On the Structure of Ordered Latent Trait Models
Gerhard Tutz2019, arXiv (Cornell University)
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
Ordered item response models that are in common use can be divided into three groups, cumulative, sequential and adjacent categories model. The derivation and motivation of the models is typically based on the assumed presence of latent traits or underlying process models. In the construction frequently binary models play an important role. The objective of this paper is to give motivations for the models and to clarify the role of the binary models for the various types of ordinal models. It is investigated which binary models are included in an ordinal model but also how the models can be constructed from a sequence of binary models. In all the models one finds a Guttman space structure, which has previously been investigated in particular for the partial credit model. The consideration of the binary models adds to the interpretation of model parameters, which is helpful, in particular, in the case of the partial credit model, for which interpretation is less straightforward than for the other models. A specific topic that is addressed is the ordering of thresholds in the partial credit model because for some researchers reversed ordering is an anomaly, others disagree. It is argued that the ordering of thresholds is not a constitutive element of the partial credit model.
Related papers
What is an Ordinal Latent Trait Model
arXiv (Cornell University), 2019
Although various polytomous item response models are considered to be ordinal models there seems no general definition of an ordinal model available. Alternative concepts of ordinal models are discussed and it is shown that they coincide for classical unidimensional models. For multidimensional models the definition of an ordinal model refers to specific traits in the multidimensional space of traits. The objective is to provide a theoretical framework for ordinal models. Practical considerations concerning the strength of the link between the latent trait and the order of categories are considered briefly.
A Taxonomy of Polytomous Item Response Models
arXiv (Cornell University), 2020
A common framework is provided that comprises classical ordinal item response models as the cumulative, sequential and adjacent categories models as well as nominal response models and item response tree models. The taxonomy is based on the ways binary models can be seen as building blocks of the various models. In particular one can distinguish between conditional and unconditional model components. Conditional models are by far the larger class of models containing the adjacent categories model and the whole class of hierarchically structured models. The latter is introduced as a class of models that comprises binary trees and hierarchically structured models that use ordinal models conditionally. The study of the binary models contained in latent trait models clarifies the relation between models and the interpretation of item parameters. It is also used to distinguish between ordinal and nominal models by giving a conceptualization of ordinal models. The taxonomy differs from previous taxonomies by focusing on the structured use of dichotomizations instead of the role of parameterizations.
New Developments of Latent Variable Models with Ordinal Data
2004
Riassunto: Nel contesto delle scienze sociali e comportamentali i modelli a variabili latenti rivestono un ruolo fondamentale nella valutazione degli atteggiamenti e delle abilit à. In tale contesto, frequentemente i dati raccolti presentano modalit à di tipo ordinale. Nel presente lavoro vengono illustrati due differenti approcci per la modellizzazione dei legami tra variabili latenti continue e variabili manifeste di natura ordinale, l’ Underlying Variable Approach(UVA) e l’ Item Response Function (IRF) approach. In prima istanza, le due soluzioni sono confrontate in termini teorici e successivamente con riferimento alle propriet̀a delle stime, attraverso uno studio di simulazione. Per la prima volta, viene proposto un algoritmo di stima per un modello di IRF con pi ù d due variabili latenti. Infine tali modelli sono applicati ad un data set reale che raccoglie i risultati di un’indagine di Customer Satisfaction degli utenti dei mezzi di trasporto pubblico.
The general causal cumulative model of ordinal response
2022
In this paper, we introduce the general causal cumulative model of ordinal response. The statistical part of this model is a new family of hierarchical or non-hierarchical generalized linear models that represent the distribution of the outcome as a thresholded latent distribution. Its defining feature is the new link functions, which are order-preserving in the sense that they allow for arbitrary effects in individual thresholds while preserving their order. We show how the model can be interpreted as a generalization of some Signal Detection Theory (SDT) models and some Item Response Theory models. We propose an approach to measurement of latent variables which seems to follow from the requirement that measurements should be interpreted in the context of a causal theory of the unobservable response process. In particular, we formulate a causal definition of measurement invariance that seems to match the examples of measurement bias found in the literature, and we show that the com...
Latent and manifest monotonicity in item response models
Applied Psychological Measurement, 2000
The monotonicity of item response functions (IRF) is a central feature of most parametric and nonparametric item response models. Monotonicity allows items to be interpreted as measuring a trait, and it allows for a general theory of nonparametric inference for traits. This theory is based on monotone likelihood ratio and stochastic ordering properties. Thus, confirming the monotonicity assumption is essential to applications of nonparametric item response models. The results of two methods of evaluating monotonicity are presented: regressing individual item scores on the total test score and on the "rest" score, which is obtained by omitting the selected item from the total test score. It was found that the item-total regressions of some familiar dichotomous item response models with monotone IRFs exhibited nonmonotonicities that persist as the test length increased. However, item-rest regressions never exhibited nonmonotonicities under the nonparametric monotone unidimensional item response model. The implications of these results for exploratory analysis of dichotomous item response data and the application of these results to polytomous item response data are discussed. Index terms: elementary symmetric functions, essential unidimensionality, latent monotonicity, manifest monotonicity, monotone homogeneity, nonparametric item response models, strict unidimensionality.
A general latent trait model for response processes
Psychometrika, 1984
The purpose of the current paper is to propose a general multicomponent latent trait model (GLTM) for response processes. The proposed model combines the linear logistic latent trait (LLTM) with the multicomponent latent trait model (MLTM). As with both LLTM and MLTM, the general multicomponent latent trait model can be used to (1) test hypotheses about the theoretical variables that underlie response difficulty and (2) estimate parameters that describe test items by basic substantive properties. However, GLTM contains both component outcomes and complexity factors in a single model and may be applied to data that neither LLTM nor MLTM can handle. Joint maximum likelihood estimators are presented for the parameters of GLTM and an application to cognitive test items is described.
Latent Variable Analysis with Ordinal Data: Comparing the Structural Equation and the Rating Scale Models
The aim of this study is to apply Rating Scale Model and Structural Equation Model to the same polytomous data in order to highlight the differences and similarities between the two models. In order to do this, a simulation study is developed. Moreover, we present a real case regarding the analysis of one of the facets of the quality of work, that is the fairness.
Cumulative or adjacent logits: Which choice for an ordinal logistic latent variable model?
Communications in Statistics - Theory and Methods, 2018
With ordinal response items, a Graded Response Model (GRM) is of cumulative logits type, while the polytomous Rasch Model (PRM) is based on adjacent logits. In this work, we compare the two approaches. We show that the PRM is superior to the GRM, with interesting properties that we prove. Note S ν the sum of item responses of individual ν and Θ ν its latent parameter, we show i) S ν is a sufficient statistic for θ ν and ii) a property of "stochastic ordering" of the conditional distributions G θ/S. The second property, less known, is, to our knowledge, nowhere satisfactorily demonstrated. monotone likelihood ratio, stochastic ordering, adjacent logit, cumulative logit, odds ratio, Rasch model, .
Latent variable models for multivariate longitudinal ordinal responses
British Journal of Mathematical & Statistical Psychology, 2009
The paper proposes af ull information maximum likelihood estimation method for modelling multivariate longitudinal ordinal variables. Tw ol atent 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 ac ommon factor.T he relationships between the time-dependent latent variables are modelled with anon-stationaryautoregressive model. The proposed models arefitted to ar eal data set.
Latent variable models for ordinal data
Contributions to Statistics, 2009
ABSTRACT Latent variable models with observed ordinal variables are particularly useful for analyzing survey data. Typical ordinal variables express attitudinal statements with response alternatives like “strongly disagree”, “disagree”, “strongly agree” or “very dissatisfied”, “dissatisfied”, “satisfied” and “very satisfied”.
Related papers
Factor models for ordinal variables with covariate eects on the manifest and latent variables: A comparison of LISREL and IRT approaches
2000
ABSTRACT We consider a general type of model for analyzing ordinal variables with covariate effects and 2 approaches for analyzing data for such models, the item response theory (IRT) approach and the PRELIS-LISREL (PLA) approach. We compare these 2 approaches on the basis of 2 examples, 1 involving only covariate effects directly on the ordinal variables and 1 involving covariate effects on the latent variables in addition.
A Class of Multidimensional Latent Class IRT Models for Ordinal Polytomous Item Responses
Communications in Statistics - Theory and Methods, 2014
We propose a class of Item Response Theory models for items with ordinal polytomous responses, which extends an existing class of multidimensional models for dichotomously-scored items measuring more than one latent trait. In the proposed approach, the random vector used to represent the latent traits is assumed to have a discrete distribution with support points corresponding to different latent classes in the population. We also allow for different parameterizations for the conditional distribution of the response variables given the latent traits-such as those adopted in the Graded Response model, in the Partial Credit model, and in the Rating Scale model-depending on both the type of link function and the constraints imposed on the item parameters. For the proposed models we outline how to perform maximum likelihood estimation via the Expectation-Maximization algorithm. Moreover, we suggest a strategy for model selection which is based on a series of steps consisting of selecting specific features, such as the number of latent dimensions, the number of latent classes, and the specific parametrization. In order to illustrate the proposed approach, we analyze data deriving from a study on anxiety and depression as perceived by oncological patients.
The Heteroscedastic Graded Response Model with a Skewed Latent Trait: Testing Statistical and Substantive Hypotheses Related to Skewed Item Category Functions
Psychometrika, 2012
The Graded Response Model (GRM; Samejima, Estimation of ability using a response pattern of graded scores, Psychometric Monograph No. 17, Richmond, VA: The Psychometric Society, 1969) can be derived by assuming a linear regression of a continuous variable, Z, on the trait, θ, to underlie the ordinal item scores (Takane & de Leeuw in Psychometrika, 52:393-408, 1987). Traditionally, a normal distribution is specified for Z implying homoscedastic error variances and a normally distributed θ. In this paper, we present the Heteroscedastic GRM with Skewed Latent Trait, which extends the traditional GRM by incorporation of heteroscedastic error variances and a skew-normal latent trait. An appealing property of the extended GRM is that it includes the traditional GRM as a special case. This enables specific tests on the normality assumption of Z. We show how violations of normality in Z can lead to asymmetrical category response functions. The ability to test this normality assumption is beneficial from both a statistical and substantive perspective. In a simulation study, we show the viability of the model and investigate the specificity of the effects. We apply the model to a dataset on affect and a dataset on alexithymia.
Constrained and Unconstrained Partial Adjacent Category Logit Models for Ordinal Response Variables
2018
Adjacent category logit models are ordered regression models that focus on comparisons of adjacent categories. These models are particularly useful for ordinal response variables with categories that are of substantive interest. In this article, we consider unconstrained and constrained versions of the partial adjacent category logit model, which is an extension of the traditional model that relaxes the proportional odds assumption for a subset of independent variables. In the unconstrained partial model, the variables without proportional odds have coefficients that freely vary across cutpoint equations, whereas in the constrained partial model two or more of these variables have coefficients that vary by common factors. We improve upon an earlier formulation of the constrained partial adjacent category model by introducing a new estimation method and conceptual justification for the model. Additionally, we discuss the connections between partial adjacent category models and other models within the adjacent approach, including
A Multivariate Latent Variable Model for Mixed Continuous and Ordinal Responses
2008
A joint model fo r multivariate mixed ordinal and continuous responses is presented. In this model the ordinal responses are intercorrelated and also are dependent on the continuous responses. The likelihood is found and modified Pearson residuals, where the correlation between multivariate responses can be taken into account, are presented to find abnormal observations. The model is applied to a
Factor Models for Ordinal Variables With Covariate Effects on the Manifest and Latent Variables: A Comparison of LISREL and IRT Approaches
Structural Equation Modeling: A Multidisciplinary Journal, 2004
... Irini Moustaki1 Statistics Department Athens University of Economics and Business Karl G Jöreskog Department of Information Science Uppsala University Dimitris Mavridis Statistics Department Athens University of Economics and Business ...
Item Response Thresholds Models: A General Class of Models for Varying Types of Items
Psychometrika
A comprehensive class of models is proposed that can be used for continuous, binary, ordered categorical and count type responses. The difficulty of items is described by difficulty functions, which replace the item difficulty parameters that are typically used in item response models. They crucially determine the response distribution and make the models very flexible with regard to the range of distributions that are covered. The model class contains several widely used models as the binary Rasch model and the graded response model as special cases, allows for simplifications, and offers a distribution free alternative to count type items. A major strength of the models is that they can be used for mixed item formats, when different types of items are combined to measure abilities or attitudes. It is an immediate consequence of the comprehensive modeling approach that allows that difficulty functions automatically adapt to the response distribution. Basic properties of the model c...
The case of dependency of responses and response times: A modeling approach based on standard latent trait models
Psychological Test and Assessment Modeling
When modeling responses and response times in tests with latent trait models, the assumption of conditional independence between responses and response times might be too strong in the case that both data are gained from reactions to the same item. In order to account for the possible dependency of responses and response times from the same item, a generalization of the model of van der Linden is proposed. The basic idea consists in the assumption of a latent continuous response that underlies the observed binary response. This latent response is assumed to be correlated with the corresponding response time. The main advantage of this approach consists in the fact that the marginal models for responses and response times follow well known, standard latent trait models. Model estimation can be accomplished by marginal maximum likelihood estimation. The adequacy of the estimation approach is demonstrated in a small scale simulation study. An empirical data application illustrates the practicability of the approach in practice.
Stochastic ordering using the latent trait and the sum score in polytomous IRT models
Psychometrika, 1997
In a restricted class of item response theory (IRT) models for polytomous items the unweighted total score has monotone likelihood ratio (MLR) in the latent trait 0. MLR implies two stochastic ordering (SO) properties, denoted SOM and SOL, which are both weaker than MLR, but very useful for measurement with IRT models. Therefore, these SO properties are investigated for a broader class of IRT models for which the MLR property does not hold.
Effects of Positively and Negatively Worded Items on Estimated Latent Trait Scores
Psychological scales, e.g., anxiety, depression, and stress inventories, tend to be a combination of positively and negatively worded items with ordered item responses using a Likert-type scale. The generalized partial credit model (GPCM) is often applied to ordinal response data, but little research uses the nominal response model (NRM) with these types of instruments. Preson, Reise, Cai, and Hays (2011) compared these models applied to psychological scales; this study focused on the item parameter estimates. We advance this study by comparing the estimated latent trait from the GPCM and the NRM to an instrument constructed with reverse-worded items. The purpose is to compare the estimated latent trait for the two models and for the subsets of positively or negatively worded items.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.