The Thurstone-Mosteller and Bradley-Terry Models are commonly used to rank items from paired comparisons experiments in which one item in each pair "wins," and to assess the importance of time-independent explanatory variables on such rankings. The first part of this thesis clarifies the use of probit and logistic regression models for such designs, including the incorporation of time-dependent explanatory variables and the analysis of unbalanced designs. In addition, likelihood inference, using the EM Algorithm, is proposed for Thurstone's Case HI Model allowing the estimation of variance parameters to account for variable item performances. The second half of this thesis presents an extension of the model to permitting the "performances" or "worths" of each competitor to be serially correlated. As an example, the performance of a basketball team in its current game is allowed to be correlated with its performance from the previous game. The Thurstone-Mosteller Model is sometimes motivated through the use of an underlying, normally-distributed performance distribution for each item or competitor, with a competitor winning a trial if a draw from its performance distribution exceeds that from its competitor's. The observed outcome is solely the win or loss for each team, but regression models, using either time-dependent or time-independent explanatory variables, may be specified for the performance means. The extension in this thesis comes from supposing the error structure for the performance distribution for each team is normal with first-order autocorrelation. The EM Algorithm is used, treating the underlying draws from the performance distributions as "missing data." This provides approximate maximum likelihood estimates; the approximation is due to the use of Monte Carlo integration in the E-step of the algorithm. Unfortunately, the heavy computational requirement and the inability to calculate the maximized likelihood function or the information matrix, make the approach unattractive for practical use. Two approximations are presented, however, which can be carried out with standard routines and some minor programming. Keywords: auto-regressive model, Bradley-Terry Model, EM Algorithm, generalized linear model, logistic regression, MCEM Algorithm, probit regression, serial correlation, Thurstone-Mosteller Model. / Graduation date: 1996

Residential real estate forecasting has become a part of the larger process of business planning and strategic management. Several studies of housing price trends recommend confining statistical analysis to repeated sales of residential property. This study presents an alternate methodology which combines information only on repeated residential sales regardless of the changes that has been made in the house in-between the sales. Additive and multiplicative models were used to forecast the residential property prices in Nelson Mandela Metropole. Data was collected from various sources and was reconciled into one data set for analysis through a process of data screening.

The method of paired comparisons is seen as a technique used to rank a set of objects with respect to an abstract or immeasurable property. To do this, the objects get to be compared two at a time. The results are input into a model, resulting in numbers known as weights being assigned to the objects. The weights are then used to rank the objects. The method of paired comparisons was first used for psychometric investigations. Various other applications of the method are also present, for example economic applications, and applications in sports statistics. This study involves taking paired comparison models and making them time-dependent. Not much research has been done in this area. Three new time series models for paired comparisons are created. Simulations are done to support the evidence obtained, and theoretical as well as practical examples are given to illustrate the results and to verify the efficiency of the new models. A literature study is given on the method of paired comparisons, as well as on the areas in which we apply our models. Our first two time series models for paired comparisons are the Linear-Trend Bradley- Terry Model and the Sinusoidal Bradley-Terry Model. We use the maximum likelihood approach to solve these models. We test our models using exact and randomly simulated data for various time periods and various numbers of objects. We adapt the Linear-Trend Bradley-Terry Model and received our third time series model for paired comparisons, the Log Linear-Trend Bradley-Terry Model. The daily maximum and minimum temperatures were received for Port Elizabeth, Uitenhage and Coega for 2005 until 2009. To evaluate the performance of the Linear-Trend Bradley-Terry Model and the Sinusoidal Bradley-Terry Model on estimating missing temperature data, we artificially remove observations of temperature from Coega’s temperature dataset for 2006 until 2008, and use various forms of these models to estimate the missing data points. The exchange rates for 2005 until 2008 between the following currencies: the Rand, Dollar, Euro, Pound and Yen, were obtained and various forms of our Log Linear-Trend Bradley-Terry Model are used to forecast the exchange rate for one day ahead for each month in 2006 until 2008. One of the features of this study is that we apply our time series models for paired comparisons to areas which comprise non-standard paired comparisons; and we want to encourage the use of the method of paired comparisons in a broader sense than what it is traditionally used for. The results of this study can be used in various other areas, like for example, in sports statistics, to rank the strength of sports players and predict their future scores; in Physics, to calculate weather risks of electricity generation, particularly risks related to nuclear power plants, and so forth, as well as in many other areas. It is hoped that this research will open the door to much more research in combining time series analysis with the method of paired comparisons.

Introduction: A typical task in quantitative data analysis is to derive estimates of population parameters based on sample statistics. For manifest variables this is usually a straightforward process utilising suitable measurement instruments and standard statistics such the mean, median and standard deviation. Latent variables on the other hand are typically more elusive, making it difficult to obtain valid and reliable measurements. One of the most widely used methods of estimating the parameter value of a latent variable is to use a summated score derived from a set of individual scores for each of the various attributes of the latent variable. A serious limitation of this method and other similar methods is that the validity and reliability of measurements depend on whether the statements included in the questionnaire cover all characteristics of the variable being measured and also on respondents’ ability to correctly indicate their perceived assessment of the characteristics on the scale provided. Methods without this limitation and that are especially useful where a set of objects/entities must be ranked based on the parameter values of one or more latent variables, are methods of paired comparisons. Although the underlying assumptions and algorithms of these methods often differ dramatically, they all rely on data derived from a series of comparisons, each consisting of a pair of specimens selected from the set of objects/entities being investigated. Typical examples of the comparison process are: subjects (judges) who have to indicate for each pair of objects which of the two they prefer; sport teams that compete against each other in matches that involve two teams at a time. The resultant data of each comparison range from a simple dichotomy to indicate which of the two objects are preferred/better, to an interval or ratio scale score for e d Bradley-Terry models, and were based on statistical theory assuming that the variable(s) being measured is either normally (Thurstone-Mosteller) or exponentially (Bradley-Terry) distributed. For many years researchers had to rely on these PCM’s when analysing paired comparison data without any idea about the implications if the distribution of the data from which their sample were obtained differed from the assumed distribution for the applicable PCM being utilised. To address this problem, PCM’s were subsequently developed to cater for discrete variables and variables with distributions that are neither normal or exponential. A question that remained unanswered is how the performance, as measured by the accuracy of parameter estimates, of PCM's are affected if they are applied to data from a range of discrete and continuous distribution that violates the assumptions on which the applicable paired comparison algorithm is based. This study is an attempt to answer this question by applying the most popular PCM's to a range of randomly derived data sets that spans typical continuous and discrete data distributions. It is hoped that the results of this study will assist researchers when selecting the most appropriate PCM to obtain accurate estimates of the parameters of the variables in their data sets.

The method of paired comparisons can be found all the way back to 1860, where Fechner made the first publication in this method, using it for his psychometric investigations [4]. Thurstone formalised the method by providing a mathematical background to it [9-11] and in 1927 the method’s birth took place with his psychometric publications, one being “a law of comparative judgment” [12-14]. The law of comparative judgment is a set of equations relating the proportion of times any stimulus k is judged greater on a given attribute than any other stimulus j to the scales and discriminal dispersions of the two stimuli on the psychological continuum. The amount of research done for discrete models of paired comparisons is not a lot. This study develops a new discrete model, the SL-model for paired comparisons. Paired comparisons data processing in which objects have an upper limit to their scores was also not yet developed, and making such a model is one of the aims of this report. The SLmodel is thus developed in this context; however, the model easily generalises to not necessarily having an upper limit on scores.

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In this thesis, we focus on the the comparison of treatments with ordered categorical responses. The three cases of treatment comparisons will all be studied. The main objective of this thesis is to develop more effective comparison methods for treatments with ordinal responses and to address some important issues involved in different comparison problems. Our major statistical approach is to consider ordinal responses as manifestations of some underlying continuous random variables. / The comparison of treatments to detect possible treatment effects is a very important topic in statistical research. It has been drawing significant interests from both academicians and practitioners. Important research work on treatment comparisons dates back several decades. For treatment comparisons, the following three cases are very common: the comparison of two independent treatments; the comparison of treatments with repeated measurements; and the multiple comparison of several treatments. For different cases, the involved research issues are usually different. In many fields of study, the level of measurement for responses of the treatments is ordinal. Many examples can be found in areas such as biostatistics, psychology, sociology, and market research, where the ordered categorical variables play an important role. / This thesis consists of three main parts. In the first part, we consider the modeling of treatments with longitudinal ordinal responses by a latent growth curve. On the basis of such a latent growth curve, we achieve a comprehensive flexible model with straightforward interpretations and a variety of applications including treatment comparison, the analysis of covariates, and equivalence test of treatments. In the second part, we consider the comparison of several treatments with a control for ordinal responses. By considering the ordinal responses as manifestations of some underlying normal random variables, a latent normal distribution model is utilized and the corresponding parameter estimation method is proposed. Further, we also derive testing procedures that compare several treatments with a control under an analytical framework. Both single-step and stepwise procedures are introduced, and these procedures are compared in terms of average power based on a simulation study. In the last part of this thesis, we establish a unified framework for treatment comparisons with ordinal responses, which allows various treatment comparison methods be comprehended using a unified perspective. The latent variable model is also utilized, but the underlying random variables are allowed to have any member of the location-scale distribution family. This latent variable model under such a specification of underlying distributions subsumes many existing models in the literature. A two-step procedure to identify the model and produce the parameter estimates is proposed. Based on this procedure, many important statistical inferences can be conveniently conducted. Furthermore, the sample size determination method based on the latent variable method is also proposed. The proposed latent variable method is compared with the existing methods in terms of power and sample size. / Lu, Tongyu. / Adviser: Wai-Yin Poon. / Source: Dissertation Abstracts International, Volume: 73-06, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 94-101). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2004. / Includes bibliographical references (p. 65).

I. A Comparison of the Effectiveness of Tournaments. A paired-comparison experiment involving t treatments is analogous to a tournament with t players. A balanced experiment, in which every possible pair occurs once per replication, is the counterpart of a round robin tournament. When the objective is to pick the best treatment, the balanced design may prove to be more expensive than necessary. The knock-out tournament has been suggested as an alternative requiring fewer units of each treatment per replication. In this paper round robin, replicated knock-out, and double elimination tournaments are investigated for their effectiveness in selecting the best one or tour players. Effectiveness is gauged in terms of the two criteria (a) the probability that the best player wins and (b) the expected number of games. For general values of the parameters involved, expressions are derived for the evaluation of the criteria. Comparisons are made on the basis of series of assigned parameter values. Possibilities for the extension of the study are briefly discussed. II. Ties in Paired-Comparison Experiments. In making paired comparisons a judge frequently is unable to express a real preference in a number of the pairs he judges. In spite of this, some or the methods in current use do not permit the judge to declare a tie. In other methods tied observations are either ignored or divided equally or randomly between the tied members. It appears that there is a need, at least in the estimation of response-scale values, for a method which takes tied observations into account. In the Thurstone-Mosteller method the standardized distribution of the difference of two stimulus responses is normal with unit variance and mean equal to the difference or the two mean stimulus responses. In prohibiting ties the assumption is in effect made that all differences, however small, are perceptible to the judge. In this paper the assumption is made that a tie will occur whenever the difference between the judge's responses to the two stimuli lie below a certain threshold, i.e. if the difference lies between -t and t the judge will declare a tie. The parameter t and the mean stimulus responses are estimated by least squares. To overcome a difficulty presented by correlated data, an angular response law is postulated for the response-scale differences. In the resulting transformed data non-homogeneity of variances is encountered. In effecting a weighted solution, weights are first determined by using a preliminary unweighted analysis, and an iterative procedure is proposed. Large-sample variances and covariances of the estimates are obtained. A test of the validity of the model is described. A computational procedure is set up, and exemplified through application to experimental data. / Ph. D.

New methods for testing hypotheses in paired-comparison experiments are presented in this dissertation. The methods are developed on the basis of a very general mathematical model and they are, in general, quite easy to employ. Two tests of the null hypothesis that all treatments have equal stimuli, against its general alternative, are proposed. One test is for the case in which it is assumed prior to the experiment that no interaction will take place between repetitions and preference probabilities (the probabilities of the possible comparison preferences). The other test is for the case in which the above assumption cannot be made. The number of times a treatment is preferred is called its score. For the “no interaction" case, the test procedure is based on a test statistic that is a function D of the corrected sum of squares of the treatment scores. In the other case, the value of D is calculated for each group of homogeneous repetitions and then the values are summed to give the new test statistic. It is established that a X²-approximation may be used to determine the critical value of the test statistic for experiments outside the range of the tabled distributions. This test procedure is shown to be simpler than other approximate tests and, in general, at least as accurate with respect to errors of the first kind. It is shown that the two test methods discussed above may be extended to ranking experiments in balanced incomplete block designs with more than two treatments per block. To test the null hypothesis of no interaction between preference probabilities and repetitions, against its general alternative, a test method based on the theory of X² homogeneity tests is introduced. Means are presented for testing whether (1) a particular treatment is better than the average of the treatment stimuli; (2) two particular treatment stimuli are not equal; and (3) the treatment receiving the highest score is better than the average. The three test procedures are based essentially on the binomial distribution of the treatment scores under the null hypothesis. In each case, the test procedure is conservative. A procedure analogous to Tukey's test based on allowances is developed to test the null hypothesis of equal treatment stimuli and to separate the significantly different treatment scores when it rejects the null hypothesis. A method for judging contrasts of treatment scores similar to Scheffe's (1953) method for judging contrasts in the analysis of variance is proposed. The test method based on D, mentioned earlier, is used in place of the F-test employed in the Scheffe method. The use of paired-comparison experiments to test factorial effects is discussed and a test method based on orthogonal contrasts of the treatment scores is suggested. Because of correlations that arise, it is necessary to restrict this method to cases in which the only factors that are allowed to appear at more than two levels are those that will not interact with the other factors in the experiment. The test methods are illustrated through application on the data from two paired-comparison experiments. / Ph. D.