Kelter, Riko
4 results
Now showing 1 - 4 of 4
- Some of the metrics are blocked by yourconsent settings
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 , 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
