Multinomial processing trees (MPTs) are a popular class of cognitive models for categorical data. Typically, researchers compare several MPTs, each equipped with many parameters, especially when the models are implemented in a hierarchical framework. A Bayesian solution is to compute posterior model probabilities and Bayes factors. Both quantities, however, rely on the marginal likelihood, a high-dimensional integral that cannot be evaluated analytically. In this case study, we show how Warp-III bridge sampling can be used to compute the marginal likelihood for hierarchical MPTs. We illustrate the procedure with two published data sets and demonstrate how Warp-III facilitates Bayesian model averaging.

In the social sciences we are often interested in comparing models specified by parametric equality or inequality constraints. For instance, when examining three group means $\{{\mathit{\mu }}_1,{\mathit{\mu }}_2,{\mathit{\mu }}_3\}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{ \mu _1, \mu _2, \mu _3\}$$\end{document} through an analysis of variance (ANOVA), a model may specify that ${\mathit{\mu }}_1<{\mathit{\mu }}_2<{\mathit{\mu }}_3$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu _1<\mu _2<\mu _3$$\end{document}, while another one may state that $\{{\mathit{\mu }}_1={\mathit{\mu }}_3\}<{\mathit{\mu }}_2$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{ \mu _1=\mu _3\} <\mu _2$$\end{document}, and finally a third model may instead suggest that all means are unrestricted. This is a challenging problem, because it involves a combination of nonnested models, as well as nested models having the same dimension. We adopt an objective Bayesian approach, requiring no prior specification from the user, and derive the posterior probability of each model under consideration. Our method is based on the intrinsic prior methodology, suitably modified to accommodate equality and inequality constraints. Focussing on normal ANOVA models, a comparative assessment is carried out through simulation studies. We also present an application to real data collected in a psychological experiment.

A Bayesian approach is developed for analyzing nonlinear structural equation models with nonignorable missing data. The nonignorable missingness mechanism is specified by a logistic regression model. A hybrid algorithm that combines the Gibbs sampler and the Metropolis–Hastings algorithm is used to produce the joint Bayesian estimates of structural parameters, latent variables, parameters in the nonignorable missing model, as well as their standard errors estimates. A goodness-of-fit statistic for assessing the plausibility of the posited nonlinear structural equation model is introduced, and a procedure for computing the Bayes factor for model comparison is developed via path sampling. Results obtained with respect to different missing data models, and different prior inputs are compared via simulation studies. In particular, it is shown that in the presence of nonignorable missing data, results obtained by the proposed method with a nonignorable missing data model are significantly better than those that are obtained under the missing at random assumption. A real example is presented to illustrate the newly developed Bayesian methodologies.

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.

The assumption of latent monotonicity in item response theory models for dichotomous data cannot be evaluated directly, but observable consequences such as manifest monotonicity facilitate the assessment of latent monotonicity in real data. Standard methods for evaluating manifest monotonicity typically produce a test statistic that is geared toward falsification, which can only provide indirect support in favor of manifest monotonicity. We propose the use of Bayes factors to quantify the degree of support available in the data in favor of manifest monotonicity or against manifest monotonicity. Through the use of informative hypotheses, this procedure can also be used to determine the support for manifest monotonicity over substantively or statistically relevant alternatives to manifest monotonicity, rendering the procedure highly flexible. The performance of the procedure is evaluated using a simulation study, and the application of the procedure is illustrated using empirical data.

Two marginal one-parameter item response theory models are introduced, by integrating out the latent variable or random item parameter. It is shown that both marginal response models are multivariate (probit) models with a compound symmetry covariance structure. Several common hypotheses concerning the underlying covariance structure are evaluated using (fractional) Bayes factor tests. The support for a unidimensional factor (i.e., assumption of local independence) and differential item functioning are evaluated by testing the covariance components. The posterior distribution of common covariance components is obtained in closed form by transforming latent responses with an orthogonal (Helmert) matrix. This posterior distribution is defined as a shifted-inverse-gamma, thereby introducing a default prior and a balanced prior distribution. Based on that, an MCMC algorithm is described to estimate all model parameters and to compute (fractional) Bayes factor tests. Simulation studies are used to show that the (fractional) Bayes factor tests have good properties for testing the underlying covariance structure of binary response data. The method is illustrated with two real data studies.

The assumption of latent monotonicity is made by all common parametric and nonparametric polytomous item response theory models and is crucial for establishing an ordinal level of measurement of the item score. Three forms of latent monotonicity can be distinguished: monotonicity of the cumulative probabilities, of the continuation ratios, and of the adjacent-category ratios. Observable consequences of these different forms of latent monotonicity are derived, and Bayes factor methods for testing these consequences are proposed. These methods allow for the quantification of the evidence both in favor and against the tested property. Both item-level and category-level Bayes factors are considered, and their performance is evaluated using a simulation study. The methods are applied to an empirical example consisting of a 10-item Likert scale to investigate whether a polytomous item scoring rule results in item scores that are of ordinal level measurement.

In relational event networks, endogenous statistics are used to summarize the past activity between actors. Typically, it is assumed that past events have equal weight on the social interaction rate in the (near) future regardless of the time that has transpired since observing them. Generally, it is unrealistic to assume that recently past events affect the current event rate to an equal degree as long-past events. Alternatively one may consider using a prespecified decay function with a prespecified rate of decay. A problem then is that the chosen decay function could be misspecified yielding biased results and incorrect conclusions. In this paper, we introduce three parametric weight decay functions (exponential, linear, and one-step) that can be embedded in a relational event model. A statistical method is presented to decide which memory decay function and memory parameter best fit the observed sequence of events. We present simulation studies that show the presence of bias in the estimates of effects of the statistics whenever the decay, as well as the memory parameter, are not properly estimated, and the ability to test different memory models against each other using the Bayes factor. Finally, we apply the methodology to two empirical case studies.

Judgment and decision making research overwhelmingly uses null hypothesis significance testing as the basis for statistical inference. This article examines an alternative, Bayesian approach which emphasizes the choice between two competing hypotheses and quantifies the balance of evidence provided by the data—one consequence of which is that experimental results may be taken to strongly favour the null hypothesis. We apply a recently-developed “Bayesian t-test” to existing studies of the anchoring effect in judgment, and examine how the change in approach affects both the tone of hypothesis testing and the substantive conclusions that one draws. We compare the Bayesian approach with Fisherian and Neyman-Pearson testing, examining its relationship to conventional p-values, the influence of effect size, and the importance of prior beliefs about the likely state of nature. The results give a sense of how Bayesian hypothesis testing might be applied to judgment and decision making research, and of both the advantages and challenges that a shift to this approach would entail.

Are difficult decisions best made after a momentary diversion of thought? Previous research addressing this important question has yielded dozens of experiments in which participants were asked to choose the best of several options (e.g., cars or apartments) either after conscious deliberation, or after a momentary diversion of thought induced by an unrelated task. The results of these studies were mixed. Some found that participants who had first performed the unrelated task were more likely to choose the best option, whereas others found no evidence for this so-called unconscious thought advantage (UTA). The current study examined two accounts of this inconsistency in previous findings. According to the reliability account, the UTA does not exist and previous reports of this effect concern nothing but spurious effects obtained with an unreliable paradigm. In contrast, the moderator account proposes that the UTA is a real effect that occurs only when certain conditions are met in the choice task. To test these accounts, we conducted a meta-analysis and a large-scale replication study (N = 399) that met the conditions deemed optimal for replicating the UTA. Consistent with the reliability account, the large-scale replication study yielded no evidence for the UTA, and the meta-analysis showed that previous reports of the UTA were confined to underpowered studies that used relatively small sample sizes. Furthermore, the results of the large-scale study also dispelled the recent suggestion that the UTA might be gender-specific. Accordingly, we conclude that there exists no reliable support for the claim that a momentary diversion of thought leads to better decision making than a period of deliberation.

The present paper studies the likelihood ratio order of posterior distributions of parameter when the same order exists between the corresponding prior of the parameter, or when the observed values of the sufficient statistic for the parameter differ. The established likelihood order allows one to compare the Bayesian estimators associated with many common and general error loss functions analytically. It can also enable one to compare the Bayes factor in hypothesis testing without using numerical computation. Moreover, using the likelihood ratio (LR) order of the posterior distributions can yield the LR order between marginal predictive distributions, and posterior predictive distributions.

Depletion of the Groningen gas field has induced earthquakes, although the north of the Netherlands is a tectonically inactive region. Increased seismic activity raised public concern which led the government to initiate a number of studies with the aim of understanding the cause(s) of the earthquakes. If the relationship between production and seismicity were understood then production could be optimized in such a way that the risk of induced seismicity would be minimal. The main question remains how production is correlated with induced seismicity. The Minister of Economic Affairs of the Netherlands decided to reduce production starting from 17 January 2014, specifically in the centre of the gas field as it has the highest rates of seismicity, the largest-magnitude events and the highest compaction values of the field.

A reduction in production could possibly lead to a reduced rate of compaction. Additionally a reduction of production rate could lead to a reduced stress rate increase on the existing faults and consequently fewer seismic events per year. One might envisage a ‘bonus effect’ in the events reduction in the sense that the total number of events will be less, with the same total production smeared out over a longer period. This is as yet unclear.

In this paper we apply different statistical methods to look for evidence supporting or disproving a decrease in the number of seismic events due to production reduction.

Differential gene expression analyses typically focus on departures across mathematical expectations (i.e. mean) from two or more groups of microarrays, without considering alternative patterns of departure. Nevertheless, recent studies in humans and great apes have suggested that differential gene expression could also be characterized in terms of heterogeneous dispersion patterns. This must be viewed as a very interesting genetic phenomenon clearly linked to the regulation mechanisms of gene transcription. Unfortunately, we completely lack information about the incidence and relevance of dispersion-specific differential gene expression in livestock species, although a specific Bayes factor (BF) for testing this kind of differential gene expression (i.e. within-probe heteroskedasticity) has been recently developed. Within this context, our main objective was to characterize the incidence of dispersion-specific differential gene expression in pigs and, if possible, providing the first evidence of this phenomenon in a livestock species. We evaluated dispersion-specific differential gene expression on ovary, uterus and hypophysis samples from 22 F2 Iberian × Meishan sows, where a total of 15 252 probes were interrogated. For each tissue, heteroskedasticity of probe-specific residual variances was evaluated by three pairwise comparisons involving three physiological stages, that is, heat, 15 days of pregnancy and 45 days of pregnancy. Between 2.9% and 37.4% of the analyzed probes provided statistical evidence of within-tissue across-physiological stages dispersion-specific differential gene expression (BF >1), and between 0.1% and 3.0% of them reported decisive evidence (BF >100). It is important to highlight that <8% of the heteroskedastic probes were also linked to differential gene expression in terms of departures among the probe-specific mathematical expectation of each physiological stage. This discarded the disturbance of scale effects in a high percentage of probes and suggested that probe-specific heteroskedasticity must be viewed as an independent phenomenon within the context of differential gene expression. As a whole, our results report a remarkable incidence of dispersion-specific differential gene expression across the whole genome of the pig, establishing a very interesting starting point for further studies focused on deciphering the genetic mechanisms underlying heteroskedasticity.

The aim of this study was to investigate chromosomal regions affecting gestation length in sows. An experimental F2 cross between Iberian and Meishan pig breeds was used for this purpose and we genotyped 119 markers covering the 18 porcine autosomal chromosomes. Within this context, we have developed a new empirical Bayes factor (BF) approach to compare between nested models, with and without the quantitative trait loci (QTL) effect, and after including the location of the QTL as an unknown parameter in the model. This empirical BF can be easily calculated from the output of a Markov chain Monte Carlo sampling by averaging conditional densities at the null QTL effects. Linkage analyses were performed in each chromosome using an animal model to account for infinitesimal genetic effects. Initially, three QTL were detected at chromosomes 6, 8 and 11 although, after correcting for multiple testing, only the additive QTL located in cM 110 of chromosome 8 remained. For this QTL, the allelic effect of substitution of the Iberian allele increased gestation length in 0.521 days, with a highest posterior density region at 95% ranged between 0.121 and 0.972 days. Although future studies are necessary to confirm if detected QTL is relevant and segregating in commercial pig populations, a hot-spot on the genetic regulation of gestation length in pigs seems to be located in chromosome 8.