Bayesian data analysis for newcomers

This article explains the foundational concepts of Bayesian data analysis using virtually no mathematical notation. Bayesian ideas already match your intuitions from everyday reasoning and from traditional data analysis. Simple examples of Bayesian data analysis are presented that illustrate how the information delivered by a Bayesian analysis can be directly interpreted. Bayesian approaches to null-value assessment are discussed. The article clarifies misconceptions about Bayesian methods that newcomers might have acquired elsewhere. We discuss prior distributions and explain how they are not a liability but an important asset. We discuss the relation of Bayesian data analysis to Bayesian models of mind, and we briefly discuss what methodological problems Bayesian data analysis is not meant to solve. After you have read this article, you should have a clear sense of how Bayesian data analysis works and the sort of information it delivers, and why that information is so intuitive and useful for drawing conclusions from data.     

An effect size measure and Bayesian analysis of single-case designs

This article describes a linear modeling approach for the analysis of single-case designs (SCDs). Effect size measures in SCDs have been defined and studied for the situation where there is a level change without a time trend. However, when there are level and trend changes, effect size measures are either defined in terms of changes in R² or defined separately for changes in slopes and intercept coefficients. We propose an alternate effect size measure that takes into account changes in slopes and intercepts in the presence of serial dependence and provides an integrated procedure for the analysis of SCDs through estimation and inference based directly on the effect size measure. A Bayesian procedure is described to analyze the data and draw inferences in SCDs. A multilevel model that is appropriate when several subjects are available is integrated into the Bayesian procedure to provide a standardized effect size measure comparable to effect size measures in a between-subjects design. The applicability of the Bayesian approach for the analysis of SCDs is demonstrated through an example.     

MorePower 6.0 for ANOVA with relational confidence intervals and Bayesian analysis

MorePower 6.0 is a flexible freeware statistical calculator that computes sample size, effect size, and power statistics for factorial ANOVA designs. It also calculates relational confidence intervals for ANOVA effects based on formulas from Jarmasz and Hollands (Canadian Journal of Experimental Psychology 63:124–138, 2009), as well as Bayesian posterior probabilities for the null and alternative hypotheses based on formulas in Masson (Behavior Research Methods 43:679–690, 2011). The program is unique in affording direct comparison of these three approaches to the interpretation of ANOVA tests. Its high numerical precision and ability to work with complex ANOVA designs could facilitate researchers’ attention to issues of statistical power, Bayesian analysis, and the use of confidence intervals for data interpretation. MorePower 6.0 is available at https://wiki.usask.ca/pages/viewpageattachments.action?pageId=420413544.     

Nonlinear Bayesian analysis for single case designs

Several authors have suggested the use of multilevel models for the analysis of data from single case designs. Multilevel models are a logical approach to analyzing such data, and deal well with the possible different time points and treatment phases for different subjects. However, they are limited in several ways that are addressed by Bayesian methods. For small samples Bayesian methods fully take into account uncertainty in random effects when estimating fixed effects; the computational methods now in use can fit complex models that represent accurately the behavior being modeled; groups of parameters can be more accurately estimated with shrinkage methods; prior information can be included; and interpretation is more straightforward. The computer programs for Bayesian analysis allow many (nonstandard) nonlinear models to be fit; an example using floor and ceiling effects is discussed here.     

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.     

Bayesian factor analysis for multilevel binary observations

Multilevel covariance structure models have become increasingly popular in the psychometric literature in the past few years to account for population heterogeneity and complex study designs. We develop practical simulation based procedures for Bayesian inference of multilevel binary factor analysis models. We illustrate how Markov Chain Monte Carlo procedures such as Gibbs sampling and Metropolis-Hastings methods can be used to perform Bayesian inference, model checking and model comparison without the need for multidimensional numerical integration. We illustrate the proposed estimation methods using three simulation studies and an application involving student's achievement results in different areas of mathematics. The authors thank Ian Westbury, University of Illinois at Urbana Champaign for kindly providing the SIMS data for the application.     

Bayesian analysis of order-statistics models for ranking data

In this paper, a class of probability models for ranking data, the order-statistics models, is investigated. We extend the usual normal order-statistics model into one where the underlying random variables follow a multivariate normal distribution. Bayesian approach and the Gibbs sampling technique are used for parameter estimation. In addition, methods to assess the adequacy of model fit are introduced. Robustness of the model is studied by considering a multivariate-t distribution. The proposed method is applied to analyze the presidential election data of the American Psychological Association (APA).The author is grateful to K. Lam, K.F. Lam, the Editor, an associate editor, and three reviewers for their valuable comments and suggestions. This research was substantially supported by the CRCG grant 335/017/0015 of the University of Hong Kong and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7169/98H). Upon completion of this paper, I became aware that similar work had been done independently by K.G. Yao and U. Böckenholt (1999).     

Statistical mediation analysis with a multicategorical independent variable

Virtually all discussions and applications of statistical mediation analysis have been based on the condition that the independent variable is dichotomous or continuous, even though investigators frequently are interested in testing mediation hypotheses involving a multicategorical independent variable (such as two or more experimental conditions relative to a control group). We provide a tutorial illustrating an approach to estimation of and inference about direct, indirect, and total effects in statistical mediation analysis with a multicategorical independent variable. The approach is mathematically equivalent to analysis of (co)variance and reproduces the observed and adjusted group means while also generating effects having simple interpretations. Supplementary material available online includes extensions to this approach and Mplus, SPSS, and SAS code that implements it.     

An irrelevant speech effect with repeated and continuous background speech

The irrelevant speech effect is the impairment of task performance by the presentation of to-be-ignored speech stimuli. Typically, the irrelevant speech comprises a variety of sounds, but previous research (e.g., Jones, Madden, & Miles, 1992) has suggested that the deleterious effect of background speech is virtually eliminated if the speech comprises repetitions of a sound (e.g., “be, be, be”) or a single continuous sound (e.g., “beeeeeee”). Four experiments are reported that challenge this finding. Experiments 1, 2, and 4 show a substantial impairment in serial recall performance in the presence of a repeated sound, and Experiments 3 and 4 show a similar impairment of serial recall in the presence of a continuous sound. The relevance of these findings to several explanations of the irrelevant speech effect is discussed.     

The effect of background familiarity in visual search: An analysis of underlying factors

The speed and accuracy of visual target search are strongly dependent on the familiarity of the background through which the search proceeds. Search through unfamiliar elements is much more difficult than search through familiar ones. This effect of background familiarity is examined in a series of three experiments. Experiment 1 suggests that the effect is not attributable to an inherently slow classification of individually unfamiliar nontargets. Experiments 2 and 3 investigate three aspects of multicharacter processing potentially sensitive to background familiarity. The results suggest that the background familiarity effect is most parsimoniously viewed as the result of slow or inaccurate segmentation of features extracted from adjacent characters. Mechanisms linking the familiarity of the background with the efficiency of character segregation are discussed in closing.     

Understanding memory impairment with memory models and hierarchical Bayesian analysis

The study of human episodic memory is a topic that interests cognitive and mathematical psychologists as well as clinicians interested in the diagnosis and assessment of Alzheimer’s disease and related disorders (ADRD). In this paper, we use simple cognitive models for the recognition and recall tasks typically applied in clinical assessments of ADRD to study memory performance in ADRD patients. Our models make use of hierarchical Bayesian methods as a way to model individual differences in patient performance and to facilitate the modeling of performance changes that occur during multiple recall tasks. We show how the models are able to account for different aspects of patient performance, and also discuss some of the predictive capabilities of the model. We conclude with a discussion on the scope to improve on our results by discussing the link between memory theory in psychology and clinical practice.     

An effect size index for comparing two independent alpha coefficients

Since Cronbach proposed the α coefficient in 1951, researchers have contributed to the derivation of its sampling distribution and the testing of related statistical hypotheses. Yet, there has been no research on effect size index relevant to coefficient α to our knowledge. Considering the importance of effect size in understanding quantitative research findings, we therefore developed an effect size index Δ for the comparison of two independent αs with equal test length based on the asymptotic distribution of under the assumptions of normality and compound symmetry. Simulations indicated that the index was applicable when the sample size was at least 100. The robustness of the derived index when the required assumptions were violated was also explored. It is suggested that the index should be applicable in most cases of unequal test lengths and could be extended to non‐normally distributed component scores. Moreover, a small simulation was conducted to explore the behaviour of Δ with correlated errors, a frequently studied situation violating the assumption of compound symmetry. The proposed index was found to be robust unless a large number of highly correlated errors were present in the data.     

Bayesian item fit analysis for unidimensional item response theory models

Assessing item fit for unidimensional item response theory models for dichotomous items has always been an issue of enormous interest, but there exists no unanimously agreed item fit diagnostic for these models, and hence there is room for further investigation of the area. This paper employs the posterior predictive model‐checking method, a popular Bayesian model‐checking tool, to examine item fit for the above‐mentioned models. An item fit plot, comparing the observed and predicted proportion‐correct scores of examinees with different raw scores, is suggested. This paper also suggests how to obtain posterior predictive p‐values (which are natural Bayesian p‐values) for the item fit statistics of Orlando and Thissen that summarize numerically the information in the above‐mentioned item fit plots. A number of simulation studies and a real data application demonstrate the effectiveness of the suggested item fit diagnostics. The suggested techniques seem to have adequate power and reasonable Type I error rate, and psychometricians will find them promising.     

Exploring an alternative in-car music background designed for driver safety

Young drivers listen to highly energetic aggressive music of a fast-tempo and accentuated beat at elevated volumes. They are not aware of the effects that music may have on perception, performance, and control of the vehicle. The crux of the matter is not the use of music per se, but rather the abuse of music that is hazardous. The current study developed a viable alternative music background for in-car listening towards improved driver safety. After a group of everyday listeners confirmed the experimental music as suitable for in-car music listening, 22 drivers each drove four trips while listening to either preferred music CDs or the experimental background, then 31 drivers each drove ten trips while listening to the alternative background. Study A demonstrated criterion related validity, although the experimental background preoccupied less attention. While Study B indicated habituation effects, drivers reported ever increasing levels of positive mood states throughout.     

Bayesian analysis of square ordinal-ordinal tables

We analyse square contingency tables with ordered categories. Assuming that the observed ordinal categorical variables are manifestations of underlying continuous variables, we formulate a model which allows the comparisons of locations and dispersions between variables. We identify the model by imposing stochastic constraints on the thresholds that define the relationship between the observed and the underlying variables. As a result, the underlying continuous variables' location and dispersion parameters which were not estimable before can be estimated by the Bayesian approach. Illustrative examples are given based on several reported data sets.     

Bayesian model selection for mixtures of structural equation models with an unknown number of components

This paper considers mixtures of structural equation models with an unknown number of components. A Bayesian model selection approach is developed based on the Bayes factor. A procedure for computing the Bayes factor is developed via path sampling, which has a number of nice features. The key idea is to construct a continuous path linking the competing models; then the Bayes factor can be estimated efficiently via grids in [0, 1] and simulated observations that are generated by the Gibbs sampler from the posterior distribution. Bayesian estimates of the structural parameters, latent variables, as well as other statistics can be produced as by‐products. The properties and merits of the proposed procedure are discussed and illustrated by means of a simulation study and a real example.