A note on the use of rank-ordered logit models for ordered response categories

Models for rankings have been shown to produce more efficient estimators than comparable models for first/top choices. The discussions and applications of these models typically only consider unordered alternatives. But these models can be usefully adapted to the case where a respondent ranks a set of ordered alternatives that are ordered response categories. This paper proposes eliciting a rank order that is consistent with the ordering of the response categories, and then modelling the observed rankings using a variant of the rank ordered logit model where the distribution of rankings has been truncated to the set of admissible rankings. This results in lower standard errors in comparison to when only a single top category is selected by the respondents. And the restrictions on the set of admissible rankings reduces the number of decisions needed to be made by respondents in comparison to ranking a set of unordered alternatives. Simulation studies and application examples featuring models based on a stereotype regression model and a rating scale item response model are provided to demonstrate the utility of this approach.     

A rasch model for partial credit scoring

A unidimensional latent trait model for responses scored in two or more ordered categories is developed. This “Partial Credit” model is a member of the family of latent trait models which share the property of parameter separability and so permit “specifically objective” comparisons of persons and items. The model can be viewed as an extension of Andrich's Rating Scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item. The difference between the parameters in this model and the “category boundaries” in Samejima's Graded Response model is demonstrated. An unconditional maximum likelihood procedure for estimating the model parameters is developed. Preparation of this paper was supported by grants from the Spencer Foundation and the National Institute for Justice. I would like to thank Professor Benjamin D. Wright of the University of Chicago for his very kind help with the various drafts of this paper.     

A latent class model for rating data

A latent class model for rating data is presented which is the analogue of Andrich's binomial Rasch model for Lazarsfeld's latent class analysis (LCA). The response probabilities for the rating categories follow a binomial distribution and depend on class-specific item parameters. The EM-algorithm for parameter estimation as well as goodness of fit tests for the model are described. An example using questionnaire items on interest in physics illustrates the use of the model as an alternative to the latent trait approach of analyzing test data.I would like to thank Clifford Clogg and the anonymous reviewers for their helpful comments.     

Fitting and Testing Conditional Multinormal Partial Credit Models

A multinormal partial credit model for factor analysis of polytomously scored items with ordered response categories is derived using an extension of the Dutch Identity (Holland in Psychometrika 55:5?C18, 1990). In the model, latent variables are assumed to have a multivariate normal distribution conditional on unweighted sums of item scores, which are sufficient statistics. Attention is paid to maximum likelihood estimation of item parameters, multivariate moments of latent variables, and person parameters. It is shown that the maximum likelihood estimates can be found without the use of numerical integration techniques. More general models are discussed which can be used for testing the model, and it is shown how models with different numbers of latent variables can be tested against each other. In addition, multi-group extensions are proposed, which can be used for testing both measurement invariance and latent population differences. Models and procedures discussed are demonstrated in an empirical data example.     

A latent class distance association model for cross‐classified data with a categorical response variable

In this paper we propose a latent class distance association model for clustering in the predictor space of large contingency tables with a categorical response variable. The rows of such a table are characterized as profiles of a set of explanatory variables, while the columns represent a single outcome variable. In many cases such tables are sparse, with many zero entries, which makes traditional models problematic. By clustering the row profiles into a few specific classes and representing these together with the categories of the response variable in a low‐dimensional Euclidean space using a distance association model, a parsimonious prediction model can be obtained. A generalized EM algorithm is proposed to estimate the model parameters and the adjusted Bayesian information criterion statistic is employed to test the number of mixture components and the dimensionality of the representation. An empirical example highlighting the advantages of the new approach and comparing it with traditional approaches is presented.     

Effects of varying numbers of Likert scale points on factor structure of the Rosenberg Self‐Esteem Scale

Likert‐type rating scales are among the most widely used tools in psychological research. Different numbers of response categories would likely affect response style, data distribution, reliability, and construct validity. There is a lack of research in factor structure invariance under Likert scales with different numbers of categories. The purpose of this study is to examine the effects of varying numbers of Likert points (4–11) on scale properties such as factor structure, external validity, and latent means based on the Rosenberg Self‐Esteem Scale (M. Rosenberg, 1989 ). The sample consists of 1,807 students from secondary schools in Macau. Confirmatory factor analysis shows that the correlated two‐factor model is the most appropriate one; longitudinal invariance analysis reveals that measurement invariance across Likert scales was satisfied at the scalar level. In addition, latent mean scores on the two factors as well as observed means on the subscales are comparable across Likert scales. Moreover, the measurement model across Likert scales exhibit similar external validity. Although psychometric properties are mostly similar among a different number of points, the 4‐point Likert scale is not recommended for its higher skewness and lower loadings; the 11‐point Likert scale from 0 to 10 is slightly preferred for its higher loadings and composite reliability.     

Variation in verbal fluency: A latent variable analysis of clustering,switching, and overall performance

Verbal fluency tasks have long been used to assess and estimate group and individual differences in executive functioning in both cognitive and neuropsychological research domains. Despite their ubiquity, however, the specific component processes important for success in these tasks have remained elusive. The current work sought to reveal these various components and their respective roles in determining performance in fluency tasks using latent variable analysis. Two types of verbal fluency (semantic and letter) were compared along with several cognitive constructs of interest (working memory capacity, inhibition, vocabulary size, and processing speed) in order to determine which constructs are necessary for performance in these tasks. The results are discussed within the context of a two-stage cyclical search process in which participants first search for higher order categories and then search for specific items within these categories.     

A Unified Framework for the Comparison of Treatments with Ordinal Responses

Different latent variable models have been used to analyze ordinal categorical data which can be conceptualized as manifestations of an unobserved continuous variable. In this paper, we propose a unified framework based on a general latent variable model for the comparison of treatments with ordinal responses. The latent variable model is built upon the location-scale family and is rich enough to include many important existing models for analyzing ordinal categorical variables, including the proportional odds model, the ordered probit-type model, and the proportional hazards model. A flexible estimation procedure is proposed for the identification and estimation of the general latent variable model, which allows for the location and scale parameters to be freely estimated. The framework advances the existing methods by enabling many other popular models for analyzing continuous variables to be used to analyze ordinal categorical data, thus allowing for important statistical inferences such as location and/or dispersion comparisons among treatments to be conveniently drawn. Analysis on real data sets is used to illustrate the proposed methods.     

A rating formulation for ordered response categories

A rating response mechanism for ordered categories, which is related to the traditional threshold formulation but distinctively different from it, is formulated. In addition to the subject and item parameters two other sets of parameters, which can be interpreted in terms of thresholds on a latent continuum and discriminations at the thresholds, are obtained. These parameters are identified with the category coefficients and the scoring function of the Rasch model for polychotomous responses in which the latent trait is assumed uni-dimensional. In the case where the threshold discriminations are equal, the scoring of successive categories by the familiar assignment of successive integers is justified. In the case where distances between thresholds are also equal, a simple pattern of category coefficients is shown to follow.This work was conducted in part in the first half of 1977 while the author was on study leave at the Danish Institute for Educational Research. The Institute provided required research facilities while The University of Western Australia provided financial support.     

A Comparison of Imputation Strategies for Ordinal Missing Data on Likert Scale Variables

This article compares a variety of imputation strategies for ordinal missing data on Likert scale variables (number of categories = 2, 3, 5, or 7) in recovering reliability coefficients, mean scale scores, and regression coefficients of predicting one scale score from another. The examined strategies include imputing using normal data models with naïve rounding/without rounding, using latent variable models, and using categorical data models such as discriminant analysis and binary logistic regression (for dichotomous data only), multinomial and proportional odds logistic regression (for polytomous data only). The result suggests that both the normal model approach without rounding and the latent variable model approach perform well for either dichotomous or polytomous data regardless of sample size, missing data proportion, and asymmetry of item distributions. The discriminant analysis approach also performs well for dichotomous data. Naïvely rounding normal imputations or using logistic regression models to impute ordinal data are not recommended as they can potentially lead to substantial bias in all or some of the parameters.     

An experimental analysis of psychometric functions in a threshold discrimination task with four response categories1

An experiment on a threshold discrimination task with four response categories was conducted to check the implicit assumptions of a threshold discrimination task with three response categories. Consideration of a task with three response categories, for example, “stronger,”“don't know,” and “weaker,” leads us to anticipate a relation between position parameters and that between the sizes of scale parameters of a model for a task with four response categories, such as, “stronger,”“probably stronger,”“probably weaker,” and “weaker.” That is, position parameters will be equidistant between adjacent categories and the middle scale parameter will be the largest. The results of the experiment on a four‐category task did not support these anticipated relation, and a possibility of biases in the estimation of point of subjective equality (PSE), using a three‐category task is pointed out. As an alternative to a three‐category task, a four‐category task is suggested when more than two categories are required.     

Using trait–state models to evaluate the longitudinal consistency of global self-esteem from adolescence to adulthood

The present research used a latent variable trait–state model to evaluate the longitudinal consistency of self-esteem during the transition from adolescence to adulthood. Analyses were based on ten administrations of the Rosenberg self-esteem scale (Rosenberg, 1965) spanning the ages of approximately 13–32 for a sample of 451 participants. Results indicated that a completely stable trait factor and an autoregressive trait factor accounted for the majority of the variance in latent self-esteem assessments, whereas state factors accounted for about 16% of the variance in repeated assessments of latent self-esteem. The stability of individual differences in self-esteem increased with age consistent with the cumulative continuity principle of personality development.     

潜变量建模的贝叶斯方法

贝叶斯统计是统计学的两大流派之一,近年来贝叶斯统计在社会及行为科学领域日益流行。鉴于国内心理学界对贝叶斯统计应用仍不广泛,本文尝试从非技术性的角度对贝叶斯统计用于潜变量建模的过程进行简要介绍。主要涉及贝叶斯与频率论在统计学基本概念上的对比;贝叶斯统计的基本原理和分析过程。最后以一个验证性因子分析为例,简要介绍贝叶斯统计用于潜变量建模的分析过程。希望本文能为国内心理学者进行潜变量建模提供新的视角。     

Additive conjoint isotonic probabilistic models (ADISOP)

The ISOP-model or model of twodimensional or bi-isotonicity (Scheiblechner, 1995) postulates that the probabilities of ordered response categories increase isotonically in the order of subject ability and item easiness. Adding a conventional cancellation axiom for the factors of subjects and items gives the ADISOP model where the c.d.f.s of response categories are functions of an additive item and subject parameter and an ordinal category parameter. Extending cancellation to the interactions of subjects and categories as well as of items and categories (independence axiom of the category factor from the subject and item factor) gives the CADISOP model (completely additive model) in which the parallel c.d.f.s are functions of the sum of subject, item and category parameters. The CADISOP model is very close to the unidimensional version of the polytomous Rasch model with the logistic item/category characteristic(s) replaced by nonparametric axioms and statistics. The axioms, representation theorems and algorithms for model fitting of the additive models are presented.     

A comparison of latent trait and latent class analyses of Likert-type data

This paper brings together and compares two developments in the analysis of Likert attitude scales. The first is the generalization of latent class models to ordered response categories. The second is the introduction of latent trait models with multiplicative parameter structures for the analysis of rating scales. Key similarities and differences between these two methods are described and illustrated by applying a latent trait model and a latent class model to the analysis of a set of life satisfaction data. The way in which the latent trait model defines a unit of measurement, takes into account the order of the response categories, and scales the latent classes, is discussed. While the latent class model provides better fit to these data, this is achieved at the cost of a logically inconsistent assignment of individuals to latent classes.The author wishes to thank Clifford C. Clogg, Otis Dudley Duncan and Benjamin D. Wright for their helpful comments on an earlier version of this paper.     

Involvement and response language effects in attitude scaling

An analytic procedure was developed for separating the effects of involvement from the number of response language categories the subject employs in judging a set of sentence stimuli. The procedure treats category usage as an independent variable and an aspect of the subject's response language. Category usage was shown to have significant influence on several attitudinal judgment processes: Informational complexity, mean judgment functions, and a successive intervals analysis testing Thurstone's assumption of the effect of involvement on equal category spacing. The decrement in scale responding typically associated with involvement is shown to depend on category usage operating either singly or in interaction with involvement. Some tentative conclusions about category usage as a response language variable are discussed.     

Information-theoretic latent distribution modeling: distinguishing discrete and continuous latent variable models

Distinguishing between discrete and continuous latent variable distributions has become increasingly important in numerous domains of behavioral science. Here, the authors explore an information-theoretic approach to latent distribution modeling, in which the ability of latent distribution models to represent statistical information in observed data is emphasized. The authors conclude that loss of statistical information with a decrease in the number of latent values provides an attractive basis for comparing discrete and continuous latent variable models. Theoretical considerations as well as the results of 2 Monte Carlo simulations indicate that information theory provides a sound basis for modeling latent distributions and distinguishing between discrete and continuous latent variable models in particular.     

IRTree models with ordinal and multidimensional decision nodes for response styles and trait-based rating responses

IRTree models decompose observed rating responses into sequences of theory-based decision nodes, and they provide a flexible framework for analysing trait-related judgements and response styles. However, most previous applications of IRTree models have been limited to binary decision nodes that reflect qualitatively distinct and unidimensional judgement processes. The present research extends the family of IRTree models for the analysis of response styles to ordinal judgement processes for polytomous decisions and to multidimensional parametrizations of decision nodes. The integration of ordinal judgement processes overcomes the limitation to binary nodes, and it allows researchers to test whether decisions reflect qualitatively distinct response processes or gradual steps on a joint latent continuum. The extension to multidimensional node models enables researchers to specify multiple judgement processes that simultaneously affect the decision between competing response options. Empirical applications highlight the roles of extreme and midpoint response style in rating judgements and show that judgement processes are moderated by different response formats. Model applications with multidimensional decision nodes reveal that decisions among rating categories are jointly informed by trait-related processes and response styles.     

Examining Nonnormal Latent Variable Distributions for Non-Ignorable Missing Data

Missing not at random (MNAR) modeling for non-ignorable missing responses usually assumes that the latent variable distribution is a bivariate normal distribution. Such an assumption is rarely verified and often employed as a standard in practice. Recent studies for “complete” item responses (i.e., no missing data) have shown that ignoring the nonnormal distribution of a unidimensional latent variable, especially skewed or bimodal, can yield biased estimates and misleading conclusion. However, dealing with the bivariate nonnormal latent variable distribution with present MNAR data has not been looked into. This article proposes to extend unidimensional empirical histogram and Davidian curve methods to simultaneously deal with nonnormal latent variable distribution and MNAR data. A simulation study is carried out to demonstrate the consequence of ignoring bivariate nonnormal distribution on parameter estimates, followed by an empirical analysis of “don’t know” item responses. The results presented in this article show that examining the assumption of bivariate nonnormal latent variable distribution should be considered as a routine for MNAR data to minimize the impact of nonnormality on parameter estimates.     

Empirical Bayes Approaches to Multivariate Fuzzy Partitions

In describing high dimensional discrete response data, mathematical and statistical issues arise that require multivariate procedures that are not based on normal distributions, that is, the mathematical representation of high dimensional discrete response data (Event Spaces) requires a representation in lower dimensional parameter spaces consistent with the discrete properties of the Event Space. Mapping discrete responses to latent discrete classes has the limitation of not representing real individual variation within the categories. The use of a fuzzy partition model is proposed which describes individuals in terms of partial membership in multiple latent categories which represents bounded discrete event spaces with significant third and higher order moments. We discuss statistical issues arising in identifying both the deterministic and the stochastic variation of data when applications involve systematic variation due to observed and unobserved variables. We present an empirical Bayes-maximum likelihood estimation scheme for the application of the fuzzy partition models.