Herbert Hoijtink

Bayesian estimation and testing of structural equation models

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, for example, output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters underidentified models, as we illustrate on a simple errors-in-variables model.

RELIABILITY AND VALIDITY OF THE DUTCH VERSION OF THE INVENTORY OF TRAUMATIC GRIEF (ITG)

The psychometric properties of the Dutch version of the Inventory of Traumatic Grief (ITG) were investigated in two studies with bereaved adults who had suffered the loss of a first-degree relative. In Study 1, exploratory factor analysis indicated that the items of the ITG clustered together into one underlying factor. In addition, the internal consistency of the ITG and its short-term temporal stability were found to be high. In Study 2 it was found that the ITG exhibited adequate discriminative, concurrent, and construct validity. Furthermore, an ITG cutoff score for a diagnosis of traumatic grief was determined, with a sensitivity of 86% and a specificity of 76%, providing evidence in favor of the predictive validity of the ITG.

The many null distributions of person fit indices

This paper deals with the situation of an investigator who has collected the scores ofn persons to a set ofk dichotomous items, and wants to investigate whether the answers of all respondents are compatible with the one parameter logistic test model of Rasch. Contrary to the standard analysis of the Rasch model, where all persons are kept in the analysis and badly fittingitems may be removed, this paper studies the alternative model in which a small minority ofpersons has an answer strategy not described by the Rasch model. Such persons are called anomalous or aberrant. From the response vectors consisting ofk symbols each equal to 0 or 1, it is desired to classify each respondent as either anomalous or as conforming to the model. As this model is probabilistic, such a classification will possibly involve false positives and false negatives. Both for the Rasch model and for other item response models, the literature contains several proposals for a person fit index, which expresses for each individual the plausibility that his/her behavior follows the model. The present paper argues that such indices can only provide a satisfactory solution to the classification problem if their statistical distribution is known under the null hypothesis that all persons answer according to the model. This distribution, however, turns out to be rather different for different values of the persons latent trait value. This value will be called “ability parameter”, although our results are equally valid for Rasch scales measuring other attributes.As the true ability parameter is unknown, one can only use its estimate in order to obtain an estimated person fit value and an estimated null hypothesis distribution. The paper describes three specifications for the latter: assuming that the true ability equals its estimate, integrating across the ability distribution assumed for the population, and conditioning on the total score, which is in the Rasch model the sufficient statistic for the ability parameter.Classification rules for aberrance will be worked out for each of the three specifications. Depending on test length, item parameters and desired accuracy, they are based on the exact distribution, its Monte Carlo estimate and a new and promising approximation based on the moments of the person fit statistic. Results for the likelihood person fit statistic are given in detail, the methods could also be applied to other fit statistics. A comparison of the three specifications results in the recommendation to condition on the total score, as this avoids some problems of interpretation that affect the other two specifications.

Redefine Statistical Significance

We propose to change the default P-value threshold for statistical significance from 0.05 to 0.005 for claims of new discoveries.

Inequality Constrained Analysis of Variance: A Bayesian approach.

Researchers often have one or more theories or expectations with respect to the outcome of their empirical research. When researchers talk about the expected relations between variables if a certain theory is correct, their statements are often in terms of one or more parameters expected to be larger or smaller than one or more other parameters. Stated otherwise, their statements are often formulated using inequality constraints. In this article, a Bayesian approach to evaluate analysis of variance or analysis of covariance models with inequality constraints on the (adjusted) means is presented. This evaluation contains two issues: estimation of the parameters given the restrictions using the Gibbs sampler and model selection using Bayes factors in the case of competing theories. The article concludes with two illustrations: a one-way analysis of covariance and an analysis of a three-way table of ordered means.

Bayesian evaluation of informative hypotheses

An introduction to bayesian evaluation of informative hypotheses. - Bayesian evaluation of inforative hypotheses. - A further study of prior distributions and the Bayes factor. - Beyond analysis of variance. - Evaluations.

The Best of Both Worlds: Factor Analysis of Dichotomous Data Using Item Response Theory and Structural Equation Modeling

Both structural equation modeling (SEM) and item response theory (IRT) can be used for factor analysis of dichotomous item responses. In this case, the measurement models of both approaches are formally equivalent. They were refined within and across different disciplines, and make complementary contributions to central measurement problems encountered in almost all empirical social science research fields. In this article (a) fundamental formal similiarities between IRT and SEM models are pointed out. It will be demonstrated how both types of models can be used in combination to analyze (b) the dimensional structure and (c) the measurement invariance of survey item responses. All analyses are conducted with Mplus, which allows an integrated application of both approaches in a unified, general latent variable modeling framework. The aim is to promote a diffusion of useful measurement techniques and skills from different disciplines into empirical social research.

A multidimensional item response model: Constrained latent class analysis using the gibbs sampler and posterior predictive checks

In this paper it will be shown that a certain class of constrained latent class models may be interpreted as a special case of nonparametric multidimensional item response models. The parameters of this latent class model will be estimated using an application of the Gibbs sampler. It will be illustrated that the Gibbs sampler is an excellent tool if inequality constraints have to be taken into consideration when making inferences. Model fit will be investigated using posterior predictive checks. Checks for manifest monotonicity, the agreement between the observed and expected conditional association structure, marginal local homogeneity, and the number of latent classes will be presented.

The Bayes factor for inequality and about equality constrained models

The Bayes factor is a useful tool for evaluating sets of inequality and about equality constrained models. In the approach described, the Bayes factor for a constrained model with the encompassing model reduces to the ratio of two proportions, namely the proportion of, respectively, the encompassing prior and posterior in agreement with the constraints. This enables easy and straightforward estimation of the Bayes factor and its Monte Carlo Error. In this set-up, the issue of sensitivity to model specific prior distributions reduces to sensitivity to one prior distribution, that is, the prior for the encompassing model. It is shown that for specific classes of inequality constrained models, the Bayes factors for the constrained with the unconstrained model is virtually independent of the encompassing prior, that is, model selection is virtually objective.

Informative hypotheses : theory and practice for behavioral and social scientists

INTRODUCTION An Introduction to Informative Hypotheses Introduction Analysis of Variance Analysis of Covariance . Multiple Regression Epistemology and Overview of the Book Appendix A: Effect Size Determination for Multiple Regression The Multivariate Normal Linear Model Introduction The Multivariate Normal Linear Model Multivariate One Sided Testing Multivariate Treatment Evaluation Multivariate Regression Repeated Measures Analysis Other Options Appendix A: Example Data for Multivariate Regression BAYESIAN EVALUATION OF INFORMATIVE HYPOTHESES An Introduction to Bayesian Evaluation of Informative Hypotheses Introduction . Density of the Data, Prior and Posterior . Bayesian Evaluation of Informative Hypotheses Specifying the Parameters of Prior Distributions Discussion . Appendix A: Density of the Data, Prior and Posterior Distribution Appendix B: Derivation of the Bayes Factor and Prior Sensitivity . Appendix C: Using BIEMS for a two group ANOVA The J Group ANOVA Model Introduction Simple Constraints One Informative Hypothesis Constraints on Combinations of Means . Ordered Means with Effect Sizes About Equality Constraints Discussion Sample Size Determination: AN(C)OVA and Multiple Regression Introduction Sample Size Determination ANOVA: Comparison of an Informative with the Null Hypothesis ANOVA: Comparison of an Informative Hypothesis with its Complement ANCOVA Signed Regression Coecients: Informative versus Null Hypothesis Signed Regression Coecients: Informative Hypothesis versus Complement Signed Regression Coecients: Including Effect Sizes Comparing Regression Coecients Discussion . Appendix A: Bayes Factors for Parameters on the Boundary of H1 and H1c Appendix B: Command Files for GenMVLData Sample Size Determination: The Multivariate Normal Linear Model Introduction Sample Size Determination: Error Probabilities Multivariate One Sided Testing Multivariate Treatment Evaluation Multivariate Regression . Repeated Measures Analysis Discussion . Appendix A: GenMVLData and BIEMS: Multivariate One Sided Testing Appendix B: GenMVLData and BIEMS: Multivariate Treatment Evaluation Appendix C: GenMVLData and BIEMS: Multivariate Regression Appendix D: GenMVLData and BIEMS: Repeated Measures Analysis OTHER MODELS, OTHER APPROACHES AND SOFTWARE Beyond the Multivariate Normal Linear Model Introduction Contingency Tables Multilevel Models Latent Class Analysis A General Frame Work Appendices: Sampling Using Winbugs Other Approaches Introduction Resume: Bayesian Evaluation of Informative Hypotheses Null Hypothesis Signi cance Testing The Order Restricted Information Criterion Discussion Appendix A: Data and Command File for Confirmatory ANOVA Software Introduction Software Packages New Developments STATISTICAL FOUNDATIONS Foundations of Bayesian Evaluation of Informative Hypotheses Introduction The Bayes Factor The Prior Distribution The Posterior Distribution Estimation of the Bayes Factor Discussion Appendix A: Density of the Data of Various Statistical Models Appendix B: Unconstrained Prior Distributions Used in Book and Software Appendix C: Probability Distributions Used in Appendices A and B References Index