Robust inference with binary data

In this paper robustness properties of the maximum likelihood estimator (MLE) and several robust estimators for the logistic regression model when the responses are binary are analysed. It is found that the MLE and the classical Rao's score test can be misleading in the presence of model misspecification which in the context of logistic regression means either misclassification's errors in the responses, or extreme data points in the design space. A general framework for robust estimation and testing is presented and a robust estimator as well as a robust testing procedure are presented. It is shown that they are less influenced by model misspecifications than their classical counterparts. They are finally applied to the analysis of binary data from a study on breastfeeding.The author is partially supported by the Swiss National Science Foundation. She would like to thank Rand Wilcox, Eva Cantoni and Elvezio Ronchetti for their helpful comments on earlier versions of the paper, as well as Stephane Heritier for providing the routine to compute the OBRE.     

CDF‐quantile distributions for modelling random variables on the unit interval

This paper introduces a two‐parameter family of distributions for modelling random variables on the (0,1) interval by applying the cumulative distribution function of one ‘parent’ distribution to the quantile function of another. Family members have explicit probability density functions, cumulative distribution functions and quantiles in a location parameter and a dispersion parameter. They capture a wide variety of shapes that the beta and Kumaraswamy distributions cannot. They are amenable to likelihood inference, and enable a wide variety of quantile regression models, with predictors for both the location and dispersion parameters. We demonstrate their applicability to psychological research problems and their utility in modelling real data.     

A maximum likelihood method for latent class regression involving a censored dependent variable

The standard tobit or censored regression model is typically utilized for regression analysis when the dependent variable is censored. This model is generalized by developing a conditional mixture, maximum likelihood method for latent class censored regression. The proposed method simultaneously estimates separate regression functions and subject membership in K latent classes or groups given a censored dependent variable for a cross-section of subjects. Maximum likelihood estimates are obtained using an EM algorithm. The proposed method is illustrated via a consumer psychology application.     

A semiparametric approach for item response function estimation to detect item misfit

When scaling data using item response theory, valid statements based on the measurement model are only permissible if the model fits the data. Most item fit statistics used to assess the fit between observed item responses and the item responses predicted by the measurement model show significant weaknesses, such as the dependence of fit statistics on sample size and number of items. In order to assess the size of misfit and to thus use the fit statistic as an effect size, dependencies on properties of the data set are undesirable. The present study describes a new approach and empirically tests it for consistency. We developed an estimator of the distance between the predicted item response functions (IRFs) and the true IRFs by semiparametric adaptation of IRFs. For the semiparametric adaptation, the approach of extended basis functions due to Ramsay and Silverman (2005) is used. The IRF is defined as the sum of a linear term and a more flexible term constructed via basis function expansions. The group lasso method is applied as a regularization of the flexible term, and determines whether all parameters of the basis functions are fixed at zero or freely estimated. Thus, the method serves as a selection criterion for items that should be adjusted semiparametrically. The distance between the predicted and semiparametrically adjusted IRF of misfitting items can then be determined by describing the fitting items by the parametric form of the IRF and the misfitting items by the semiparametric approach. In a simulation study, we demonstrated that the proposed method delivers satisfactory results in large samples (i.e., N ≥ 1,000).     

Robust estimation of the hierarchical model for responses and response times

Van der Linden's (2007, Psychometrika, 72, 287) hierarchical model for responses and response times in tests has numerous applications in psychological assessment. The success of these applications requires the parameters of the model to have been estimated without bias. The data used for model fitting, however, are often contaminated, for example, by rapid guesses or lapses of attention. This distorts the parameter estimates. In the present paper, a novel estimation approach is proposed that is robust against contamination. The approach consists of two steps. In the first step, the response time model is fitted on the basis of a robust estimate of the covariance matrix. In the second step, the item response model is extended to a mixture model, which allows for a proportion of irregular responses in the data. The parameters of the mixture model are then estimated with a modified marginal maximum likelihood estimator. The modified marginal maximum likelihood estimator downweights responses of test-takers with unusual response time patterns. As a result, the estimator is resistant to several forms of data contamination. The robustness of the approach is investigated in a simulation study. An application of the estimator is demonstrated with real data.     

Probabilistic Disjoint Principal Component Analysis

One of the most relevant problems in principal component analysis and factor analysis is the interpretation of the components/factors. In this paper, disjoint principal component analysis model is extended in a maximum-likelihood framework to allow for inference on the model parameters. A coordinate ascent algorithm is proposed to estimate the model parameters. The performance of the methodology is evaluated on simulated and real data sets.     

教育和心理研究中的多层线性模型

多层线性模型是分析具有层次结构数据的一种新型统计分析技术,与传统统计方法相比,具有模型假设与实际更吻合、结果解释更合理等特点。近年来这一方法的应用逐渐在社会科学的研究中受到重视。文章从多层线性模型的基本假设入手,较系统地介绍了模型参数估计和假设检验的方法,并通过一个具体例子将这一方法与传统回归分析方法相比,进一步说明了多层线性模型在分析具有层次结构数据时的优点。     

Model modification

An analysis of empirical data often leads to a rejection of a hypothesized model, even if the researcher has spent considerable efforts in including all available information in the formulation of the model. Thus, the researcher must reformulate the model in some way, but in most instances there is, at least theoretically, an overwhelming number of possible actions that could be taken. In this paper a modification index will be discussed which should serve as a guide in the search for a better model. In statistical terms, the index measures how much we will be able to reduce the discrepancy between model and data, as defined by a general fit function, when one parameter is added or freed or when one equality constraint is relaxed. The modification index discussed in this paper is an improvement of the one incorporated in the LISREL V computer program in that it takes into account changes in all the parameters of the model when one particular parameter is freed.The research reported in this paper has been supported by The Swedish Council for Research in the Humanities and Social Sciences under Research Program Multivariate Statistical Analysis, Project Director Karl G Jöreskog.     

An accumulator model for responses and response times in tests based on the proportional hazards model

Latent trait models for responses and response times in tests often lack a substantial interpretation in terms of a cognitive process model. This is a drawback because process models are helpful in clarifying the meaning of the latent traits. In the present paper, a new model for responses and response times in tests is presented. The model is based on the proportional hazards model for competing risks. Two processes are assumed, one reflecting the increase in knowledge and the second the tendency to discontinue. The processes can be characterized by two proportional hazards models whose baseline hazard functions correspond to the temporary increase in knowledge and discouragement. The model can be calibrated with marginal maximum likelihood estimation and an application of the ECM algorithm. Two tests of model fit are proposed. The amenability of the proposed approaches to model calibration and model evaluation is demonstrated in a simulation study. Finally, the model is used for the analysis of two empirical data sets.     

A sequential exploratory diagnostic model using a Pólya-gamma data augmentation strategy

Cognitive diagnostic models provide a framework for classifying individuals into latent proficiency classes, also known as attribute profiles. Recent research has examined the implementation of a Pólya-gamma data augmentation strategy binary response model using logistic item response functions within a Bayesian Gibbs sampling procedure. In this paper, we propose a sequential exploratory diagnostic model for ordinal response data using a logit-link parameterization at the category level and extend the Pólya-gamma data augmentation strategy to ordinal response processes. A Gibbs sampling procedure is presented for efficient Markov chain Monte Carlo (MCMC) estimation methods. We provide results from a Monte Carlo study for model performance and present an application of the model.     

A Thurstonian Analysis of Preference Change

This paper presents a Thurstonian model for the analysis of preference change. Preferences are expressed in the form of rankings, possibly with ties. A vector-autoregression framework is used to investigate relationships between past and current rankings. It is shown that this approach yields a parsimonious and easily interpretable representation of individual preference differences in time-dependent ranking data. A detailed analysis of the 1992 National Election Study illustrates the proposed approach.     

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 bayesian alternative to least squares and equal weighting coefficients in regression

This paper details a Bayesian alternative to the use of least squares and equal weighting coefficients in regression. An equal weight prior distribution for the linear regression parameters is described with regard to the conditional normal regression model, and resulting posterior distributions for these parameters are detailed. Some interesting connections between this Bayesian procedure and several other methods for estimating optimal weighting coefficients are discussed. In addition, results are presented of a Monte Carlo investigation which compared the effectiveness of the Bayesian procedure relative to least squares, equal weight, ridge, and Bayesian exchangeability estimations.     

Stevens’ Direct Scaling Methods and the Uniqueness Problem

Stevens postulated that we can use the responses of a participant in a ratio scaling experiment directly to construct a psychophysical function representing the participant's sensations. Although Stevens' methods of constructing measurement scales are widely used in the behavioral sciences, the problem of which scale type is appropriate to describe ratio scaling data is still unresolved. To deal with this problem, we develop a theoretical framework to specify the scale type attained by Stevens' direct scaling methods. It is shown, under fairly mild background assumptions, that the behavioral axioms presented in this paper are necessary and sufficient for the psychophysical functions to be ordinal-, interval-, log-interval-, or ratio-scales. Furthermore, suggestions on how to test these behavioral axioms are provided. Requests for reprints should be sent to thomas.     

Analysis of Interactions and Nonlinear Effects with Missing Data: A Factored Regression Modeling Approach Using Maximum Likelihood Estimation

Abstract

When estimating multiple regression models with incomplete predictor variables, it is necessary to specify a joint distribution for the predictor variables. A convenient assumption is that this distribution is a multivariate normal distribution, which is also the default in many statistical software packages. This distribution will in general be misspecified if predictors with missing data have nonlinear effects (e.g., x²) or are included in interaction terms (e.g., x·z). In the present article, we introduce a factored regression modeling approach for estimating regression models with missing data that is based on maximum likelihood estimation. In this approach, the model likelihood is factorized into a part that is due to the model of interest and a part that is due to the model for the incomplete predictors. In three simulation studies, we showed that the factored regression modeling approach produced valid estimates of interaction and nonlinear effects in regression models with missing values on categorical or continuous predictor variables under a broad range of conditions. We developed the R package mdmb, which facilitates a user-friendly application of the factored regression modeling approach, and present a real-data example that illustrates the flexibility of the software.     

A reformulation of the general Euclidean model for the external analysis of preference data

This paper discusses least squares methods for fitting a reformulation of the general Euclidean model for the external analysis of preference data. The reformulated subject weights refer to a common set of reference vectors for all subjects and hence are comparable across subjects. If the rotation of the stimulus space is fixed, the subject weight estimates in the model are uniquely determined. Weight estimates can be guaranteed nonnegative. While the reformulation is a metric model for single stimulus data, the paper briefly discusses extensions to nonmetric, pairwise, and logistic models. The reformulated model is less general than Carroll's earlier formulation.The author is grateful to Christopher J. Nachtsheim for his helpful suggestions.     

A marginalization model for the multidimensional unfolding analysis of ranking data

A marginalization model for the multidimensional unfolding analysis of ranking data is presented. A subject samples one of a number of random points that are multivariate normally distributed. The subject perceives the distances from the point to all the stimulus points fixed in the same multidimensional space. The distances are error perturbed in this perception process. He/she produces a ranking dependent on these error-perturbed distances. The marginal probability of a ranking is obtained according to this ranking model and by integrating out the subject (ideal point) parameters, assuming the above distribution. One advantage of the model is that the individual differences are captured using the posterior probabilities of subject points. Three sets of ranking data are analyzed by the model.     

Identifiability and statistical properties of two-stage learning with no successes in the initial stage

A model is considered for learning where there is an initial stage with only errors, an intermediate state with errors and correct responses, and an absorbing state with only correct responses. A model with observable states is constructed, and sufficient identifiable parameters are given for the original model. Distribution statistics and other properties of data are derived as functions of the identifiable parameters. Informal estimates and sufficient statistics are given for the identifiable parameters. A number of testable hypotheses about the theoretical parameters are described. Some identifying restrictions for the model are given, along with the estimates for the unrestricted free parameters which follow. An application is made to data from a pairedassociate learning experiment.This research was sponsored by NSF Grant GB2791 and by PHS Grant MH12717. Part of the work was done while the author held a visiting faculty appointment at Stanford University.     

On the relationship between item response theory and factor analysis of discretized variables

Equivalence of marginal likelihood of the two-parameter normal ogive model in item response theory (IRT) and factor analysis of dichotomized variables (FA) was formally proved. The basic result on the dichotomous variables was extended to multicategory cases, both ordered and unordered categorical data. Pair comparison data arising from multiple-judgment sampling were discussed as a special case of the unordered categorical data. A taxonomy of data for the IRT and FA models was also attempted.The work reported in this paper has been supported by Grant A6394 to the first author from the Natural Sciences and Engineering Research Council of Canada.