensitive quantitative question with a known probability distrim assignment unknown to the researcher.While the researcher ndents' responses, the respondents' confidentiality would none ause the researcher would not know which question each indiswering.ts to Warner's and Greenberg's models were made over the al (2018) proposed a means of estimating the sensitivity level of [7].Different kinds of scrambling techniques were explored by ingh et al. ( 2018), and Priyanka and Trisandhya (2019) [8][9][10].d that adding optionality to RRT models (respondents may opt cording to whether they personally find the "sensitive" question ly improved model efficiency [11].This same concept of optiont of the level of a quantitative question's sensitivity.Additiontroduced an enhanced trust feature that enabled respondents to rambling if they felt their responses were not being sufficiently mbling alone [6].respondent privacy was formally recognized during this model attribute, necessary for motivating respondent truthful-8) developed a "unified measure" designed to evaluate quantion a single statistic that incorporated the competing elements of ].Vishwakarma et al. ( 2023) developed a two-stage unrelated del [13].Narjis and Shabir (2021) proposed an RRT model for ttributes using a Poisson distribution [14].The concept of "mixr-based and Greenberg-based constructs into a single model ndomly assigned to one technique or the other-was incorpodels by Lovig (2021), but "mixture" has not been incorporated prior to this study [15].se in this study, which we call the Mixture Optional Enhanced rporates optionality, enhanced trust, and mixture into a single ing many of the advantages of predecessor models into a single dy, we will explore the key metrics that we will use to evaluate .In Section 3, we will propose the Mixture Optional Enhanced derive estimators for all of its key attributes.Section 4 will be lations.We will observe the simulations' output both as a way el's behavior and in contrast to the OET model.
ill discuss the methods used to measure the key characteristics ble the comparisons between models shown later in the study.ean Squared Error, MSE), a measure of privacy (), and a unirates both efficiency and privacy () are provided.All three of mmonly used by other researchers to quantify and compare the hese are the metrics we will use to quantify key attributes of the e the same metrics to compare the characteristics of the MOET model.
ibution of the scrambling variable, could then be used across the group of backsolve for the group level mean response to the sensitive question.rg (1971) developed another Quantitative RRT model based on an entirely hanism [5].Rather than asking each respondent the sensitive question and ing them to scramble their response, Greenberg suggested that each respondnswer one of two questions-either the sensitive quantitative question or ted and nonsensitive quantitative question with a known probability distrid on a random assignment unknown to the researcher.While the researcher l of the respondents' responses, the respondents' confidentiality would none aintained because the researcher would not know which question each indindent was answering.advancements to Warner's and Greenberg's models were made over the , and Aggarwal (2018) proposed a means of estimating the sensitivity level of nary question [7].Different kinds of scrambling techniques were explored by erri (2011), Singh et al. (2018), and Priyanka and Trisandhya (2019) [8][9][10].2002) showed that adding optionality to RRT models (respondents may opt of the RRT according to whether they personally find the "sensitive" question e) significantly improved model efficiency [11].This same concept of optionmeasurement of the level of a quantitative question's sensitivity.Additiont al. (2022) introduced an enhanced trust feature that enabled respondents to er levels of scrambling if they felt their responses were not being sufficiently additive scrambling alone [6].portance of respondent privacy was formally recognized during this a key RRT model attribute, necessary for motivating respondent truthfulpta et al. (2018) developed a "unified measure" designed to evaluate quantiodels based on a single statistic that incorporated the competing elements of efficiency [12].Vishwakarma et al. ( 2023) developed a two-stage unrelated response model [13].Narjis and Shabir (2021) proposed an RRT model for re sensitive attributes using a Poisson distribution [14].The concept of "mixining Warner-based and Greenberg-based constructs into a single model ndents are randomly assigned to one technique or the other-was incorponary RRT models by Lovig (2021), but "mixture" has not been incorporated tative model prior to this study [15].del we propose in this study, which we call the Mixture Optional Enhanced ) model incorporates optionality, enhanced trust, and mixture into a single by consolidating many of the advantages of predecessor models into a single n 2 of this study, we will explore the key metrics that we will use to evaluate RRT models.In Section 3, we will propose the Mixture Optional Enhanced ) model and derive estimators for all of its key attributes.Section 4 will be mputer simulations.We will observe the simulations' output both as a way ding the model's behavior and in contrast to the OET model.

and Methods
ection, we will discuss the methods used to measure the key characteristics ls, which enable the comparisons between models shown later in the study.f efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unithat incorporates both efficiency and privacy () are provided.All three of have been commonly used by other researchers to quantify and compare the T models.These are the metrics we will use to quantify key attributes of the l.We will use the same metrics to compare the characteristics of the MOET se of the OET model.
PEER REVIEW 2 of 13 known distribution of the scrambling variable, could then be used across the group of responses to backsolve for the group level mean response to the sensitive question.Greenberg (1971) developed another Quantitative RRT model based on an entirely different mechanism [5].Rather than asking each respondent the sensitive question and then instructing them to scramble their response, Greenberg suggested that each respondent should answer one of two questions-either the sensitive quantitative question or some unrelated and nonsensitive quantitative question with a known probability distribution-based on a random assignment unknown to the researcher.While the researcher would see all of the respondents' responses, the respondents' confidentiality would none the less be maintained because the researcher would not know which question each individual respondent was answering.
Several advancements to Warner's and Greenberg's models were made over the years.Metha, and Aggarwal (2018) proposed a means of estimating the sensitivity level of a sensitive binary question [7].Different kinds of scrambling techniques were explored by Diana and Perri (2011), Singh et al. (2018), and Priyanka and Trisandhya (2019) [8][9][10].Gupta et al. (2002) showed that adding optionality to RRT models (respondents may opt in or opt out of the RRT according to whether they personally find the "sensitive" question to be sensitive) significantly improved model efficiency [11].This same concept of optionality enabled measurement of the level of a quantitative question's sensitivity.Additionally, Gupta et al. ( 2022) introduced an enhanced trust feature that enabled respondents to opt for greater levels of scrambling if they felt their responses were not being sufficiently obscured by additive scrambling alone [6].
The importance of respondent privacy was formally recognized during this timeframe as a key RRT model attribute, necessary for motivating respondent truthfulness, and Gupta et al. (2018) developed a "unified measure" designed to evaluate quantitative RRT models based on a single statistic that incorporated the competing elements of privacy and efficiency [12].Vishwakarma et al. ( 2023) developed a two-stage unrelated randomized response model [13].Narjis and Shabir (2021) proposed an RRT model for estimating rare sensitive attributes using a Poisson distribution [14].The concept of "mixture"-combining Warner-based and Greenberg-based constructs into a single model where respondents are randomly assigned to one technique or the other-was incorporated into binary RRT models by Lovig (2021), but "mixture" has not been incorporated into a quantitative model prior to this study [15].
The model we propose in this study, which we call the Mixture Optional Enhanced Trust (MOET) model incorporates optionality, enhanced trust, and mixture into a single model, thereby consolidating many of the advantages of predecessor models into a single model.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.pose the Mixture Optional Enhanced of its key attributes.Section 4 will be the simulations' output both as a way rast to the OET model.sed to measure the key characteristics een models shown later in the study.), a measure of privacy (), and a uniprivacy () are provided.All three of searchers to quantify and compare the ill use to quantify key attributes of the pare the characteristics of the MOET Reducing further, we have the following expression, which represents privacy for the MOET model.The assumption of equal-split sampling underlies this formula, but a similar formula could easily be developed to represent unequal splitting.
the Mixture Optional Enhanced trust, and mixture into a single redecessor models into a single trics that we will use to evaluate the Mixture Optional Enhanced key attributes.Section 4 will be ulations' output both as a way the OET model.
measure the key characteristics odels shown later in the study.asure of privacy (), and a uniy () are provided.All three of ers to quantify and compare the to quantify key attributes of the the characteristics of the MOET )

MOET: Sensitivity Estimator
Recall that the two samples used to estimate µ Y yield the equations Estimating E(Z i ) by Z i and µ Y by µ Y , we have: Solving the above expression for W in samples 1 and 2 and then combining estimates, we have: Inserting our estimator for µ Y , this expression reduces to From Equation ( 19) one can see that this estimator becomes unstable when µ R is close to µ Y .

Discussion
In this section of the study, we provide two tables that represent theoretical and simulated values and illustrate important characteristics of the MOET model.Each row of each table represents the output from a particular scenario (combination of A, W, and α values).Each scenario represents a sampling of n = 500 respondents within an RRT sampling scenario, and each simulation sampling is conducted N = 10,000 times in R software.Table 1 shows that the estimators developed in this study perform in line with theoretical expectations.Table 1 also establishes the value of the mixture model by showing that the MOET model-which is fundamentally a mixture of a Greenberg-based and a Warner-based model-yields performance preferable to either model on its own.Table 2 compares MOET model statistics to published OET model statistics in Gupta et al. (2022), and thereby demonstrates the MSE and unified measure superiority of the MOET model to the OET model [6].
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.( level mean response to the sensitive question.ther Quantitative RRT model based on an entirely asking each respondent the sensitive question and ir response, Greenberg suggested that each respondtions-either the sensitive quantitative question or antitative question with a known probability distrient unknown to the researcher.While the researcher onses, the respondents' confidentiality would none searcher would not know which question each indier's and Greenberg's models were made over the oposed a means of estimating the sensitivity level of nt kinds of scrambling techniques were explored by (2018), and Priyanka and Trisandhya (2019) [8][9][10].g optionality to RRT models (respondents may opt hether they personally find the "sensitive" question d model efficiency [11].This same concept of optionel of a quantitative question's sensitivity.Additionenhanced trust feature that enabled respondents to they felt their responses were not being sufficiently e [6].t privacy was formally recognized during this ute, necessary for motivating respondent truthfuled a "unified measure" designed to evaluate quantitatistic that incorporated the competing elements of arma et al. ( 2023) developed a two-stage unrelated arjis and Shabir (2021) proposed an RRT model for ing a Poisson distribution [14].The concept of "mixd Greenberg-based constructs into a single model igned to one technique or the other-was incorpoig (2021), but "mixture" has not been incorporated s study [15].udy, which we call the Mixture Optional Enhanced tionality, enhanced trust, and mixture into a single f the advantages of predecessor models into a single l explore the key metrics that we will use to evaluate 3, we will propose the Mixture Optional Enhanced mators for all of its key attributes.Section 4 will be will observe the simulations' output both as a way or and in contrast to the OET model.he methods used to measure the key characteristics parisons between models shown later in the study.ed Error, MSE), a measure of privacy (), and a unifficiency and privacy () are provided.All three of ed by other researchers to quantify and compare the metrics we will use to quantify key attributes of the metrics to compare the characteristics of the MOET a ) T and ( he group level mean response to the sensitive question.loped another Quantitative RRT model based on an entirely ther than asking each respondent the sensitive question and amble their response, Greenberg suggested that each respondtwo questions-either the sensitive quantitative question or sitive quantitative question with a known probability distriassignment unknown to the researcher.While the researcher ents' responses, the respondents' confidentiality would none use the researcher would not know which question each indiering.to Warner's and Greenberg's models were made over the l (2018) proposed a means of estimating the sensitivity level of 7].Different kinds of scrambling techniques were explored by gh et al. (2018), and Priyanka and Trisandhya (2019) [8][9][10].that adding optionality to RRT models (respondents may opt rding to whether they personally find the "sensitive" question improved model efficiency [11].This same concept of optionof the level of a quantitative question's sensitivity.Additionoduced an enhanced trust feature that enabled respondents to mbling if they felt their responses were not being sufficiently bling alone [6].espondent privacy was formally recognized during this odel attribute, necessary for motivating respondent truthful-) developed a "unified measure" designed to evaluate quantia single statistic that incorporated the competing elements of Vishwakarma et al. ( 2023) developed a two-stage unrelated el [13].Narjis and Shabir (2021) proposed an RRT model for ributes using a Poisson distribution [14].The concept of "mixbased and Greenberg-based constructs into a single model domly assigned to one technique or the other-was incorpoels by Lovig (2021), but "mixture" has not been incorporated rior to this study [15]. in this study, which we call the Mixture Optional Enhanced porates optionality, enhanced trust, and mixture into a single g many of the advantages of predecessor models into a single y, we will explore the key metrics that we will use to evaluate In Section 3, we will propose the Mixture Optional Enhanced erive estimators for all of its key attributes.Section 4 will be ations.We will observe the simulations' output both as a way l's behavior and in contrast to the OET model.discuss the methods used to measure the key characteristics le the comparisons between models shown later in the study.an Squared Error, MSE), a measure of privacy (), and a unites both efficiency and privacy () are provided.All three of monly used by other researchers to quantify and compare the se are the metrics we will use to quantify key attributes of the the same metrics to compare the characteristics of the MOET odel.
a ) E represent theoretical and empirical model privacy.(δ a ) T and (δ a ) E represent theoretical and empirical unified measure.
Table 1 shows the results from the MOET model run according to the parameter values listed.The close fit between the empirical and theoretical values speaks to the veracity of the formulae proposed in this study, as well as the accuracy of the empirical simulations.
We draw attention to three particularly important aspects of the scenario presented in Table 1.First, observe that in the shown scenario, maximum efficiency (minimum MSE) occurs when α = 0 for any pairing of A and W values. Mathematically, this does not have to be true in every scenario.For example, in the scenario underlying the table above, the extreme choice of σ 2 R = 7 with all other assumptions left unchanged will lead to a circumstance where maximum efficiency is found at α = 1 rather than α = 0. To study this important relationship closely, we let f (a) = Var( µ Y ), per Equation ( 13), and then take the derivative of this expression with respect to a. Expressing the result in the form below, we can see that f ′(a) will always be positive under certain conditions.
ating many of the advantages of predecessor models into a single tudy, we will explore the key metrics that we will use to evaluate ls.In Section 3, we will propose the Mixture Optional Enhanced d derive estimators for all of its key attributes.Section 4 will be ulations.We will observe the simulations' output both as a way del's behavior and in contrast to the OET model.

s
ill discuss the methods used to measure the key characteristics able the comparisons between models shown later in the study.(Mean Squared Error, MSE), a measure of privacy (), and a uniorates both efficiency and privacy () are provided.All three of ommonly used by other researchers to quantify and compare the These are the metrics we will use to quantify key attributes of the se the same metrics to compare the characteristics of the MOET T model.
model, thereby consolidating many of the advantages of predecessor models into a single model.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.Priyanka and Trisandhya (2019) [8][9][10].d that adding optionality to RRT models (respondents may opt cording to whether they personally find the "sensitive" question ly improved model efficiency [11].This same concept of optiont of the level of a quantitative question's sensitivity.Additiontroduced an enhanced trust feature that enabled respondents to rambling if they felt their responses were not being sufficiently mbling alone [6].respondent privacy was formally recognized during this odel attribute, necessary for motivating respondent truthful-8) developed a "unified measure" designed to evaluate quantin a single statistic that incorporated the competing elements of ].Vishwakarma et al. (2023) developed a two-stage unrelated del [13].Narjis and Shabir (2021) proposed an RRT model for ttributes using a Poisson distribution [14].The concept of "mixr-based and Greenberg-based constructs into a single model ndomly assigned to one technique or the other-was incorpodels by Lovig (2021), but "mixture" has not been incorporated prior to this study [15].se in this study, which we call the Mixture Optional Enhanced rporates optionality, enhanced trust, and mixture into a single ing many of the advantages of predecessor models into a single dy, we will explore the key metrics that we will use to evaluate .In Section 3, we will propose the Mixture Optional Enhanced derive estimators for all of its key attributes.Section 4 will be lations.We will observe the simulations' output both as a way el's behavior and in contrast to the OET model.
ill discuss the methods used to measure the key characteristics ble the comparisons between models shown later in the study.ean Squared Error, MSE), a measure of privacy (), and a unirates both efficiency and privacy () are provided.All three of mmonly used by other researchers to quantify and compare the hese are the metrics we will use to quantify key attributes of the e the same metrics to compare the characteristics of the MOET model.a ) T , and (δ a ) T represent the theoretical mean squared error, privacy and unified measure for the respective models.Green indicates model superiority.
Specifically, under the following two sufficient but not necessary conditions, f ′(a) will be positive across all a values, so Var( µ Y ) will be an increasing function of a.That is, maximum efficiency (minimum MSE) will always occur when α = 0 if The first condition sets µ R equal to µ Y , which seems both reasonable and intuitively appealing, as the response to the unrelated question should be designed to have a similar In a real-life scenario, a researcher could choose µ R to accommodate their research needs, choosing µ R far from µ Y if an estimate of W is needed.
The Table 2 scenario shows the MOET model's efficiency to be superior to that of the OET model.For example, in the circled row, when A = 0.95 and W = 0.9, we can see that (MSE = 0.0089) < (MSE = 0.0454), implying that the MOET has superior efficiency; this efficiency superiority in fact holds true for all A and W values in the table.It follows that in scenarios where efficiency is significantly more important than privacy, the MOET model will often be the better choice of model.However, the OET model outperforms the MOET in terms of privacy.See Figures 2 and 3 below that illustrate the tabular data above: Axioms 2024, 12, x FOR PEER REVIEW 10 of 13 The Table 2 scenario shows the MOET model's efficiency to be superior to that of the OET model.For example, in the circled row, when A = 0.95 and W = 0.9, we can see that (MSE = 0.0089) < (MSE = 0.0454), implying that the MOET has superior efficiency; this efficiency superiority in fact holds true for all A and W values in the table.It follows that in scenarios where efficiency is significantly more important than privacy, the MOET model will often be the better choice of model.However, the OET model outperforms the MOET in terms of privacy.See Figures 2 and 3 below that illustrate the tabular data above:  Parameter values underlying these exhibits are those cited in Table 2.
Axioms 2024, 12, x FOR PEER REVIEW 10 of 13 The Table 2 scenario shows the MOET model's efficiency to be superior to that of the OET model.For example, in the circled row, when A = 0.95 and W = 0.9, we can see that (MSE = 0.0089) < (MSE = 0.0454), implying that the MOET has superior efficiency; this efficiency superiority in fact holds true for all A and W values in the table.It follows that in scenarios where efficiency is significantly more important than privacy, the MOET model will often be the better choice of model.However, the OET model outperforms the MOET in terms of privacy.See Figures 2 and 3 below that illustrate the tabular data above:   While in the illustrated scenario MOET outperforms OET in terms of efficiency, OET outperforms MOET in terms of privacy.When we combine the two measures, we see that the MOET model outperforms the OET model according to the unified measure.For example, when A = 1 and W = 1, we can see in Table 2 that (δ a = 0.0056) < (δ a = 0.0121); the superiority of the MOET model in terms of unified measure holds true for all A and W values.

Conclusions
The Mixture-Optional Enhanced Trust model proposed in this study has important advantages over the OET model, which in turn was shown to be superior to the basic Warner model (Gupta et al. 2022) [6].Specifically, the MOET model can yield lower MSE (therefore higher efficiency) than the OET model when MOET parameters are elected that favor efficiency over privacy.But the OET model generally achieves superior privacy, so it will usually be the better model in cases where the need for privacy is the overriding concern.This lower MSE of the MOET model offsets the higher privacy offered by the OET model, as indicated by superior unified measure values.In this study, we have shown, furthermore, that the MOET model's mixture capability (captured by model parameter α) causes MOET to be superior to either a full Warner-based or a full Greenberg-based model.This mixture model will balance the competing concerns of privacy and efficiency, which will always be preferable to fully sacrificing one of these important characteristics for the other.2022) provided estimators for the OET's mean, variance of mean estimator (efficiency), and privacy, as given by [6]:

Optional Enhanced Trust Model
Mean Estimator: Variance of Mean Estimator (Efficiency): If W and/or A are unknown, the variance of the mean estimator can be estimated by where s 2 Z is the variance in sample responses.Privacy: tribution of the scrambling variable, could then be used across the group of to backsolve for the group level mean response to the sensitive question.berg (1971) developed another Quantitative RRT model based on an entirely echanism [5].Rather than asking each respondent the sensitive question and cting them to scramble their response, Greenberg suggested that each respondanswer one of two questions-either the sensitive quantitative question or lated and nonsensitive quantitative question with a known probability distrised on a random assignment unknown to the researcher.While the researcher all of the respondents' responses, the respondents' confidentiality would none maintained because the researcher would not know which question each indiondent was answering.al advancements to Warner's and Greenberg's models were made over the ha, and Aggarwal (2018) proposed a means of estimating the sensitivity level of binary question [7].Different kinds of scrambling techniques were explored by Perri (2011), Singh et al. (2018), and Priyanka and Trisandhya (2019) [8][9][10].l. (2002) showed that adding optionality to RRT models (respondents may opt t of the RRT according to whether they personally find the "sensitive" question tive) significantly improved model efficiency [11].This same concept of optioned measurement of the level of a quantitative question's sensitivity.Additionet al. ( 2022) introduced an enhanced trust feature that enabled respondents to ater levels of scrambling if they felt their responses were not being sufficiently y additive scrambling alone [6].mportance of respondent privacy was formally recognized during this as a key RRT model attribute, necessary for motivating respondent truthfulupta et al. (2018) developed a "unified measure" designed to evaluate quantimodels based on a single statistic that incorporated the competing elements of d efficiency [12].Vishwakarma et al. ( 2023) developed a two-stage unrelated d response model [13].Narjis and Shabir (2021) proposed an RRT model for rare sensitive attributes using a Poisson distribution [14].The concept of "mixbining Warner-based and Greenberg-based constructs into a single model ondents are randomly assigned to one technique or the other-was incorpobinary RRT models by Lovig (2021), but "mixture" has not been incorporated titative model prior to this study [15].odel we propose in this study, which we call the Mixture Optional Enhanced ET) model incorporates optionality, enhanced trust, and mixture into a single reby consolidating many of the advantages of predecessor models into a single tion 2 of this study, we will explore the key metrics that we will use to evaluate re RRT models.In Section 3, we will propose the Mixture Optional Enhanced ET) model and derive estimators for all of its key attributes.Section 4 will be computer simulations.We will observe the simulations' output both as a way anding the model's behavior and in contrast to the OET model.

ls and Methods
s section, we will discuss the methods used to measure the key characteristics dels, which enable the comparisons between models shown later in the study. of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unire that incorporates both efficiency and privacy () are provided.All three of ics have been commonly used by other researchers to quantify and compare the RRT models.These are the metrics we will use to quantify key attributes of the del.We will use the same metrics to compare the characteristics of the MOET ose of the OET model.
Unified Measure: known distribution of the scrambling variable, could then be used across the group of responses to backsolve for the group level mean response to the sensitive question.Greenberg (1971) developed another Quantitative RRT model based on an entirely different mechanism [5].Rather than asking each respondent the sensitive question and then instructing them to scramble their response, Greenberg suggested that each respondent should answer one of two questions-either the sensitive quantitative question or some unrelated and nonsensitive quantitative question with a known probability distribution-based on a random assignment unknown to the researcher.While the researcher would see all of the respondents' responses, the respondents' confidentiality would none the less be maintained because the researcher would not know which question each individual respondent was answering.
Several advancements to Warner's and Greenberg's models were made over the years.Metha, and Aggarwal (2018) proposed a means of estimating the sensitivity level of a sensitive binary question [7].Different kinds of scrambling techniques were explored by Diana and Perri (2011), Singh et al. (2018), and Priyanka and Trisandhya (2019) [8][9][10].Gupta et al. (2002) showed that adding optionality to RRT models (respondents may opt in or opt out of the RRT according to whether they personally find the "sensitive" question to be sensitive) significantly improved model efficiency [11].This same concept of optionality enabled measurement of the level of a quantitative question's sensitivity.Additionally, Gupta et al. (2022) introduced an enhanced trust feature that enabled respondents to opt for greater levels of scrambling if they felt their responses were not being sufficiently obscured by additive scrambling alone [6].
The importance of respondent privacy was formally recognized during this timeframe as a key RRT model attribute, necessary for motivating respondent truthfulness, and Gupta et al. (2018) developed a "unified measure" designed to evaluate quantitative RRT models based on a single statistic that incorporated the competing elements of privacy and efficiency [12].Vishwakarma et al. ( 2023) developed a two-stage unrelated randomized response model [13].Narjis and Shabir (2021) proposed an RRT model for estimating rare sensitive attributes using a Poisson distribution [14].The concept of "mixture"-combining Warner-based and Greenberg-based constructs into a single model where respondents are randomly assigned to one technique or the other-was incorporated into binary RRT models by Lovig (2021), but "mixture" has not been incorporated into a quantitative model prior to this study [15].
The model we propose in this study, which we call the Mixture Optional Enhanced Trust (MOET) model incorporates optionality, enhanced trust, and mixture into a single model, thereby consolidating many of the advantages of predecessor models into a single model.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.known distribution of the scrambling variable, could then be used across the group of responses to backsolve for the group level mean response to the sensitive question.Greenberg (1971) developed another Quantitative RRT model based on an entirely different mechanism [5].Rather than asking each respondent the sensitive question and then instructing them to scramble their response, Greenberg suggested that each respondent should answer one of two questions-either the sensitive quantitative question or some unrelated and nonsensitive quantitative question with a known probability distribution-based on a random assignment unknown to the researcher.While the researcher would see all of the respondents' responses, the respondents' confidentiality would none the less be maintained because the researcher would not know which question each individual respondent was answering.
Several advancements to Warner's and Greenberg's models were made over the years.Metha, and Aggarwal (2018) proposed a means of estimating the sensitivity level of a sensitive binary question [7].Different kinds of scrambling techniques were explored by Diana and Perri (2011), Singh et al. (2018), and Priyanka and Trisandhya (2019) [8][9][10].Gupta et al. (2002) showed that adding optionality to RRT models (respondents may opt in or opt out of the RRT according to whether they personally find the "sensitive" question to be sensitive) significantly improved model efficiency [11].This same concept of optionality enabled measurement of the level of a quantitative question's sensitivity.Additionally, Gupta et al. (2022) introduced an enhanced trust feature that enabled respondents to opt for greater levels of scrambling if they felt their responses were not being sufficiently obscured by additive scrambling alone [6].
The importance of respondent privacy was formally recognized during this timeframe as a key RRT model attribute, necessary for motivating respondent truthfulness, and Gupta et al. (2018) developed a "unified measure" designed to evaluate quantitative RRT models based on a single statistic that incorporated the competing elements of privacy and efficiency [12].Vishwakarma et al. ( 2023) developed a two-stage unrelated randomized response model [13].Narjis and Shabir (2021) proposed an RRT model for estimating rare sensitive attributes using a Poisson distribution [14].The concept of "mixture"-combining Warner-based and Greenberg-based constructs into a single model where respondents are randomly assigned to one technique or the other-was incorporated into binary RRT models by Lovig (2021), but "mixture" has not been incorporated into a quantitative model prior to this study [15].
The model we propose in this study, which we call the Mixture Optional Enhanced Trust (MOET) model incorporates optionality, enhanced trust, and mixture into a single model, thereby consolidating many of the advantages of predecessor models into a single model.
In Section 2 of this study, we will explore the key metrics that we will use to evaluate and compare RRT models.In Section 3, we will propose the Mixture Optional Enhanced Trust (MOET) model and derive estimators for all of its key attributes.Section 4 will be devoted to computer simulations.We will observe the simulations' output both as a way of understanding the model's behavior and in contrast to the OET model.

Materials and Methods
In this section, we will discuss the methods used to measure the key characteristics of RRT models, which enable the comparisons between models shown later in the study.A measure of efficiency (Mean Squared Error, MSE), a measure of privacy (), and a unified measure that incorporates both efficiency and privacy () are provided.All three of these metrics have been commonly used by other researchers to quantify and compare the efficacy of RRT models.These are the metrics we will use to quantify key attributes of the MOET model.We will use the same metrics to compare the characteristics of the MOET model to those of the OET model.(A6) a
), Singh et al. (2018), and Priyanka and Trisandhya (2019) [8-10].al. ( could then be used across the group of up level mean response to the sensitive question.another Quantitative RRT model based on an entirely han asking each respondent the sensitive question and their response, Greenberg suggested that each responduestions-either the sensitive quantitative question or quantitative question with a known probability distrinment unknown to the researcher.While  the researcher responses, the respondents' confidentiality would none e researcher would not know which question each indi-.arner's and Greenberg's models were made over the ) proposed a means of estimating the sensitivity level of fferent kinds of scrambling techniques were explored by al. (2018), and Priyanka and Trisandhya (2019) [8-10].

Figure 2 .
Figure 2. MOET vs OET comparison across trust levels.(a) Efficiency results for MOET and OET models across trust levels (A).(b) Privacy results for MOET and OET models across trust levels (A).Parameter values underlying these exhibits are those cited in Table2.

Figure 3 .*Figure 2 .
Figure 3. MOET vs OET comparison across sensitivity levels.(a) Efficiency results for MOET and OET models across sensitivity levels (W).(b) Privacy results for MOET and OET models across sensitivity levels (W).Parameter values underlying these exhibits are those cited in Table 2.

Figure 2 .
Figure 2. MOET vs OET comparison across trust levels.(a) Efficiency results for MOET and OET models across trust levels (A).(b) Privacy results for MOET and OET models across trust levels (A).Parameter values underlying these exhibits are those cited in Table2.

Figure 3 .
Figure 3. MOET vs OET comparison across sensitivity levels.(a) Efficiency results for MOET and OET models across sensitivity levels (W).(b) Privacy results for MOET and OET models across sensitivity levels (W).Parameter values underlying these exhibits are those cited in Table 2.

Table 2 .
* Low MSE values are preferred.
Figure 3. MOET vs. OET comparison across sensitivity levels.(a) Efficiency results for MOET and OET models across sensitivity levels (W).(b) Privacy results for MOET and OET models across sensitivity levels (W).Parameter values underlying these exhibits are those cited in Table 2.