ON the use of probit based models for ranking data ANALYSIS



ON the use of probit based models for ranking data ANALYSISGopindra S. NairThe University of Texas at AustinDepartment of Civil, Architectural and Environmental Engineering301 E. Dean Keeton St. Stop C1761, Austin TX 78712Tel: 512-471-4535; Email: gopindra.s.nair@ Chandra R. Bhat (corresponding author)The University of Texas at AustinDepartment of Civil, Architectural and Environmental Engineering301 E. Dean Keeton St. Stop C1761, Austin TX 78712Tel: 512-471-4535; Email: bhat@mail.utexas.eduandThe Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong KongRam M. PendyalaArizona State UniversitySchool of Sustainable Engineering and the Built Environment660 S. College Avenue, Tempe, AZ 85287-3005Tel: 480-727-4587; Email: ram.pendyala@asu.edu Becky P.Y. LooThe University of Hong KongDepartment of GeographyThe Jockey Club Tower, Centennial CampusPokfulam Road, Hong KongTel: +852-3917-7024; Email: bpyloo@hku.hkWilliam H.K. LamThe Hong Kong Polytechnic UniversityDepartment of Civil and Environmental Engineering9/F, Block Z, 181 Chatham Road South, Hung HomKowloon, Hong KongTel : +852-2766-6045; Email: william.lam@polyu.edu.hkAbstractIn consumer surveys, more information per response regarding preferences of alternatives may be obtained if individuals are asked to rank alternatives instead of being asked to select only the most preferred alternative. However, the latter method continues to be the common method of preference elicitation. This is because of the belief that ranking of alternatives is cognitively burdensome. In addition, the limited research on modeling ranking data has been based on the rank ordered logit (ROL) model. In this paper, we show that a rank ordered probit (ROP) model can better utilize ranking data information, and that the prevalent view of ranking data as not being reliable (due to the attenuation of model coefficients with rank depth) may be traced to the use of a misspecified ROL model rather than to any cognitive burden considerations. Keywords: ranking; rank ordered probit (ROP); rank ordered logit (ROL); heteroscedastic ROL (HROL); heteroscedastic ROP (HROP); coefficient attenuation; stated preference.IntroductionThe preferences of individuals regarding market goods and services are typically imputed in choice models using consumer survey data. In the context of transportation planning, surveys have been extensively used to explore individuals’ preferences for travel modes, vehicle type choices and route choices, among many other activity-travel choice dimensions. It is common practice to elicit only the most preferred alternative in these surveys. However, it is possible to elicit not just the preferred alternative but a ranking of all the available alternatives. Although it would appear that the additional information available from the ranking of alternatives should prove beneficial in producing more precise estimates of model coefficients, several past studies ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"SogBtag4","properties":{"formattedCitation":"(Chapman and Staelin, 1982; Foster and Mourato, 2002; Hausman and Ruud, 1987)","plainCitation":"(Chapman and Staelin, 1982; Foster and Mourato, 2002; Hausman and Ruud, 1987)","noteIndex":0},"citationItems":[{"id":27,"uris":[""],"uri":[""],"itemData":{"id":27,"type":"article-journal","title":"Exploiting Rank Ordered Choice Set Data within the Stochastic Utility Model","container-title":"Journal of Marketing Research","page":"288-301","volume":"19","issue":"3","source":"JSTOR","abstract":"The authors report on a procedure for exploiting the information content of rank ordered choice sets to estimate efficiently the parameters of the multinomial logit model formulation of the stochastic utility model of choice behavior. The availability of rank ordered choice set data leads to an \"explosion\" or decomposition procedure for exploiting such extra information. This \"explosion\" process involves the decomposition of a ranked choice set into a series of unranked and statistically independent choice sets. In relation to explosion strategies, several heuristics and an analytical procedure for determining the \"optimal\" explosion depth are discussed in detail. The results of a Monté Carlo study of the small sample properties of the conditional logit estimation procedure (the maximum likelihood estimation procedure used to develop parameter estimates of the multinomial logit model formulation of the stochastic utility model) are reported and interpreted. A college choice empirical application illustrates the procedures developed.","DOI":"10.2307/3151563","ISSN":"0022-2437","author":[{"family":"Chapman","given":"Randall G."},{"family":"Staelin","given":"Richard"}],"issued":{"date-parts":[["1982"]]}}},{"id":90,"uris":[""],"uri":[""],"itemData":{"id":90,"type":"article-journal","title":"Testing for Consistency in Contingent Ranking Experiments","container-title":"Journal of Environmental Economics and Management","page":"309-328","volume":"44","issue":"2","source":"RePEc - Econpapers","ISSN":"0095-0696","author":[{"family":"Foster","given":"Vivien"},{"family":"Mourato","given":"Susana"}],"issued":{"date-parts":[["2002"]]}}},{"id":19,"uris":[""],"uri":[""],"itemData":{"id":19,"type":"article-journal","title":"Specifying and testing econometric models for rank-ordered data","container-title":"Journal of Econometrics","page":"83-104","volume":"34","issue":"1","source":"ScienceDirect","abstract":"The rank-ordered logit model is used as the basic specification for rank-ordered consumer choice data. Two specification tests are proposed for this specification. The first is a Hausman specification test for the independence from irrelevant alternatives hypothesis. The second test examines the possibility that the estimates of equivalent prices are consistent. Two alternative estimators are also proposed. One generalizes the rank-ordered logit specification to allow for a form of heteroscedasticity that permits top ranked choices to be more precisely ranked than bottom ranked choices. The other estimator is an application of a weighted M-estimator that yields consistent equivalent price estimators despite any misspecification of the distribution in the rank-ordered logit model.","DOI":"10.1016/0304-4076(87)90068-6","ISSN":"0304-4076","journalAbbreviation":"Journal of Econometrics","author":[{"family":"Hausman","given":"Jerry A."},{"family":"Ruud","given":"Paul A."}],"issued":{"date-parts":[["1987",1,1]]}}}],"schema":""} (1–3) have shown that the rankings provided among the less preferred alternatives appear to be less reliable than the rankings provided among the more preferred alternatives. This unreliability was assumed to be a result of an increased cognitive burden placed on respondents when ranking alternatives with lower preference. In other words, individuals are assumed to be more uncertain when ranking alternatives with lower preferences.The most commonly used model for ranking data has been the Rank Ordered Logit (ROL) model and its variants. The ROL model is a random utility maximization model that assumes a type 1 extreme value distribution for its utility error kernel term. To account for the hypothesis that rankings of less preferred alternatives, i.e., rankings at higher rank depths, are less reliable, “error scaling parameters” have been introduced into the ROL model to capture the varying uncertainty of individuals at each ranking level. The scaling up of the error terms at higher rank depths would represent increasing uncertainty in the rankings among the lesser preferred alternatives. A study by ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"WGJe2dWO","properties":{"formattedCitation":"(Yan and Yoo, 2014)","plainCitation":"(Yan and Yoo, 2014)","dontUpdate":true,"noteIndex":0},"citationItems":[{"id":62,"uris":[""],"uri":[""],"itemData":{"id":62,"type":"webpage","title":"The seeming unreliability of rank-ordered data as a consequence of model misspecification","genre":"MPRA Paper","abstract":"The rank-ordered logit model's coefficients often vary\nsignificantly with the depth of rankings used in the estimation process. The common interpretation of the unstable coefficients across ranks is that survey\nrespondents state their more and less preferred alternatives in an incoherent manner. We point out another source of the same empirical regularity: stochastic misspecification of the random utility function. An example is provided to show how the well-known symptoms of incoherent ranking behavior can result from stochastic misspecification, followed by\nMonte Carlo evidence. Our finding implies that the empirical regularity can be addressed by the development of robust estimation methods.","URL":"","language":"en","author":[{"family":"Yan","given":"Jin"},{"family":"Yoo","given":"Hong Il"}],"issued":{"date-parts":[["2014",5]]},"accessed":{"date-parts":[["2018",6,18]]}}}],"schema":""} Yan and Yoo (4) has questioned the notion that the increasing scale as one goes down the rankings (that is, at higher ranking depths) is due to cognitive burden or less reliability in the ranking. They show that the perceived unreliability of rankings of less preferred alternatives can be a result of model misspecification in the ROL model. Since the ROL model assumes its error kernel to follow a type 1 extreme value distribution, the model has a special property that the alternative chosen at any rank level among the unranked alternatives is independent of the rank ordering of the higher ranked alternatives. If the true error kernel is any distribution other than the type 1 extreme value distribution, this property would not hold true. In particular, when a generic distribution of the error kernel is incorrectly constrained to be of the type 1 extreme value distribution, the estimated parameters can mimic a situation of coefficient attenuation across rank depths (that is, increasing uncertainty in utility preferences as the rank depth increases). The findings by ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"1C1DmBuM","properties":{"formattedCitation":"(Yan and Yoo, 2014)","plainCitation":"(Yan and Yoo, 2014)","dontUpdate":true,"noteIndex":0},"citationItems":[{"id":62,"uris":[""],"uri":[""],"itemData":{"id":62,"type":"webpage","title":"The seeming unreliability of rank-ordered data as a consequence of model misspecification","genre":"MPRA Paper","abstract":"The rank-ordered logit model's coefficients often vary\nsignificantly with the depth of rankings used in the estimation process. The common interpretation of the unstable coefficients across ranks is that survey\nrespondents state their more and less preferred alternatives in an incoherent manner. We point out another source of the same empirical regularity: stochastic misspecification of the random utility function. An example is provided to show how the well-known symptoms of incoherent ranking behavior can result from stochastic misspecification, followed by\nMonte Carlo evidence. Our finding implies that the empirical regularity can be addressed by the development of robust estimation methods.","URL":"","language":"en","author":[{"family":"Yan","given":"Jin"},{"family":"Yoo","given":"Hong Il"}],"issued":{"date-parts":[["2014",5]]},"accessed":{"date-parts":[["2018",6,18]]}}}],"schema":""} Yan and Yoo (4) suggest that it may be worthwhile to explore ranking models that do not rely on the IIA property. The Rank Ordered Probit (ROP) model is another random utility maximization model for ranking data which assumes a normal distribution on its error kernel. In this paper, we perform simulation experiments on the ROP model to evaluate how the model performance varies with rank depth and then compare the ROP model results with the ROL model results. We also extend the ROP model similar to the manner in which the ROL model was extended to incorporate scaling of error terms at different conditional rank depths. The performance of the extended ROP model is compared against the performance of other models in the simulated and empirical datasets.Literature ReviewThe ROL model was first developed by ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"1QETAi2l","properties":{"formattedCitation":"(Beggs et al., 1981)","plainCitation":"(Beggs et al., 1981)","noteIndex":0},"citationItems":[{"id":9,"uris":[""],"uri":[""],"itemData":{"id":9,"type":"article-journal","title":"Assessing the potential demand for electric cars","container-title":"Journal of Econometrics","page":"1-19","volume":"17","issue":"1","source":"ScienceDirect","abstract":"An ordered logit specification for use on ranked individual data is used to analyze survey data on potential consumer demand for electric cars. In many situations in economics and marketing we would like to be able to forecast consumer demands for goods which have not yet appeared in actual markets. By defining goods as a bundle of underlying attributes, we can use discrete choice models to estimate consumer evaluations. Then new good demand is forecast by use of the estimated coefficients to compare consumer evaluation of the new good to existing choices. When ranked individual data are available, we can estimate separate coefficients for each individual rather than assuming identical coefficients as is usual with logit models. Our results indicate considerable dispersion in individual coefficients. This finding can have important implications for new product analysis.","DOI":"10.1016/0304-4076(81)90056-7","ISSN":"0304-4076","journalAbbreviation":"Journal of Econometrics","author":[{"family":"Beggs","given":"S"},{"family":"Cardell","given":"S"},{"family":"Hausman","given":"J"}],"issued":{"date-parts":[["1981",9,1]]}}}],"schema":""} Beggs et al. (5). This model follows a random utility maximization framework and assumes that the error kernels of the utility functions follow an independent and identically distributed (IID) type 1 extreme value (EV) distribution across alternatives. A consequence of this assumption is that the distribution of utility of the most preferred alternative is independent of the ordering of utilities of less preferred alternatives. This property is a manifestation of the independence of irrelevant alternatives (IIA) property that is associated with logit models. Therefore, if the utilities of alternatives are type 1 EV distributed, the probability of any rank ordering of alternatives can be written as the product of the sequence of probabilities of choosing the most preferred alternative among all the unranked alternatives.Logically, the coefficients estimated using the ROL model should be the same irrespective of the rank depth that is used in estimation. In other words, the coefficients estimated using only the most preferred choice must be around the same as the coefficients obtained when the estimation is undertaken using the first ranked choices. However, several previous studies have shown that this is not the case. In fact, it is observed that the coefficients of the ROL model tend to attenuate (move closer to zero) when more rank levels are used for estimation ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"Kvtkdz1Y","properties":{"formattedCitation":"(Foster and Mourato, 2002; Hausman and Ruud, 1987)","plainCitation":"(Foster and Mourato, 2002; Hausman and Ruud, 1987)","noteIndex":0},"citationItems":[{"id":90,"uris":[""],"uri":[""],"itemData":{"id":90,"type":"article-journal","title":"Testing for Consistency in Contingent Ranking Experiments","container-title":"Journal of Environmental Economics and Management","page":"309-328","volume":"44","issue":"2","source":"RePEc - Econpapers","ISSN":"0095-0696","author":[{"family":"Foster","given":"Vivien"},{"family":"Mourato","given":"Susana"}],"issued":{"date-parts":[["2002"]]}}},{"id":19,"uris":[""],"uri":[""],"itemData":{"id":19,"type":"article-journal","title":"Specifying and testing econometric models for rank-ordered data","container-title":"Journal of Econometrics","page":"83-104","volume":"34","issue":"1","source":"ScienceDirect","abstract":"The rank-ordered logit model is used as the basic specification for rank-ordered consumer choice data. Two specification tests are proposed for this specification. The first is a Hausman specification test for the independence from irrelevant alternatives hypothesis. The second test examines the possibility that the estimates of equivalent prices are consistent. Two alternative estimators are also proposed. One generalizes the rank-ordered logit specification to allow for a form of heteroscedasticity that permits top ranked choices to be more precisely ranked than bottom ranked choices. The other estimator is an application of a weighted M-estimator that yields consistent equivalent price estimators despite any misspecification of the distribution in the rank-ordered logit model.","DOI":"10.1016/0304-4076(87)90068-6","ISSN":"0304-4076","journalAbbreviation":"Journal of Econometrics","author":[{"family":"Hausman","given":"Jerry A."},{"family":"Ruud","given":"Paul A."}],"issued":{"date-parts":[["1987",1,1]]}}}],"schema":""} (2, 3). This has fostered a belief that progressively higher cognitive demands are placed on individuals when ranking less preferred alternatives, because of which rankings become less reliable and not consistent with the individual’s true underlying utilities as the rank depth increases. To accommodate this higher unreliability of rankings, some studies (such as Hausman and Ruud (3) and ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"Y3vr6X31","properties":{"formattedCitation":"(Foster and Mourato, 2002; Hausman and Ruud, 1987)","plainCitation":"(Foster and Mourato, 2002; Hausman and Ruud, 1987)","noteIndex":0},"citationItems":[{"id":90,"uris":[""],"uri":[""],"itemData":{"id":90,"type":"article-journal","title":"Testing for Consistency in Contingent Ranking Experiments","container-title":"Journal of Environmental Economics and Management","page":"309-328","volume":"44","issue":"2","source":"RePEc - Econpapers","ISSN":"0095-0696","author":[{"family":"Foster","given":"Vivien"},{"family":"Mourato","given":"Susana"}],"issued":{"date-parts":[["2002"]]}}},{"id":19,"uris":[""],"uri":[""],"itemData":{"id":19,"type":"article-journal","title":"Specifying and testing econometric models for rank-ordered data","container-title":"Journal of Econometrics","page":"83-104","volume":"34","issue":"1","source":"ScienceDirect","abstract":"The rank-ordered logit model is used as the basic specification for rank-ordered consumer choice data. Two specification tests are proposed for this specification. The first is a Hausman specification test for the independence from irrelevant alternatives hypothesis. The second test examines the possibility that the estimates of equivalent prices are consistent. Two alternative estimators are also proposed. One generalizes the rank-ordered logit specification to allow for a form of heteroscedasticity that permits top ranked choices to be more precisely ranked than bottom ranked choices. The other estimator is an application of a weighted M-estimator that yields consistent equivalent price estimators despite any misspecification of the distribution in the rank-ordered logit model.","DOI":"10.1016/0304-4076(87)90068-6","ISSN":"0304-4076","journalAbbreviation":"Journal of Econometrics","author":[{"family":"Hausman","given":"Jerry A."},{"family":"Ruud","given":"Paul A."}],"issued":{"date-parts":[["1987",1,1]]}}}],"schema":""} Foster and Mourato (2)) have suggested the use of models where the utility function for alternatives changes with each ranking level. These models make the use of scaling parameters to alter the variance of the error kernel with rank levels. The error kernel is assumed to capture an individual’s inability to assess utilities reliably. Therefore, scaling up of the error kernel at higher rank depths is considered to be a sign of decreased ability to rank reliably. This extension of the ROL model that makes use of scale parameters to capture coefficient variations across rank levels is called the heteroscedastic ROL (HROL) model.In contrast to the prevalent view of cognitive burden considerations with ranking data (based on the observation of coefficient attenuation across conditional rank levels in the ROL), ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"44B5zRgC","properties":{"formattedCitation":"(Yan and Yoo, 2014)","plainCitation":"(Yan and Yoo, 2014)","dontUpdate":true,"noteIndex":0},"citationItems":[{"id":62,"uris":[""],"uri":[""],"itemData":{"id":62,"type":"webpage","title":"The seeming unreliability of rank-ordered data as a consequence of model misspecification","genre":"MPRA Paper","abstract":"The rank-ordered logit model's coefficients often vary\nsignificantly with the depth of rankings used in the estimation process. The common interpretation of the unstable coefficients across ranks is that survey\nrespondents state their more and less preferred alternatives in an incoherent manner. We point out another source of the same empirical regularity: stochastic misspecification of the random utility function. An example is provided to show how the well-known symptoms of incoherent ranking behavior can result from stochastic misspecification, followed by\nMonte Carlo evidence. Our finding implies that the empirical regularity can be addressed by the development of robust estimation methods.","URL":"","language":"en","author":[{"family":"Yan","given":"Jin"},{"family":"Yoo","given":"Hong Il"}],"issued":{"date-parts":[["2014",5]]},"accessed":{"date-parts":[["2018",6,18]]}}}],"schema":""} Yan and Yoo (4) show, through simulation experiments and computational analyses, that estimates produced by the ROL model can show coefficient attenuation if the true distribution of the utility error term deviates even slightly from the type 1 EV distribution. This is because, if the error term does not follow a type 1 EV distribution, the probability of a ranking pattern can no longer be written as the product (across rank depths) of the probabilities of choosing the most preferred alternative among the unranked alternatives at each rank depth. For a generic distribution of the error term, the probability for selecting an alternative at a rank level must be conditioned on the ordering of alternatives that have already been ranked. Assuming a type 1 EV distribution for the error kernel and using the resulting exploded logit structure ignores this conditioning (except that, when the error kernel is truly a type 1 EV, this conditioning becomes mute). The solution proposed in ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"6KjYygy6","properties":{"formattedCitation":"(Yan and Yoo, 2014)","plainCitation":"(Yan and Yoo, 2014)","dontUpdate":true,"noteIndex":0},"citationItems":[{"id":62,"uris":[""],"uri":[""],"itemData":{"id":62,"type":"webpage","title":"The seeming unreliability of rank-ordered data as a consequence of model misspecification","genre":"MPRA Paper","abstract":"The rank-ordered logit model's coefficients often vary\nsignificantly with the depth of rankings used in the estimation process. The common interpretation of the unstable coefficients across ranks is that survey\nrespondents state their more and less preferred alternatives in an incoherent manner. We point out another source of the same empirical regularity: stochastic misspecification of the random utility function. An example is provided to show how the well-known symptoms of incoherent ranking behavior can result from stochastic misspecification, followed by\nMonte Carlo evidence. Our finding implies that the empirical regularity can be addressed by the development of robust estimation methods.","URL":"","language":"en","author":[{"family":"Yan","given":"Jin"},{"family":"Yoo","given":"Hong Il"}],"issued":{"date-parts":[["2014",5]]},"accessed":{"date-parts":[["2018",6,18]]}}}],"schema":""} Yan and Yoo (4) to avoid the problem of attenuation of coefficients because of misspecification of the error term distribution is to increase the flexibility of the ROL model by introducing mixing of coefficients and latent classes in addition to scaling parameters at each ranking level. With such a flexible specification, the kernel error will play a lower role in determining the ranking sequence, and therefore the problem of attenuation of coefficients will not be as severe.The rank ordered probit (ROP) model was developed based on the assumption that the true distribution followed by the utility error terms of alternatives is normal. This model is discussed in ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"Gg8Z4nSL","properties":{"formattedCitation":"(Train, 2009)","plainCitation":"(Train, 2009)","noteIndex":0},"citationItems":[{"id":24,"uris":[""],"uri":[""],"itemData":{"id":24,"type":"book","title":"Discrete Choice Methods with Simulation","publisher":"Cambridge University Press","publisher-place":"Cambridge ; New York","number-of-pages":"400","edition":"2 edition","source":"Amazon","event-place":"Cambridge ; New York","abstract":"This book describes the new generation of discrete choice methods, focusing on the many advances that are made possible by simulation. Researchers use these statistical methods to examine the choices that consumers, households, firms, and other agents make. Each of the major models is covered: logit, generalized extreme value, or GEV (including nested and cross-nested logits), probit, and mixed logit, plus a variety of specifications that build on these basics. Simulation-assisted estimation procedures are investigated and compared, including maximum stimulated likelihood, method of simulated moments, and method of simulated scores. Procedures for drawing from densities are described, including variance reduction techniques such as anithetics and Halton draws. Recent advances in Bayesian procedures are explored, including the use of the Metropolis-Hastings algorithm and its variant Gibbs sampling. The second edition adds chapters on endogeneity and expectation-maximization (EM) algorithms. No other book incorporates all these fields, which have arisen in the past 25 years. The procedures are applicable in many fields, including energy, transportation, environmental studies, health, labor, and marketing.","ISBN":"978-0-521-74738-7","language":"English","author":[{"family":"Train","given":"Kenneth E."}],"issued":{"date-parts":[["2009",6,30]]}}}],"schema":""} Train (6). While the ROL model and its variations have been developed upon extensively and used in several empirical contexts ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"ugLHBEDd","properties":{"formattedCitation":"(Beaumais et al., 2016; Bogue et al., 2017; Oviedo and Yoo, 2017)","plainCitation":"(Beaumais et al., 2016; Bogue et al., 2017; Oviedo and Yoo, 2017)","noteIndex":0},"citationItems":[{"id":43,"uris":[""],"uri":[""],"itemData":{"id":43,"type":"article-journal","title":"Why Not Allow Individuals to Rank Freely? A Scaled Rank-Ordered Logit Approach Applied to Waste Management in Corsica","container-title":"Annals of Economics and Statistics","page":"187-212","issue":"121/122","source":"JSTOR","abstract":"Since the introduction of the rank-ordered logit model in the eighties, the cognitive effort involved in a ranking task has been the source of concern. Despite the fact that the rank-ordered logit model provides efficiency gains when compared to the basic multinomial logit model, respondents may not all be able to provide a reliable full ranking of the alternatives they face. Unreliable or 'noisy' rankings result in estimate biases, so that it has even been suggested that only the first three ranks be used for estimation. In order to deal with this ranking capability issue, we propose a survey design which allows the respondents to provide freely incomplete rankings in accordance with their actual heterogeneous ranking capabilities. Using the full-ranking of the alternatives and the accurate sub-ranking of the alternatives, we first estimate a basic rank-ordered logit model. After testing for heteroscedasticity, we also estimate a heteroscedastic rank-ordered logit à la Hausman and Ruud (1987) and introduce a new scaled rank-ordered logit which allows us to model further the sources of heteroscedasticity in the data. The methodology is applied to the issue of waste management in Corsica. Using rankings of waste management options given by a representative sample of the Corsican population (530 respondents) we provide estimates of the willingness-to-pay for various options of waste management calculated from models estimated on the full-ranking and on the sub-ranking data. We find strong evidence that estimations on the full-ranking data set and on the accurate sub-ranking data set differ widely. Allowing individuals to provide freely incomplete rankings eliminates a large part of the heteroscedasticity stemming from heterogeneous ranking capabilities. JEL: Q51, Q53. / KEY WORDS: Choice Experiment, Heterogeneous Ranking Capabilities, Scaled Rank-Ordered Logit Model, Monetary Valuation, Corsica, Solid Waste Management.","DOI":"10.15609/annaeconstat2009.121-122.187","ISSN":"2115-4430","shortTitle":"Why Not Allow Individuals to Rank Freely?","author":[{"family":"Beaumais","given":"Olivier"},{"family":"Casabianca","given":"Anne"},{"family":"Pieri","given":"Xavier"},{"family":"Prunetti","given":"Dominique"}],"issued":{"date-parts":[["2016"]]}}},{"id":40,"uris":[""],"uri":[""],"itemData":{"id":40,"type":"article-journal","title":"A Modified Rank Ordered Logit model to analyze injury severity of occupants in multivehicle crashes","container-title":"Analytic Methods in Accident Research","page":"22-40","volume":"14","source":"ScienceDirect","abstract":"The current study developed a simultaneous model of injury severity outcomes of all occupants in multi-vehicle crashes including all the drivers and the passengers of all vehicles involved in a crash. Specifically, a Modified Rank Ordered Logit (MROL) methodology that can predict the relative order of occupant injury severity as well as the actual injury severity was developed. The final model captures the effects of several key occupant, vehicle, and accident level variables on four possible levels of injury severity. The results indicate the presence of accident-specific unobserved factors that influence the severity outcomes of all people involved in the crash as well as unobserved heterogeneity in the effect of key covariates including occupant’s gender and speed limit. The performance of the MROL model was compared with the traditional mixed multinomial logit (MMNL) model that is the most commonly used model for injury severity analysis. Overall, the results demonstrate superior predictive ability of the MROL model in comparison to the MMNL model. The traditional MMNL model performed satisfactory in terms of replicating the simple shares of different injury severity levels across all occupants. However, the performance of the MMNL model dropped significantly when the observed and predicted shares were compared for combinations of injury severity levels among crashes involving multiple occupants. Lastly, elasticity effects were computed to demonstrate considerably different policy implications of the MROL and MMNL models.","DOI":"10.1016/j.amar.2017.03.001","ISSN":"2213-6657","journalAbbreviation":"Analytic Methods in Accident Research","author":[{"family":"Bogue","given":"Shelley"},{"family":"Paleti","given":"Rajesh"},{"family":"Balan","given":"Lacramioara"}],"issued":{"date-parts":[["2017",6,1]]}}},{"id":44,"uris":[""],"uri":[""],"itemData":{"id":44,"type":"article-journal","title":"A Latent Class Nested Logit Model for Rank-Ordered Data with Application to Cork Oak Reforestation","container-title":"Environmental and Resource Economics","page":"1021-1051","volume":"68","issue":"4","source":"link.","abstract":"We analyze stated ranking data collected from recreational visitors to the Alcornocales Natural Park (ANP) in Spain. The ANP is a large protected area which comprises mainly cork oak woodlands. The visitors ranked cork oak reforestation programs delivering different sets of environmental (reforestation technique, biodiversity, forest surface) and social (jobs and recreation sites created) outcomes. We specify a novel latent class nested logit model for rank-ordered data to estimate the distribution of willingness-to-pay for each outcome. Our modeling approach jointly exploits recent advances in discrete choice methods. The results suggest that prioritizing biodiversity would increase certainty over public support for a reforestation program. In addition, a substantial fraction of the visitor population are willing to pay more for the social outcomes than the environmental outcomes, whereas the existing reforestation subsidies are often justified by the environmental outcomes alone.","DOI":"10.1007/s10640-016-0058-7","ISSN":"0924-6460, 1573-1502","journalAbbreviation":"Environ Resource Econ","language":"en","author":[{"family":"Oviedo","given":"José L."},{"family":"Yoo","given":"Hong Il"}],"issued":{"date-parts":[["2017",12,1]]}}}],"schema":""} (7–9), few studies have made use of the ROP model ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"78XkGWAI","properties":{"formattedCitation":"(Nair et al., 2018; Schechter, 2010; Tamiya et al., 2012)","plainCitation":"(Nair et al., 2018; Schechter, 2010; Tamiya et al., 2012)","noteIndex":0},"citationItems":[{"id":54,"uris":[""],"uri":[""],"itemData":{"id":54,"type":"article-journal","title":"An application of a rank ordered probit modeling approach to understanding level of interest in autonomous vehicles","page":"16","source":"Zotero","language":"en","author":[{"family":"Nair","given":"Gopindra S."},{"family":"Astroza","given":"Sebastian"},{"family":"Bhat","given":"Chandra R."},{"family":"Khoeini","given":"Sara"},{"family":"Pendyala","given":"Ram M."}],"issued":{"date-parts":[["2018"]]}}},{"id":47,"uris":[""],"uri":[""],"itemData":{"id":47,"type":"article-journal","title":"The apple and your eye: Visual and taste rank-ordered probit analysis with correlated errors","container-title":"Food Quality and Preference","page":"112-120","volume":"21","issue":"1","source":"ScienceDirect","abstract":"I look at data from an experiment in which people rank apples according to how they think they will taste. They are then blindfolded and rank how they actually taste. I estimate a multinomial rank-ordered probit model with correlated errors between the taste and visual rankings. I find that the errors for visual characteristics are correlated based on coloring, while the errors for taste are correlated based on sweetness and tartness. Allowing for correlation between the errors in the two regressions shows that, although people often misperceive apple taste based upon visual cues, they do so in systematic ways. People who prefer the looks of an apple they think to be tart (Granny Smith), will like the taste of other apples which are also tart but less well-known (Jonagold).","DOI":"10.1016/j.foodqual.2009.08.009","ISSN":"0950-3293","shortTitle":"The apple and your eye","journalAbbreviation":"Food Quality and Preference","author":[{"family":"Schechter","given":"Laura"}],"issued":{"date-parts":[["2010",1,1]]}}},{"id":50,"uris":[""],"uri":[""],"itemData":{"id":50,"type":"article-journal","title":"Second to fourth digit ratio and the sporting success of sumo wrestlers","container-title":"Evolution and Human Behavior","page":"130-136","volume":"33","issue":"2","source":"ScienceDirect","abstract":"The second (index finger) to fourth (ring finger) digit length ratio (2D:4D) is known to be a putative marker of prenatal exposure to testosterone. It has been reported that fetal and adult testosterone may be critical for development of physical and mental traits such as cardiovascular system, reaction time, aggressiveness and masculinity. Testosterone-driven attributes are associated with success in male-to-male physical competition, which may be proxied by ability in sports. Many researchers have found that 2D:4D is sexually dimorphic and is a negative correlate of athletic performance. This study aims to investigate the associations of 2D:4D with measures of power as another possible testosterone-associated trait using ability in sumo wrestling as a proxy for male physical competitiveness. The measures of sumo performance comprised the sumo ranks and winning percentages of 142 Japanese professional sumo wrestlers. We found that sumo wrestlers with low 2D:4D had higher sumo ranks and better winning records. The significant negative associations between 2D:4D and the athletic prowess of sumo wrestlers provide further evidence of the possible link between high testosterone levels and muscle strength. The relatively small effect sizes found in this study, however, imply that 2D:4D may be a weaker predictor for sports requiring explosive power than for those requiring endurance.","DOI":"10.1016/j.evolhumbehav.2011.07.003","ISSN":"1090-5138","journalAbbreviation":"Evolution and Human Behavior","author":[{"family":"Tamiya","given":"Rie"},{"family":"Lee","given":"Sun Youn"},{"family":"Ohtake","given":"Fumio"}],"issued":{"date-parts":[["2012",3,1]]}}}],"schema":""} (10–12). To the authors’ knowledge, ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"6YKnTJYY","properties":{"formattedCitation":"(Schechter, 2010)","plainCitation":"(Schechter, 2010)","noteIndex":0},"citationItems":[{"id":47,"uris":[""],"uri":[""],"itemData":{"id":47,"type":"article-journal","title":"The apple and your eye: Visual and taste rank-ordered probit analysis with correlated errors","container-title":"Food Quality and Preference","page":"112-120","volume":"21","issue":"1","source":"ScienceDirect","abstract":"I look at data from an experiment in which people rank apples according to how they think they will taste. They are then blindfolded and rank how they actually taste. I estimate a multinomial rank-ordered probit model with correlated errors between the taste and visual rankings. I find that the errors for visual characteristics are correlated based on coloring, while the errors for taste are correlated based on sweetness and tartness. Allowing for correlation between the errors in the two regressions shows that, although people often misperceive apple taste based upon visual cues, they do so in systematic ways. People who prefer the looks of an apple they think to be tart (Granny Smith), will like the taste of other apples which are also tart but less well-known (Jonagold).","DOI":"10.1016/j.foodqual.2009.08.009","ISSN":"0950-3293","shortTitle":"The apple and your eye","journalAbbreviation":"Food Quality and Preference","author":[{"family":"Schechter","given":"Laura"}],"issued":{"date-parts":[["2010",1,1]]}}}],"schema":""} Schechter (11) is the only paper that applied the ROP model to a stated preference ranking dataset. The reason for the dearth of literature on the ROP model may be because, until the past decade, computing cumulative distribution functions of multivariate normal distributions for evaluation of the ROP model was much more cumbersome than computing logistic distributions for evaluation of the ROL model. However, with recent advancements in analytical ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"qpqR73o1","properties":{"formattedCitation":"(Bhat, 2018)","plainCitation":"(Bhat, 2018)","noteIndex":0},"citationItems":[{"id":58,"uris":[""],"uri":[""],"itemData":{"id":58,"type":"article-journal","title":"New matrix-based methods for the analytic evaluation of the multivariate cumulative normal distribution function","container-title":"Transportation Research Part B: Methodological","page":"238-256","volume":"109","issue":"C","source":"RePEc - Econpapers","abstract":"In this paper, we develop a new matrix-based implementation of the Mendell and Elston (ME) analytic approximation to evaluate the multivariate normal cumulative distribution (MVNCD) function, using an LDLT decomposition method followed by a rank 1 update of the LDLT factorization. Our implementation is easy to code for individuals familiar with matrix-based coding. Further, our new matrix-based implementation for the ME algorithm allows us to efficiently write the analytic matrix-based gradients of the approximated MVNCD function with respect to the abscissae and correlation parameters, an issue that is important in econometric model estimation. In addition, we propose four new analytic methods for approximating the MVNCD function. The paper then evaluates the ability of the multiple approximations for individual MVNCD evaluations as well as multinomial probit model estimation. As expected, in our tests for evaluating individual MVNCD functions, we found that the traditional GHK approach degrades rapidly as the dimensionality of integration increases. Concomitant with this degradation in accuracy is a rapid increase in computational time. The analytic approximation methods are also much more stable across different numbers of dimensions of integration, and even the simplest of these methods is superior to the GHK-500 beyond seven dimensions of integration. Based on all the evaluation results in this paper, we recommend the new Two-Variate Bivariate Screening (TVBS) method proposed in this paper as the evaluation approach for MVNCD function evaluation.","ISSN":"0191-2615","author":[{"family":"Bhat","given":"Chandra R."}],"issued":{"date-parts":[["2018"]]}}}],"schema":""} (13) and simulation methods ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"j8fD4uVm","properties":{"formattedCitation":"(Pace and LeSage, 2016)","plainCitation":"(Pace and LeSage, 2016)","noteIndex":0},"citationItems":[{"id":60,"uris":[""],"uri":[""],"itemData":{"id":60,"type":"chapter","title":"Fast Simulated Maximum Likelihood Estimation of the Spatial Probit Model Capable of Handling Large Samples","container-title":"Spatial Econometrics: Qualitative and Limited Dependent Variables","collection-title":"Advances in Econometrics","collection-number":"37","publisher":"Emerald Group Publishing Limited","page":"3-34","volume":"37","number-of-volumes":"0","source":" (Atypon)","URL":"","note":"DOI: 10.1108/S0731-905320160000037008","author":[{"family":"Pace","given":"R. Kelley"},{"family":"LeSage","given":"James P."}],"issued":{"date-parts":[["2016",12,1]]},"accessed":{"date-parts":[["2018",6,18]]}}}],"schema":""} (14) for the approximation of cumulative multivariate normal distribution functions, estimation of a ROP model for the usual modeling contexts encountered in practice should no longer be intractable.In addition to the dearth of ROP applications, there has been no study that we are aware of that investigates if the problem of unstable coefficients which afflicts the ROL model also prevails for the ROP model. The findings of ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"BPsDcwfu","properties":{"formattedCitation":"(Yan and Yoo, 2014)","plainCitation":"(Yan and Yoo, 2014)","dontUpdate":true,"noteIndex":0},"citationItems":[{"id":62,"uris":[""],"uri":[""],"itemData":{"id":62,"type":"webpage","title":"The seeming unreliability of rank-ordered data as a consequence of model misspecification","genre":"MPRA Paper","abstract":"The rank-ordered logit model's coefficients often vary\nsignificantly with the depth of rankings used in the estimation process. The common interpretation of the unstable coefficients across ranks is that survey\nrespondents state their more and less preferred alternatives in an incoherent manner. We point out another source of the same empirical regularity: stochastic misspecification of the random utility function. An example is provided to show how the well-known symptoms of incoherent ranking behavior can result from stochastic misspecification, followed by\nMonte Carlo evidence. Our finding implies that the empirical regularity can be addressed by the development of robust estimation methods.","URL":"","language":"en","author":[{"family":"Yan","given":"Jin"},{"family":"Yoo","given":"Hong Il"}],"issued":{"date-parts":[["2014",5]]},"accessed":{"date-parts":[["2018",6,18]]}}}],"schema":""} Yan and Yoo (4) suggest that, since the ROP model takes into consideration the dependencies of choices between (the conditioned) rank levels, the effects of unstable coefficients because of misspecification must be much less severe or even non-existent. In this paper, we compare the performance of the ROL and ROP models on simulated datasets in terms of robustness of coefficients to misspecification and goodness of fit across rank depths. Further, we generalize the econometric approach to introduce heteroscedasticity with rank depth through the use of scaling parameters. This approach is used to develop a heteroscedastic version of the rank ordered probit (HROP) model. The tendency for the ROL and ROP models to show coefficient variation across rank levels is studied for the case of two empirical datasets by comparing estimates of the scaling parameters produced by the HROL and HROP models respectively.The remainder of this paper is organized as follows. Section REF _Ref516386906 \r \h 3 provides a background of the traditional non-heteroscedastic or homoscedastic ranking models. Section REF _Ref520632017 \r \h 4 describes the concept behind the development of heteroscedastic ranking models. Further, the ranking probability functions for the HROL and HROP models are derived. Section REF _Ref516387098 \r \h 5 provides details regarding the simulation experiments conducted to evaluate the different ranking models in terms of coefficient variation and goodness of fit. In Section 6, the ranking models are estimated on two different empirical datasets to investigate whether the insights gained from the simulation studies carry over to the empirical datasets as well. Section REF _Ref516387400 \r \h 7 concludes the paper.Ranking ModelsConsider the case of an individual who ranks a set of different alternatives. Let be the individual’s utility for an alternative expressed as:( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 1)where is a column vector (of dimension ) of individual-level attributes specific to the alternative, β is the corresponding column vector of coefficients, and is the idiosyncratic random error term. Let r be a specific rank ordering of the alternatives. That is, is the first alternative, is the second alternative and so on. denotes the event that the alternatives are ranked in the order r by the individual. According to the random utility maximization framework, the probability of can be expressed as:( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 2)The above probability can also be considered as the likelihood value for a given ranking observation r. When using a likelihood framework for estimating the coefficients of utilities, it is common to consider only a part of the ranking sequence. If only the top d alternatives of an individual are considered during estimation, the resulting probability function of the partial ranking sequence up to a rank depth of d can be expressed as follows:( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 3)If the rank depth d is set as (because the individual’s least preferred alternative is implicitly known once all other alternatives are ranked), Equation ( GOTOBUTTON ZEqnNum168180 \* MERGEFORMAT 3) is equivalent to Equation ( GOTOBUTTON ZEqnNum713400 \* MERGEFORMAT 2).In the ROL model, which assumes the kernel error term to follow a type 1 EV distribution, the probability of a ranking sequence up to a rank depth of d reduces to the following equation:( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 4)which is the product of probabilities (across ranks up to the rank depth of d) of selecting the most preferred alternative among all the unranked alternatives. The reader is referred to ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"5b1YNGOK","properties":{"formattedCitation":"(Beggs et al., 1981)","plainCitation":"(Beggs et al., 1981)","noteIndex":0},"citationItems":[{"id":9,"uris":[""],"uri":[""],"itemData":{"id":9,"type":"article-journal","title":"Assessing the potential demand for electric cars","container-title":"Journal of Econometrics","page":"1-19","volume":"17","issue":"1","source":"ScienceDirect","abstract":"An ordered logit specification for use on ranked individual data is used to analyze survey data on potential consumer demand for electric cars. In many situations in economics and marketing we would like to be able to forecast consumer demands for goods which have not yet appeared in actual markets. By defining goods as a bundle of underlying attributes, we can use discrete choice models to estimate consumer evaluations. Then new good demand is forecast by use of the estimated coefficients to compare consumer evaluation of the new good to existing choices. When ranked individual data are available, we can estimate separate coefficients for each individual rather than assuming identical coefficients as is usual with logit models. Our results indicate considerable dispersion in individual coefficients. This finding can have important implications for new product analysis.","DOI":"10.1016/0304-4076(81)90056-7","ISSN":"0304-4076","journalAbbreviation":"Journal of Econometrics","author":[{"family":"Beggs","given":"S"},{"family":"Cardell","given":"S"},{"family":"Hausman","given":"J"}],"issued":{"date-parts":[["1981",9,1]]}}}],"schema":""} Beggs et al. (5) for the complete derivation.To compute the probability for the ROP model, construct the matrix. Let U ( vector) be the vector of utility values of alternatives and let ( vector) be the vector of idiosyncratic error terms associated with the alternatives. Then the vector of utilities U can be expressed as: ( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 5)Here, is assumed to follow a multivariate normal distribution, with mean 0 and variance Ω. Let M be the mask matrix of size which when pre-multiplied with U produces the vector of utility differences that should be less than zero according to Equation ( GOTOBUTTON ZEqnNum168180 \* MERGEFORMAT 3). To generate the mask matrix M corresponding to a ranking sequence r and rank depth d, first generate a matrix of size filled with zeros. Then, in the first row, place a value of ‘–1’ at the column corresponding to the first ranked alternative and ‘1’ at the column corresponding to the second ranked alternative. Similarly, in the second row, place a value of ‘–1’ at the column corresponding to the second ranked alternative and ‘1’ at the column corresponding to the third ranked alternative. Continue this procedure for d rows. After row d, place ‘–1’ on all rows at the column corresponding to the alternative with rank d. But continue placing ‘1’s at the columns corresponding to the alternative that is ranked one more than the row index. An illustration of the mask matrix for the ranking sequence and rank depth 3 is given below. Then the probability of a ranking sequence r up to a rank depth d when using an ROP model is expressed as follows:( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 6)where, is the K–1 dimensional cumulative multivariate normal distribution function computed at the truncation point vector (a vector of zeros of dimension K–1) with mean μ and variance-covariance matrix Σ.Heteroscedastic Ranking ModelsThe process of ranking alternatives can be considered as a series of choice decisions in which an individual selects, conditional on earlier choices, the best alternative among all the unranked alternatives ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"3LjNqWCh","properties":{"formattedCitation":"(Luce and Suppes, 1965)","plainCitation":"(Luce and Suppes, 1965)","noteIndex":0},"citationItems":[{"id":18,"uris":[""],"uri":[""],"itemData":{"id":18,"type":"article-journal","title":"Preference, utility and subjective probability","author":[{"family":"Luce","given":"R Duncan"},{"family":"Suppes","given":"Patrick"}],"issued":{"date-parts":[["1965"]]}}}],"schema":""} (15). Let denote an event where the individual selects alternative from an ordered set of alternatives r. ( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 7)Let denote the vector of alternatives between and including the and alternatives in a ranking r of the K alternatives, with the convention that Equation ( GOTOBUTTON ZEqnNum168180 \* MERGEFORMAT 3) can be written in the following manner to better reflect the ranking process as a series of conditional single choice decisions. ( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 8)In Equation ( GOTOBUTTON ZEqnNum322236 \* MERGEFORMAT 8), denotes the probability that is selected as the best alternative among the unranked alternatives , given that the individual has selected as the first alternatives. The conditioning is required because the unobserved factors that affect the individual’s choice of the first alternatives may also affect the choice of the lth alternative.To account for the phenomenon of attenuation of coefficients with rank depth, it is hypothesized that individuals find it difficult to rank alternatives at higher rank depths reliably. This hypothesis is incorporated into ranking models by multiplying scaling parameters with the attribute coefficients at each rank level. The resulting ranking models are referred to as heteroscedastic ranking models and its general form is as shown below.( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 9)The scaling parameter effectively controls the relative contributions of the exogenous parameters and the idiosyncratic error term in the overall utility function. A lower value for would introduce higher variability in the utilities of alternatives at rank level l, while a higher value of would make the utilities more deterministic. Therefore, if the hypothesis that individuals rank alternatives at higher rank depths less reliably is true, the value of should be lower for larger l. To avoid issues with identification of coefficients, is fixed to 1 during estimation. To ensure that the scaling parameter remains positive, it may be reparametrized as . Note that multiplying the utility coefficients by has the same effect as dividing the covariance matrix of the error terms by .The conditional probability of selection of an unranked alternative at any given rank depth can be expressed in terms of the ranking probabilities as follows:( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 10)Equation ( GOTOBUTTON ZEqnNum842764 \* MERGEFORMAT 9) on application of Equation ( GOTOBUTTON ZEqnNum348036 \* MERGEFORMAT 10) becomes:( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 11)Equation ( GOTOBUTTON ZEqnNum569821 \* MERGEFORMAT 11) can be used to extend the ROL model to produce the heteroscedastic ROL (HROL). Substituting the generic ranking probability function in Equation ( GOTOBUTTON ZEqnNum569821 \* MERGEFORMAT 11) with that of the ROL model, (from Equation ( GOTOBUTTON ZEqnNum484970 \* MERGEFORMAT 4)), the probability function for the HROL model is as follows:( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 12)As mentioned in Section REF _Ref520637248 \r \h 2, Equation ( GOTOBUTTON ZEqnNum903681 \* MERGEFORMAT REF ZEqnNum903681 \* Charformat \! \* MERGEFORMAT 12) can be viewed as the product of probabilities of choosing the best alternative among the unranked alternatives independent of the alternatives that have already been chosen. This is an artifact of the IIA property of logit based models. In other words, if the errors are assumed to be type 1 EV distributed,( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 13)where, denotes the probability of selection of from alternatives when the utility coefficients are and the error terms follow a type 1 EV distribution.To the authors’ knowledge, heteroscedasticity using scale parameters has not been introduced to any ranking model other than ROL. However, the same concept can also be extended to the ROP model to produce the Heteroscedastic ROP (HROP) model. Replacing the generic ranking probability function in Equation ( GOTOBUTTON ZEqnNum569821 \* MERGEFORMAT 11) with that of the ROP model, (from Equation ( GOTOBUTTON ZEqnNum172451 \* MERGEFORMAT 6)), the probability function for the HROP model is as follows:( MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT SEQ MTEqn \c \* Arabic \* MERGEFORMAT 14)Unlike the HROL model, the HROP model cannot be simplified to a sequence of independent single choice decisions.All the heteroscedastic and homoscedastic ranking models discussed in this paper were estimated using the GAUSS matrix programming language, and are available at for coefficient attenuation in ROP modelsIn this section, we compare the extent of coefficient attenuation and goodness of fit for the ROL and ROP models. The comparison is performed on simulated datasets. Details regarding the development of the simulated datasets are provided in Section REF _Ref516389897 \r \h \* MERGEFORMAT 5.1, and the results and interpretations from the experiments are presented in Sections REF _Ref516389945 \r \h \* MERGEFORMAT 5.2 and REF _Ref529127113 \r \h 5.3.Experimental SetupThe objective of the simulation experiments is to understand the robustness of coefficients obtained from the different ranking models to misspecification of the kernel error term. By robustness of coefficients, we refer to the property of lack of variation in estimated coefficients across rank depths. To focus on the issue of kernel distribution misspecification, we consider strictly IID error terms across alternatives (so that covariance across error terms of alternatives does not add another dimension of misspecification impacting variation in estimated coefficients across rank depths). The data generation process used for the generation of test datasets is the same as Equation ( GOTOBUTTON ZEqnNum894153 \* MERGEFORMAT REF ZEqnNum894153 \* Charformat \! \* MERGEFORMAT 1). In all our experiments we assume 8 alternatives and 2 attributes for each alternative. The alternative specific constant term for the first alternative is set to zero, and that for all other alternatives is set to one. The value of coefficients for the two attributes are set as 1 and –1. We consider 4 different distributions for the IID kernel error terms. They are the normal distribution, type 1 extreme value distribution, uniform distribution and logistic distribution. The parameters of these distributions were set in a way that the mean is zero and the variance is for all distributions. The values of attributes for each observation are drawn from normal distributions with a variance of one. The mean of normal distributions for the first attribute of the 8 alternatives ranged linearly from 0.5 for the first alternative to –0.5 for the last alternative. This trend was reversed for the distributions of the second attribute. The distributions of attributes were set in this manner to ensure that there are variations in the number of observations that select each alternative as their first choice. All coefficients are estimated using the maximum likelihood method. The one variate univariate screening method (13) is used to evaluate the cumulative multivariate normal distribution function for the probit based models.Experimental Results: Robustness of CoefficientsTo evaluate the degree of coefficient attenuation, 50 datasets of size 500 were generated for each distribution of the utility error term. For each dataset, coefficients were estimated using the different ranking models at rank depths varying from 1 to 7. In other words, a total of 4 (error distributions) × 50 (datasets) × 4 (ranking models) × 7 (rank depths) = 5600 models were estimated. A box plot of the estimated coefficient of the first attribute (with a coefficient of 1 in the data generation process) is provided in Figure 1. The box plots for the second attribute (with a coefficient of –1 in the data generation process) is not shown here as it is almost the same as that for the first attribute except for a reversal in sign of the estimated coefficients. Note that the variation of coefficient values presented in the box plot in Figure 1 represent the different estimates obtained across the 50 datasets.For the models generated from datasets having error distributions as normal, uniform or logistic, the extent of coefficient attenuation with the ROP model seems to be much lower than that of the ROL model. For these datasets, the advantage of using a heteroscedastic ranking model over the ROP model seems to be relatively less. The coefficient attenuation shown by the ROL model appears similar to what would be expected if the rankings at higher rank depths were made less reliably. This suggests that it may be possible to rectify the problem of attenuation of coefficients observed in previous literature by using the ROP model instead of the ROL model. In the datasets where the utility follows type 1 EV distribution, the coefficients of the ROL model does not attenuate, but that of the ROP model is amplified. None of the heteroscedastic ranking models show any variation in coefficient parameters with rank depth. This is expected because the variation of coefficients is captured by the scaling parameters in these models. However, the robustness of coefficients in heteroscedastic ranking models comes at the cost of higher variance (determined by the length of the boxes and whiskers) at higher rank depths. The reduction of variance of heteroscedastic models with rank depth seems to be lower than that of homoscedastic models.Overall, among the homoscedastic models, the ROP model seems to be a better alternative to the ROL model if one is not sure about the distribution of the error term. The probit kernel is much more accommodative and robust to misspecification of the kernel error term. Both the heteroscedastic models do not show significant coefficient attenuation for any distribution of the kernel error term.Experimental Results: Goodness of FitCoefficient attenuation by itself is not necessarily problematic if it improves the predictive power of a model. The maximum likelihood procedure ensures that the coefficients maximize the probability of observed rankings up to the rank depth used for estimation. However, in most cases where ranking data is used, the true objective is not to be able to predict the probability of an individual’s ranking of all alternatives, but to predict the probability of an alternative being selected as the most preferred alternative. The idea behind using the ranking data is to use information of choices at lower rank levels to produce more precise estimates of the probability of selection of the most preferred alternative. Therefore, in this section, we assess the ability of the different models to predict the probability of the most preferred alternative when different rank depths are used.To evaluate the predictive power of the models estimated in the previous step, test datasets with 1000 observations were generated for each of the utility error distributions. The likelihood of the most preferred alternative in the test dataset was computed for the 5600 models estimated in the previous step. The models were tested on the dataset having the same distribution as the dataset on which the model is estimated. Figure 2 shows a plot of the computed likelihood values. The likelihood value plotted is the average of the likelihoods produced by ranking models estimated using the 50 datasets. The likelihood of the most preferred alternatives in the test dataset is considered to be a metric for goodness of fit or predictive capability.A comparison between the corresponding plots in Figure 1 and Figure 2 indicates the goodness of fit improves with rank depth for models that showed robust coefficients in Figure 1. The ROL model shows a drop in the goodness of fit when the error terms follow normal, uniform or logistic distribution. The likelihood value of the ROP model deteriorates when the error term is type 1 EV distributed. All heteroscedastic ranking models show improvement in goodness of fit with rank depth. Overall, the plots of goodness of fit reinforce our finding that the ROP model is more robust to the misspecification of distribution of error terms. It should not be surprising that the ROL model performs better than the ROP model when the utilities in the datasets are type 1 EV distributed since this is the same as the utility distribution assumed by the ROL model. In the broader context, these results further corroborate ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"qtYAuAUF","properties":{"formattedCitation":"(Yan and Yoo, 2014)","plainCitation":"(Yan and Yoo, 2014)","dontUpdate":true,"noteIndex":0},"citationItems":[{"id":62,"uris":[""],"uri":[""],"itemData":{"id":62,"type":"webpage","title":"The seeming unreliability of rank-ordered data as a consequence of model misspecification","genre":"MPRA Paper","abstract":"The rank-ordered logit model's coefficients often vary\nsignificantly with the depth of rankings used in the estimation process. The common interpretation of the unstable coefficients across ranks is that survey\nrespondents state their more and less preferred alternatives in an incoherent manner. We point out another source of the same empirical regularity: stochastic misspecification of the random utility function. An example is provided to show how the well-known symptoms of incoherent ranking behavior can result from stochastic misspecification, followed by\nMonte Carlo evidence. Our finding implies that the empirical regularity can be addressed by the development of robust estimation methods.","URL":"","language":"en","author":[{"family":"Yan","given":"Jin"},{"family":"Yoo","given":"Hong Il"}],"issued":{"date-parts":[["2014",5]]},"accessed":{"date-parts":[["2018",6,18]]}}}],"schema":""} Yan and Yoo (4) that the attenuation of coefficients observed in past studies on the ROL model may be a result of misspecification and not because of inconsistent ranking of alternatives at higher rank depths. The ROP model did not show attenuation of coefficients in three of the four error distributions that were tested. The robustness of the ROP model in our simulated datasets means that, in the context of empirical datasets of stated preference surveys, if the ROP model shows coefficient attenuation with rank depth, or equivalently if estimated values of scale parameters in the HROP model are significantly less than one, this would be a better indication that individuals in fact rank alternatives with lower preference less reliably (while making this same conclusion based on a ROL model is more dubious because of confounding of misspecification in the kernel error distribution with less reliability in the rankings at higher depth).ApplicationIn this section, we compare the extent of coefficient attenuation and goodness of fit of logit based models and probit based models when the estimation is undertaken on empirical datasets. Coefficient attenuation is captured in the heteroscedastic ranking models through the logarithm of the scale parameter at each rank depth except the last (that is, where K is the number of alternatives and as an innocuous normalization) (see Section 4). This term is an indicator of the extent of coefficient variation at each rank level. A negative value for the log scale parameter would indicate coefficient attenuation (that is lower reliability of ranking at higher depths), while a positive value implies that the coefficients are amplified (or higher reliability of ranking at higher depths). Insignificance or a value close to zero for this term at a particular rank level indicates that the coefficient at that rank level is the same as the coefficient at the first rank level where the log scale parameter is fixed to zero. The variance of the error term is fixed to in all models.For each of the estimated ranking models, the log-likelihood value at convergence and the adjusted likelihood ratio index (ADLRI) are computed. In computing the ADLRI, we use the likelihood at convergence of the constants only model for the ROL model as the common basis to evaluate the alternative models. The performance of the probit based models are compared against the corresponding logit based models through the use of the non-nested adjusted likelihood ratio test, which determines if the ADLRIs of two non-nested models are significantly different. In other words, this statistic gives the probability that the difference between the ADLRI statistic of two models occurred because of random chance. Within each of the probit-based and logit-based model categories, the performance of the homoscedastic model is compared against its corresponding heteroscedastic counterpart using a nested likelihood ratio test. Additionally, the goodness of fit in predicting the highest and lowest ranked alternatives is computed. That is, using the estimates from each ranked model estimation, the probability of the observed first ranked choice and the probability of the observed last ranked choice for each individual in the sample is computed. Then, the corresponding log-likelihood values and the average probability of correct predictions (across all individuals) are computed. These additional exercises are undertaken to obtain a sense of how the ranked model estimation performs in predicting only the first-ranked choice and only the last-ranked choice (discarding performance at the intermediate ranks). Note, however, that it is not possible to use any rigorous statistical tests to compare performance for the first-ranked choice and last-ranked choice predictions, because the model estimations themselves are undertaken using the full ranking order. But the log-likelihood values and the average probability of correct prediction values serve as informal measures of fit. Empirical Example 1In this section, we analyze the data on ranking of gaming consoles by 91 Dutch students. The students were asked to consider buying a new gaming platform on which to play computer games. The six gaming platforms available were Xbox, PlayStation, Gamecube, PlayStation Portable, Gameboy or a regular personal computer. For more information on this dataset, the reader is referred to ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"UmAlYYdT","properties":{"formattedCitation":"(van Dijk et al., 2007)","plainCitation":"(van Dijk et al., 2007)","dontUpdate":true,"noteIndex":0},"citationItems":[{"id":31,"uris":[""],"uri":[""],"itemData":{"id":31,"type":"report","title":"A rank-ordered logit model with unobserved heterogeneity in ranking capabilities","publisher":"Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute","genre":"Econometric Institute Research Paper","source":"RePEc - Econpapers","abstract":"In this paper we consider the situation where one wants to study the preferences of individuals over a discrete choice set through a survey. In the classical setup respondents are asked to select their most preferred option out of a (selected) set of alternatives. It is well known that, in theory, more information can be obtained if respondents are asked to rank the set of alternatives instead. In statistical terms, the preferences can then be estimated more efficiently. However, when individuals are unable to perform (part of) this ranking task, using the complete ranking may lead to a substantial bias in parameter estimates. In practice, one usually opts to only use a part of the reported ranking. In this paper we introduce a latent-class rank-ordered logit model in which we use latent segments to endogenously identify the ranking capabilities of individuals. Each segment corresponds to a different assumption on the ranking capability. Using simulations and an empirical application, we show that using this model for parameter estimation results in a clear efficiency gain over a multinomial logit model in case some individuals are able to rank. At the same time it does not suffer from biases due to ranking inabilities of some of the respondents.","URL":"","number":"EI 2007-07","author":[{"family":"Dijk","given":"Bram","non-dropping-particle":"van"},{"family":"Fok","given":"Dennis"},{"family":"Paap","given":"Richard"}],"issued":{"date-parts":[["2007",2,6]]},"accessed":{"date-parts":[["2018",6,18]]}}}],"schema":""} van Dijk et al. (16). The dataset is available as part of the R package mlogit ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"K4KyGto8","properties":{"formattedCitation":"(Yves Croissant, 2018)","plainCitation":"(Yves Croissant, 2018)","noteIndex":0},"citationItems":[{"id":100,"uris":[""],"uri":[""],"itemData":{"id":100,"type":"book","title":"mlogit: Multinomial Logit Models. R package version 0.3-0","URL":"","author":[{"literal":"Yves Croissant"}],"issued":{"date-parts":[["2018"]]}}}],"schema":""} (17).The specification used for modeling is the same as that used by van Dijk et al. (16). The exogenous variables are the number of hours spent playing and a binary variable indicating whether the individual owns (or not) the gaming platform under consideration. To conserve on space, and also because the substantive variable effects themselves are not of primary importance in this paper, we do not present the complete model results (suffice it to say though that all the variable effects for all models were as expected and intuitive). Readers interested in the complete model results will find these in an online supplement to this paper available at . But we should state here that, in terms of the estimated logarithms of the ranking scale parameters, these were statistically significant at the 0.10 level of significance or lower at rank levels two, four, and five for the HROL model, but statistically insignificant at literally any reasonable level for the HROP model. Also, the magnitudes of the estimated logarithm of scale parameters were all negative at each rank level and generally increasing in magnitude with increasing rank depth for both the HROL and HROP models, indicating potentially less reliability for lower ranked alternatives. In addition, the magnitudes at each rank level were higher in the HROL model compared to the HROP model. All of these results, taken together, very strongly imply that the coefficient attenuation observed in the ROL model is a result of model misspecification rather than unreliable rankings. The model fit statistics and the statistics for comparison between the different models is provided in Table 1. The non-nested likelihood ratio tests between the ROL and ROP, and the HROL and HROP, show that the superiority in performance of the probit kernel over the logit kernel is significant. The nested likelihood ratio test between the ROL and HROL models indicates that the superior performance of the latter is statistically significant at about the 0.075 level of significance. There is no difference between the ROP and HROP models at any reasonable level of significance. In fact, the ADLRI value for the HROP is worse than that of the ROP, because the HROP adds another four parameters to the ROP with little benefit in prediction. With regard to the prediction of the observed first choice, once again, the ROP model dominates the ROL model. There is literally no difference in performance between the HROL and HROP models, and also literally no difference in performance between the ROP and HROP models. For the prediction of the last observed choice, the probit-based models perform better than their logit-based counterparts on the log-likelihood measure, though the ROL performs just a little better than the ROP based on the average probability of correct prediction. More important to note is that the heteroscedastic models perform worse here than their homoscedastic counterparts, especially when comparing the two logit-based models. This poor performance of the heteroscedastic models for the last choice prediction is not surprising, given the larger error variance at increasing rank depths. Empirical Example 2In this section, we explore the ranked preferences for buses of public light bus operators obtained from a stated preference (SP) survey conducted in 2002 in Hong Kong. The objective of the SP survey was to gauge the interest of operators in buying LPG powered buses. The survey included an SP game in which each respondent was asked to rank between four hypothetical alternatives for buses. The alternatives were described using the following attributes, fuel type (diesel or LPG powered), fuel price, vehicle price, distance to the nearest refueling station, maximum distance the vehicle can travel between refueling stops (vehicle range), life of the vehicle, number of seats and horsepower of the vehicle. The dataset consisted of 903 valid observations. The reader is referred to ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"s45xXK4i","properties":{"formattedCitation":"(Loo et al., 2006)","plainCitation":"(Loo et al., 2006)","noteIndex":0},"citationItems":[{"id":83,"uris":[""],"uri":[""],"itemData":{"id":83,"type":"article-journal","title":"Introducing alternative fuel vehicles in Hong Kong: views from the public light bus industry","container-title":"Transportation","page":"605-619","volume":"33","issue":"6","source":"link.","abstract":"Hong Kong was the first place in the world to implement a trial scheme to convert all public light buses (PLBs) on the road from diesel to alternative fuel vehicles (AFVs). The scheme, however, did not receive much support from PLB operators. At present, there is a rich literature on households’ demand for AFVs (especially in the USA). However, there have not been many studies about the demand for commercial AFVs in the business and public transport sectors. Since light buses running on alternative fuels are not widely available in the Hong Kong market, a stated preference (SP) survey was conducted to solicit the preferences of PLB operators on eight commercial vehicle attributes and seven forms of government support. The SP data are analyzed by multinomial logit (MNL) models. Detailed analyses on market segmentation and price elasticities follow. The results are of theoretical and practical significance.","DOI":"10.1007/s11116-006-7947-5","ISSN":"0049-4488, 1572-9435","shortTitle":"Introducing alternative fuel vehicles in Hong Kong","journalAbbreviation":"Transportation","language":"en","author":[{"family":"Loo","given":"Becky P. Y."},{"family":"Wong","given":"S. C."},{"family":"Hau","given":"Timothy D."}],"issued":{"date-parts":[["2006",11,1]]}}}],"schema":""} Loo et al. (18) for further details regarding the survey and ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"6xuVxjuf","properties":{"formattedCitation":"(Loo et al., 2008)","plainCitation":"(Loo et al., 2008)","noteIndex":0},"citationItems":[{"id":86,"uris":[""],"uri":[""],"itemData":{"id":86,"type":"article-journal","title":"Choice or Rank Data in Stated Preference Surveys?","container-title":"The Open Transportation Journal","page":"74-79","volume":"2","source":"ResearchGate","abstract":"Should researchers collect choice or rank data in stated preference (SP) surveys? Answer to this question can have significant implications on survey costs and modeling outputs available for policy analysis. In particular, the ex-ploded rank multinomial logit model (MNL) is compared with the ordinary choice-based MNL model. Using the empiri-cal SP rank data collected among the public light bus operators in Hong Kong, the selected modeling approaches are compared in terms of model assumptions, model fit, modeling outputs and policy implications. Besides, the reliability of the exploded rank data is tested. The mixed results suggest that extra care must be exercised in the design of SP ranking tasks.","DOI":"10.2174/1874447800802010074","author":[{"family":"Loo","given":"Becky"},{"family":"Wang","given":"Shuai"},{"family":"Hau","given":"Timothy"}],"issued":{"date-parts":[["2008",10,30]]}}}],"schema":""} Loo et al. (19) for a detailed analysis of the dataset. The coefficients estimated by the different ranking models are provided in an online supplement available at . In this second empirical example (and similar to the first), the estimated logarithms of the ranking scale parameters were negative and increasing in magnitude at higher rank depths. But, unlike the first empirical example, the logarithm of scale parameters were statistically significant at the 0.05 level of significance even in the HROP model (and at a much higher level of significance for the HROL model). But, similar to the first example, the magnitudes of these scale parameters were much higher in the HROL model relative to the HROP model. Overall, the implication is that there may be some degradation in reliability at higher rank depths, but the HROL substantially overestimates any such degradation. The model fit statistics and the statistics for comparison between the different models are presented in Table 1. The non-nested likelihood ratio tests show again that the probit-based models outperform their logit counterparts. Also, introducing heteroscedasticity helps in both the probit and logit cases, though it improves the ROL model much more than the ROP model. In terms of the ability of the models to predict the most preferred alternative, the probit-based models outperform their logit counterparts, and introducing heteroscedasticity substantially helps in the logit case but relatively less so for the probit case. For predicting the least preferred alternative, the ROP model is superior to all other models in terms of log-likelihood, although the probability of correct prediction is slightly higher for the ROL model.Across both the datasets, for the full ranking and first choice predictions, our results indicate that the simple ROP model does better than or almost as well as the HROL model. Further, there is not much difference in data fit between the ROP and HROP models. In predicting the least preferred choice, the homoscedastic models fare better, and the ROP model outperforms the ROL model in terms of log-likelihood and the average probability of correct prediction. Taken together, the implication is that the ROP model appears to be far more robust to error term misspecification than the ROL model. Earlier studies that use the ROL model and suggest that ranking data is not reliable due to cognitive burden considerations may be misplaced. This is particularly so because, when using the probit kernel, there was little need to introduce heteroscedasticity, while introducing heteroscedasticity in the ROL model improved fit considerably. ConclusionIn this paper, we demonstrate through simulation experiments that the coefficient attenuation with rank depth observed with ROL models in past literature may be a result of model misspecification and not unreliability of the ranking data. Our simulation experiments indicate that the ROP model may be a superior option to the ROL model in terms of robustness of coefficients across rank depths and goodness of fit. The different ranking models are also estimated on two different empirical stated preference survey datasets. In both datasets, on almost all metrics, the probit based models were superior to the logit based models. The ROP model appears to be a particularly robust one to error misspecification, and there was relatively little attenuation observed in the parameters at different conditional rank depths. Our analysis suggests that it would be far superior to use a probit kernel to analyze ranked data, and that ranked data may be quite reliable after all. This is an important result for survey data collection and preference elicitation, and suggests that researchers seriously consider ranking data as a way to elicit preferences. Rank-ordered data are as easy to collect as the most preferred alternative, and also have the distinct advantage of providing the ability to exploit the additional information to achieve a certain desired precision in choice model estimation with a much smaller sample size. Thus, ranked data surveys are more cost-effective for a specified precision level of parameters than purely choice (or first preference) data surveys. Besides, ranked data estimation allows the simultaneous prediction of both first choice and last choice, both of which may be helpful in practical situations. Again, the advantage of the ROP model (compared to all other models) comes through clearly in this context from our empirical results. The transportation industry has recently witnessed the advent of several new technologies such as autonomous vehicles and affordable electric vehicles. For these technologies, there is a growing need to study and identify early adopters as well as the product features that would be most appealing to individuals. Ranking models applied to stated preference data can be a powerful tool for undertaking such studies because there exists little revealed preference data on these new technologies. We hope that researchers and practitioners will reconsider the use of ranking data in modeling choice behavior, rather than inappropriately and summarily dismissing this type of data collection as being unreliable. ACKNOWLEDGEMENTThis research was partially supported by the Center for Teaching Old Models New Tricks (TOMNET) (Grant No. 69A3551747116) as well as the Data-Supported Transportation Operations and Planning (D-STOP) Center (Grant No. DTRT13GUTC58), both of which are Tier 1 University Transportation Centers sponsored by the US Department of Transportation. The authors are grateful to Lisa Macias for her help in formatting this document, and to three anonymous reviewers for excellent suggestions on improving the paper. Author contribution statementThe authors confirm contribution to the paper as follows: study conception and design: G.S. Nair, C.R. Bhat, B.P.Y. Loo, W.H.K. Lam, R.M. Pendyala; data collection: D. Fok, R. Paap, B. van Dijk (first dataset), B.P.Y. Loo (second dataset); analysis and interpretation of results: G.S. Nair, C.R. Bhat, B.P.Y. Loo, W.H.K. Lam, R.M. Pendyala; draft manuscript preparation: G.S. Nair, C.R. Bhat, B.P.Y. Loo, W.H.K. Lam, R.M. Pendyala. All authors reviewed the results and approved the final version of the manuscript.ReferencesChapman, R.G., and R. Staelin. Exploiting Rank Ordered Choice Set Data within the Stochastic Utility Model. Journal of Marketing Research, 1982. 19: 288–301. Foster, V., and S. Mourato. Testing for Consistency in Contingent Ranking Experiments. 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Transportation Research Part B, 2018. 109: 238–256.Pace, R.K., and J.P. LeSage. Fast Simulated Maximum Likelihood Estimation of the Spatial Probit Model Capable of Handling Large Samples. In: Spatial Econometrics: Qualitative and Limited Dependent Variables, Advances in Econometrics. Emerald Group Publishing Limited, 2016, pp. 3–34. , R.D., and P. Suppes. Preference, Utility and Subjective Probability. In: Handbook of Mathematical Psychology (R.D. Luce, R.R. Bush, E.H. Galanter, eds.), Wiley, New York, 1965, Vol. 3, pp. 249–410.van Dijk, B., D. Fok, and R. Paap. A Rank-ordered Logit Model with Unobserved Heterogeneity in Ranking Capabilities. Econometric Institute Research Paper No. EI 2007-07, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute, 2007.Croissant, Y. mlogit: Multinomial Logit Models. R package version 0.3-0, 2018.Loo, B.P.Y., S.C. Wong, and T.D. Hau. Introducing Alternative Fuel Vehicles in Hong Kong: Views from the public light bus industry. Transportation, 2006. 33: 605–619. , B., S. Wang, and T. Hau. Choice or Rank Data in Stated Preference Surveys? The Open Transportation Journal, 2008. 2: 74–79. ADDIN ZOTERO_BIBL {"uncited":[],"omitted":[],"custom":[]} CSL_BIBLIOGRAPHY LIST OF FIGURESFigure 1 Coefficient estimated by ROL, ROP, HROL, and HROP models for different kernel error distributions.Figure 2 Likelihood of test dataset for the different error distributions when using ROL, ROP, HROL, and HROP models.LIST OF TABLESTable SEQ Table \* ARABIC 1 Goodness of fit and comparison statistics for models estimated using ROL, ROP, HROL and HROP modelsFIGURE 1 Coefficient estimated by ROL, ROP, HROL, and HROP models for different kernel error distributions.FIGURE 2 Likelihood of test dataset for the different error distributions when using ROL, ROP, HROL, and HROP models.TABLE 1 Goodness of fit and comparison statistics for models estimated using ROL, ROP, HROL and HROP modelsSummary StatisticsModelROLROPHROLHROPMeasures of fit for empirical example 1 – Gaming console datasetNo. of variables in full specification11111515No. of variables in constants model5555For complete rankingsLog-likelihood at constants for the ROL model-546.82Log-Likelihood at convergence-517.37-507.34-513.13-506.41ADLRI0.04290.06120.04330.0556Non-nested comparison between modelsROL and ROPHROL and HROPp valueNested comparison between models ROL and HROLROP and HROPp valueFor observed first choiceLog-Likelihood value-131.30-124.77-124.53-124.05Average probability of correct prediction0.280.320.320.32For observed last choiceLog-Likelihood value-133.65-130.38-138.02-131.26Average probability of correct prediction0.300.290.260.28Measures of fit for empirical example 2 – LPG bus utility datasetNo. of variables in full specification881010No. of variables in constants model0000For complete rankingsLog-likelihood at constants for the ROL model-2869.78Log-Likelihood at convergence-2489.54-2416.67-2419.88-2402.49ADLRI0.12970.15510.15330.1593Non-nested comparison between modelsROL and ROPHROL and HROPp valueNested comparison between models ROL and HROLROP and HROPp valueFor observed first choiceLog-Likelihood of full specification-928.69-878.79-869.37-866.50Average probability of correct prediction0.410.460.480.48For observed last choiceLog-Likelihood of full specification-1141.847-1135.127-1161.127-1147.029Average probability of correct prediction0.370.360.320.34 ................
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