Bayesian One-Sided Variable Selection.

This paper presents a novel Bayesian variable selection approach that accounts for the sign of the regression coefficients based on multivariate one-sided tests. We propose a truncated g prior to specify a prior distribution of coefficients with anticipated signs in a given model. Informative priors for the direction of the effects can be incorporated into prior model probabilities. The best subset of variables is selected by comparing the posterior probabilities of the possible models. The new Bayesian one-sided variable selection procedure has higher chance to include relevant variables and therefore select the best model, if the anticipated direction is accurate. For a large number of candidate variables, we present an adaptation of a Bayesian model search method for the one-sided variable selection problem to ensure fast computation. In addition, a fully Bayesian approach is used to adjust the prior inclusion probability of each one-sided model to correct for multiplicity. The performance of the proposed method is investigated using several simulation studies and two real data examples.

Sample-size determination for the Bayesian t test and Welch's test using the approximate adjusted fractional Bayes factor.

When two independent means µ1 and µ2 are compared, H0 : µ1 = µ2, H1 : µ1≠µ2, and H2 : µ1 > µ2 are the hypotheses of interest. This paper introduces the R package SSDbain, which can be used to determine the sample size needed to evaluate these hypotheses using the approximate adjusted fractional Bayes factor (AAFBF) implemented in the R package bain. Both the Bayesian t test and the Bayesian Welch's test are available in this R package. The sample size required will be calculated such that the probability that the Bayes factor is larger than a threshold value is at least η if either the null or alternative hypothesis is true. Using the R package SSDbain and/or the tables provided in this paper, psychological researchers can easily determine the required sample size for their experiments.

Parental Age and Offspring Childhood Mental Health: A Multi-Cohort, Population-Based Investigation.

To examine the contributions of maternal and paternal age on offspring externalizing and internalizing problems, this study analyzed problem behaviors at age 10-12 years from four Dutch population-based cohorts (N = 32,892) by a multiple informant design. Bayesian evidence synthesis was used to combine results across cohorts with 50% of the data analyzed for discovery and 50% for confirmation. There was evidence of a robust negative linear relation between parental age and externalizing problems as reported by parents. In teacher-reports, this relation was largely explained by parental socio-economic status. Parental age had limited to no association with internalizing problems. Thus, in this large population-based study, either a beneficial or no effect of advanced parenthood on child problem behavior was observed.

All for one or some for all? Evaluating informative hypotheses using multiple N = 1 studies.

Analyses are mostly executed at the population level, whereas in many applications the interest is on the individual level instead of the population level. In this paper, multiple N = 1 experiments are considered, where participants perform multiple trials with a dichotomous outcome in various conditions. Expectations with respect to the performance of participants can be translated into so-called informative hypotheses. These hypotheses can be evaluated for each participant separately using Bayes factors. A Bayes factor expresses the relative evidence for two hypotheses based on the data of one individual. This paper proposes to "average" these individual Bayes factors in the gP-BF, the average relative evidence. The gP-BF can be used to determine whether one hypothesis is preferred over another for all individuals under investigation. This measure provides insight into whether the relative preference of a hypothesis from a pre-defined set is homogeneous over individuals. Two additional measures are proposed to support the interpretation of the gP-BF: the evidence rate (ER), the proportion of individual Bayes factors that support the same hypothesis as the gP-BF, and the stability rate (SR), the proportion of individual Bayes factors that express a stronger support than the gP-BF. These three statistics can be used to determine the relative support in the data for the informative hypotheses entertained. Software is available that can be used to execute the approach proposed in this paper and to determine the sensitivity of the outcomes with respect to the number of participants and within condition replications.

P300 amplitude variations, prior probabilities, and likelihoods: A Bayesian ERP study.

The capability of the human brain for Bayesian inference was assessed by manipulating probabilistic contingencies in an urn-ball task. Event-related potentials (ERPs) were recorded in response to stimuli that differed in their relative frequency of occurrence (.18 to .82). A veraged ERPs with sufficient signal-to-noise ratio (relative frequency of occurrence > .5) were used for further analysis. Research hypotheses about relationships between probabilistic contingencies and ERP amplitude variations were formalized as (in-)equality constrained hypotheses. Conducting Bayesian model comparisons, we found that manipulations of prior probabilities and likelihoods were associated with separately modifiable and distinct ERP responses. P3a amplitudes were sensitive to the degree of prior certainty such that higher prior probabilities were related to larger frontally distributed P3a waves. P3b amplitudes were sensitive to the degree of likelihood certainty such that lower likelihoods were associated with larger parietally distributed P3b waves. These ERP data suggest that these antecedents of Bayesian inference (prior probabilities and likelihoods) are coded by the human brain.

Why Bayesian Psychologists Should Change the Way They Use the Bayes Factor.

The discussion following Bem's ( 2011 ) psi research highlights that applications of the Bayes factor in psychological research are not without problems. The first problem is the omission to translate subjective prior knowledge into subjective prior distributions. In the words of Savage ( 1961 ): "they make the Bayesian omelet without breaking the Bayesian egg." The second problem occurs if the Bayesian egg is not broken: the omission to choose default prior distributions such that the ensuing inferences are well calibrated. The third problem is the adherence to inadequate rules for the interpretation of the size of the Bayes factor. The current paper will elaborate these problems and show how to avoid them using the basic hypotheses and statistical model used in the first experiment described in Bem ( 2011 ). It will be argued that a thorough investigation of these problems in the context of more encompassing hypotheses and statistical models is called for if Bayesian psychologists want to add a well-founded Bayes factor to the tool kit of psychological researchers.

Bayes Factors Have Frequency Properties-This Should Not Be Ignored: A Rejoinder to Morey, Wagenmakers, and Rouder.

Hoijtink, van Kooten, and Hulsker ( 2016 ) outline a research agenda for Bayesian psychologists: evaluate and use the frequency properties of Bayes factors. Morey, Wagenmakers, and Rouder ( 2016 ) respond that Bayes factors calibrated using frequency properties should not be used. This paper contains the response of Hoijtink, van Kooten, and Hulsker to the criticism of Morey, Wagenmakers, and Rouder ( 2016 ).

Bayesian evaluation of diverging theories of episodic and affective memory distortions in dysphoria.

People suffering from dysphoria retrieve autobiographical memories distorted in content and affect, which may contribute to the aetiology and maintenance of depression. However, key memory difficulties in dysphoria remain elusive because theories disagree how memories of different valence are altered. Here, we assessed the psychophysiological expression of affect and retrieved episodic detail while participants with dysphoria (but without a diagnosed mental illness) and participants without dysphoria relived positive, negative, and neutral memories. We show that participants with dysphoria retrieve positive memories with diminished episodic detail and negative memories with enhanced detail, compared to participants without dysphoria. This is in line with negativity bias but not overgeneral memory bias theories. According to confirmatory analyses, participants with dysphoria also express diminished positive affect and enhanced negative affect when retrieving happy memories, but exploratory analyses suggest that this increase in negative affect may not be robust. Further confirmatory analyses showed that affective responses to memories are not related to episodic detail and already present during the experience of new emotional events. Our results indicate that affective memory distortions may not emerge from mnemonic processes but from general distortions in positive affect, which challenges assumptions of memory theories and therapeutics. Protocol registration: The Stage 1 protocol for this Registered Report was accepted in principle on the 18rd of March 2021. The protocol, as accepted by the journal, can be found at https://doi.org/10.6084/m9.figshare.14605374.v1 .

Any indication for bias due to participation? Comparison of the effects of a parent management training rated by participating and non-participating parents.

Ratings of parents that have participated in a parent training for child externalizing behaviour problems might be biased (e.g., they may report symptom reduction to reward their own endeavours for attending the training). The potential for bias in parent ratings was investigated in a secondary analysis of an effectiveness study of a parent management training for children (aged 3-10 years) with externalizing behaviour problems under routine care conditions. For the 56 families included in the current analysis, we compared the ratings of training participants (predominately mothers) and training non-participants (predominately fathers). A 3-month waiting period prior to treatment served as the control condition. Outcome measures were attention problems and conduct problems of the children and perceived parental self-efficacy rated by both the mothers and fathers. Child attention problems and conduct problems both decreased significantly during the treatment period for participating and non-participating parents, and the changes in ratings during treatment were of a similar magnitude for participants compared with non-participants. Taking into account the methodological limitations of the current analysis, no indication was found that treatment effects are strongly biased due to participation in the training. KEY PRACTITIONER MESSAGES: Previous positive findings of mothers' and fathers' ratings on parent management training for children with externalizing problem behaviour have mostly come from parents who also attended the training. For child behavioural problems, the results of the current analysis demonstrate similar perceived changes by parents who did and did not participate in the training. Our findings underscore the relevance and importance of parent ratings for treatment evaluation.

A default Bayes factor for testing null hypotheses about the fixed effects of linear two-level models.

Testing null hypotheses of the form "ß = 0," by the use of various Null Hypothesis Significance Tests (rendering a dichotomous reject/not reject decision), is considered standard practice when evaluating the individual parameters of statistical models. Bayes factors for testing these (and other) hypotheses allow users to quantify the evidence in the data that is in favor of a hypothesis. Unfortunately, when testing equality-contained hypotheses, the Bayes factors are sensitive to the specification of prior distributions, which may be hard to specify by applied researchers. The paper proposes a default Bayes factor with clear operating characteristics when used for testing whether the fixed parameters of linear two-level models are equal to zero. This is achieved by generalizing an already existing approach for linear regression. The generalization requires: (a) the sample size for which a new estimator for the effective sample size in two-level models containing random slopes is proposed; (b) the effect size for the fixed effects for which the so-called marginal R² for the fixed effects is used. Implementing the aforementioned requirements in a small simulation study shows that the Bayes factor yields clear operating characteristics regardless of the value for sample size and the estimation method. The paper gives practical examples and access to an easy-to-use wrapper function to calculate Bayes factors for hypotheses with respect to the fixed coefficients of linear two-level models by using the R package bain. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

A review of applications of the Bayes factor in psychological research.

The last 25 years have shown a steady increase in attention for the Bayes factor as a tool for hypothesis evaluation and model selection. The present review highlights the potential of the Bayes factor in psychological research. We discuss six types of applications: Bayesian evaluation of point null, interval, and informative hypotheses, Bayesian evidence synthesis, Bayesian variable selection and model averaging, and Bayesian evaluation of cognitive models. We elaborate what each application entails, give illustrative examples, and provide an overview of key references and software with links to other applications. The article is concluded with a discussion of the opportunities and pitfalls of Bayes factor applications and a sketch of corresponding future research lines. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Prior sensitivity of null hypothesis Bayesian testing.

Researchers increasingly use Bayes factor for hypotheses evaluation. There are two main applications: null hypothesis Bayesian testing (NHBT) and informative hypothesis Bayesian testing (IHBT). As will be shown in this article, NHBT is sensitive to the specification of the scale parameter of the prior distribution, while IHBT is not. As will also be shown in this article, for NHBT using four different Bayes factors, use of the recommended default values for the scaling parameters results in unpredictable operating characteristics, that is, the Bayes factor will usually be biased against or in favor of the null hypothesis. As will furthermore be shown in this article, this problem can be addressed by choosing the scaling parameter such that the Bayes factor is 19 in favor of the null hypothesis over the alternative hypothesis if the observed effect size is equal to zero, because this renders a Bayes factor with clearly specified operating characteristics. However, this does not solve all problems regarding NHBT. The discussion of this article contains elaborations with respect to: the multiverse of Bayes factors; the choice of "19"; Bayes factor calibration outside the context of the univariate normal linear model; and, reporting the results of NHBT. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Sample size determination for Bayesian ANOVAs with informative hypotheses.

Researchers can express their expectations with respect to the group means in an ANOVA model through equality and order constrained hypotheses. This paper introduces the R package SSDbain, which can be used to calculate the sample size required to evaluate (informative) hypotheses using the Approximate Adjusted Fractional Bayes Factor (AAFBF) for one-way ANOVA models as implemented in the R package bain. The sample size is determined such that the probability that the Bayes factor is larger than a threshold value is at least η when either of the hypotheses under consideration is true. The Bayesian ANOVA, Bayesian Welch's ANOVA, and Bayesian robust ANOVA are available. Using the R package SSDbain and/or the tables provided in this paper, researchers in the social and behavioral sciences can easily plan the sample size if they intend to use a Bayesian ANOVA.

Evaluation of inequality constrained hypotheses using a generalization of the AIC.

In the social and behavioral sciences, it is often not interesting to evaluate the null hypothesis by means of a p-value. Researchers are often more interested in quantifying the evidence in the data (as opposed to using p-values) with respect to their own expectations represented by equality and/or inequality constrained hypotheses (as opposed to the null hypothesis). This article proposes an Akaike-type information criterion (AIC; Akaike, 1973, 1974) called the generalized order-restricted information criterion approximation (GORICA) that evaluates (in)equality constrained hypotheses under a very broad range of statistical models. The results of five simulation studies provide empirical evidence showing that the performance of the GORICA on selecting the best hypothesis out of a set of (in)equality constrained hypotheses is convincing. To illustrate the use of the GORICA, the expectations of researchers are investigated in a logistic regression, multilevel regression, and structural equation model. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

One-year follow-up of a parent management training for children with externalizing behaviour problems in the real world.

BACKGROUND: The long-term effectiveness of parent training for children with externalizing behaviour problems under routine care within the German health care system is unclear. We report the 1-year follow-up results of the parent training component of the Prevention Program for Externalizing Problem Behaviour (PEP) for 270 children aged 3-10 years with externalizing behaviour problems. METHOD: Outcome measures included child behaviour problems (externalizing behaviour problems, Attention-Deficit/Hyperactivity Disorder symptoms and Oppositional Defiant Disorder symptoms) and parenting (self efficacy of parenting and perceived ability to solve difficult parenting situations). Data were analysed using multilevel modelling. RESULTS: Comparison of the changes during the 3-month waiting and treatment periods revealed significantly stronger treatment effects on all outcome measures, indicating a substantial decrease in child behaviour problems and a significant increase in parenting due to treatment. At 1-year follow-up, initial treatment effects on child behaviour problems were maintained, while parenting continued to improve. CONCLUSIONS: Families whose children exhibited externalizing problem behaviour profit from PEP and improvements are maintained for at least one year.

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.

Perception of action-outcomes is shaped by life-long and contextual expectations.

The way humans perceive the outcomes of their actions is strongly colored by their expectations. These expectations can develop over different timescales and are not always complementary. The present work examines how long-term (structural) expectations - developed over a lifetime - and short-term (contextual) expectations jointly affect perception. In two studies, including a pre-registered replication, participants initiated the movement of an ambiguously rotating sphere by operating a rotary switch. In the absence of any learning, participants predominantly perceived the sphere to rotate in the same direction as their rotary action. This bias toward structural expectations was abolished (but not reversed) when participants were exposed to incompatible action-effect contingencies (e.g., clockwise actions causing counterclockwise percepts) during a preceding learning phase. Exposure to compatible action-effect contingencies, however, did not add to the existing structural bias. Together, these findings reveal that perception of action-outcomes results from the combined influence of both long-term and immediate expectations.

A tutorial on testing hypotheses using the Bayes factor.

Learning about hypothesis evaluation using the Bayes factor could enhance psychological research. In contrast to null-hypothesis significance testing it renders the evidence in favor of each of the hypotheses under consideration (it can be used to quantify support for the null-hypothesis) instead of a dichotomous reject/do-not-reject decision; it can straightforwardly be used for the evaluation of multiple hypotheses without having to bother about the proper manner to account for multiple testing; and it allows continuous reevaluation of hypotheses after additional data have been collected (Bayesian updating). This tutorial addresses researchers considering to evaluate their hypotheses by means of the Bayes factor. The focus is completely applied and each topic discussed is illustrated using Bayes factors for the evaluation of hypotheses in the context of an ANOVA model, obtained using the R package bain. Readers can execute all the analyses presented while reading this tutorial if they download bain and the R-codes used. It will be elaborated in a completely nontechnical manner: what the Bayes factor is, how it can be obtained, how Bayes factors should be interpreted, and what can be done with Bayes factors. After reading this tutorial and executing the associated code, researchers will be able to use their own data for the evaluation of hypotheses by means of the Bayes factor, not only in the context of ANOVA models, but also in the context of other statistical models. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Computing Bayes factors from data with missing values.

The Bayes factor is increasingly used for the evaluation of hypotheses. These may be traditional hypotheses specified using equality constraints among the parameters of the statistical model of interest or informative hypotheses specified using equality and inequality constraints. Thus far, no attention has been given to the computation of Bayes factors from data with missing values. A key property of such a Bayes factor should be that it is only based on the information in the observed values. This article will show that such a Bayes factor can be obtained using multiple imputations of the missing values. After introduction of the general framework elaborations for Bayes factors based on default or subjective prior distributions and Bayes factors based on priors specified using training data will be given. It will be illustrated that the approach proposed can be applied using R packages for multiple imputation in combination with the Bayes factor packages Bain and BayesFactor. It will furthermore be illustrated that Bayes factors computed using a single imputation of the data are very inaccurate approximations of the correct Bayes factor. (PsycINFO Database Record (c) 2019 APA, all rights reserved).