Automatic Bayes Factors for Testing Equality- and Inequality-Constrained Hypotheses on Variances

In comparing characteristics of independent populations, researchers frequently expect a certain structure of the population variances. These expectations can be formulated as hypotheses with equality and/or inequality constraints on the variances. In this article, we consider the Bayes factor for testing such (in)equality-constrained hypotheses on variances. Application of Bayes factors requires specification of a prior under every hypothesis to be tested. However, specifying subjective priors for variances based on prior information is a difficult task. We therefore consider so-called automatic or default Bayes factors. These methods avoid the need for the user to specify priors by using information from the sample data. We present three automatic Bayes factors for testing variances. The first is a Bayes factor with equal priors on all variances, where the priors are specified automatically using a small share of the information in the sample data. The second is the fractional Bayes factor, where a fraction of the likelihood is used for automatic prior specification. The third is an adjustment of the fractional Bayes factor such that the parsimony of inequality-constrained hypotheses is properly taken into account. The Bayes factors are evaluated by investigating different properties such as information consistency and large sample consistency. Based on this evaluation, it is concluded that the adjusted fractional Bayes factor is generally recommendable for testing equality- and inequality-constrained hypotheses on variances.     

Approximated adjusted fractional Bayes factors: A general method for testing informative hypotheses

Informative hypotheses are increasingly being used in psychological sciences because they adequately capture researchers’ theories and expectations. In the Bayesian framework, the evaluation of informative hypotheses often makes use of default Bayes factors such as the fractional Bayes factor. This paper approximates and adjusts the fractional Bayes factor such that it can be used to evaluate informative hypotheses in general statistical models. In the fractional Bayes factor a fraction parameter must be specified which controls the amount of information in the data used for specifying an implicit prior. The remaining fraction is used for testing the informative hypotheses. We discuss different choices of this parameter and present a scheme for setting it. Furthermore, a software package is described which computes the approximated adjusted fractional Bayes factor. Using this software package, psychological researchers can evaluate informative hypotheses by means of Bayes factors in an easy manner. Two empirical examples are used to illustrate the procedure.     

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.     

A caveat on the Savage–Dickey density ratio: The case of computing Bayes factors for regression parameters

The Savage–Dickey density ratio is a simple method for computing the Bayes factor for an equality constraint on one or more parameters of a statistical model. In regression analysis, this includes the important scenario of testing whether one or more of the covariates have an effect on the dependent variable. However, the Savage–Dickey ratio only provides the correct Bayes factor if the prior distribution of the nuisance parameters under the nested model is identical to the conditional prior under the full model given the equality constraint. This condition is violated for multiple regression models with a Jeffreys–Zellner–Siow prior, which is often used as a default prior in psychology. Besides linear regression models, the limitation of the Savage–Dickey ratio is especially relevant when analytical solutions for the Bayes factor are not available. This is the case for generalized linear models, non-linear models, or cognitive process models with regression extensions. As a remedy, the correct Bayes factor can be computed using a generalized version of the Savage–Dickey density ratio.     

How to quantify the evidence for the absence of a correlation

We present a suite of Bayes factor hypothesis tests that allow researchers to grade the decisiveness of the evidence that the data provide for the presence versus the absence of a correlation between two variables. For concreteness, we apply our methods to the recent work of Donnellan et al. (in press) who conducted nine replication studies with over 3,000 participants and failed to replicate the phenomenon that lonely people compensate for a lack of social warmth by taking warmer baths or showers. We show how the Bayes factor hypothesis test can quantify evidence in favor of the null hypothesis, and how the prior specification for the correlation coefficient can be used to define a broad range of tests that address complementary questions. Specifically, we show how the prior specification can be adjusted to create a two-sided test, a one-sided test, a sensitivity analysis, and a replication test.     

Treatment effects on count outcomes with non-normal covariates

The effects of a treatment or an intervention on a count outcome are often of interest in applied research. When controlling for additional covariates, a negative binomial regression model is usually applied to estimate conditional expectations of the count outcome. The difference in conditional expectations under treatment and under control is then defined as the (conditional) treatment effect. While traditionally aggregates of these conditional treatment effects (e.g., average treatment effects) are computed by averaging over the empirical distribution, a recently proposed moment-based approach allows for computing aggregate effects as a function of distribution parameters. The moment-based approach makes it possible to control for (latent) multivariate normally distributed covariates and provides more reliable inferences under certain conditions. In this paper we propose three different ways to account for non-normally distributed continuous covariates in this approach: an alternative, known non-normal distribution; a plausible factorization of the joint distribution; and an approximation using finite Gaussian mixtures. A saturated model is used for categorical covariates, making a distributional assumption obsolete. We further extend the moment-based approach to allow for multiple treatment conditions and the computation of conditional effects for categorical covariates. An illustrative example highlighting the key features of our extension is provided.     

Hierarchical modeling of sequential behavioral data: an empirical Bayesian approach

The authors review the common methods for measuring strength of contingency between 2 behaviors in a behavioral sequence, the binomial z score and the adjusted cell residual, and point out a number of limitations of these approaches. They present a new approach using log odds ratios and empirical Bayes estimation in the context of hierarchical modeling, an approach not constrained by these limitations. A series of hierarchical models is presented to test the stationarity of behavioral sequences, the homogeneity of sequences across a sample of episodes, and whether covariates can account for variation in sequences across the sample. These models are applied to observational data taken from a study of the behavioral interactions of 254 couples to illustrate their use.     

Future Expectations Among Adolescents: A Latent Class Analysis

Future expectations have been important predictors of adolescent development and behavior. Its measurement, however, has largely focused on single dimensions and misses potentially important components. This analysis investigates whether an empirically-driven, multidimensional approach to conceptualizing future expectations can substantively contribute to our understanding of adolescent risk behavior. We use data from the National Longitudinal Survey of Youth 1997 to derive subpopulations of adolescents based on their future expectations with latent class analysis. Multinomial regression then determines which covariates from Bronfenbrenner's ecological systems theory are associated with class membership. After modeling these covariates, we examine whether future expectations is associated with delinquency, substance use, and sexual experience. Our analysis suggests the emergence of four distinct classes labeled the Student Expectations, Student/Drinking Expectations, Victim Expectations, and Drinking/Arrest Expectations classes according to their indicator profiles. These classes differ with respect to covariates associated with membership; furthermore, they are all statistically and differentially associated with at least one adolescent risk behavior. This analysis demonstrates the additional benefit derived from using this multidimensional approach for studying future expectations. Further research is needed to investigate its stability and role in predicting adolescent risk behavior over time.     

A Propensity Score Adjustment for Multiple Group Structural Equation Modeling

In the behavioral and social sciences, quasi-experimental and observational studies are used due to the difficulty achieving a random assignment. However, the estimation of differences between groups in observational studies frequently suffers from bias due to differences in the distributions of covariates. To estimate average treatment effects when the treatment variable is binary, Rosenbaum and Rubin (1983a) proposed adjustment methods for pretreatment variables using the propensity score. However, these studies were interested only in estimating the average causal effect and/or marginal means. In the behavioral and social sciences, a general estimation method is required to estimate parameters in multiple group structural equation modeling where the differences of covariates are adjusted. We show that a Horvitz–Thompson-type estimator, propensity score weighted M estimator (PWME) is consistent, even when we use estimated propensity scores, and the asymptotic variance of the PWME is shown to be less than that with true propensity scores. Furthermore, we show that the asymptotic distribution of the propensity score weighted statistic under a null hypothesis is a weighted sum of independent χ² ₁ variables. We show the method can compare latent variable means with covariates adjusted using propensity scores, which was not feasible by previous methods. We also apply the proposed method for correlated longitudinal binary responses with informative dropout using data from the Longitudinal Study of Aging (LSOA). The results of a simulation study indicate that the proposed estimation method is more robust than the maximum likelihood (ML) estimation method, in that PWME does not require the knowledge of the relationships among dependent variables and covariates.     

A tutorial on Bayes factor estimation with the product space method

The Bayes factor is an intuitive and principled model selection tool from Bayesian statistics. The Bayes factor quantifies the relative likelihood of the observed data under two competing models, and as such, it measures the evidence that the data provides for one model versus the other. Unfortunately, computation of the Bayes factor often requires sampling-based procedures that are not trivial to implement. In this tutorial, we explain and illustrate the use of one such procedure, known as the product space method (Carlin & Chib, 1995). This is a transdimensional Markov chain Monte Carlo method requiring the construction of a “supermodel” encompassing the models under consideration. A model index measures the proportion of times that either model is visited to account for the observed data. This proportion can then be transformed to yield a Bayes factor. We discuss the theory behind the product space method and illustrate, by means of applied examples from psychological research, how the method can be implemented in practice.     

Generalized linear models with ordinally‐observed covariates

An ordinally‐observed variable is a variable that is only partially observed through an ordinal surrogate. Although statistical models for ordinally‐observed response variables are well known, relatively little attention has been given to the problem of ordinally‐observed regressors. In this paper I show that if surrogates to ordinally‐observed covariates are used as regressors in a generalized linear model then the resulting measurement error in the covariates can compromise the consistency of point estimators and standard errors for the effects of fully‐observed regressors. To properly account for this measurement error when making inferences concerning the fully‐observed regressors, I propose a general modelling framework for generalized linear models with ordinally‐observed covariates. I discuss issues of model specification, identification, and estimation, and illustrate these with examples.     

Empirical Bayes Derivative Estimates

Abstract

A dynamic system is a set of interacting elements characterized by changes occurring over time. The estimation of derivatives is a mainstay for exploring dynamics of constructs, particularly when the dynamics are complicated or unknown. The presence of measurement error in many social science constructs frequently results in poor estimates of derivatives, as even modest proportions of measurement error can compound when estimating derivatives. Given the overlap in the specification of latent differential equation models and latent growth curve models, and the equivalence of latent growth curve models and mixed models under some conditions, derivatives could be estimated from estimates of random effects. This article proposes a new method for estimating derivatives based on calculating the Empirical Bayes estimates of derivatives from a mixed model. Two simulations compare four derivative estimation methods: Generalized Local Linear Approximation, Generalized Orthogonal Derivative Estimates, Functional Data Analysis, and the proposed Empirical Bayes Derivative Estimates. The simulations consider two data collection scenarios: short time series (≤10 observations) from many individuals or occasions, and long individual time series (25–500 observations). A substantive example visualizing the dynamics of intraindividual positive affect time series is also presented.     

Beta‐binomial ANOVA for multivariate randomized response data

There is much empirical evidence that randomized response methods improve the cooperation of the respondents when asking sensitive questions. The traditional methods for analysing randomized response data are restricted to univariate data and only allow inferences at the group level due to the randomized response sampling design. Here, a novel beta‐binomial model is proposed for analysing multivariate individual count data observed via a randomized response sampling design. This new model allows for the estimation of individual response probabilities (response rates) for multivariate randomized response data utilizing an empirical Bayes approach. A common beta prior specifies that individuals in a group are tied together and the beta prior parameters are allowed to be cluster‐dependent. A Bayes factor is proposed to test for group differences in response rates. An analysis of a cheating study, where 10 items measure cheating or academic dishonesty, is used to illustrate application of the proposed model.     

Nonlinear Random-Effects Mixture Models for Repeated Measures

A mixture model for repeated measures based on nonlinear functions with random effects is reviewed. The model can include individual schedules of measurement, data missing at random, nonlinear functions of the random effects, of covariates and of residuals. Individual group membership probabilities and individual random effects are obtained as empirical Bayes predictions. Although this is a complicated model that combines a mixture of populations, nonlinear regression, and hierarchical models, it is straightforward to estimate by maximum likelihood using SAS PROC NLMIXED. Many different models can be studied with this procedure. The model is more general than those that can be estimated with most special purpose computer programs currently available because the response function is essentially any form of nonlinear regression. Examples and sample code are included to illustrate the method.     

Bayesian evaluation of informative hypotheses for multiple populations

The software package Bain can be used for the evaluation of informative hypotheses with respect to the parameters of a wide range of statistical models. For pairs of hypotheses the support in the data is quantified using the approximate adjusted fractional Bayes factor (BF). Currently, the data have to come from one population or have to consist of samples of equal size obtained from multiple populations. If samples of unequal size are obtained from multiple populations, the BF can be shown to be inconsistent. This paper examines how the approach implemented in Bain can be generalized such that multiple-population data can properly be processed. The resulting multiple-population approximate adjusted fractional Bayes factor is implemented in the R package Bain.     

The deviant citizen: Measuring potential positive relations between counterproductive work behaviour and organizational citizenship behaviour

Studies have shown a strong negative correlation between counterproductive work behaviour (CWB) and organizational citizenship behaviour (OCB), and opposite correlations with hypothesized antecedents. Such observed correlations may have been erroneously caused by three measurement artefacts: items measuring absence of CWBs, rather than behaviours that exceed requirements or expectations in OCB scales; supervisory halo; and agreement rather than frequency response format. A new OCB scale, the OCB‐checklist (OCB‐C) was used that did not have these artefacts. Contrary to prior expectations from the literature, positive relations were found between CWB and OCB, and stressors and OCB. Theoretical explanations for positive CWB/OCB relations (demand‐elicited OCB, social loafing, work process problems, rater perceptions and attributions, and aggravated job stress processes) are discussed.     

A Multivariate Multilevel Approach to the Modeling of Accuracy and Speed of Test Takers

Response times on test items are easily collected in modern computerized testing. When collecting both (binary) responses and (continuous) response times on test items, it is possible to measure the accuracy and speed of test takers. To study the relationships between these two constructs, the model is extended with a multivariate multilevel regression structure which allows the incorporation of covariates to explain the variance in speed and accuracy between individuals and groups of test takers. A Bayesian approach with Markov chain Monte Carlo (MCMC) computation enables straightforward estimation of all model parameters. Model-specific implementations of a Bayes factor (BF) and deviance information criterium (DIC) for model selection are proposed which are easily calculated as byproducts of the MCMC computation. Both results from simulation studies and real-data examples are given to illustrate several novel analyses possible with this modeling framework. The authors thank Steven Wise, James Madison University, and Pere Joan Ferrando, Universitat Rovira i Virgili, for generously making available their data sets for the empirical examples in this paper.     

School System Evaluation by Value Added Analysis Under Endogeneity

Value added is a common tool in educational research on effectiveness. It is often modeled as a (prediction of a) random effect in a specific hierarchical linear model. This paper shows that this modeling strategy is not valid when endogeneity is present. Endogeneity stems, for instance, from a correlation between the random effect in the hierarchical model and some of its covariates. This paper shows that this phenomenon is far from exceptional and can even be a generic problem when the covariates contain the prior score attainments, a typical situation in value added modeling. Starting from a general, model-free definition of value added, the paper derives an explicit expression of the value added in an endogeneous hierarchical linear Gaussian model. Inference on value added is proposed using an instrumental variable approach. The impact of endogeneity on the value added and the estimated value added is calculated accurately. This is also illustrated on a large data set of individual scores of about 200,000 students in Chile.     

Model comparison of nonlinear structural equation models with fixed covariates

Recently, it has been recognized that the commonly used linear structural equation model is inadequate to deal with some complicated substantive theory. A new nonlinear structural equation model with fixed covariates is proposed in this article. A procedure, which utilizes the powerful path sampling for computing the Bayes factor, is developed for model comparison. In the implementation, the required random observations are simulated via a hybrid algorithm that combines the Gibbs sampler and the Metropolis-Hastings algorithm. It is shown that the proposed procedure is efficient and flexible; and it produces Bayesian estimates of the parameters, latent variables, and their highest posterior density intervals as by-products. Empirical performances of the proposed procedure such as sensitivity to prior inputs are illustrated by a simulation study and a real example.This research is fully supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 4346/01H). The authors are thankful to the Editor, the Associate Editor, and anonymous reviewers for valuable comments which improve the paper significantly, and grateful to ICPSR and the relevant funding agency for allowing use of the data in the example. The assistance of Michael K.H. Leung and Esther L.S. Tam is gratefully acknowledged.     

Bayesian data analysis as a tool for behavior analysts

Bayesian approaches to data analysis are considered within the context of behavior analysis. The paper distinguishes between Bayesian inference, the use of Bayes Factors, and Bayesian data analysis using specialized tools. Given the importance of prior beliefs to these approaches, the review addresses those situations in which priors have a big effect on the outcome (Bayes Factors) versus a smaller effect (parameter estimation). Although there are many advantages to Bayesian data analysis from a philosophical perspective, in many cases a behavior analyst can be reasonably well‐served by the adoption of traditional statistical tools as long as the focus is on parameter estimation and model comparison, not null hypothesis significance testing. A strong case for Bayesian analysis exists under specific conditions: When prior beliefs can help narrow parameter estimates (an especially important issue given the small sample sizes common in behavior analysis) and when an analysis cannot easily be conducted using traditional approaches (e.g., repeated measures censored regression).