An application of a biased version of the Bradley-Terry-Luce model to professional basketball results

Some extensions of the Bradley-Terry-Luce model for paired comparisons are presented and used to assess the existence of a home court advantage for professional basketball teams. It is shown how maximum likelihood estimates for these models can be obtained from a simple iterative scaling algorithm. Won-loss records for the regular season are used to make predictions for playoff series. The models are extended to include possible differences among divisions. These models can be applied in many other situations to incorporate group effects and order effects into analysis of paired comparisons.     

Item Response Modeling of Paired Comparison and Ranking Data

The comparative format used in ranking and paired comparisons tasks can significantly reduce the impact of uniform response biases typically associated with rating scales. Thurstone's (1927, 1931) model provides a powerful framework for modeling comparative data such as paired comparisons and rankings. Although Thurstonian models are generally presented as scaling models, that is, stimuli-centered models, they can also be used as person-centered models. In this article, we discuss how Thurstone's model for comparative data can be formulated as item response theory models so that respondents' scores on underlying dimensions can be estimated. Item parameters and latent trait scores can be readily estimated using a widely used statistical modeling program. Simulation studies show that item characteristic curves can be accurately estimated with as few as 200 observations and that latent trait scores can be recovered to a high precision. Empirical examples are given to illustrate how the model may be applied in practice and to recommend guidelines for designing ranking and paired comparisons tasks in the future.     

An investigation of data collection methods for auditory stimuli: Paired comparisons versus a computer sorting task

Two experiments were performed to determine whether a computer sorting task could be used to collect data for auditory stimuli. Within each experiment, subjects were assigned to either a paired comparison task or a computer sorting task. For the first experiment, the stimuli were 18 graph representations constructed to vary on three parameters (shape, frequency, and amplitude). The stimuli for the second experiment were 30 sentences produced by actors portraying five emotions (anger, fear, happiness, sadness, and neutral). Multidimensional scaling solutions were produced and comparisons were made between the paired comparison and computer sorting task in each experiment. The results suggest that the computer sorting task is a viable alternative to the paired comparison task traditionally used for the investigation of the perception of auditory stimuli.     

Successive intervals scaling of paired comparisons

In this paper a method is developed for analyzing data resulting from the use of a modification of the usual paired comparisons procedure which allows for ratings of size of difference. The scaling model is essentially an extension of the Thurstonian successive intervals model. The method is applied to scaling of a set of nine immoral actions and the results agree rather well with those from conventional successive intervals scaling of the same stimuli.This study was supported by a grant from the Swedish Council for Social Science Research.     

An Analysis of the Status and Stability of the Behaviors of Students with Emotional and Behavioral Difficulties

Students with emotional and behavioral problems are placed in a variety of classroom settings, presumably where their needs can be best served. Little attention, however, has been dedicated to monitoring their behavioral progress in these settings. This study analyzed data on student engagement, disruption, and destruction during 10 types of instructional activities. Data were collected at two times across one academic year in Pennsylvania and California in 90 classrooms, ranging from grades 1 to 12. Tests of mean comparisons (i.e., independent samples and paired t-tests) and the association between variables (i.e., bivariate correlation analysis) were conducted to address a number of concerns regarding the status and stability of classrooms for students with EBD. Independent paired sample t-tests showed a lack of significant improvement in behaviors across time, and the correlational data reflect a significant relationship between student engagement and classroom activity. Findings are discussed in light of limitations and directions for future research. In addition, practical implications are noted.     

A choice model for paired comparison data based on imperfectly nested sets

A ratio scale of subjective magnitude is developed from paired-comparison data. The model attempts to combine arguments of Restle, Ekman, and Luce relating data obtained from paired comparisons and corresponding data from direct psychophysical scaling methods at the ratio level. The basic set-theoretical model involves the use of imperfectly nested sets. A numerical example illustrates the application of the theory.This research was supported by the Swedish Social Science Research Council and by the Swedish Board of Computing Machinery. I am indebted to Mr. U. Forsberg for computational assistance.     

A new scaling technique for absolute judgments

The results of an experiment using the method of absolute judgments can be viewed as a matrix of conditional probabilities in which the rows represent stimuli and the columns responses. The cosine of the angle between two row vectors is a measure of the similarity of the corresponding stimuli. This cosine provides the basis for a method of scaling the stimuli. Unlike the method of paired comparisons, this new technique does not require arbitrary fixing of a unit of measurement. A numerical example is given.     

A stochastic multidimensional scaling procedure for the empirical determination of convex indifference curves for preference/choice analysis

The vast majority of existing multidimensional scaling (MDS) procedures devised for the analysis of paired comparison preference/choice judgments are typically based on either scalar product (i.e., vector) or unfolding (i.e., ideal-point) models. Such methods tend to ignore many of the essential components of microeconomic theory including convex indifference curves, constrained utility maximization, demand functions, et cetera. This paper presents a new stochastic MDS procedure called MICROSCALE that attempts to operationalize many of these traditional microeconomic concepts. First, we briefly review several existing MDS models that operate on paired comparisons data, noting the particular nature of the utility functions implied by each class of models. These utility assumptions are then directly contrasted to those of microeconomic theory. The new maximum likelihood based procedure, MICROSCALE, is presented, as well as the technical details of the estimation procedure. The results of a Monte Carlo analysis investigating the performance of the algorithm as a number of model, data, and error factors are experimentally manipulated are provided. Finally, an illustration in consumer psychology concerning a convenience sample of thirty consumers providing paired comparisons judgments for some fourteen brands of over-the-counter analgesics is discussed.     

A covariance analysis of multiple paired comparisons

In the method of multiple paired comparisons the dominance of objectj over objectk is observed uponp attributes. The present paper develops a covariance analysis for these paired comparisons in terms of a linear model which includes scale, bias, and interaction effects, along withs covariants upon which the comparisons are presumably dependent. The covariance model gives rise to adjusted parameter estimates and hypothesis tests for the residual pairwise layout from which the effects of thes covariants have been removed. These estimation and testing procedures are illustrated with an analysis of political judgment data, and their relevance to the general problem of residual scaling is discussed.This research was supported by National Institute of Mental Health grants MH 12972 and MH 15506. The author would like to express his appreciation to Jerry Solomon and Wei-Ching Chang of the Oregon Research Institute for the data analysis reported at the end of the paper. Computing assistance was obtained from the Health Sciences Computing Facility, UCLA, sponsored by N.I.H. Grant FR-3.     

Intrapersonal and social comparison determinants of happiness: a range-frequency analysis

Examined whether intrapersonal comparisons and social comparisons operate in similar ways to determine ratings of happiness. Events were varied to create positively and negatively skewed distributions. The events in each distribution were ascribed to either a single person or a group of people; Ss rated how happy they would feel if they experienced specific events within the distribution. Ratings for both intrapersonal and social comparisons were fit well by Parducci's (1984) range-frequency theory. Individual events received higher ratings when presented within the positively skewed context. Overall happiness, as measured by both the mean of the happiness ratings as well as direct ratings, was highest for the negatively skewed distributions. The effects of skewing were more pronounced for intrapersonal comparisons, but ratings were more closely defined by the range of experimental stimuli for social comparisons.     

An Alternate Method of Deriving Priorities and Related Inferences for Group Decision Making in Analytic Hierarchy Process

Some work has been carried out in the past on statistically deriving priorities in Analytic Hierarchy Process (AHP). In AHP, the aggregated worths of the alternatives, when compared with respect to several criteria, are estimated in a hierarchical comparisons model introduced by Saaty. In this setup, statistical models are used for Saaty's method of scaling in paired comparisons experiments in any level of the hierarchy. At the end, the final priority weights of the alternatives and related inferences are developed with appropriate statistical methods. Existing statistical methods in the literature assume independence of the entries of the paired comparison matrix. However, these entries are highly dependent among themselves. In this article, we propose a statistical method that allows for the dependence among the entries of the pairwise comparisons matrix. The proposed method is then illustrated with a numerical example. Copyright © 2011 John Wiley & Sons, Ltd.     

Tests of the psychological meaning of the power law.

The purpose of this study was to establish a theoretical framework for Stevens' empirically derived power law. Three models were proposed to explain the power law. They respectively outline how sensory, stimulus, and response variables determine the judgmental behavior in a psychophysical task. A correlational study on individual differences in exponents was carried out to test the predictions derived from each model. The use of four different sensory continua and four scaling procedures provided the experimental means of manipulating the sensory, stimulus, and response variables in the scaling situation. The results showed that response variables are important determinants of judgmental behavior in psychophysical scaling. These findings suggest that subjects' responses to stimulus intensities in a scaling task are largely cognitive.     

An internal consistency check for scale values determined by the method of successive intervals

The method of successive intervals is a psychological scaling procedure in which stimuli are classified into successive intervals according to the degree of some defined attribute which they are judged to possess. A psychological continuum is defined and the scale values are then taken as the medians of the distributions of judgments on the psychological continuum. It is assumed that the distributions of judgments for each stimulus are normal on the psychological continuum as defined.An internal consistency check indicates that the cumulative distributions of empirical judgments for the various stimuli can be reproduced by means of a limited number of parameters with an average error that compares favorably with that usually reported for paired comparison data. Furthermore, the scale values obtained by successive interval scaling, for the data reported, are shown to be linearly related to those obtained by the method of paired comparisons.This paper was written while the first author was a post-doctoral Research Training Fellow of the Social Science Research Council studying at the University of Chicago. It reports research undertaken in cooperation with the Quartermaster Food and Container Institute for the Armed Forces. The views or conclusions contained in this report are those of the authors. They are not to be construed as necessarily reflecting the views or endorsement of the Department of Defense.     

Increased conformity to a fellow deviant as a function of prior deviation

A subject was induced to publically report his opinions on a variety of issues. A group of confederates disagreed with the subject; one confederate agreed with him. Thus, the subject deviated from the majority of the group, but received social support from one other individual. Each subject was then paired with one individual from the previous group, and regardless of his pairing, was subjected to a constant amount of conformity pressure from the confederate on a new set of judgments. Deviant subjects conformed more to the confederate who had previously given them social support than they did to a previous majority member. This increased conformity was found in two studies, (1) females judging physical realities and, (2) males reporting their opinions on attitudinal issues. Ratings taken in the second study demonstrated that the deviant subject felt increased liking for and similarity to the confederate who initially provided him with social support, but comparisons between various experimental groups indicated conformity to a confederate was not completely predictable from the subject's liking for that confederate. Instead, conformity was greatest following an experience of deviation from the majority.     

Perceived trait-co-occurrences according to managerial role: A structural representation

Fifty-eight middle managers were presented with three check lists containing effective and ineffective managerial traits previously obtained as free responses from another sample of middle managers. Each manager in the present study was asked to describe a superior, a peer, and a subordinate. Multidimensional scaling solutions were then calculated from these check list data. Ratings, obtained on five properties, were used to interpret the solutions.The findings demonstrated that managers in a highly competitive organization having frequent, formalized evaluative procedures (1) ascribe highly negative attributes to those peers perceived as effective, (2) have relatively undifferentiated views of their ineffective peers and ineffective subordinates, but (3) not their superiors who are viewed critically whether effective or ineffective.     

The assumed effects of positive feedback paired with success on motivation to do a task: The cases of characters with high and low initial levels of interest

This study investigated the effects of positive feedback paired with success on motivation to do a task, and examined whether this differed according to whether the person performing the task had a high or low initial level of interest in that task. Subjects (448 junior college students) were asked to rate (from 0–10) the level of motivation of characters in four different stories. Subjects believed that positive feedback would encourage the character to do a task regardless of initial level of interest. However, when positive feedback could not be expected, ratings for motivation fell. Ratings for motivation were further analyzed when no positive feedback could be expected, either after an earlier experience of positive feedback paired with success or after success only (no feedback). In the case of a low initial level of interest, it was assumed that motivation would be increased after positive feedback, but in the case of a high initial level of interest motivation was assumed to be decreased by an earlier experience of positive feedback paired with success. These results suggest that the effect of positive feedback paired with success will differ according to the level of a person's initial interest.     

The scaling of test scores by the method of paired comparisons

The purpose of this study is to investigate, by the method of paired comparisons, a possible scaling of individuals who have made certain test scores, such that the additive property will be satisfied and such that a stability in scaling will be maintained,—in other words, a scaling such that the scaled score of an individual will remain relatively the same regardless of the grouping of individuals in which he may be placed. The results show that it is possible to utilize psychophysical methods in psychological and educational test situations. Among the major findings are that Case V of the Law of Comparative Judgment is applicable to the data in this problem, the method of dividing the intermediate category equally between the greater and the less was the best of three possible methods, internal consistency was satisfied, and, finally, when a new test of stability was applied, it was found that the distances between the hypothetical individuals remain the same.The writer wishes to express appreciation for the invaluable help and guidance of Professor Harold O. Gulliksen and also to Professor Marion W. Richardson who suggested the problem and made valuable suggestions.     

Individual loudness functions determined from direct comparisons of loudness intervals

Five subjects were required in each trial to directly compare two pairs of tones and indicate which pair of tones had the greater loudness difference. Ten 1,200-Hz tones differing only in intensity were employed. Subjects made binary comparisons among the 45 tone pairs that can be formed from these 10 tones. The loudness difference comparisons of each subject were found to satisfy two properties (transitivity and monotonicity) that are required for an interval scale representation of loudness. Therefore, individual loudness scales were constructed using a nonmetric scaling technique designed for comparisons of sensory intervals. These loudness scales differed significantly from subject to subject. Since a nonnumerical scaling procedure was employed, these individual differences could not be attributed to biases in the way in which observers use numbers or numerical concepts to describe the loudness of tones. Hence, they suggest strong individual differences in the coding of sound intensity.     

Loudness “ratios” and “differences” involve the same psychophysical operation

Subjects judged both “atios” of loudness and “differences” in loudness between pairs of tones that varied in intensity. The pairs were constructed from factorial designs, permitting separation of stimulus and response scaling for each subject. Ratings of “differences” and estimations of “ratios” were monotonically related, inconsistent with the hypothesis that subjects perform both subtractive and ratio operations on a common scale. Instead, the data suggest that both tasks involve the same psychophysical comparison operation with different response transformations. If the operation can be represented by the subtractive model, then category ratings involve a nearly linear transformation and magnitude estimations involve a nearly exponential transformation.