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The internal validity of a causal inference made based on an observational study is often subject to debate. The potential outcomes framework of causal inference stipulates that causal inference is essentially a missing data problem, and we follow this spirit to define the ideal sample as the combination of the observed data and the missing/counterfactual data for regression models. The robustness of a causal inference can be quantified by the probability of a robust inference for internal validity in regression, i.e., the PIVR, which is the probability of rejecting the null hypothesis again for the ideal sample provided the same null hypothesis has been already rejected for the observed sample. Drawing on the relationship between the PIVR and the mean counterfactual outcomes, we formalize a conceptual framework of quantifying the robustness of a regression-based causal inference based on a joint distribution about the mean counterfactual outcomes, holding the observed sample fixed. Interpretatively, the PIVR is the statistical power of the null hypothesis significance testing that is thought to be built on the ideal sample. We demonstrate the conceptual framework of quantifying the robustness of a regression-based causal inference with an empirical example. [ABSTRACT FROM AUTHOR]

Amid the growing emphasis on the environmental and health benefits of walking, enhanced network connectivity emerges as a significant determinant in promoting pedestrian activity, as proven by scholars statistically and theoretically. Recent academic endeavors have introduced metrics such as pedestrian route directness (PRD) to measure connectivity, which maps the directness of routes between critical origins and destinations of the urban layout. However, there have been concerns from scholars about the limitations of PRD in theory, especially its sensitivity to larger radii of analysis. Addressing this theoretical inconsistency, this paper employs Pearson's correlation and linear regression analysis to explore the correlation between the radius of analysis and variance of results, the significance of this correlation for subsequent research, and the geographical context's influence on metric selection. The findings have revealed an r-value of −0.82, suggesting a strong and negative correlation. Moreover, the p-value of 0.0003 demonstrates the significance of the correlation and the rejection of the null hypothesis. These results bridge the gap between theoretical discussions and empirical analysis, revealing that, as the analysis radius expands, the sensitivity of results diminishes. The findings of this study hold significant implications for policy development and regulation of PRD, offering crucial insights that particularly advance the field of street connectivity. [ABSTRACT FROM AUTHOR]

Hydraulic fracturing (HF) can trigger induced seismicity, but documented occurrences tend to be localized compared with the regional extent of industry operations. Factors that determine intrinsic geological susceptibility of a given region to induced seismicity remain incompletely understood. To address this uncertainty, we have developed a stochastic modeling approach to enable statistical testing of hypotheses regarding the distribution of induced seismicity. For reference, we adopted a null hypothesis that HF-induced seismic events are randomly associated with HF wells. Realizations of synthetic induced-seismicity catalogs are generated based on the Gutenberg–Richter relationship for magnitudes and explicit assumed spatial relationship(s) between HF wells and other known features, such as mapped structural corridors. Uncertainties in observed event locations and magnitudes are also considered. Based on 1000 independent realizations for each test scenario, normalized correlation coefficients, Bayesian information criteria and other statistical measures are used to quantify the similarity of synthetic catalogs to the observed seismicity distribution. We applied this approach to induced seismicity associated with HF operations within the Montney Formation, in western Canada. Three hypotheses were tested, each showing a statistically significant improvement over the null hypothesis. A previous machine-learning-based model for Seismogenic Activation Potential (SAP) showed the highest correlation between observed and synthetically generated seismicity catalogs. Our method has been developed using cloud-based computing and is easily adapted to other regions and data types. [ABSTRACT FROM AUTHOR]

In particular, the data allow us to calculate the I p i -value, which is defined as " I the probability under a specified statistical model that a statistical summary of the data would be equal to or more extreme than its observed value i " [[1]], and practically reflects the degree of data compatibility with the null hypothesis. (2) " I p-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone. i " They can only tell us how consistent the data are with the null hypothesis. Conventionally, if the I p i -value is lower than the 0.05 significance level, we reject the null hypothesis and accept the alternative hypothesis. At the beginning of our research training, we learned about hypothesis testing, I p i -values, and statistical inference. [Extracted from the article]

The full Bayesian significance test (FBST) for precise hypotheses is a Bayesian alternative to the traditional significance tests based on p-values. The FBST is characterized by the e-value as an evidence index in favor of the null hypothesis (H). An important practical issue for the implementation of the FBST is to establish how small the evidence against H must be in order to decide for its rejection. In this work, we present a method to find a cutoff value for the e-value in the FBST by minimizing the linear combination of the averaged type-I and type-II error probabilities for a given sample size and also for a given dimensionality of the parameter space. Furthermore, we compare our methodology with the results obtained from the test with adaptive significance level, which presents the capital-P P-value as a decision-making evidence measure. For this purpose, the scenario of linear regression models with unknown variance under the Bayesian approach is considered. [ABSTRACT FROM AUTHOR]

This paper deals with a new Bayesian approach to the one-sample test for proportion. More specifically, let x = (x 1 , ... , x n) be an independent random sample of size n from a Bernoulli distribution with an unknown parameter θ. For a fixed value θ 0 , the goal is to test the null hypothesis H 0 : θ = θ 0 against all possible alternatives. The proposed approach is based on using the well-known formula of the Kullback–Leibler divergence between two binomial distributions chosen in a certain way. Then, the difference of the distance from a priori to a posteriori is compared through the relative belief ratio (a measure of evidence). Some theoretical properties of the method are developed. Examples and simulation results are included. [ABSTRACT FROM AUTHOR]

Measurement-device-independent quantum key distribution (MDI-QKD) is innately immune to all detection-side attacks. Due to the limitations of technology, most MDI-QKD protocols use weak coherent photon sources (WCPs), which may suffer from a photon-number splitting (PNS) attack from eavesdroppers. Therefore, the existing MDI-QKD protocols also need the decoy-state method, which can resist PNS attacks very well. However, the existing decoy-state methods do not attend to the existence of PNS attacks, and the secure keys are only generated by single-photon components. In fact, multiphoton pulses can also form secure keys if we can confirm that there is no PNS attack. For simplicity, we only analyze the weaker version of a PNS attack in which a legitimate user's pulse count rate changes significantly after the attack. In this paper, under the null hypothesis of no PNS attack, we first determine whether there is an attack or not by retrieving the missing information of the existing decoy-state MDI-QKD protocols via statistical hypothesis testing, extract a normal distribution statistic, and provide a detection method and the corresponding Type I error probability. If the result is judged to be an attack, we use the existing decoy-state method to estimate the secure key rate. Otherwise, all pulses with the same basis leading to successful Bell state measurement (BSM) events including both single-photon pulses and multiphoton pulses can be used to generate secure keys, and we give the formula of the secure key rate in this case. Finally, based on actual experimental data from other literature, the associated experimental results (e.g., the significance level is 5%) show the correctness of our method. [ABSTRACT FROM AUTHOR]

Null hypothesis significance testing is a commonly used tool for making statistical inferences in empirical studies, but its use has always been controversial. In this manuscript, I argue that even more problematic is that significance testing, and other abstract statistical benchmarks, often are used as tools for interpreting study data. This is problematic because interpreting data requires domain knowledge of the scientific topic and sensitivity to the study context, something that significance testing and other purely statistical approaches are not. By using simple examples, I demonstrate that researchers must first use their domain knowledge—professional expertise, clinical experience, practical insight—to interpret the data in their study and then use inferential statistics to provide some reasonable estimates about what can be generalized from the study data. Moving beyond the current focus on abstract statistical benchmarks will encourage researchers to measure their phenomena in more meaningful ways, transparently convey their data, and communicate their intellectual reasons for interpreting the data as they do, a shift that will better foster a scientific forum for cumulative science. [ABSTRACT FROM AUTHOR]

We discuss hypothesis testing and compare different theories in light of observed or experimental data as fundamental endeavors in the sciences. Issues associated with the p-value approach and null hypothesis significance testing are reviewed, and the Bayesian alternative based on the Bayes factor is introduced, along with a review of computational methods and sensitivity related to prior distributions. We demonstrate how Bayesian testing can be practically implemented in several examples, such as the t-test, two-sample comparisons, linear mixed models, and Poisson mixed models by using existing software. Caveats and potential problems associated with Bayesian testing are also discussed. We aim to inform researchers in the many fields where Bayesian testing is not in common use of a well-developed alternative to null hypothesis significance testing and to demonstrate its standard implementation. [ABSTRACT FROM AUTHOR]