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Article Using Maxwell Distribution to Handle Selector’s Indecisiveness in Choice Data: A New Latent Bayesian Choice Model 1 2 3,4, 3 3 Muhammad Arshad , Tanveer Kifayat , Juan L. G. Guirao *, Juan M. Sánchez and Adrián Valverde Department of Applied Sciences, School of Science, National Textile University, Faisalabad 37610, Pakistan; muhammadarshad@ntu.edu.pk Department of Computer Science, SZABIST Islamabad, Islamabad 44000, Pakistan; tanveerkifayat.qau@gmail.com Department of Applied Mathematics and Statistics, Hospital de Marina, Technical University of Cartagena, 30203 Cartagena, Spain; juanmasanchezparra@gmail.com (J.M.S.); adrian_valverde12@hotmail.com (A.V.) Nonlinear Analysis and Applied Mathematics (NAAM)‐Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia * Correspondence: juan.garcia@upct.es Abstract: This research primarily aims at the development of new pathways to facilitate the resolv‐ ing of the long debated issue of handling ties or the degree of indecisiveness precipitated in com‐ parative information. The decision chaos is accommodated by the elegant application of the choice axiom ensuring intact utility when imperfect choices are observed. The objectives are facilitated by inducing an additional parameter in the probabilistic set up of Maxwell to retain the extent of inde‐ cisiveness prevalent in the choice data. The operational soundness of the proposed model is eluci‐ dated through the rigorous employment of Gibbs sampling—a popular approach of the Markov chain Monte Carlo methods. The outcomes of this research clearly substantiate the applicability of Citation: Arshad, M.; Kifayat, T.; the proposed scheme in retaining the advantages of discrete comparative data when the freedom of Guirao, J.L.G.; Sánchez, J.M.; no indecisiveness is permitted. The legitimacy of the devised mechanism is enumerated on multi‐ Valverde, A. Using Maxwell fronts such as the estimation of preference probabilities and assessment of worth parameters, and Distribution to Handle Selector’s through the quantification of the significance of choice hierarchy. The outcomes of the research Indecisiveness in Choice Data: A New Latent Bayesian Choice Model. highlight the effects of sample size and the extent of indecisiveness exhibited in the choice data. The Appl. Sci. 2022, 12, 6337. estimation efficiency is estimated to be improved with the increase in sample size. For the largest https://doi.org/10.3390/app12136337 considered sample of size 100, we estimated an average confidence width of 0.0097, which is notably more compact than the contemporary samples of size 25 and 50. Academic Editors: Zhenglei He, Yi Man and Kim Phuc Tran Keywords: Bayesian approach; choices; comparative models; Maxwell distribution; preference Received: 24 May 2022 ordering Accepted: 17 June 2022 Published: 22 June 2022 Publisher’s Note: MDPI stays neu‐ tral with regard to jurisdictional 1. Introduction claims in published maps and institu‐ Competent decision making requires a variety of cognitive skills assisting the notion tional affiliations. of the search for value information to enhance the working potentials, especially when dealing with a complex multifaceted environment [1]. This process demands comparing and mastering the available choices while simultaneously dealing with the practical lim‐ itations [2]. Therefore, the enchanted status of analyzing and modeling choice behaviors Copyright: © 2022 by the authors. Li‐ in the multidisciplinary research literature is of no surprise. The well‐directed historic censee MDPI, Basel, Switzerland. tour of [3] traced the roots of the comparative notions in seventeenth century France, This article is an open access article where [4] advocated the use of comparative models (in a very abstract form) as a method distributed under the terms and con‐ to ensure higher levels of fairness in the electoral process. However, it was the seminal ditions of the Creative Commons At‐ contributions of [5,6] that laid the foundational blocks communicable through mathemat‐ tribution (CC BY) license (https://cre‐ ativecommons.org/licenses/by/4.0/). ical rigors to encapsulate individual differences and associated choice behaviors. The Appl. Sci. 2022, 12, 6337. https://doi.org/10.3390/app12136337 www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 6337 2 of 13 aforementioned efforts instigated the idea of paired comparison (PC) experiments and brought related models into the lime light. Since then, PC methodologies have attracted the attention of many researchers from diverse fields of enquiry ranging from health sur‐ veillance to sport analysis. For example, in the past, [7] explored the applicability of PC models to evaluate the performance of industrial accessories. Furthermore, [8,9] eluci‐ dated the use of the PC approach to analyze sporting events and predict their outcomes. In the recent past, [10] argued the PC schemes were an alternative to the Likert scale for the ranking of psychological markers and indicators’ evaluation. Moreover, [11,12] com‐ petently elaborated on the interlinkages coupling the choice modelling strategies and the numerous variants of rational choice theory governing the preference attitudes in political science, sociology and criminology. For comprehensive accounts documenting the utility of PC methodologies in investigative pursuit, one may also consult [13–15]. In general, comparative experiments peruse the complexity of rational decision mak‐ ing by providing comprehension of the vital ingredients, such as attribute‐level combina‐ tions, repeated choices and utility‐based trade‐off, by offering mutually exclusive choice alternatives to the selectors or judges. The inherent randomness of individual choices is explained by assuming the probabilistic model, whereas the associated utility is deline‐ ated through the stimuli governing the overall choice dynamics. In its simplest form the judges, say 𝑛 , are asked about their preferences while pairwise comparing, say 𝑚 , items, objects or individuals through a simple question, that is, “Do you prefer item 𝑖 over item 𝑗 ?”. One may notice the immediate relevance of the inquiry in all sorts of human behav‐ ioral assessment mechanisms. Thus, in a complete two factorial setup, each judge provides responses in regard to the above inquiry. Table 1 presents the binary string of com‐ parative choice hierarchy recorded by a single selector or judge while conducting a paired comparison experiment. Table 1. Hypothetical choice matrix involving single selector and 𝑚 objects, Y = yes and N = No. Objects 1 2 3 4 5 ‐ 𝒎 1 ‐ Y Y Y N ‐ Y 2 ‐ ‐ N N N ‐ N 3 ‐ ‐ N Y ‐ Y 4 ‐ ‐ Y ‐ Y 5 ‐ ‐ ‐ N ‐ ‐ ‐ ‐ 𝒎 ‐ It is trivial to extend the above reported structure to the incorporation of comparative information accumulated over 𝑛 selectors. The numerous delicacies have been introduced through a rich stream of ongoing ef‐ forts in the above given simple structure, enabling choice models to deal with the com‐ plexities of real phenomena. For more recent, interesting and knowledgeable accounts of the more notable contributions, one may consult [16–19]. This article fundamentally fo‐ cuses on the entertainment of selectors’ indecisiveness in choice reporting, at methodo‐ logical and modeling levels. The issue of handling ties in preference data has long been debated and remains a primary component of the available literature focusing on the anal‐ ysis of discrete choices. The opinion chaos has rightly been summarized by [20] who noted “the key point is that modelling of ties explicitly can be important, although there is no consensus on how this should be done; no approach apart from ignoring ties appears to be in widespread use”. This article is mainly divided into five partitions. Section 2 is dedicated to document‐ ing the mathematical foundations of the proposed procedure, whereas Section 3 provides a rigorous account of the simulation‐based evaluations of the proposal while mimicking Appl. Sci. 2022, 12, 6337 3 of 13 numerous experimental states. Section 4 delineates the applicability of the suggested ap‐ proach while analyzing drinking water brands’ choice data. Lastly, the main findings are discussed along with some future possible research avenues in Section 5. 2. Methods and Materials 2.1. Preliminaries Let us consider that full factorial pairwise comparison set up is launched to generate comparative information among 𝑚 objects by 𝑛 judges, where pair of stimuli elicits a continuous discriminal process. The latent preference hierarchy between 𝑖′𝑡ℎ object and object 𝑗 is then thought to follow one dimensional Maxwell distribution over the con‐ sistent support in the population, such as 𝑓 𝑥 𝑥 , 𝑥 0,𝜃 0, and (1) 𝑓 𝑥 , 𝑥 0,𝜃 0. Here, 𝜃 and 𝜃 are scale parameters of the hallmark structure connecting the prob‐ ability density of the particle’s kinetic energies to the temperature of the system while taking into account the configurational fluctuations of the system [21]. It is noteworthy that the traditional competency of Maxwell formation in encapsulating the atomic velocity distribution with the assumption of lacking potentials provides natural foundations to model preference stimuli while considering utility as a latent phenomenon. In the case of binary string of choice alternatives, the interest lies in the deduction of preference proba‐ bilities as a function of worth parameters dictating the comparative utility precipitation of competing objects. Mathematically, probability of preferring object 𝑖 over object 𝑗 that is, 𝑖→𝑗 , remains quantifiable, such as 𝑝 𝑃𝑋 𝑋 . Similarly, 𝑝 . . 𝑃𝑋 𝑋 represents the preference probability of 𝑗→𝑖 , as a function of estimated worth parameters [22]. Recently, [23] provided the simplified form of preference proba‐ bilities such as 2 𝜃 𝜃 𝜃 𝜃 𝑝 1 𝑎𝑐𝑡𝑛𝑎𝑟 𝜋 𝜃 and 2 𝜃 𝜃 𝜃 𝜃 𝑝 1 𝑎𝑐𝑡𝑛𝑎𝑟 (2) 𝜋 𝜃 2.2. Proposed Model In preference data, when ties are permitted, selectors are indeed offered three poten‐ tial responses while confronting the task of choosing 𝑖→𝑗 , those are “yes”, “no” or “no preference”. Thereby, it is to be noted that in the case of a balanced factorial comparative experiment, the selector responses distinguishing each pair follow trinomial distribution. For notational purposes, the preference probability of 𝑖→𝑗 is denoted as 𝑝 , where 𝑝 represents preference for 𝑗→𝑖 . The probability highlighting the extent of indistinc‐ tiveness or indecisiveness while comparing both competing objects is reported as 𝑝 , that is 𝑖𝑗 when no preference is given between the paired items under comparison. The accommodation of ties is proceeded in accordance with [24] proposition ensuring in‐ tact utility of [25] choice axiom allowing the occurrence of imperfect choices as follows Appl. Sci. 2022, 12, 6337 4 of 13 𝑝 𝜏 (3) . . . where 𝜏 0 is the constant of proportionality representing the tie parameter and inde‐ pendent of the ,𝑗 pair, whereas preference probabilities, 𝑝 and 𝑝 , are as given in . . Equation (2). Moreover, 𝑝 and 𝑝 0,1 and 𝑝 𝑝 𝑝 1. It is noteworthy . . . . . that the proportionality functional in Equation (3) ensures that the probability of no pref‐ erence is dependent upon the extent of distinguishability of pairs. Furthermore, the use of geometric formulation permits representation of the compared item on a linear scale when logarithmic function is applied. Thus, by using tie adjusted formation supported by the liberty of imperfect choices and preference probability sum, it remains verifiable that the preference probability of 𝑖→𝑗 remains simplified as 𝜋𝜃 𝜃 2 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 𝑝 (4) 𝜃 𝜃 𝜋𝜃 𝜃 𝜏 𝜋𝜃 𝜃 2 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 2 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 Similarly, governed by the one dimensional Maxwell distribution, the preference probability of 𝑗→𝑖 is calculated as 2 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 𝑝 (5) 𝜃 𝜃 𝜋𝜃 𝜃 𝜏 𝜃 2 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 2 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 Furthermore, the extent of indistinguishability is estimable such as 𝜃 𝜃 𝜏 𝜃 2 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 2 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 𝑝 . (6) 𝜃 𝜃 𝜋𝜃 𝜃 𝜏 𝜃 𝜃 2 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 2 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 𝜃 In the case of 𝑚 competing objects and 𝑛 selectors, let 𝒘𝑤 ,𝑤 ,𝑤 repre‐ , , , sent the vector comprehending the observed preferences when object 𝑖 is competing with object 𝑗 . Additionally, 𝑛 denotes the total number of possible comparisons in 𝑟 repli‐ cations by 𝑛 selectors, where 𝑖𝑗 ;𝑖 1,𝑗𝑚 . Then the likelihood function of realized choice data generated through the complete factorial set up consists of w trials with the permission of ties, is written as . . . . 𝑙 𝒘 ,𝜽 ,𝜏 ∏ 𝑝 𝑝 𝑝 , 0𝜃 1 and 0𝜏 1. (7) . . . ! ! ! . . . . Here, the issue of identifiability is resolved by ensuring ∑ 𝜃 1, and 𝜃 is asso‐ ciated worth parameter attached with 𝑖′𝑡ℎ object dictating the degree of preference of the object, where 𝑖 1, 2,.. .,𝑚 . 2.3. Incorporating Prior Information In this era of next generation computing hardware, the utility of prior information for the execution of more sound and knowledgeable policy interventions has gain unprec‐ edented momentum in scientific rigors. It is noteworthy that under the considered for‐ mation, two priors are required, one to explain stochastic behavior of worth parameters and second for the capsulation of tie parameter. For demonstration, we consider informa‐ tive prior in the form of Dirichlet prior for the elaboration of worth parameters as follows 𝑝 𝜽 𝜃 , 0𝜃 1 (8) 𝑡𝑎𝑛 𝑡𝑎𝑛 𝜋 𝑡𝑎𝑛 𝑡𝑎𝑛 𝜃𝜋 𝑡𝑎𝑛 𝑡𝑎𝑛 𝜃𝜋 𝑡𝑎𝑛 𝑡𝑎𝑛 𝑡𝑎𝑛 𝑡𝑎𝑛 𝑝 𝑝 Appl. Sci. 2022, 12, 6337 5 of 13 Similarly, Gamma prior is considered for the conceptualization of tie parameter as under 𝑝 𝜏 𝜏 𝑒 , 0𝜏 1 (9) Here, 𝑑 , a and a are the hyperparameters of the prior structure. The motivation behind the use of Dirichlet prior remains intact under the notion of parsimony, as it em‐ ploys fewer number of hyperparameters and thereby is thought to be providing more concise estimates. Furthermore, the Gamma prior remains attractive for tie parameter as both distributional spaces are bounded over 0, 1 range, and thus offers natural support to the estimation efforts. 2.4. Posterior Distribution and Estimation of the Worth Parameters By using the prior distributions given in Equations (8) and (9) along with likelihood function of Equation (7), the joint posterior distribution is deducted as below . . . . 𝑝 𝜃 ,𝜃 ,…,𝜃 |𝒘 𝜃 𝜏 𝑒 𝑝 𝑝 𝑝 , (10) . . . where 𝑘 1 1 1 ∑ . . . . … ∏ 𝜃 𝜏 𝑒 𝑝 𝑝 𝑝 𝑑𝜃 …𝑑𝜃 𝑑𝜃 and 1 2 1 . . . 0 0 0 represents the normalizing constant. The deduction of marginal posterior distributions requires the resolve of complex integrations involved in the expression of joint posterior distribution. The objective is attained by the launch of Gibbs sampling—popular approach of Markov chain Monte Carlo methods [26]. In general, Gibbs sampling proceeds by as‐ suming 𝑃𝜃 ;𝑥 be the joint posterior density, where 𝜃 𝜃 ,𝜃 ,…,𝜃 . The conditional 1 2 densities of worth parameters are then given by 𝑃 𝜃 I𝜃 ,𝜃 …,𝜃 ,𝑃 𝜃 I𝜃 ,𝜃 …,𝜃 …𝑃 𝜃 I𝜃 ,𝜃 …,𝜃 . The Gibbs sampler now initi‐ 1 2 3 2 1 3 1 2 1 0 0 0 ates by assuming initial values upon worth parameters such as ,𝜃 ,…,𝜃 and 2 3 1 0 0 0 conceptualizes the conditional distribution of 𝜃 such that 𝑃𝜃 I𝜃 ,𝜃 ,…,𝜃 . 1 1 2 3 The iterative procedure continues until the convergence occurs. For demonstration pur‐ poses, the expression for the marginal posterior distribution of worth parameter associ‐ ated with 𝑚 ′𝑡ℎ object, that is 𝜃 , is solved as follows . . . . 𝑝 𝜃 |𝒘 … ∏ 𝜃 𝜏 𝑒 𝑝 𝑝 𝑝 𝑑𝜃 …𝑑𝜃 , . . . (11) 0𝜃 1 The 1𝛼 100% credible intervals attached with 𝜃 , say 𝐶 , are obtained numeri‐ cally by solving the given expression 𝑝 𝜃 :𝒙 ,𝑤 ,𝜏 𝑑𝜃 1𝛼 , (12) which is a subset of 𝜃 ′𝑠 parametric space. Figure 1 below presents the flow diagram summarizing the working of the proposed scheme along with algorithmic advancements. Appl. Sci. 2022, 12, 6337 6 of 13 Figure 1. Flow diagram of the proposed scheme. 2.5. Bayes Hypothesis Testing Now, we proceed towards the evaluation of statistical significance of the underlying comparative hierarchy. In Bayesian framework, the task is accomplished by quantifying the posterior probabilities and resultant Bayes factors associated with concerned hypoth‐ esis. The complementary hypothesis streaming pair of objects is given as 𝐻 :𝜃 𝜃 vs. 𝐻 :𝜃 𝜃 . The posterior probabilities deciding upon the existent discrepancies remain calcula‐ ble as 𝜙 𝑃 𝜁 ,𝜂𝛪𝜔 𝑑𝜂𝑑𝜁 , (13) Appl. Sci. 2022, 12, 6337 7 of 13 where 𝜂𝜃 , 𝜁𝜃 𝜃 . It is trivial to show that 𝜙 1𝜙 . The Bayes factor now remains estimable with straightforward operation such that 𝐵𝐹 𝜙 ⁄𝜙 . Generally ac‐ cepted criterion nominating the degree of significance while employing Bayes factor is given as 𝐵𝐹 1, support 𝐻 10 BF 1, minimal evidence against 𝐻 10 𝐵𝐹 10 , substantial evidence against 𝐻 10 𝐵𝐹 10 , strong evidence against 𝐻 𝐵𝐹 10 , decisive evidence against 𝐻 . 2.6. Limitations of the Proposed Scheme It is important to note the limitations of the proposed mechanism at this stage. Our newly developed model is demonstrated to be workable for the adjustment of the re‐ sponse of “no choice or tie”. However, the existence of ties in comparative data is not the only challenge. The choice data may suffer from two other sources of contamination that are: (i) the order of presenting the objects, most commonly known as order effect, and (ii) the tendency to report socially desirable responses while hiding the true status of the mat‐ ter. It is noteworthy that the origin of these complications is different from the docu‐ mented response of no choice. The indecisiveness arises from either indistinguishable na‐ ture of the competing objects or from the judge(s) lacking the capability to distinguish the objects, whereas desirability bias generates due to the fondness of the respondent(s) to‐ wards being socially acceptable or approved. On the other hand, order effect indicates the lack of consistency of the judge(s). Keeping the inherent differences in mind, it is antici‐ pated that new post hoc strategy capable of entertaining aforementioned complexities, one by one or simultaneously, is desirable in future. It can be reported that the devised scheme is capable of catering ties in its present formation. However, the treatment of aforementioned challenges is an attractive future research scope. 3. Simulation‐Based Evaluation At first, rigorous simulation‐based investigation is launched to explore the dynamics of the proposed scheme. The performance of the devised mechanism is studied while con‐ sidering a wide range of parametric settings, including varying sample sizes and the pa‐ rameter defining the extent of indecisiveness. We consider three samples as, 𝑛 25, 50 and 100, for two values of tie parameters, that is, 𝜏 0.1 and 0.2. These states are then studied for three competing objects, that is, 𝑚 3, under a preset preference order‐ ing where 𝜃 𝜃 𝜃 and 𝜃 0.24,𝜃 0.36 and 𝜃 0.40. Table 2 presents the ar‐ tificially generated data sets resulting from the aforementioned parametric settings. Table 2. Artificial data under the preset experimental states. 𝒏 𝒘 𝒘 𝒘 𝒘 𝒘 𝒘 𝒘 𝒘 𝒘 𝟏 . 𝟐 .𝟏𝟐 𝟎 .𝟏𝟐 𝟏 . 𝟑 .𝟏𝟑 𝟎 .𝟏𝟑 𝟐 .𝟐𝟑 𝟑 .𝟐𝟑 𝟎 .𝟐𝟑 𝜏 0.10 25 8 16 1 6 18 1 5 19 1 50 11 37 2 9 39 2 16 33 1 100 27 72 1 19 73 8 44 51 5 𝜏 0.20 25 5 18 2 6 17 2 10 12 3 50 15 27 8 12 34 4 19 26 5 100 19 72 9 24 73 3 37 52 11 Tables 3–5 demonstrate the relevant summaries highlighting the various perfor‐ mance aspects of the schemes. Table 3 provides the Bayes estimates of worth parameters along with the associated 95% credible intervals. 𝟏𝟑 𝟏𝟐 Appl. Sci. 2022, 12, 6337 8 of 13 Table 3. Estimates of worth parameters and associated 95% credible intervals (in parenthesis) for pre‐defined experimental settings. 𝒏 𝜽 𝜽 𝜽 𝝉 𝟏 𝟐 𝟑 𝜏 0.10 0.2549 0.3014 0.4437 0.0865 25 (0.2448, 0.2651) (0.2905, 0.3122) (0.4301, 0.4571) (0.0728, 0.1002) 0.2414 0.3403 0.4183 0.0911 50 (0.2330, 0.2498) (0.3389, 0.3415) (0.4091, 0.4275) (0.0806, 0.1016) 0.2491 0.3516 0.3992 0.0986 100 (0.2337, 0.2494) (0.3508, 0.3603) (0.3964, 0.4022) (0.0933, 0.1039) 𝜏 0.20 0.2689 0.3739 0.3832 0.1881 25 (0.2640, 0.2738) (0.3619, 0.3860) (0.3610, 0.3954) (0.1715, 0.2048) 0.2428 0.3510 0.3958 0.2128 50 (0.2327, 0.2528) (0.3489, 0.3647) (0.3896, 0.4275) (0.1956, 0.2199) 0.2421 0.3594 0.3969 0.1926 100 (0.2390, 0.2454) (0.3556, 0.3624) (0.3939, 0.4098) (0.1894, 0.2086) Table 4. Posterior probabilities and resultant Bayes factors associated with competing hypothesis. Posterior Probabilities Bayes Factor 𝑯 𝑯 𝑯 𝑩 𝑩 𝑩 𝟏𝟐 𝟏𝟑 𝟐𝟑 𝟏𝟐 𝟏𝟑 𝟐𝟑 𝜏 0.10 25 0.1085 0.0001 0.0026 0.1217 0.0001 0.0026 50 0.0377 0.0002 0.0234 0.0392 0.0002 0.0240 100 0.0126 0.0001 0.0527 0.0128 0.0001 0.0557 𝜏 0.20 25 0.0006 0.0004 0.3549 0.0006 0.0004 0.5502 50 0.0039 0.0001 0.0414 0.0039 0.0001 0.0432 100 0.0005 0.0001 0.1876 0.0005 0.0001 0.2309 Table 5. Posterior preference probabilities of pairwise competing objects when ties are permitted. 𝒏 𝒑 𝒑 𝒑 𝒑 𝒑 𝒑 𝒑 𝒑 𝒑 𝟏 .𝟏𝟐 𝟐 .𝟏𝟐 𝟎 .𝟏𝟐 𝟏 .𝟏𝟑 𝟑 .𝟏𝟑 𝟎 .𝟏𝟑 𝟐 .𝟐𝟑 𝟑 .𝟐𝟑 𝟎 .𝟐𝟑 𝜏 0.10 25 0.3789 0.5804 0.0405 0.1871 0.7797 0.0330 0.2607 0.7021 0.0370 50 0.2819 0.6782 0.0398 0.1885 0.7766 0.0348 0.3555 0.6022 0.0421 100 0.2802 0.6768 0.0429 0.2201 0.7399 0.0398 0.4001 0.5534 0.0464 𝜏 0.20 25 0.2295 0.6953 0.0751 0.2189 0.7069 0.0740 0.4427 0.4712 0.0859 50 0.3229 0.5846 0.0924 0.2523 0.6607 0.0869 0.3708 0.5344 0.0947 100 0.2757 0.6431 0.0811 0.2197 0.7044 0.0757 0.3857 0.5273 0.0868 The summaries presented in the above table competently indicate the legitimacy of the proposed model in retaining the predefined underlying ordering of the competing objects, that is, 𝜃 →𝜃 →𝜃 , when ties in the discrete choice data are permitted. Addi‐ tionally, it is noteworthy that the proposition competently captures the degree of indeci‐ siveness precipitated in the comparative information. This fact is realized regardless of the varying sample sizes and different values of the tie parameter. However, a more pro‐ found performance of the approach under discussion is witnessed with an increased sam‐ ple size. For example, the closest and most precise estimation of both delicacies, that is, the worth parameters and tie parameter, are witnessed for the case of 𝑛 100, where the proposed scheme closely estimates the values of both the worth parameters and tie pa‐ rameter. Moreover, the decreased width of the credible interval shows the precision of the procedure with which it remains capable of estimating the preset experimental states. This realization is further highlighted in Figure 2 depicting the prevalent variability in the Appl. Sci. 2022, 12, 6337 9 of 13 Bayes estimates through side‐by‐side box plots. One may notice that minimal variation is attributed with a larger sample size. (a) 𝜃 (b) 𝜃 (c) 𝜃 (d) 𝜃 (e) 𝜃 (f) 𝜃 Figure 2. Side‐by‐side box plots depicting the extent of variability observed in the estimation of worth parameters; top panel projects the variation for 𝜏 0.10, whereas lower panel shows the behavior for 𝜏 0.20. The * indicates the presence of outliers. The significance of the utility differences of the competing objects are demonstrated by quantifying the posterior probabilities and related Bayes factors for the complementary hypothesis. The results are summarized in Table 4. The establishment of a predefined preference ranking and its associated significance is observable from the calculated pos‐ terior probabilities and Bayes factors. Regardless of the varying sample sizes and the ex‐ tent of indecisiveness, we witnessed maintained ordering such as, 𝜃 →𝜃 →𝜃 ; how‐ ever, with different extents of associated significance. In general, we estimate that for 𝜏 0.10, substantial evidence exists indicating that 𝜃 →𝜃 , and decisive evidence is ob‐ served highlighting 𝜃 →𝜃 , whereas strong evidence establishes the significance of 𝜃 →𝜃 . This observation is realized for all considered sample sizes. Furthermore, in the case of 𝜏 0.20, strong evidence is attached with 𝜃 →𝜃 and 𝜃 →𝜃 along with strong indications of the instance of 𝜃 →𝜃 . The estimated posterior probabilities of the preferences while pairwise comparing all three objects are assembled in Table 5. The outcomes seal the consistent behavior of the proposed mechanism. 4. Application—Preference of Drinking Water Brands We now proceed by demonstrating the applicability of the devised model by study‐ ing the choice data of three drinking water brands commonly available in market. The pairwise comparative data with permitted ties were collected from fifty local residents of Islamabad, Pakistan, by inquiring about their preferred brand among (i)—Aquafina (AQ), (ii)—Nestle (NL) and (iii)—Kinley (KN). One may notice that in this situation 𝑛 50 and Appl. Sci. 2022, 12, 6337 10 of 13 𝑚 3. Table 6 displays the collected data, whereas Figure 3 depicts the distribution of counts in the relevant predefined classifications. Table 6. Preference counts pairwise choices of the drinking water brands. Pairs 𝒊 ,𝒋 𝒘 𝒘 𝒘 𝒊 . 𝒋 . 𝟎 . AQ, NT 17 27 6 AQ, KL 26 19 5 NT, KL 28 18 4 Figure 3. The distribution of discrete preferences along with counts of indecisiveness for drinking water brands’ data. We start the exploration by first eliciting the hyperparameters using confidence lev‐ els by the use of the joint posterior distribution of Equation (10) and defining the prior predictive distribution as under . . . . 𝑝𝑤 ,𝑤 𝑄 𝜃 1𝜃 𝜏 𝑒 𝑝 𝑝 𝑝 𝜃 . . . . . Here, 𝑄 . [27] proposed the elicitation of hy‐ ! ! ! . . . . perparameters through the function as follows Ψ 𝒄 𝑎𝑖𝑚𝑛𝑔𝑟 | 𝐶𝐶𝐿 𝐸𝐶𝐿 |, where 𝒄 is the set of elicited hyperparameters, whereas 𝑘 represents the number inter‐ val required to elicit the hyperparameters. Additionally, 𝐶𝐶𝐿 and 𝐸𝐶𝐿 are the con‐ fidence level and elicited confidence level, respectively, characterized with the specific hyperparameter. By exploiting the joint posterior distribution, the confidence levels are given as ∑∑ 𝑝 𝑤 ,𝑤 0.05 ,∑∑ 𝑝 𝑤 ,𝑤 0.07, . . . . . . . . ∑∑ ∑∑ 𝑝 𝑤 ,𝑤 0.05 , 𝑝 𝑤 ,𝑤 0.07, . . . . . . . . ∑∑ 𝑝 𝑤 ,𝑤 0.05 ,∑∑ 𝑝 𝑤 ,𝑤 0.07. . . . . . . . . The elicited values of the hyperparameters for both priors, that is, Dirichlet prior for worth parameters and Gamma prior for the tie parameter, are 𝑑 2.5012, 𝑑 2.6595, 𝑑 2.7001, 𝑎 2.1086 and 𝑎 5.4596. Table 7 presents the Bayes estimates of the worth parameters dictating the choice hierarchy of the comparative data gathered from the drinking water experiment along with the associated evidence of significance. The proposed scheme establishes the choice ranking as such: 𝜃 →𝜃 →𝜃 , indicating that most of the individuals selected in the sample preferred the Nestle brand, followed by Aquafina, whereas Kinley was the least preferred. This hierarchy can also be anticipated from the observed data. Moreover, the preference ordering is observed to be statistically significant with strong evidence associated with the instances of 𝜃 →𝜃 and 𝜃 → 𝑑𝜏𝑑 𝒊𝒋 𝒊𝒋 𝒊𝒋 Appl. Sci. 2022, 12, 6337 11 of 13 𝜃 , whereas we witnessed substantial evidence attached with 𝜃 →𝜃 . These realiza‐ tions are further supported by the posterior probability estimates of the comparative pref‐ erences, given in Table 8. Table 7. Summaries of the analysis of the drinking water choice data. Estimates Bayes Factor 𝝉 𝑩 𝜽 𝒊 𝒋 𝒊 𝒋 0.37153 𝜏 ̂ 0.12504 𝐵 16.181 𝜃 , , 𝜃 0.32643 𝜏 ̂ 0.13252 𝐵 56.306 , , 𝜏 ̂ 𝐵 𝜃 0.30204 0.12878 3.062 , , Table 8. Posterior preference probabilities of comparative choices. Preference Probabilities 𝑝 0.54696 𝑝 0.59222 𝑝 0.51621 . , . , . , 𝑝 𝑝 𝑝 0.39331 0.34921 0.42357 . , . , . , 𝑝 0.05973 𝑝 0.05856 𝑝 0.06022 . , . , . , 5. Discussion and Conclusions The realization and confrontation of choices is unavoidable in every aspect of daily life. Thereby, the methods facilitating the fundamentals of choice dynamics have a long and well cherished history in the multidisciplinary research literature. It has been compe‐ tently argued in the existing literature that the foundations of rational decision making stand on the essential elements of (i) utility: the latent factor derived by the choice axiom [28] and (ii) consistency: the extent of judgment following the axiom using inferential soundness [29]. Therefore, the search for such methods capable of entertaining both fun‐ damentals simultaneously, has attracted noticeable attention in research circles [30]. How‐ ever, the issue of handling the indecisiveness of selector(s) in reporting choice data re‐ mains long standing. There is undoubted consensus over the misleading nature of inde‐ cisive responses, but, unfortunately, the available literature lacks in its ability to demon‐ strate feasible solutions, especially at the methodological level. Motivated by the afore‐ mentioned factors, this article proposes a new choice model in conjunction with the Bayes‐ ian paradigm when the judge or selectors have the opportunity of reporting a “no prefer‐ ence” response. The proposed scheme is fundamentally advantageous in reducing the forced response bias. It is anticipated that by permitting the occurrence of ties while re‐ porting the preferences, the selector(s) are offered extended flexibility to be able to report their true status. Moreover, by devising a workable approach, the scheme enables the in‐ vestigator to estimate the influence of ties in the reported data through methodological sound pathways. The applicability of the suggested approach is affirmed on multiple fronts, such as through mathematically derived expressions, and by the launch of rigorous simulations and being demonstrated empirically. The outcomes of the research substan‐ tiate the legitimacy of the proposed mechanism, especially on four frontiers. Firstly, it is witnessed that the newly suggested model delicately maintains the inherent ordered structure of the observed choice data. This is observed with respect to all considered sam‐ ple sizes and the varying extent of choice parameters. Secondly, the proposed scheme en‐ ables us to estimate the degree of indecisiveness prevalent in the comparative information. Furthermore, in concordance with asymptotic theory, the estimated subtitles become more obvious with an increased sample size. Lastly, the suggested model assists the ra‐ tional decision making notions by providing sound inferential aspects facilitated through the Bayesian framework. At this stage, it is obligatory to report the limitations of the newly developed model that offers attractive research pursuits for the future. Along with ties, the choice data show numerous concerns of practical significance such as the order of presentation of the com‐ peting objects (order effect) and socially‐motivated preferences (desirability bias). If not Appl. Sci. 2022, 12, 6337 12 of 13 treated appropriately, the aforementioned contaminations pose serious threats to the va‐ lidity of the modeling strategies by producing misleading results. It is noteworthy that this research encapsulates the issue of ties and is not capable of entertaining other docu‐ mented complexities. In future, it will be interesting to further elaborate the proposed procedure for the accommodation of order effects and desirability bias. 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Applied Sciences – Multidisciplinary Digital Publishing Institute
Published: Jun 22, 2022
Keywords: Bayesian approach; choices; comparative models; Maxwell distribution; preference ordering
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