Search

Many people have flipped coins but few have stopped to ponder the statistical and physical intricacies of the process. In a preregistered study we collected $350{,}757$ coin flips to test the counterintuitive prediction from a physics model of human coin tossing developed by Diaconis, Holmes, and Montgomery (DHM; 2007). The model asserts that when people flip an ordinary coin, it tends to land on the same side it started -- DHM estimated the probability of a same-side outcome to be about 51%. Our data lend strong support to this precise prediction: the coins landed on the same side more often than not, $\text{Pr}(\text{same side}) = 0.508$, 95% credible interval (CI) [$0.506$, $0.509$], $\text{BF}_{\text{same-side bias}} = 2359$. Furthermore, the data revealed considerable between-people variation in the degree of this same-side bias. Our data also confirmed the generic prediction that when people flip an ordinary coin -- with the initial side-up randomly determined -- it is equally likely to land heads or tails: $\text{Pr}(\text{heads}) = 0.500$, 95% CI [$0.498$, $0.502$], $\text{BF}_{\text{heads-tails bias}} = 0.182$. Furthermore, this lack of heads-tails bias does not appear to vary across coins. Additional exploratory analyses revealed that the within-people same-side bias decreased as more coins were flipped, an effect that is consistent with the possibility that practice makes people flip coins in a less wobbly fashion. Our data therefore provide strong evidence that when some (but not all) people flip a fair coin, it tends to land on the same side it started. Our data provide compelling statistical support for the DHM physics model of coin tossing.

After Bayes, the oldest Bayesian account of enumerative induction is given by Laplace's so-called rule of succession: if all $n$ observed instances of a phenomenon to date exhibit a given character, the probability that the next instance of that phenomenon will also exhibit the character is $\frac{n+1}{n+2}$. Laplace's rule however has the apparently counterintuitive mathematical consequence that the corresponding "universal generalization" (every future observation of this type will also exhibit that character) has zero probability. In 1932, the British scientist J. B. S. Haldane proposed an alternative rule giving a universal generalization the positive probability $\frac{n+1}{n+2} \times \frac{n+3}{n+2}$. A year later Harold Jeffreys proposed essentially the same rule in the case of a finite population. A related variant rule results in a predictive probability of $\frac{n+1}{n+2} \times \frac{n+4}{n+3}$. These arguably elegant adjustments of the original Laplacean form have the advantage that they give predictions better aligned with intuition and common sense. In this paper we discuss J. B. S. Haldane's rule and its variants, placing them in their historical context, and relating them to subsequent philosophical discussions.

The delayed and incomplete availability of historical findings and the lack of integrative and user-friendly software hampers the reliable interpretation of new clinical data. We developed a free, open, and user-friendly clinical trial aggregation program combining a large and representative sample of existing trial data with the latest classical and Bayesian meta-analytical models, including clear output visualizations. Our software is of particular interest for (post-graduate) educational programs (e.g., medicine, epidemiology) and global health initiatives. We demonstrate the database, interface, and plot functionality with a recent randomized controlled trial on effective epileptic seizure reduction in children treated for a parasitic brain infection. The single trial data is placed into context and we show how to interpret new results against existing knowledge instantaneously. Our program is of particular interest to those working on the contextualizing of medical findings. It may facilitate the advancement of global clinical progress as efficiently and openly as possible and simulate further bridging clinical data with the latest biostatistical models.

Bayesian model-averaged meta-analysis allows quantification of evidence for both treatment effectiveness $\mu$ and across-study heterogeneity $\tau$. We use the Cochrane Database of Systematic Reviews to develop discipline-wide empirical prior distributions for $\mu$ and $\tau$ for meta-analyses of binary and time-to-event clinical trial outcomes. First, we use 50% of the database to estimate parameters of different required parametric families. Second, we use the remaining 50% of the database to select the best-performing parametric families and explore essential assumptions about the presence or absence of the treatment effectiveness and across-study heterogeneity in real data. We find that most meta-analyses of binary outcomes are more consistent with the absence of the meta-analytic effect or heterogeneity while meta-analyses of time-to-event outcomes are more consistent with the presence of the meta-analytic effect or heterogeneity. Finally, we use the complete database - with close to half a million trial outcomes - to propose specific empirical prior distributions, both for the field in general and for specific medical subdisciplines. An example from acute respiratory infections demonstrates how the proposed prior distributions can be used to conduct a Bayesian model-averaged meta-analysis in the open-source software R and JASP.

Publication selection bias undermines the systematic accumulation of evidence. To assess the extent of this problem, we survey over 68,000 meta-analyses containing over 700,000 effect size estimates from medicine (67,386/597,699), environmental sciences (199/12,707), psychology (605/23,563), and economics (327/91,421). Our results indicate that meta-analyses in economics are the most severely contaminated by publication selection bias, closely followed by meta-analyses in environmental sciences and psychology, whereas meta-analyses in medicine are contaminated the least. After adjusting for publication selection bias, the median probability of the presence of an effect decreased from 99.9% to 29.7% in economics, from 98.9% to 55.7% in psychology, from 99.8% to 70.7% in environmental sciences, and from 38.0% to 29.7% in medicine. The median absolute effect sizes (in terms of standardized mean differences) decreased from d = 0.20 to d = 0.07 in economics, from d = 0.37 to d = 0.26 in psychology, from d = 0.62 to d = 0.43 in environmental sciences, and from d = 0.24 to d = 0.13 in medicine.

The ongoing replication crisis in science has increased interest in the methodology of replication studies. We propose a novel Bayesian analysis approach using power priors: The likelihood of the original study's data is raised to the power of $\alpha$, and then used as the prior distribution in the analysis of the replication data. Posterior distribution and Bayes factor hypothesis tests related to the power parameter $\alpha$ quantify the degree of compatibility between the original and replication study. Inferences for other parameters, such as effect sizes, dynamically borrow information from the original study. The degree of borrowing depends on the conflict between the two studies. The practical value of the approach is illustrated on data from three replication studies, and the connection to hierarchical modeling approaches explored. We generalize the known connection between normal power priors and normal hierarchical models for fixed parameters and show that normal power prior inferences with a beta prior on the power parameter $\alpha$ align with normal hierarchical model inferences using a generalized beta prior on the relative heterogeneity variance $I^2$. The connection illustrates that power prior modeling is unnatural from the perspective of hierarchical modeling since it corresponds to specifying priors on a relative rather than an absolute heterogeneity scale.

We present a novel and easy-to-use method for calibrating error-rate based confidence intervals to evidence-based support intervals. Support intervals are obtained from inverting Bayes factors based on a parameter estimate and its standard error. A $k$ support interval can be interpreted as "the observed data are at least $k$ times more likely under the included parameter values than under a specified alternative". Support intervals depend on the specification of prior distributions for the parameter under the alternative, and we present several types that allow different forms of external knowledge to be encoded. We also show how prior specification can to some extent be avoided by considering a class of prior distributions and then computing so-called minimum support intervals which, for a given class of priors, have a one-to-one mapping with confidence intervals. We also illustrate how the sample size of a future study can be determined based on the concept of support. Finally, we show how the bound for the type I error rate of Bayes factors leads to a bound for the coverage of support intervals. An application to data from a clinical trial illustrates how support intervals can lead to inferences that are both intuitive and informative.

Power priors are used for incorporating historical data in Bayesian analyses by taking the likelihood of the historical data raised to the power $\alpha$ as the prior distribution for the model parameters. The power parameter $\alpha$ is typically unknown and assigned a prior distribution, most commonly a beta distribution. Here, we give a novel theoretical result on the resulting marginal posterior distribution of $\alpha$ in case of the the normal and binomial model. Counterintuitively, when the current data perfectly mirror the historical data and the sample sizes from both data sets become arbitrarily large, the marginal posterior of $\alpha$ does not converge to a point mass at $\alpha = 1$ but approaches a distribution that hardly differs from the prior. The result implies that a complete pooling of historical and current data is impossible if a power prior with beta prior for $\alpha$ is used.

Null hypothesis statistical significance testing (NHST) is the dominant approach for evaluating results from randomized controlled trials. Whereas NHST comes with long-run error rate guarantees, its main inferential tool -- the $p$-value -- is only an indirect measure of evidence against the null hypothesis. The main reason is that the $p$-value is based on the assumption the null hypothesis is true, whereas the likelihood of the data under any alternative hypothesis is ignored. If the goal is to quantify how much evidence the data provide for or against the null hypothesis it is unavoidable that an alternative hypothesis be specified (Goodman & Royall, 1988). Paradoxes arise when researchers interpret $p$-values as evidence. For instance, results that are surprising under the null may be equally surprising under a plausible alternative hypothesis, such that a $p=.045$ result (`reject the null') does not make the null any less plausible than it was before. Hence, $p$-values have been argued to overestimate the evidence against the null hypothesis. Conversely, it can be the case that statistically non-significant results (i.e., $p>.05)$ nevertheless provide some evidence in favor of the alternative hypothesis. It is therefore crucial for researchers to know when statistical significance and evidence collide, and this requires that a direct measure of evidence is computed and presented alongside the traditional $p$-value.

A staple of Bayesian model comparison and hypothesis testing, Bayes factors are often used to quantify the relative predictive performance of two rival hypotheses. The computation of Bayes factors can be challenging, however, and this has contributed to the popularity of convenient approximations such as the BIC. Unfortunately, these approximations can fail in the case of informed prior distributions. Here we address this problem by outlining an approximation to informed Bayes factors for a focal parameter $\theta$. The approximation is computationally simple and requires only the maximum likelihood estimate $\hat\theta$ and its standard error. The approximation uses an estimated likelihood of $\theta$ and assumes that the posterior distribution for $\theta$ is unaffected by the choice of prior distribution for the nuisance parameters. The resulting Bayes factor for the null hypothesis $\mathcal{H}_0: \theta = \theta_0$ versus the alternative hypothesis $\mathcal{H}_1: \theta \sim g(\theta)$ is then easily obtained using the Savage--Dickey density ratio. Three real-data examples highlight the speed and closeness of the approximation compared to bridge sampling and Laplace's method. The proposed approximation facilitates Bayesian reanalyses of standard frequentist results, encourages application of Bayesian tests with informed priors, and alleviates the computational challenges that often frustrate both Bayesian sensitivity analyses and Bayes factor design analyses. The approximation is shown to suffer under small sample sizes and when the posterior distribution of the focal parameter is substantially influenced by the prior distributions on the nuisance parameters. The proposed methodology may also be used to approximate the posterior distribution for $\theta$ under $\mathcal{H}_1$.

The Jeffreys-Lindley paradox exposes a rift between Bayesian and frequentist hypothesis testing that strikes at the heart of statistical inference. Contrary to what most current literature suggests, the paradox was central to the Bayesian testing methodology developed by Sir Harold Jeffreys in the late 1930s. Jeffreys showed that the evidence against a point-null hypothesis $\mathcal{H}_0$ scales with $\sqrt{n}$ and repeatedly argued that it would therefore be mistaken to set a threshold for rejecting $\mathcal{H}_0$ at a constant multiple of the standard error. Here we summarize Jeffreys's early work on the paradox and clarify his reasons for including the $\sqrt{n}$ term. The prior distribution is seen to play a crucial role; by implicitly correcting for selection, small parameter values are identified as relatively surprising under $\mathcal{H}_1$. We highlight the general nature of the paradox by presenting both a fully frequentist and a fully Bayesian version. We also demonstrate that the paradox does not depend on assigning prior mass to a point hypothesis, as is commonly believed., Comment: To be published in Archive for History of Exact Sciences

We outline a Bayesian model-averaged meta-analysis for standardized mean differences in order to quantify evidence for both treatment effectiveness $\delta$ and across-study heterogeneity $\tau$. We construct four competing models by orthogonally combining two present-absent assumptions, one for the treatment effect and one for across-study heterogeneity. To inform the choice of prior distributions for the model parameters, we used 50% of the Cochrane Database of Systematic Reviews to specify rival prior distributions for $\delta$ and $\tau$. The relative predictive performance of the competing models and rival prior distributions was assessed using the remaining 50\% of the Cochrane Database. On average, $\mathcal{H}_1^r$ -- the model that assumes the presence of a treatment effect as well as across-study heterogeneity -- outpredicted the other models, but not by a large margin. Within $\mathcal{H}_1^r$, predictive adequacy was relatively constant across the rival prior distributions. We propose specific empirical prior distributions, both for the field in general and for each of 46 specific medical subdisciplines. An example from oral health demonstrates how the proposed prior distributions can be used to conduct a Bayesian model-averaged meta-analysis in the open-source software R and JASP. The preregistered analysis plan is available at https://osf.io/zs3df/.

Testing the equality of two proportions is a common procedure in science, especially in medicine and public health. In these domains it is crucial to be able to quantify evidence for the absence of a treatment effect. Bayesian hypothesis testing by means of the Bayes factor provides one avenue to do so, requiring the specification of prior distributions for parameters. The most popular analysis approach views the comparison of proportions from a contingency table perspective, assigning prior distributions directly to the two proportions. Another, less popular approach views the problem from a logistic regression perspective, assigning prior distributions to logit-transformed parameters. Reanalyzing 39 null results from the New England Journal of Medicine with both approaches, we find that they can lead to markedly different conclusions, especially when the observed proportions are at the extremes (i.e., very low or very high). We explain these stark differences and provide recommendations for researchers interested in testing the equality of two proportions and users of Bayes factors more generally. The test that assigns prior distributions to logit-transformed parameters creates prior dependence between the two proportions and yields weaker evidence when the observations are at the extremes. When comparing two proportions, we argue that this test should become the new default., Comment: 16 pages, 7 figures

A perennial objection against Bayes factor point-null hypothesis tests is that the point-null hypothesis is known to be false from the outset. We examine the consequences of approximating the sharp point-null hypothesis by a hazy `peri-null' hypothesis instantiated as a narrow prior distribution centered on the point of interest. The peri-null Bayes factor then equals the point-null Bayes factor multiplied by a correction term which is itself a Bayes factor. For moderate sample sizes, the correction term is relatively inconsequential; however, for large sample sizes the correction term becomes influential and causes the peri-null Bayes factor to be inconsistent and approach a limit that depends on the ratio of prior ordinates evaluated at the maximum likelihood estimate. We characterize the asymptotic behavior of the peri-null Bayes factor and briefly discuss suggestions on how to construct peri-null Bayes factor hypothesis tests that are also consistent., Comment: Accepted for publication in Test

Testing the (in)equality of variances is an important problem in many statistical applications. We develop default Bayes factor tests to assess the (in)equality of two or more population variances, as well as a test for whether the population variances equal a specific value. The resulting test can be used to check assumptions for commonly used procedures such as the $t$-test or ANOVA, or test substantive hypotheses concerning variances directly. We show that our Bayes factor fulfills a number of desiderata. Researchers may have directed hypotheses such as $\sigma_{1}^{2} > \sigma_{2}^{2}$, they may want to extend $\mathcal{H}_{0}$ to have a null-region, or wish to combine hypotheses about equality with hypotheses about inequality, for example $\sigma_{1}^{2} = \sigma_{2}^{2} > (\sigma_{3}^{2}, \sigma_{4}^{2})$. We extend our Bayes factor test to allow for these deviations from our proposed default and illustrate it on a number of practical examples. Our procedure is implemented in the R package $bfvartest$.

There has been a tremendous methodological development of Bayes factors for hypothesis testing in the social and behavioral sciences, and related fields. This development is due to the flexibility of the Bayes factor for testing multiple hypotheses simultaneously, the ability to test complex hypotheses involving equality as well as order constraints on the parameters of interest, and the interpretability of the outcome as the weight of evidence provided by the data in support of competing scientific theories. The available software tools for Bayesian hypothesis testing are still limited however. In this paper we present a new R-package called BFpack that contains functions for Bayes factor hypothesis testing for the many common testing problems. The software includes novel tools (i) for Bayesian exploratory testing (null vs positive vs negative effects), (ii) for Bayesian confirmatory testing (competing hypotheses with equality and/or order constraints), (iii) for common statistical analyses, such as linear regression, generalized linear models, (multivariate) analysis of (co)variance, correlation analysis, and random intercept models, (iv) using default priors, and (v) while allowing data to contain missing observations that are missing at random., Comment: 52 pages, 4 tables

Booming in business and a staple analysis in medical trials, the A/B test assesses the effect of an intervention or treatment by comparing its success rate with that of a control condition. Across many practical applications, it is desirable that (1) evidence can be obtained in favor of the null hypothesis that the treatment is ineffective; (2) evidence can be monitored as the data accumulate; (3) expert prior knowledge can be taken into account. Most existing approaches do not fulfill these desiderata. Here we describe a Bayesian A/B procedure based on Kass and Vaidyanathan (1992) that allows one to monitor the evidence for the hypotheses that the treatment has either a positive effect, a negative effect, or, crucially, no effect. Furthermore, this approach enables one to incorporate expert knowledge about the relative prior plausibility of the rival hypotheses and about the expected size of the effect, given that it is non-zero. To facilitate the wider adoption of this Bayesian procedure we developed the abtest package in R. We illustrate the package options and the associated statistical results with a fictitious business example and a real data medical example.

Current developments in the statistics community suggest that modern statistics education should be structured holistically, that is, by allowing students to work with real data and to answer concrete statistical questions, but also by educating them about alternative frameworks, such as Bayesian inference. In this article, we describe how we incorporated such a holistic structure in a Bayesian research project on ordered binomial probabilities. The project was conducted with a group of three undergraduate psychology students who had basic knowledge of Bayesian statistics and programming, but lacked formal mathematical training. The research project aimed to (1) convey the basic mathematical concepts of Bayesian inference; (2) have students experience the entire empirical cycle including collection, analysis, and interpretation of data and (3) teach students open science practices.

Bayesian inference for rank-order problems is frustrated by the absence of an explicit likelihood function. This hurdle can be overcome by assuming a latent normal representation that is consistent with the ordinal information in the data: the observed ranks are conceptualized as an impoverished reflection of an underlying continuous scale, and inference concerns the parameters that govern the latent representation. We apply this generic data-augmentation method to obtain Bayes factors for three popular rank-based tests: the rank sum test, the signed rank test, and Spearman's $\rho_s$., Comment: 33 pages, 12 figures

Statistical procedures such as Bayes factor model selection and Bayesian model averaging require the computation of normalizing constants (e.g., marginal likelihoods). These normalizing constants are notoriously difficult to obtain, as they usually involve high-dimensional integrals that cannot be solved analytically. Here we introduce an R package that uses bridge sampling (Meng & Wong, 1996; Meng & Schilling, 2002) to estimate normalizing constants in a generic and easy-to-use fashion. For models implemented in Stan, the estimation procedure is automatic. We illustrate the functionality of the package with three examples.