Assessing goodness-of-fit of categorical regression models based on case-control data

Demonstrated equivalence between a categorical regression model based on case‐control data and an I‐sample semiparametric selection bias model leads to a new goodness‐of‐fit test. The proposed test statistic is an extension of an existing Kolmogorov–Smirnov‐type statistic and is the weighted average of the absolute differences between two estimated distribution functions in each response category. The paper establishes an optimal property for the maximum semiparametric likelihood estimator of the parameters in the I‐sample semiparametric selection bias model. It also presents a bootstrap procedure, some simulation results and an analysis of two real datasets.     

Ordinal utility models of decision making under uncertainty

This paper studies two models of rational behavior under uncertainty whose predictions are invariant under ordinal transformations of utility. The quantile utility model assumes that the agent maximizes some quantile of the distribution of utility. The utility mass model assumes maximization of the probability of obtaining an outcome whose utility is higher than some fixed critical value. Both models satisfy weak stochastic dominance. Lexicographic refinements satisfy strong dominance.The study of these utility models suggests a significant generalization of traditional ideas of riskiness and risk preference. We define one action to be riskier than another if the utility distribution of the latter crosses that of the former from below. The single crossing property is equivalent to a minmax spread of a random variable. With relative risk defined by the single crossing criterion, the risk preference of a quantile utility maximizer increases with the utility distribution quantile that he maximizes. The risk preference of a utility mass maximizer increases with his critical utility value.     

资本下乡：进入模式选择及其影响因素

进入模式是经济组织配置资源的重要手段。以工商资本对农户经营活动的控制权程度为依据,将工商资本投资农业的进入模式划分为“松散型”、“半紧密型”、“紧密型”三种类型。利用对广东、湖北、江苏以及云南四省151个投资主体的调查数据,采用有序逻辑回归模型分析生产因素、市场因素、制度因素以及生产技术传递能力对工商资本投资农业的进入模式选择的影响。结果表明：耕地资源优势越弱,工商资本投资农业选择紧密型进入模式的可能性越高;市场潜力越大,越倾向于选择紧密型进入模式;非正式制度越不利,选择紧密型进入模式的可能性越低;生产技术的传递能力越弱,越倾向于选择紧密型进入模式。     

The Box–Cox Transformation and Non‐Iterative Estimation Methods for Ordinal Log‐Linear Models

Recently Beh and Farver investigated and evaluated three non‐iterative procedures for estimating the linear‐by‐linear parameter of an ordinal log‐linear model. The study demonstrated that these non‐iterative techniques provide estimates that are, for most types of contingency tables, statistically indistinguishable from estimates from Newton's unidimensional algorithm. Here we show how two of these techniques are related using the Box–Cox transformation. We also show that by using this transformation, accurate non‐iterative estimates are achievable even when a contingency table contains sampling zeros.     

Evidence of bias in the Eurovision song contest: modelling the votes using Bayesian hierarchical models

The Eurovision Song Contest is an annual musical competition held among active members of the European Broadcasting Union since 1956. The event is televised live across Europe. Each participating country presents a song and receive a vote based on a combination of tele-voting and jury. Over the years, this has led to speculations of tactical voting, discriminating against some participants and thus inducing bias in the final results. In this paper we investigate the presence of positive or negative bias (which may roughly indicate favouritisms or discrimination) in the votes based on geographical proximity, migration and cultural characteristics of the participating countries through a Bayesian hierarchical model. Our analysis found no evidence of negative bias, although mild positive bias does seem to emerge systematically, linking voters to performers.     

Multiple-class classification: Ordinal and categorical labels

We study multiple-class classification problems. Both ordinal and categorical labeled cases are discussed. The common approaches for multiple-class classification are built on binary classifiers, in which one-versus-one and one-versus-rest are typical approaches. When the number of classes is large, then these binary-classifier-based methods may suffer from either computational costs or the highly imbalanced sample sizes in their training stage. In order to alleviate the computational burden and the imbalanced training data issue in multiple-class classification problems, we propose a method that has competitive performance and retains the ease of model interpretation, which is essential for a prognostic/predictive model.     

Bivariate versus univariate ordinal categorical data with reference to an ophthalmologic study

The Wisconsin Epidemiologic Study of Diabetic Retinopathy is a population-based epidemiological study carried out in Southern Wisconsin during the 1980s. The resulting data were analysed by different statisticians and ophthalmologists during the last two decades. Most of the analyses were carried out on the baseline data, although there were two follow-up studies on the same population. A Bayesian analysis of the first follow-up data, taken four years after the baseline study, was carried out by Angers and Biswas [Angers, J.-F. and Biswas, A., 2004, A Bayesian analysis of the four-year follow-up data of theWisconsin epidemiologic study of diabetic retinopathy. Statistics in Medicine, 23, 601–615.], where the choice of the best model in terms of the covariate inclusion is done, and estimates of the associated covariate effects were obtained using the baseline data to set the prior for the parameters. In the present article we consider an univariate transformation of the bivariate ordinal data, and a parallel analysis with the much simpler univariate data is carried out. The results are then compared with the results of Angers and Biswas (2004). In conclusion, our analyses suggest that the univariate analysis fails to detect features of the data found by the bivariate analysis. Even an univariate transformation of our data with quite high correlation with both left and right eyes is inadequate.     

Cumulative correspondence analysis using orthogonal polynomials

Taguchi's statistic has long been known to be a more appropriate measure of association of the dependence for ordinal variables compared to the Pearson chi-squared statistic. Therefore, there is some advantage in using Taguchi's statistic in the correspondence analysis context when a two-way contingency table consists at least of an ordinal categorical variable. The aim of this paper, considering the contingency table with two ordinal categorical variables, is to show a decomposition of Taguchi's index into linear, quadratic and higher-order components. This decomposition has been developed using Emerson's orthogonal polynomials. Moreover, two case studies to explain the methodology have been analyzed.     

Indirect membership function assignment based on ordinal regression

In many fuzzy sets applications, fuzzy membership functions are commonly developed based on empirical or expert knowledge. The equation of a membership function is usually determined somewhat arbitrarily. This paper explores a novel membership function design method based on ordinal regression analysis. The estimated thresholds between ordinal measurement categories are applied to calculate the intersection points between fuzzy sets. These intersection points are further applied to determine the equations of the membership functions. Information distortion due to empirical guess can thus be reduced and more latent information in the fuzzy responses can therefore be captured. A case study investigating the relationship between foster mothers’ satisfaction and the foster time and information provided has been conducted in this research. The applicability and effectiveness of the proposed membership function assignment approach have been demonstrated through several case studies.     

Goodness-of-fit tests of generalized linear mixed models for repeated ordinal responses

Categorical longitudinal data are frequently applied in a variety of fields, and are commonly fitted by generalized linear mixed models (GLMMs) and generalized estimating equations models. The cumulative logit is one of the useful link functions to deal with the problem involving repeated ordinal responses. To check the adequacy of the GLMMs with cumulative logit link function, two goodness-of-fit tests constructed by the unweighted sum of squared model residuals using numerical integration and bootstrap resampling technique are proposed. The empirical type I error rates and powers of the proposed tests are examined by simulation studies. The ordinal longitudinal studies are utilized to illustrate the application of the two proposed tests.