Estimating social and nonsocial utility functions from ordinal data

This paper reports an experiment testing two hypotheses. The first is that the value or utility associated with a payment to one's self and a payment to a co-worker can be represented as an additive function of a utility for own payment (nonsocial utility) and a utility for the difference between own and other's payment (social utility). The second hypothesis is that changes in the amount of work accomplished by one's self and/or the other should influence the social, but not the the nonsocial utilities. Support for both hypotheses is reported.     

The ABC of cardinal and ordinal number representations

Numerical cognition encompasses the concepts of quantity ('how many?') and serial order ('which position?'). Yet, although numbers can convey different meanings, a recent imaging study by Fias and coworkers showed that ranking letters in the alphabet is subserved by a cortical network highly similar to that involved in judging magnitudes. In terms of neural processing, quantity and rank might just be two sides of the same coin.     

Inducing ordinal and cardinal representations of the first five natural numbers

The prediction that the ordinal property of natural number symbols (using these symbols to represent the terms in an ordered progression) is more easily learned than the cardinal property of natural number symbols (using these symbols to represent the manyness of collections) was examined in this experiment. Preschoolers who evidenced no proficiency with either the ordinal or cardinal properties of natural number symbols were trained to acquire these properties via simple feedback. Both properties proved to be trainable. The most important findings were that the ordinal property was much easier to train than the cardinal property, ordinal training effects were more durable across a 1-week interval than cardinal training effects, and ordinal training appeared to transfer better than cardinal training.     

Decomposition of multivariate utility functions in non-additive expected utility theory

In expected utility many results have been derived that give necessary and/or sufficient conditions for a multivariate utility function to be decomposable into lower-dimensional functions. In particular, multilinear, multiplicative and additive decompositions have been widely discussed. These utility functions can be more easily assessed in practical situations. In this paper we present a theory of decomposition in the context of nonadditive expected utility such as anticipated utility or Choquet expected utility. We show that many of the results used in conventional expected utility carry over to these more general frameworks. If preferences over lotteries depend only on the marginal probability distributions, then in expected utility the utility function is additively decomposable. We show that in anticipated utility the marginality condition implies not only that the utility function is additively decomposable but also that the distortion function is the identity function. We further demonstrate that a decision maker who is bivariate risk neutral has a utility function that is additively decomposable and a distortion function q for which q(½) = ½.     

Positive-difference structures and bilinear utility functions

Let (M₁, f), (M₂, g) be mixture sets and let ? be a binary preference relation on M₁ × M₂. By using the concept of positive-difference structures, necessary and sufficient conditions are given for the existence of a real-valued utility function u on M₁ × M₂ which represents ? and possesses the bilinearity property

$\text{u(?(α, x}{}_1\text{,x}{}_2\text{),g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{αu(x}{}_1\text{, g(βy}{}_1\text{, y}{}_2\text{))}\text{+}\text{(1 ? α) u(x}{}_2\text{, g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{βu(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_1\text{)}\text{+}\text{(1 ? β) u(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_2\text{)}$

, for all α, β ∈ [0, 1], all x₁, x₂ ∈ M₁ and all y₁, y₂ ∈ M₂. Moreover, uniqueness up to positive linear transformations can be proved for those utility functions. Finally an outline is given of applications of these results in expected utility theory.     

Validation and error in multiplicative utility functions

An approach to the concept of error in utility assessment is proposed. Four kinds of errors are considered and each kind is related to four separate elicitation methods - all in the context of a general multiplicative multiattribute utility model. The methods are a Keeney-Raiffa (1976) procedure; SMART, for Simple Multi-Attribute Rating Technique (Edwards 1977); SJT, for a Social Judgment Theory based regression model (Hammond et al. 1975); and HOPE, for Holistic Orthogonal Parameter Estimation (Barron and Person 1979).The individual judgments elicited are either holistic - in which the entity to be evaluated is considered as a whole - or decomposed - in which attention is directed to one or two aspects at a time.If a general multiplicative model can be assumed to be an appropriate representation of the decision maker's basic preference structure, error can occur in the direct estimation of the scaling constants and univariate utility functions for decomposition methods (Keeney-Raiffa and SMART), or in the holistic assessments for holistic methods (SJT and HOPE). Individual estimates may be merely subject to random noise or may be substantially incorrect. The utility model may be incorrectly specified; finally, all four methods may be subject to systematic error. The four assessment methods are considered in conjunction with errors of each kind.     

Seven independence concepts and continuous multiattribute utility functions

This paper examines seven independence concepts based on a preference relation on the set of simple probability measures defined on a set of multiattribute consequences. Three of the independence relations involve gambles and the other four are based on riskless preferences over the n-tuples in the consequence set. The main theorems state conditions under which one or more of the risky independence relations can be derived from a riskless independence relation in conjunction with other conditions. The other conditions include a risky independence condition which differs from the one(s) to be derived, the assumption that the consequence set is a convex subset of a finite-dimensional Euclidean space, and the assumption that the individual's von Neumann-Morgenstern utility function on the consequence set is continuous.     

Accounting for nonlinear utility functions in composite measures of productivity and performance

Composite measures of productivity and performance combine different indicators of work activities into a single index. The term nonlinearities refers to the phenomenon that the relationship between the scores on an indicator of productivity and the value of that level of the indicator to the organization is not always linear. For example, more is not always better. This research evaluated whether accounting for such nonlinear relationships in productivity measures adds new information to the composite measures. Data from five organizations showed that the correlations between a composite measure which includes nonlinear relationships and one which does not were very high, but treatment effect sizes and decisions resulting from the two measurement systems were substantially different. It was concluded that inclusion of nonlinearities added important information to composite measures of productivity and that accounting for nonlinearities should result in more valid composites. Applications of this issue for other topics such as performance appraisal and promotions were discussed.     

The role of instructions in the transfer of ordinal functions through equivalence classes

In two experiments, adult subjects completed match-to-sample training and testing to establish four equivalence classes of four figures each. Then the subjects were taught one three-position sequence consisting of one stimulus from Class 1, one from Class 2, and one from Class 3. Inclusion of Class 4 stimuli in sequences was never reinforced, but two different stimuli from Class 4 appeared as distractors on each sequence trial. Tests assessed whether subjects would produce novel three-position sequences composed of members of Classes 1 through 3 that had not been used in sequence training. Three subjects in Experiment 1 received instructions about the match-to-sample and sequencing tasks, in addition to training contingencies. All 3 demonstrated equivalence class formation after match-to-sample training. After they were taught one sequence with one member of Classes 1 through 3, none of these subjects produced untrained sequences with other equivalence class members reliably. One additional sequence was trained directly; thereafter 1 subject showed some evidence of transfer of the trained ordinal functions across the remaining members of the equivalence classes, but the other 2 did not. Following a review of equivalence class training and testing and a review of the original sequence training, all 3 subjects produced most of the predicted, untrained sequences on tests. Experiment 2 replicated Experiment 1 with 2 adults but omitted all instructions except the minimal ones necessary to initiate responding. Unlike the subjects in Experiment 1, both of these subjects demonstrated virtually complete transfer of ordinal functions through the equivalence classes after direct training on just one sequence composed of one member of Classes 1 through 3.     

Evolutionary preference/utility functions: A dynamic perspective

The collection of repeated measures in psychological research is one of the most common data collection formats employed in survey and experimental research. The behavioral decision theory literature documents the existence of the dynamic evolution of preferences that occur over time and experience due to learning, exposure to additional information, fatigue, cognitive storage limitations, etc. We introduce a Bayesian dynamic linear methodology employing an empirical Bayes estimation framework that permits the detection and modeling of such potential changes to the underlying preference utility structure of the respondent. An illustration of revealed stated preference analysis (i.e., conjoint analysis) is given involving students’ preferences for apartments and their underlying attributes and features. We also present the results of several simulations demonstrating the ability of the proposed procedure to recover a variety of different sources of dynamics that may surface with preference elicitation over repeated sequential measurement. Finally, directions for future research are discussed.The authors wish to acknowledge and thank the Editor, the Associate Editor, and two anonymous reviewers for their constructive and insightful comments. Duncan K.H. Fong’s work was sponsored in part by a research grant from the Smeal College.This revised article was published online in August 2005 with the PDF paginated correctly.     

An empirical assessment of the form of utility functions

Utility functions, which relate subjective value to physical attributes of experience, are fundamental to most decision theories. Seven experiments were conducted to test predictions of the most widely assumed mathematical forms of utility (power, log, and negative exponential), and a function proposed by Rachlin (1992). For pairs of gambles for real monetary gains, undergraduate and nonstudent subjects either reported an equalizing amount for 1 outcome that made the gambles subjectively equal or chose between gambles where the amounts were varied across trial, which allowed the equalizing amount to be estimated from their pattern of choices. Using a novel method that eliminates several limitations of previous research, I manipulated the outcomes across trials such that each type of utility function predicted a linear relationship between the equalizing amounts and the amounts of the other outcomes, and made point predictions for either the slope or intercept of that relationship. In a meta-analysis across experiments, systematic departures from the point predictions were observed for each type of utility function. Thus, the data imply that despite their historical importance and incorporation in many psychological and economic decision theories, the most widely assumed models of utility are incorrect.     

One-sided nonparametric tests for ordinal data

Baumgartner, Weiss, and Schindler (1998) introduced a novel non-parametric test for the two-sample comparison that is superior to commonly used tests such as the Wilcoxon rank-sum test. A modification of the novel test statistic can be used for one-sided comparisons based on ordinal data. Such comparisons frequently occur in psychological research, and the Wilcoxon test is often recommended for their analysis. Here, the two tests were compared in a simulation study. According to this study the tests have a similar type I error rate, but the modified Baumgartner-Weiss-Schindler test is more powerful than the Wilcoxon test.     

Models for ordinal hierarchical classes analysis

This paper proposes an ordinal generalization of the hierarchical classes model originally proposed by De Boeck and Rosenberg (1998). Any hierarchical classes model implies a decomposition of a two-way two-mode binary arrayM into two component matrices, called bundle matrices, which represent the association relation and the set-theoretical relations among the elements of both modes inM. Whereas the original model restricts the bundle matrices to be binary, the ordinal hierarchical classes model assumes that the bundles are ordinal variables with a prespecified number of values. This generalization results in a classification model with classes ordered along ordinal dimensions. The ordinal hierarchical classes model is shown to subsume Coombs and Kao's (1955) model for nonmetric factor analysis. An algorithm is described to fit the model to a given data set and is subsequently evaluated in an extensive simulation study. An application of the model to student housing data is discussed.     

Coefficients of feasibility for ordinal response scales

Cronbach's alpha is systematically used in the social sciences to estimate internal consistency. However, this coefficient assumes the continuity of the variables and this assumption is not met by ordinal response items or Likert scales. This work shows two alternatives to the alpha coefficient, the ordinal alpha and beta coefficients. Both of them take into account the ordinal nature of the data. The new coefficients were applied to several scales and compared with the traditional estimation of alpha. This paper also shows how to estimate those coefficients.     

Confidence intervals for gamma-family measures of ordinal association

This research focused on confidence intervals (CIs) for 10 measures of monotonic association between ordinal variables. Standard errors (SEs) were also reviewed because more than 1 formula was available per index. For 5 indices, an element of the formula used to compute an SE is given that is apparently new. CIs computed with different SEs were compared in simulations with small samples (N = 25, 50, 75, or 100) for variables with 4 or 5 categories. With N > 25, many CIs performed well. Performance was best for consistent CIs due to N. Cliff and colleagues (N. Cliff, 1996; N. Cliff & V. Charlin, 1991; J. D. Long & N. Cliff, 1997). CIs for Spearman's rank correlation were also examined: Parameter coverage was erratic and sometimes egregiously underestimated.     

Latent variable models for multivariate longitudinal ordinal responses

The paper proposes a full information maximum likelihood estimation method for modelling multivariate longitudinal ordinal variables. Two latent variable models are proposed that account for dependencies among items within time and between time. One model fits item‐specific random effects which account for the between time points correlations and the second model uses a common factor. The relationships between the time‐dependent latent variables are modelled with a non‐stationary autoregressive model. The proposed models are fitted to a real data set.     

Methods for describing ordinal data with cardinal models

Several methods, both new and old, for describing ordinal data with a cardinal model are discussed in a newly developed and general context which broadens their applicability beyond their traditional use in multidimensional scaling. The relationships between the methods is investigated. It is shown that the two most commonly used methods (Guttman's rank-image principle and Kruskal's least-square monotonic transformation) are the boundary conditions of a newly proposed single parameter family of methods. An additional method is proposed which is shown to yield an equal-density model space. The commonly made distinction between “transformation” methods and “transformation-free” methods is shown to be a pseudodistinction. It is observed that all the methods can be stated as matrix operations on the model space with the conclusion that they try to optimize a linear combination of the model space.