Conceptually plausible Bayesian inference in interval timing

In a world that is uncertain and noisy, perception makes use of optimization procedures that rely on the statistical properties of previous experiences. A well-known example of this phenomenon is the central tendency effect observed in many psychophysical modalities. For example, in interval timing tasks, previous experiences influence the current percept, pulling behavioural responses towards the mean. In Bayesian observer models, these previous experiences are typically modelled by unimodal statistical distributions, referred to as the prior. Here, we critically assess the validity of the assumptions underlying these models and propose a model that allows for more flexible, yet conceptually more plausible, modelling of empirical distributions. By representing previous experiences as a mixture of lognormal distributions, this model can be parametrized to mimic different unimodal distributions and thus extends previous instantiations of Bayesian observer models. We fit the mixture lognormal model to published interval timing data of healthy young adults and a clinical population of aged mild cognitive impairment patients and age-matched controls, and demonstrate that this model better explains behavioural data and provides new insights into the mechanisms that underlie the behaviour of a memory-affected clinical population.

Systematic Parameter Reviews in Cognitive Modeling: Towards a Robust and Cumulative Characterization of Psychological Processes in the Diffusion Decision Model

Parametric cognitive models are increasingly popular tools for analyzing data obtained from psychological experiments. One of the main goals of such models is to formalize psychological theories using parameters that represent distinct psychological processes. We argue that systematic quantitative reviews of parameter estimates can make an important contribution to robust and cumulative cognitive modeling. Parameter reviews can benefit model development and model assessment by providing valuable information about the expected parameter space, and can facilitate the more efficient design of experiments. Importantly, parameter reviews provide crucial-if not indispensable-information for the specification of informative prior distributions in Bayesian cognitive modeling. From the Bayesian perspective, prior distributions are an integral part of a model, reflecting cumulative theoretical knowledge about plausible values of the model's parameters (Lee, 2018). In this paper we illustrate how systematic parameter reviews can be implemented to generate informed prior distributions for the Diffusion Decision Model (DDM; Ratcliff and McKoon, 2008), the most widely used model of speeded decision making. We surveyed the published literature on empirical applications of the DDM, extracted the reported parameter estimates, and synthesized this information in the form of prior distributions. Our parameter review establishes a comprehensive reference resource for plausible DDM parameter values in various experimental paradigms that can guide future applications of the model. Based on the challenges we faced during the parameter review, we formulate a set of general and DDM-specific suggestions aiming to increase reproducibility and the information gained from the review process.

The Importance of Prior Sensitivity Analysis in Bayesian Statistics: Demonstrations Using an Interactive Shiny App

The current paper highlights a new, interactive Shiny App that can be used to aid in understanding and teaching the important task of conducting a prior sensitivity analysis when implementing Bayesian estimation methods. In this paper, we discuss the importance of examining prior distributions through a sensitivity analysis. We argue that conducting a prior sensitivity analysis is equally important when so-called diffuse priors are implemented as it is with subjective priors. As a proof of concept, we conducted a small simulation study, which illustrates the impact of priors on final model estimates. The findings from the simulation study highlight the importance of conducting a sensitivity analysis of priors. This concept is further extended through an interactive Shiny App that we developed. The Shiny App allows users to explore the impact of various forms of priors using empirical data. We introduce this Shiny App and thoroughly detail an example using a simple multiple regression model that users at all levels can understand. In this paper, we highlight how to determine the different settings for a prior sensitivity analysis, how to visually and statistically compare results obtained in the sensitivity analysis, and how to display findings and write up disparate results obtained across the sensitivity analysis. The goal is that novice users can follow the process outlined here and work within the interactive Shiny App to gain a deeper understanding of the role of prior distributions and the importance of a sensitivity analysis when implementing Bayesian methods. The intended audience is broad (e.g., undergraduate or graduate students, faculty, and other researchers) and can include those with limited exposure to Bayesian methods or the specific model presented here.

Weyl Prior and Bayesian Statistics

When using Bayesian inference, one needs to choose a prior distribution for parameters. The well-known Jeffreys prior is based on the Riemann metric tensor on a statistical manifold. Takeuchi and Amari defined the α -parallel prior, which generalized the Jeffreys prior by exploiting a higher-order geometric object, known as a Chentsov-Amari tensor. In this paper, we propose a new prior based on the Weyl structure on a statistical manifold. It turns out that our prior is a special case of the α -parallel prior with the parameter α equaling - n , where n is the dimension of the underlying statistical manifold and the minus sign is a result of conventions used in the definition of α -connections. This makes the choice for the parameter α more canonical. We calculated the Weyl prior for univariate Gaussian and multivariate Gaussian distribution. The Weyl prior of the univariate Gaussian turns out to be the uniform prior.

Quantifying posterior effect size distribution of susceptibility loci by common summary statistics

Testing millions of single nucleotide polymorphisms (SNPs) in genetic association studies has become a standard routine for disease gene discovery. In light of recent re-evaluation of statistical practice, it has been suggested that p-values are unfit as summaries of statistical evidence. Despite this criticism, p-values contain information that can be utilized to address the concerns about their flaws. We present a new method for utilizing evidence summarized by p-values for estimating odds ratio (OR) based on its approximate posterior distribution. In our method, only p-values, sample size, and standard deviation for ln(OR) are needed as summaries of data, accompanied by a suitable prior distribution for ln(OR) that can assume any shape. The parameter of interest, ln(OR), is the only parameter with a specified prior distribution, hence our model is a mix of classical and Bayesian approaches. We show that our method retains the main advantages of the Bayesian approach: it yields direct probability statements about hypotheses for OR and is resistant to biases caused by selection of top-scoring SNPs. Our method enjoys greater flexibility than similarly inspired methods in the assumed distribution for the summary statistic and in the form of the prior for the parameter of interest. We illustrate our method by presenting interval estimates of effect size for reported genetic associations with lung cancer. Although we focus on OR, the method is not limited to this particular measure of effect size and can be used broadly for assessing reliability of findings in studies testing multiple predictors.

Incorporating external evidence on between-trial heterogeneity in network meta-analysis

In a network meta‐analysis, between‐study heterogeneity variances are often very imprecisely estimated because data are sparse, so standard errors of treatment differences can be highly unstable. External evidence can provide informative prior distributions for heterogeneity and, hence, improve inferences. We explore approaches for specifying informative priors for multiple heterogeneity variances in a network meta‐analysis.

First, we assume equal heterogeneity variances across all pairwise intervention comparisons (approach 1); incorporating an informative prior for the common variance is then straightforward. Models allowing unequal heterogeneity variances are more realistic; however, care must be taken to ensure implied variance‐covariance matrices remain valid. We consider three strategies for specifying informative priors for multiple unequal heterogeneity variances. Initially, we choose different informative priors according to intervention comparison type and assume heterogeneity to be proportional across comparison types and equal within comparison type (approach 2). Next, we allow all heterogeneity variances in the network to differ, while specifying a common informative prior for each. We explore two different approaches to this: placing priors on variances and correlations separately (approach 3) or using an informative inverse Wishart distribution (approach 4).

Our methods are exemplified through application to two network metaanalyses. Appropriate informative priors are obtained from previously published evidence‐based distributions for heterogeneity.

Relevant prior information on between‐study heterogeneity can be incorporated into network meta‐analyses, without needing to assume equal heterogeneity across treatment comparisons. The approaches proposed will be beneficial in sparse data sets and provide more appropriate intervals for treatment differences than those based on imprecise heterogeneity estimates.

Use of a random effects meta-analysis in the design and analysis of a new clinical trial

In designing a randomized controlled trial, it has been argued that trialists should consider existing evidence about the likely intervention effect. One approach is to form a prior distribution for the intervention effect based on a meta‐analysis of previous studies and then power the trial on its ability to affect the posterior distribution in a Bayesian analysis. Alternatively, methods have been proposed to calculate the power of the trial to influence the “pooled” estimate in an updated meta‐analysis. These two approaches can give very different results if the existing evidence is heterogeneous, summarised using a random effects meta‐analysis. We argue that the random effects mean will rarely represent the trialist's target parameter, and so, it will rarely be appropriate to power a trial based on its impact upon the random effects mean. Furthermore, the random effects mean will not generally provide an appropriate prior distribution. More appropriate alternatives include the predictive distribution and shrinkage estimate for the most similar study. Consideration of the impact of the trial on the entire random effects distribution might sometimes be appropriate. We describe how beliefs about likely sources of heterogeneity have implications for how the previous evidence should be used and can have a profound impact on the expected power of the new trial. We conclude that the likely causes of heterogeneity among existing studies need careful consideration. In the absence of explanations for heterogeneity, we suggest using the predictive distribution from the meta‐analysis as the basis for a prior distribution for the intervention effect.

An introduction to Bayesian statistics in health psychology

The aim of the current article is to provide a brief introduction to Bayesian statistics within the field of health psychology. Bayesian methods are increasing in prevalence in applied fields, and they have been shown in simulation research to improve the estimation accuracy of structural equation models, latent growth curve (and mixture) models, and hierarchical linear models. Likewise, Bayesian methods can be used with small sample sizes since they do not rely on large sample theory. In this article, we discuss several important components of Bayesian statistics as they relate to health-based inquiries. We discuss the incorporation and impact of prior knowledge into the estimation process and the different components of the analysis that should be reported in an article. We present an example implementing Bayesian estimation in the context of blood pressure changes after participants experienced an acute stressor. We conclude with final thoughts on the implementation of Bayesian statistics in health psychology, including suggestions for reviewing Bayesian manuscripts and grant proposals. We have also included an extensive amount of online supplementary material to complement the content presented here, including Bayesian examples using many different software programmes and an extensive sensitivity analysis examining the impact of priors.

Bayesian bivariate meta-analysis of diagnostic test studies with interpretable priors

In a bivariate meta-analysis, the number of diagnostic studies involved is often very low so that frequentist methods may result in problems. Using Bayesian inference is particularly attractive as informative priors that add a small amount of information can stabilise the analysis without overwhelming the data. However, Bayesian analysis is often computationally demanding and the selection of the prior for the covariance matrix of the bivariate structure is crucial with little data. The integrated nested Laplace approximations method provides an efficient solution to the computational issues by avoiding any sampling, but the important question of priors remain. We explore the penalised complexity (PC) prior framework for specifying informative priors for the variance parameters and the correlation parameter. PC priors facilitate model interpretation and hyperparameter specification as expert knowledge can be incorporated intuitively. We conduct a simulation study to compare the properties and behaviour of differently defined PC priors to currently used priors in the field. The simulation study shows that the PC prior seems beneficial for the variance parameters. The use of PC priors for the correlation parameter results in more precise estimates when specified in a sensible neighbourhood around the truth. To investigate the usage of PC priors in practice, we reanalyse a meta-analysis using the telomerase marker for the diagnosis of bladder cancer and compare the results with those obtained by other commonly used modelling approaches. Copyright © 2017 John Wiley & Sons, Ltd.

Bayesian bivariate meta-analysis of correlated effects: Impact of the prior distributions on the between-study correlation, borrowing of strength, and joint inferences

Multivariate random-effects meta-analysis allows the joint synthesis of correlated results from multiple studies, for example, for multiple outcomes or multiple treatment groups. In a Bayesian univariate meta-analysis of one endpoint, the importance of specifying a sensible prior distribution for the between-study variance is well understood. However, in multivariate meta-analysis, there is little guidance about the choice of prior distributions for the variances or, crucially, the between-study correlation, ρ_B; for the latter, researchers often use a Uniform(−1,1) distribution assuming it is vague. In this paper, an extensive simulation study and a real illustrative example is used to examine the impact of various (realistically) vague prior distributions for ρ_B and the between-study variances within a Bayesian bivariate random-effects meta-analysis of two correlated treatment effects. A range of diverse scenarios are considered, including complete and missing data, to examine the impact of the prior distributions on posterior results (for treatment effect and between-study correlation), amount of borrowing of strength, and joint predictive distributions of treatment effectiveness in new studies. Two key recommendations are identified to improve the robustness of multivariate meta-analysis results. First, the routine use of a Uniform(−1,1) prior distribution for ρ_B should be avoided, if possible, as it is not necessarily vague. Instead, researchers should identify a sensible prior distribution, for example, by restricting values to be positive or negative as indicated by prior knowledge. Second, it remains critical to use sensible (e.g. empirically based) prior distributions for the between-study variances, as an inappropriate choice can adversely impact the posterior distribution for ρ_B, which may then adversely affect inferences such as joint predictive probabilities. These recommendations are especially important with a small number of studies and missing data.

The choice of prior distribution for a covariance matrix in multivariate meta-analysis: a simulation study

Bayesian meta-analysis is an increasingly important component of clinical research, with multivariate meta-analysis a promising tool for studies with multiple endpoints. Model assumptions, including the choice of priors, are crucial aspects of multivariate Bayesian meta-analysis (MBMA) models. In a given model, two different prior distributions can lead to different inferences about a particular parameter. A simulation study was performed in which the impact of families of prior distributions for the covariance matrix of a multivariate normal random effects MBMA model was analyzed. Inferences about effect sizes were not particularly sensitive to prior choice, but the related covariance estimates were. A few families of prior distributions with small relative biases, tight mean squared errors, and close to nominal coverage for the effect size estimates were identified. Our results demonstrate the need for sensitivity analysis and suggest some guidelines for choosing prior distributions in this class of problems. The MBMA models proposed here are illustrated in a small meta-analysis example from the periodontal field and a medium meta-analysis from the study of stroke. Copyright © 2015 John Wiley & Sons, Ltd.

Predictive distributions for between-study heterogeneity and simple methods for their application in Bayesian meta-analysis

Numerous meta-analyses in healthcare research combine results from only a small number of studies, for which the variance representing between-study heterogeneity is estimated imprecisely. A Bayesian approach to estimation allows external evidence on the expected magnitude of heterogeneity to be incorporated.

The aim of this paper is to provide tools that improve the accessibility of Bayesian meta-analysis. We present two methods for implementing Bayesian meta-analysis, using numerical integration and importance sampling techniques. Based on 14 886 binary outcome meta-analyses in the Cochrane Database of Systematic Reviews, we derive a novel set of predictive distributions for the degree of heterogeneity expected in 80 settings depending on the outcomes assessed and comparisons made. These can be used as prior distributions for heterogeneity in future meta-analyses.

The two methods are implemented in R, for which code is provided. Both methods produce equivalent results to standard but more complex Markov chain Monte Carlo approaches. The priors are derived as log-normal distributions for the between-study variance, applicable to meta-analyses of binary outcomes on the log odds-ratio scale. The methods are applied to two example meta-analyses, incorporating the relevant predictive distributions as prior distributions for between-study heterogeneity.

We have provided resources to facilitate Bayesian meta-analysis, in a form accessible to applied researchers, which allow relevant prior information on the degree of heterogeneity to be incorporated. © 2014 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.