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Bradley-Terry choice probability in maximum likelihood and eigenproblem solutions

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Abstract

The Bradley–Terry model (BT) is commonly used for evaluation of choice preferences by paired comparison data in various areas of applied psychology, advertising, and marketing research. The estimation of BT parameters of preference is usually achieved in an iterative procedure based on the maximum likelihood approach. In this paper an easier way of finding these parameters via an eigenproblem is considered. This approach corresponds to solving a Chapman–Kolmogorov system of equations to estimate the steady-state probabilities of the compared items. Both techniques produce very similar results, but the eigenvector solution is simpler for applications and suggests an interpretation of BT preferences as the choice probabilities. The suggested approach can facilitate the paired comparison estimations and be utilized in various practical aims of managerial decision making.

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... Thurstone scaling is a method of priority evaluation among items by the frequency of their empirical pairwise preferences (Thurstone 1927(Thurstone , 1959Thurstone & Jones, 1957). This technique is widely used in fields of applied psychology, particularly, in marketing and advertising research (Edwards, 1957;Torgerson, 1958;Bock & Jones, 1968;Green & Tull, 1978;Conklin & Lipovetsky, 1999, 2004a, 2004bLipovetsky, 2007aLipovetsky, , 2007b. Thurstone scaling transforms ranked or paired comparison data into a scale that is used for displaying the results of a ranking procedure. ...
... Thurstone scaling is a method of priority evaluation among items by the frequency of their empirical pairwise preferences (Thurstone 1927(Thurstone , 1959Thurstone & Jones, 1957). This technique is widely used in fields of applied psychology, particularly, in marketing and advertising research (Edwards, 1957;Torgerson, 1958;Bock & Jones, 1968;Green & Tull, 1978;Conklin & Lipovetsky, 1999, 2004a, 2004bLipovetsky, 2007aLipovetsky, , 2007b. Thurstone scaling transforms ranked or paired comparison data into a scale that is used for displaying the results of a ranking procedure. ...
... The Thurstone model defines a scale of differences; standardizing to zero-one range corresponds to the interval scale. Together with the TMD model, the Bradley-Terry-Luce (BTL) model is also considered for pair comparison (Bradley & Terry, 1952;Luce (1959); Luce & Suppes, 1965;Lipovetsky, 2008) that corresponds to applying the logistic as opposed to the normal probability function. ...
Article
Thurstone scaling is widely used in marketing and advertising research where various methods of applied psychology are utilized. This article considers several analytical tools usefulfor positioning a set of items on a Thurstone scale via regression modeling and Markov stochastic processing in the form of Chapman- Kolmogorov equations. These approaches produce intervaland ratio scales of preferences and enrich the possibilities of paired comparison estimation applied for solving practical problems of prioritization and probability of choice modeling.
... Discrete choice modeling (DCM) is nowadays one of the main tools for estimating preferences among multiple alternatives widely used in applied economics and psychology, transportation and management, social and marketing research. Estimation of the utility parameters and choice probabilitiesisusuallyperformedviathemultinomial-logit(MNL)modelingoriginatedbyMcFadden (1973,1981)andfurtherdevelopedinnumerousworks (McFaddenandRichter,1990;Louviere, Hensher,&Swait,2000;Train,2003;Orme,2010).Oneofthemostpopulartechniquesbasedon DCM is the Best-Worst Scaling (BWS), also called Maximum Difference (MaxDiff) which is a modern marketing research approach to evaluation probability of choice among many compared items.ThismethodhasbeenproposedbyLouviere (1991,1993),anddevelopedinvariousworks (MarleyandLouviere,2005;Bacon,Lenk,Seryakova,&Veccia,2007,2008.BWScanbeseenas extensionofscalingbyThurstoneandBradley-Terrymodelsfrompairedcomparisons (Thurstone, 1927;Bradley and Terry, 1952;David, 1988;Lipovetsky and Conklin, 2004;Lipovetsky, 2008) tosimultaneouscomparisonsamongthreeandmoreitemsinabalancedplanwhereeachitemis representedapproximatelythesamenumberoftimesacrossthesample,andtherespondentsindicate whichitemsarethebestandworst,withfollowingestimationofchoiceprobabilitiesinMNLbyvarious availablesoftwareoranalytically (Louviere,Flynn,&Marley,2015;Marley,Flynn,&Louviere, 2008;Marley,Islam,&Hawkins,2016;LipovetskyandConklin,2014aLipovetskyandConklin, ,2014bLipovetsky,2018). ...
... Discrete choice modeling (DCM) is nowadays one of the main tools for estimating preferences among multiple alternatives widely used in applied economics and psychology, transportation and management, social and marketing research. Estimation of the utility parameters and choice probabilitiesisusuallyperformedviathemultinomial-logit(MNL)modelingoriginatedbyMcFadden (1973,1981)andfurtherdevelopedinnumerousworks (McFaddenandRichter,1990;Louviere, Hensher,&Swait,2000;Train,2003;Orme,2010).Oneofthemostpopulartechniquesbasedon DCM is the Best-Worst Scaling (BWS), also called Maximum Difference (MaxDiff) which is a modern marketing research approach to evaluation probability of choice among many compared items.ThismethodhasbeenproposedbyLouviere (1991,1993),anddevelopedinvariousworks (MarleyandLouviere,2005;Bacon,Lenk,Seryakova,&Veccia,2007,2008.BWScanbeseenas extensionofscalingbyThurstoneandBradley-Terrymodelsfrompairedcomparisons (Thurstone, 1927;Bradley and Terry, 1952;David, 1988;Lipovetsky and Conklin, 2004;Lipovetsky, 2008) tosimultaneouscomparisonsamongthreeandmoreitemsinabalancedplanwhereeachitemis representedapproximatelythesamenumberoftimesacrossthesample,andtherespondentsindicate whichitemsarethebestandworst,withfollowingestimationofchoiceprobabilitiesinMNLbyvarious availablesoftwareoranalytically (Louviere,Flynn,&Marley,2015;Marley,Flynn,&Louviere, 2008;Marley,Islam,&Hawkins,2016;LipovetskyandConklin,2014aLipovetskyandConklin, ,2014bLipovetsky,2018). ...
Article
Discrete choice modeling is one of the main tools of estimation utilities and preference probabilities among multiple alternatives in economics, psychology, social sciences, and marketing research. One of popular DCM tools is the Best-Worst Scaling, also known as Maximum Difference. Data for such modeling is given by respondents presented with several items, and each respondent chooses the best alternative. Estimation of utilities is usually performed in a multinomial-logit modeling which produces utilities and choice probabilities. This article describes how to obtain probability estimation adjusted to possible absence of items in actual purchasing. We apply Markov chain modeling in the form of Chapman-Kolmogorov equations and its steady-state solution for stochastic matrix can be obtained analytically. An adjustment to choice probability with network effects is also considered. Numerical example by marketing research data is used.
... Maximum Difference scaling, or MaxDiff, also known as best-worst scaling, is a contemporary method for the prioritization of items proposed by Jordan Louviere [1991,1993], and developed and applied in numerous works [for a few examples, see Cohen and Orme (2004); Marley and Louviere (2005); Orme (2003Orme ( , 2009]. MaxDiff is based on scaling methods known in Thurstone, Bradley-Terry and other paired comparison models [for instance, see Thurstone (1927Thurstone ( , 1959; Bradley and Terry (1952); Green and Tull (1978); David (1988); Conklin and Lipovetsky (1999a); Lipovetsky (2007Lipovetsky ( , 2008], and also on discrete choice modeling (DCM) which permits the simultaneous presentation of three, four or more items to the respondents and estimation of the utility parameters and choice probabilities using multinomiallogit (MNL) modeling [McFadden (1973); Ben-Akiva and Lerman (1985); McFadden and Richter (1990); Conklin and Lipovetsky (1999b); Louviere et al. (2000); Train (2003); Lipovetsky (2011Lipovetsky ( , 2014Lipovetsky ( , 2015]. ...
... These formulae are very convenient for numerical and analytical consideration of the choice probabilities, for instance they can be easily used in bootstrap estimations of the item choice. MaxDiff data can also be used for priority evaluation in other methods: In Thurstone scaling, SVD, Bradley-Terry estimation in ML and Markov stochastic modeling in Chapman-Kolmogorov equations for finding steady-state probabilities [Conklin and Lipovetsky (1999a); Conklin (2003, 2004); Lipovetsky (2007Lipovetsky ( , 2008]. ...
Article
Full-text available
Maximum Difference (MaxDiff) is a discrete choice modeling approach widely used in marketing research for finding utilities and preference probabilities among multiple alternatives. It can be seen as an extension of the paired comparison in Thurstone and Bradley-Terry techniques for the simultaneous presenting of three, four, or more items to respondents. A respondent identifies the best and the worst ones, so the remaining are deemed intermediate by preference alternatives. Estimation of individual utilities is usually performed in a Hierarchical Bayesian (HB) multinomial-logit (MNL) modeling. MNL can be reduced to a logit model by the data composed of two specially constructed design matrices of the prevalence from the best and the worst sides. The composed data can be of a large size which makes logistic modeling less precise and very consuming in computer time and memory. This paper describes how the results for utilities and choice probabilities can be obtained from the raw data, and instead of HB the empirical Bayes techniques can be applied. This approach enriches MaxDiff and is useful for estimations on large data sets. The results of analytical approach are compared with HB-MNL and several other techniques.
... The current work concentrates specifically on the so called Supercritical Pitchfork Bifurcation (SPB) model. The main reason is that this model can be obtained from a wide set of the sigmoid functions extensively used in applied regression modeling -for several examples, see the models of logistic and algebraic share and choice probability functions described in [41][42][43][44][45]. Another reason -in a diagram this model looks like a regular linear dependence, which at some point diverges into two additional streams. ...
... The characteristic function ) ( y f can be described as utility, usefulness, worth, importance, etc. In the assumptions of algebraic, linear or quadratic exponential behaviour of ) ( y f , the model (4) reduces to a power share model, Bradley-Terry model of paired comparison, or logit model, respectively [41][42][43][44][45][71][72][73]. Convenient for analytical investigation, simple quadratic share model (4) can be expressed explicitly as: ...
... Comparison of priorities in Thurstone scaling with the steady-state probabilities in Markov stochastic modeling was performed in (Lipovetsky, 2013a). Another approach used for evaluation of choice preferences by paired comparisons is the Bradley-Terry choice probability modeling via the maximum likelihood, and a similar estimation can be obtained via the eigenproblem solution related to the Chapman-Kolmogorov system of equations for the steady-state probabilities described in (Lipovetsky, 2008a). ...
Article
The work considers statistical techniques developed for solving various special marketing research problems. These approaches include items comparisons in Thurstone and Bradley-Terry scaling, total unduplicated reach and frequency and Shapley value, sample balancing and price sensitivity analysis, customer satisfaction and identification of key drivers, best-worst and max-diff priority estimation, items cannibalization and synergy, and different other methods. The described techniques have been developed and employed in multiple marketing research projects, and they are helpful for successful solving various practical problems.
... Various other approaches to the choice modeling and decision making solved with logistic and MNL techniques include, for example, van Westendrop price sensitivity meter, and Bradley-Terry choice model (Lipovetsky, 2006b(Lipovetsky, , 2008b. In the large area of the multiple-criteria decision making, for example, in the Analytic Hierarchy Process (AHP) originated by T. Saaty (1980Saaty ( , 2005, the new extensions can be achieved with logit and MNL modeling as well (Lipovetsky, 2021c, d). ...
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... The characteristic function ) ( y f can be described as utility, usefulness, worth, importance, and reduced to a paired comparison logistic model [33][34]. Convenient for analytical investigation, simple quadratic share model (4) can be expressed explicitly as: ...
Article
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... Discrete choice modeling by multiple predictors is widely used in regression analysis. Dichotomy response is often performed in the logistic approach [Long (1997); McCullagh and Nelder (1997); Ripley (1997); Lloid (1999); Lipovetsky (2006Lipovetsky ( , 2008aLipovetsky ( , 2010a]. Categorical variables with several outcomes have been developed in conditional and multinomial logits (MNLs) modeling and used in various applications [McFadden (1973[McFadden ( , 1981; Hausman and McFadden (1984); McFadden and Richter (1990); Ben-Akiva and Lerman (1985); Arminger et al. (1995); Wedel and Kamakura (1999); Louviere et al. (2000); Hastie et al. (2001); Train (2003); Berry et al. (2004); Bishop (2006); Lipovetsky (2008bLipovetsky ( , 2009a; Greene and Hensher (2010)]. ...
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This paper is concerned with the problems of the development of a subjective metric for the study of values. Measurement areas discussed are, social attitudes, propaganda, moral values, experimental semantics, market research, economic research, and aesthetics. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Article
This article introduces novel statistical models for the sequence analysis of events. The models are formulated to analyze occurrence, association, and sequencing among events as an extension of log-linear models. A set of parameters characterizes marginal odds and odds ratios of frequencies summed across sequence patterns for each combination of the occurrence/nonoccurrence of events. These parameters are used for the analysis of the occurrence and association of events. Another set of parameters characterizes conditional odds and odds ratios among sequence patterns within each combination of the occurrence/nonoccurrence of events. These parameters are used for the analysis of sequencing of events. The models permit a decomposition of the likelihood function into a marginal likelihood component that includes only parameters for occurrence and association among events and a conditional likelihood component that includes only parameters for sequencing among events. The models are then extended further for regressions with covariates. An application analyzes gender and racial/ethnic differences in patterns of drug use progression. Sequential patterns of initiations and association among initiations are analyzed for three groups of drugs: alcoholic beverages, cigarettes, and marijuana. Findings that cross-validate previous findings based on different datasets and findings that are novel are reported.
& Decision Making2
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& Decision Making1
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& Decision Making5
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