Kelter, Riko
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Publication Access status: Open Access , A toolbox to demystify probabilistic and statistical paradoxes(2023); ; There is a variety of empirical evidence that the coverage of paradoxes in mathematics education helps to support thorough understanding of probabilistic and statistical concepts. However, existing literature often focuses on extensive analysis of a specific paradox, provides new perspectives or an analysis from a different angle. Often neglected aspects in this context are common features between different paradoxes and the fact, that the same situation might look paradoxical to different people for entirely different reasons. We develop a toolbox to demystify paradoxes in probability and statistics. Therefore, we first analyze in which steps of stochastic modeling one might be faced with a paradoxical situation. Secondly, we build on a representative selection of well-known paradoxes and isolate the techniques and methods which help to explain why people find the paradox surprising, identify the class of scenarios where the paradox may occur and make a choice between the seemingly contradictory conclusions. Thirdly, we present the toolbox, which helps to demystify various paradoxical situations. This helps teachers to chose appropriate problems and students to find the right method to resolve these problems. While the developed toolbox is not exhaustive, it helps to dissect the anatomy of probabilistic and statistical paradoxes.Source Type:Article1 3 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Open Access , Bayesian identification of structural coefficients in causal models and the causal false-positive risk of confounders and colliders in linear Markovian models(2022)Background: Causal inference has seen an increasing popularity in medical research. Estimation of causal effects from observational data allows to draw conclusions from data when randomized controlled trials cannot be conducted. Although the identification of structural causal models (SCM) and the calculation of structural coefficients has received much attention, a key requirement for valid causal inference is that conclusions are drawn based on the true data-generating model. Methods: It remains widely unknown how large the probability is to reject the true structural causal model when observational data from it is sampled. The latter probability – the causal false-positive risk – is crucial, as rejection of the true causal model can induce bias in the estimation of causal effects. In this paper, the widely used causal models of confounders and colliders are studied regarding their causal false-positive risk in linear Markovianmodels. A simulation study is carried out which investigates the causal false-positive risk in Gaussian linear Markovian models. Therefore, the testable implications of the DAG corresponding to confounders and colliders are analyzed from a Bayesian perspective. Furthermore, the induced bias in estimating the structural coefficients and causal effects is studied. Results: Results show that the false-positive risk of rejecting a true SCM of even simple building blocks like confounders and colliders is substantial. Importantly, estimation of average, direct and indirect causal effects can become strongly biased if a true model is rejected. The causal false-positive risk may thus serve as an indicator or proxy for the induced bias. Conclusion: While the identification of structural coefficients and testable implications of causal models have been studied rigorously in the literature, this paper shows that causal inference also must develop new concepts for controlling the causal false-positive risk. Although a high risk cannot be equated with a substantial bias, it is indicative of the induced bias. The latter fact calls for the development of more advanced risk measures for committing a causal type I error in causal inference.Source Type:Article1 2 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Open Access , bayest: an R-package for effect-size targeted Bayesian two-sample t-tests(2020)Typical situations in research include the comparison of two groups regarding a metric variable, in which case usually the two-sample t-test is applied. While common frequentist two-sample t-tests focus on the difference of means of both groups via a p-value, the quantity of interest in applied research most often is the effect size. Existing Bayesian alternatives of the two-sample t-test replace frequentist significance thresholds like the p-value with the Bayes factor, taking the same testing stance. The R package bayest implements a Markov-Chain-Monte-Carlo algorithm to conduct a Bayesian two-sample t-test which estimates the effect size between two groups, while also providing detailed visualization and analysis of all parameters of interest. Because of its focus on the ease of use and interpretability, clinicians and other users can run this t-test within a few lines of code and find out if differences between two groups are scientifically meaningful, instead of significant.Source Type:Article534 181 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Open Access , New perspectives on statistical data analysis: challenges and possibilities of digitalization for hypothesis testing in quantitative research(2020)p-values, the 'gold standard' of statistical validity are not as reliable as many scientists assume. In the last decade, severe problems have been observed regarding the validity of highly reputable research. Additionally, the growing availability of big data challenges the design and statistical analysis of studies and experiments across science. Therefore, it is more important than ever to make the best use of available computational tools, software and possibilities digitalization offers to improve the validity of research results. In this paper, we focus on an essential procedure often carried out in quantitative research, which is directly related to the experienced problems: Statistical hypothesis testing. First, we show that the traditional way of hypothesis testing has severe logical problems. Second, it is shown that due to the increasing availability of computational resources, highly sophisticated methods from the area of computational statistics - namely Bayesian data analysis - can complement and even replace traditional hypothesis testing. Third, we highlight how digitalization helps in making these technologies available to a vast range of researchers in the form of the novel and free software package JASP. Together, this paper shows that considering a change in perspective on statistical data analysis, in particular on hypothesis testing, provides the possibility to improve the transparency and reliability of research in the medical, social and natural sciences.Source Type:InProceedings1 8 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Metadata only , Reducing the false discovery rate of preclinical animal research with Bayesian statistical decision criteria(2023)The success of preclinical research hinges on exploratory and confirmatory animal studies. Traditional null hypothesis significance testing is a common approach to eliminate the chaff from a collection of drugs, so that only the most promising treatments are funneled through to clinical research phases. Balancing the number of false discoveries and false omissions is an important aspect to consider during this process. In this paper, we compare several preclinical research pipelines, either based on null hypothesis significance testing or based on Bayesian statistical decision criteria. We build on a recently published large-scale meta-analysis of reported effect sizes in preclinical animal research and elicit a non-informative prior distribution under which both approaches are compared. After correcting for publication bias and shrinkage of effect sizes in replication studies, simulations show that (i) a shift towards statistical approaches which explicitly incorporate the minimum clinically important difference reduces the false discovery rate of frequentist approaches and (ii) a shift towards Bayesian statistical decision criteria can improve the reliability of preclinical animal research by reducing the number of false-positive findings. It is shown that these benefits hold while keeping the number of experimental units low which are required for a confirmatory follow-up study. Results show that Bayesian statistical decision criteria can help in improving the reliability of preclinical animal research and should be considered more frequently in practice.Source Type:Article1 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Metadata only , The Bayesian Group-Sequential Predictive Evidence Value Design for Phase II Clinical Trials with Binary Endpoints(2024); In clinical research, the initial efficacy of a new agent is typically assessed in a phase IIA study. Bayesian group-sequential designs are often based on predictive probability of trial success. In this paper, the novel Bayesian group-sequential predictive evidence value design is introduced, and we prove that the predictive probability approach is a special case of it. A comparison with Simon’s two-stage and competing Bayesian designs based on phase IIA cancer trials is provided. Results show that the novel design can improve operating characteristics such as the false-positive rate, probability of early stopping for futility and expected sample size of the trial. Given these advantages, the predictive evidence value design constitutes an important addition to the biostatistician’s toolbelt when planning a phase IIA trial the Bayesian way, in particular, when small sample sizes and a large probability for early termination under the null hypothesis are desired.Source Type:Article1 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Metadata only , The Bayesian simulation study (BASIS) framework for simulation studies in statistical and methodological research(2023)Statistical simulation studies are becoming increasingly popular to demonstrate the performance or superiority of new computational procedures and algorithms. Despite this status quo, previous surveys of the literature have shown that the reporting of statistical simulation studies often lacks relevant information and structure. The latter applies in particular to Bayesian simulation studies, and in this paper the Bayesian simulation study framework (BASIS) is presented as a step towards improving the situation. The BASIS framework provides a structured skeleton for planning, coding, executing, analyzing, and reporting Bayesian simulation studies in biometrical research and computational statistics. It encompasses various features of previous proposals and recommendations in the methodological literature and aims to promote neutral comparison studies in statistical research. Computational aspects covered in the BASIS include algorithmic choices, Markov–chain‐Monte‐Carlo convergence diagnostics, sensitivity analyses, and Monte Carlo standard error calculations for Bayesian simulation studies. Although the BASIS framework focuses primarily on methodological research, it also provides useful guidance for researchers who rely on the results of Bayesian simulation studies or analyses, as current state‐of‐the‐art guidelines for Bayesian analyses are incorporated into the BASIS.Source Type:Article1 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Metadata only , The Calibrated Bayesian Hypothesis Test for Directional Hypotheses of the Odds Ratio in $$2\times 2$$ Contingency Tables(2024)The $$\chi ^{2}$$ test is among the most widely used statistical hypothesis tests in medical research. Often, the statistical analysis deals with the test of row-column independence in a $$2\times 2$$ contingency table, and the statistical parameter of interest is the odds ratio. A novel Bayesian analogue to the frequentist $$\chi ^{2}$$ test is introduced. The test is based on a Dirichlet-multinomial model under a joint sampling scheme and works with balanced and unbalanced randomization. The test focusses on the quantity of interest in a variety of medical research, the odds ratio in a $$2\times 2$$ contingency table. A computational implementation of the test is developed and R code is provided to apply the test. To meet the demands of regulatory agencies, a calibration of the Bayesian test is introduced which allows to calibrate the false-positive rate and power. The latter provides a Bayes-frequentist compromise which ensures control over the long-term error rates of the test. Illustrative examples using clinical trial data and simulations show how to use the test in practice. In contrast to existing Bayesian tests for $$2\times 2$$ tables, calibration of the acceptance threshold for the hypothesis of interest allows to achieve a bound on the false-positive rate and minimum power for a prespecified odds ratio of interest. The novel Bayesian test provides an attractive choice for Bayesian biostatisticians who face the demands of regulatory agencies which usually require formal control over false-positive errors and power under the alternative. As such, it constitutes an easy-to-apply addition to the arsenal of already existing Bayesian tests.Source Type:Article3 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Metadata only , The Case of the Jeffreys-Lindley-paradox as a Bayes-frequentist Compromise: A Perspective Based on the Rao-Lovric-Theorem(2023)Testing a precise hypothesis can lead to substantially different results in the frequentist and Bayesian approach, a situation which is highlighted by the Jeffreys-Lindley paradox. While there exist various explanations why the paradox occurs, this Article extends prior work by placing the less well-studied point-null-zero-probability paradox at the center of the analysis. The relationship between the two paradoxes is analyzed based on accepting or rejecting the existence of precise hypotheses. The perspective provided in this paper aims at demonstrating how the Bayesian and frequentist solutions can be reconciled when paying attention to the assumption of the point-null-zero-probability paradox. As a result, the Jeffreys-Lindley-paradox can be reinterpreted as a Bayes-frequentist compromise. The resolution shows that divergences between Bayesian and frequentist modes of inference stem from (a) accepting the existence of a precise hypothesis or not, (b) the assignment of positive measure to a null set and (c) the use of unstandardized p-values or p-values standardized to tail-area probabilities.Source Type:Article1 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Open Access , The evolution of statistical hypothesis testing(2021)Statistical hypothesis testing is a central method for the judgement of empirical findings in the medical, social and natural sciences. In recent years, the ongoing problems with null hypothesis significance testing and p-values have shown that the underlying paradigm for quantifying statistical evidence about a research hypothesis is highly problematic, and the situation has been termed a replication crisis. In this thesis, the evolution of statistical hypothesis testing is reconstructed, and it is shown that various of the recently observed problems with the reproducibility of research can be attributed to the underlying statistical theory of widely used inferential statistical methods. In the first part, the development of an inconsistent hybrid approach to statistical hypothesis testing which emerged out of Fisher’s theory of significance tests, p-values and the Neyman-Pearson theory is analyzed. In part two, the evolution of Bayesian approaches to hypothesis testing is detailed with a focus on the Bayes factor. Part three discusses the development of modern Markov-Chain-Monte-Carlo algorithms and their impact on Bayesian hypothesis testing. Part four then provides an axiomatic analysis of the concept of statistical evidence in the context of statistical hypothesis testing and it is shown that various substantial problems which were observed in the replication crisis can be attributed to purely axiomatic inconsistencies and conflicts with the likelihood principle. Based on the axiomatic analysis, it is shown that robust Bayesian methods, in particular robust Bayesian hypothesis tests provide a solution to some substantial problems with the reproducibility of research. Bayesian statistical solutions to the replication crisis are provided in the fifth part with a focus on widely used Bayesian statistical models in the biomedical sciences. New results demonstrate that the implicit error control of Bayesian hypothesis tests is comparable to frequentist tests based on p-values, and that a variety of Bayesian evidence measures attains reasonable type I error control and power in practice. Also, a shift towards the Hodges-Lehmann paradigm which advocates testing small interval instead of point null hypotheses is explored, and new theoretical results show that such a shift may be an appealing additional step towards increasing the reproducibility of science which has not received enough attention in the discussion about the validity of statistical hypotheses and the reproducibility of scientific research.Source Type:Doctoral Thesis2 7
