A default Bayesian hypothesis test for correlations and partial correlations

We propose a default Bayesian hypothesis test for the presence of a correlation or a partial correlation. The test is a direct application of Bayesian techniques for variable selection in regression models. The test is easy to apply and yields practical advantages that the standard frequentist tests lack; in particular, the Bayesian test can quantify evidence in favor of the null hypothesis and allows researchers to monitor the test results as the data come in. We illustrate the use of the Bayesian correlation test with three examples from the psychological literature. Computer code and example data are provided in the journal archives.     

When two limbs are weaker than one: Sensorimotor syncopation with alternating hands

This study addresses the demands of alternating bimanual syncopation, a coordination mode in which the two hands move in alternation while tapping in antiphase with a metronomic tone sequence. Musically trained participants were required to engage in alternating bimanual syncopation and five other coordination modes: unimanual syncopation where taps are made (with the left or right hand) after every tone; unimanual syncopation where taps are made after every other tone; bimanual synchronization with alternating hands; unimanual synchronized tapping with every tone; and unimanual tapping with every other tone. Variability in tap timing was greatest overall for alternating bimanual syncopation, indicating that it is the most difficult. This appears to be due to instability arising from the simultaneous presence of two levels of antiphase coordination (one between the pacing sequence and the hands, the other between the two hands) rather than factors relating to movement frequency or dexterity limits of the nonpreferred hand.     

The neural evidence for simulation is weaker than I think you think it is

Simulation theory accounts of mind-reading propose that the observer generates a mental state that matches the state of the target and then uses this state as the basis for an attribution of a similar state to the target. The key proposal is thus that mechanisms that are primarily used online, when a person experiences a kind of mental state, are then co-opted to run Simulations of similar states in another person. Here I consider the neuroscientific evidence for this view. I argue that there is substantial evidence for co-opted mechanisms, leading from one individual’s mental state to a matching state in an observer, but there is no evidence that the output of these co-opted mechanisms serve as the basis for mental state attributions. There is also substantial evidence for attribution mechanisms that serve as the basis for mental state attributions, but there is no evidence that these mechanisms receive their input from co-opted mechanisms.     

Theoretical correlations and measured correlations: relating recognition and recall in four distributed memory models

This article addresses the relation between item recognition and associative (cued) recall. Going beyond measures of performance on each task, the analysis focuses on the degree to which the contingency between successful recognition and successful recall of a studied item reflects the commonality of memory processes underlying the recognition and recall tasks. Specifically, 4 classes of distributed memory models are assessed for their ability to account for the relatively invariant correlation (approximately .5) between successive recognition and recall. Basic versions of each model either under- or overpredict the intertask correlation. Introducing variability in goodness-of-encoding and response criteria, as well as output encoding, enabled all 4 models to reproduce the moderate intertask correlation and the increase in correlation observed in 2 mixed-list experiments. This model-based analysis provides a general theoretical framework for interpreting contingencies between successive memory tests.     

Determining the number of components from the matrix of partial correlations

A common problem for both principal component analysis and image component analysis is determining how many components to retain. A number of solutions have been proposed, none of which is totally satisfactory. An alternative solution which employs a matrix of partial correlations is considered. No components are extracted after the average squared partial correlation reaches a minimum. This approach gives an exact stopping point, has a direct operational interpretation, and can be applied to any type of component analysis. The method is most appropriate when component analysis is employed as an alternative to, or a first-stage solution for, factor analysis.     

The relation between factor loadings and biserial correlations in item analysis

The first centroid factor loadings obtained from various interitem relations are compared with item discrimination indices commonly used in item analysis. Depending upon what type of matrix is factored, the factor loadings are shown to be related to point biserial and biserial correlations.     

Bayesian inference for graphical factor analysis models

We generalize factor analysis models by allowing the concentration matrix of the residuals to have nonzero off-diagonal elements. The resulting model is named graphical factor analysis model. Allowing a structure of associations gives information about the correlation left unexplained by the unobserved variables, which can be used both in the confirmatory and exploratory context. We first present a sufficient condition for global identifiability of this class of models with a generic number of factors, thereby extending the results in Stanghellini (1997) and Vicard (2000). We then consider the issue of model comparison and show that fast local computations are possible for this purpose, if the conditional independence graphs on the residuals are restricted to be decomposable and a Bayesian approach is adopted. To achieve this aim, we propose a new reversible jump MCMC method to approximate the posterior probabilities of the considered models. We then study the evolution of political democracy in 75 developing countries based on eight measures of democracy in two different years. We acknowledge support from M.U.R.S.T. of Italy and from the European Science Foundation H.S.S.S. Network. We are grateful to the referees and the Editor for many useful suggestions and comments which led to a substantial improvement of the paper. We also thank Nanny Wermuth for stimulating discussions and Kenneth A. Bollen for kindly providing us with the data-set.     

Investigating population heterogeneity with factor mixture models

Sources of population heterogeneity may or may not be observed. If the sources of heterogeneity are observed (e.g., gender), the sample can be split into groups and the data analyzed with methods for multiple groups. If the sources of population heterogeneity are unobserved, the data can be analyzed with latent class models. Factor mixture models are a combination of latent class and common factor models and can be used to explore unobserved population heterogeneity. Observed sources of heterogeneity can be included as covariates. The different ways to incorporate covariates correspond to different conceptual interpretations. These are discussed in detail. Characteristics of factor mixture modeling are described in comparison to other methods designed for data stemming from heterogeneous populations. A step-by-step analysis of a subset of data from the Longitudinal Survey of American Youth illustrates how factor mixture models can be applied in an exploratory fashion to data collected at a single time point.     

Stochastic search item selection for factor analytic models

In this paper we implement a Markov chain Monte Carlo algorithm based on the stochastic search variable selection method of George and McCulloch (1993) for identifying promising subsets of manifest variables (items) for factor analysis models. The suggested algorithm is constructed by embedding in the usual factor analysis model a normal mixture prior for the model loadings with latent indicators used to identify not only which manifest variables should be included in the model but also how each manifest variable is associated with each factor. We further extend the suggested algorithm to allow for factor selection. We also develop a detailed procedure for the specification of the prior parameters values based on the practical significance of factor loadings using ideas from the original work of George and McCulloch (1993). A straightforward Gibbs sampler is used to simulate from the joint posterior distribution of all unknown parameters and the subset of variables with the highest posterior probability is selected. The proposed method is illustrated using real and simulated data sets.     

Three common factor models for groups of variables

For the case of two or more groups of variables (batteries), generalizations on three common factor models, the Equal Residual Variances model, the Image model of Jöreskog, and the Canonical Factor Analysis model, are described and illustrated with empirical examples.The author is indebted to Dr. E. J. Burr for his valuable criticisms.     

Heterogeneous factor analysis models: A bayesian approach

Multilevel factor analysis models are widely used in the social sciences to account for heterogeneity in mean structures. In this paper we extend previous work on multilevel models to account for general forms of heterogeneity in confirmatory factor analysis models. We specify various models of mean and covariance heterogeneity in confirmatory factor analysis and develop Markov Chain Monte Carlo (MCMC) procedures to perform Bayesian inference, model checking, and model comparison.We test our methodology using synthetic data and data from a consumption emotion study. The results from synthetic data show that our Bayesian model perform well in recovering the true parameters and selecting the appropriate model. More importantly, the results clearly illustrate the consequences of ignoring heterogeneity. Specifically, we find that ignoring heterogeneity can lead to sign reversals of the factor covariances, inflation of factor variances and underappreciation of uncertainty in parameter estimates. The results from the emotion study show that subjects vary both in means and covariances. Thus traditional psychometric methods cannot fully capture the heterogeneity in our data.     

A note on the factor analysis of partial covariance matrices

A relationship is given between the joint common factor structure of two sets of variables, and the factor structure of the partial covariance matrix of one of the sets with the other partialled out.     

Numerical methods for polynomial models in nonlinear factor analysis

The basic concepts of nonlinear factor analysis are introduced and some extensions of the general theory are developed. An elementary account of the class of multiple-factor polynomial models is presented, using more elementary algebraic methods than have been employed in earlier accounts of this theory. Working formulas are developed for the multiple-factor polynomial model without product terms.Some empirical results are presented.This work was carried out while the author was a Visiting Research Fellow at the Educational Testing Service, Princeton, N. J. It made use of computer facilities at Princeton University that are supported in part by National Science Foundation Grant NSF-Gp579.     

Confirmatory factor analysis of ITPA models with language-impaired children

The major objective of the current investigation was to provide an evaluation of the validity of the ITPA for language-impaired children using confirmatory factor analysis. In addition to an examination of the validity of the traditional ITPA models, a new model is tested. The new model is based on considerations of the structural complexity of the ITPA subtests. and allows one to interpret ITPA results in processing terms. The results suggest that four-factor models provided the best representation of the ITPA scores in language-impaired children. Compared with the traditional ITPA models, the new model showed either an improved or equal goodness of fit. Language-impaired children tested with the ITPA (n = 100) exhibited a profile pattern suggesting that subtests with low processing demands are "easier" than tests with high processing demands. Thus, the present study supports the contention that language-impaired children suffer from constraints on information processing.     

The effect of additional variables on factor indeterminacy in models with a single common factor

Determinate solutions for the indeterminate common factor ofp variables satisfying the single common factor model are not unique. Therefore an infinite sequence of additional variables that conform jointly with the originalp variables to the original single common factor model does not determine a unique solution for the indeterminate factor of thep variables (although the solution is unique for the factor of the infinite sequence). Other infinite sequences may be found to determine different solutions for the factor of the originalp variables. The paper discusses a number of theorems about the effects of additional variables on factor indeterminacy in a model with a single common factor and draws conclusions from them for factor theory in general.     

On local influence analysis of full information item factor models

The full information item factor (FIIF) model is very useful for analyzing relations of dichotomous variables. In this article, we present a feasible procedure to assess local influence of minor perturbations for identifying influence aspects of the FIIF model. The development is based on a Q-displacement function which is closely related with the Monte Carlo EM algorithm in the ML estimation. In the E-step of this algorithm, the conditional expectations are approximated by sample means of observations simulated by the Gibbs sampler from the appropriate conditional distributions. It turns out that these observations can be utilized for computing the building blocks of the proposed diagnostic measures. The diagnoses are based on the conformal normal curvature that can be computed easily. A number of interesting perturbation schemes are considered. The methodology is illustrated with two real examples.The research is fully supported by a grant (CUHK 4356/00H) from the Research Grant Council of the Hong Kong Special Administration Region. The authors are thankful to the Editor, Associate Editor, anonymous reviewers, and W.Y. Poon for valuable comments for improving the paper, and to ICPSR and the relevant founding agency for allowing us to use of their data. The assistance of Michael Leung and Esther Tam is gratefully acknowledged.     

Comparison of factor structure models for the Beck Depression Inventory--II

Factor analyses of the Beck Depression Inventory--II (A. T. Beck, R. A. Steer, & G. K. Brown, 1996) have frequently produced 2 different 2-factor oblique structures. The author used confirmatory factor analyses to compare these structures with a general-factor model with 2 orthogonal group factors. The general-factor model fit as well as or better than the 2-factor models when applied to item data from previous studies (3 clinical and 2 college samples). Communalities associated with the General Depression factor ranged from 71% to 82%. Cognitive and Somatic group factors were indicative of intropunitiveness and fatigue. It was concluded that the general-factor model gives an acceptable empirical explanation of item covariance structure and offers a conceptual interpretation that is well suited to clinical practice and research.     

Heteroscedastic one‐factor models and marginal maximum likelihood estimation

In the present paper, a general class of heteroscedastic one‐factor models is considered. In these models, the residual variances of the observed scores are explicitly modelled as parametric functions of the one‐dimensional factor score. A marginal maximum likelihood procedure for parameter estimation is proposed under both the assumption of multivariate normality of the observed scores conditional on the single common factor score and the assumption of normality of the common factor score. A likelihood ratio test is derived, which can be used to test the usual homoscedastic one‐factor model against one of the proposed heteroscedastic models. Simulation studies are carried out to investigate the robustness and the power of this likelihood ratio test. Results show that the asymptotic properties of the test statistic hold under both small test length conditions and small sample size conditions. Results also show under what conditions the power to detect different heteroscedasticity parameter values is either small, medium, or large. Finally, for illustrative purposes, the marginal maximum likelihood estimation procedure and the likelihood ratio test are applied to real data.