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Almost all publications in biomedical literature have employed statistical tests, with p-values being considered of particular importance in the assessment of the presence of a link between two variables. However, these tests and p-values have been the subject of considerable criticism. It may appear paradoxical that tools utilised by the scientific community for nearly a century could possess all the flaws attributed to them. This paradox can partially be explained by the counterintuitive nature of p-values and the fact that the test that generates them is the result of a combination of two tests that were developed to answer statistical questions of a very different nature. The respective characteristics of these two tests are essentially unknown to the majority of users of p-values. The aforementioned paradox can be partially explained by the paucity of publications that seek to elucidate these concepts for users of p-values, the majority of whom are not statisticians. The recently introduced Bayesian methods have properties that enable us to understand the limitations of traditional methods. In Bayesian methods, the use of a specific interpretation of probability allows for better exploitation of clinical research data. The aim of this article is to highlight the limits of non-Bayesian methods and explain the principles and functioning of Bayesian methods to a non-statistical audience.

We present a Bayesian approach for making statistical inference about the accuracy (or any other score) of two competing algorithms which have been assessed via cross-validation on multiple data sets. The approach is constituted by two pieces. The first is a novel correlated Bayesian t test for the analysis of the cross-validation results on a single data set which accounts for the correlation due to the overlapping training sets. The second piece merges the posterior probabilities computed by the Bayesian correlated t test on the different data sets to make inference on multiple data sets. It does so by adopting a Poisson-binomial model. The inferences on multiple data sets account for the different uncertainty of the cross-validation results on the different data sets. It is the first test able to achieve this goal. It is generally more powerful than the signed-rank test if ten runs of cross-validation are performed, as it is anyway generally recommended.

Whale-watching is one of the fastest growing ecotourism industries and involves the observation of endangered wild cetacean species. However, this growth has raised concerns because of the negative effects this activity may have on the behavior and survival of focal species. Hence, detecting the effects of this activity requires sensitive analytical methods allowing the implementation of regulations to protect cetacean welfare. We compared the performance of different hypothesis tests from classical and Bayesian approaches to detect whale-watching effects on humpback whale (Megaptera novaeangliae) behavior. From a cliff located 31 m above sea level in northern Peru, we measured breathing frequency, surface time, long dive duration, directness index (i.e., path linearity), and swimming speed of humpback whales before, during, and after encounters with whale-watching boats. During 167 hours of observation, we tracked 180 humpback whale groups; 43% of groups had calves and 57% did not. Inference by null-hypothesis testing indicated significant changes only in directness index after boat encounters in groups with a calf. Other methods of inference detected moderate behavior responses as increments in the number of adult breaths, swimming speed, and dive intervals for adults and calves. Whale-watching regulations must be implemented in Peru to regulate number of boats, distance to whales, approximate speed, and time observing humpback whales. Whale-watching of humpback whales with calves should be avoided.

Despite their wide use in scientific journals such as The Journal of Wildlife Management, statistical hypothesis tests add very little value to the products of research. Indeed, they frequently confuse the interpretation of data. This paper describes how statistical hypothesis tests are often viewed, and then contrasts that interpretation with the correct one. I discuss the arbitrariness of P-values, conclusions that the null hypothesis is true, power analysis, and distinctions between statistical and biological significance. Statistical hypothesis testing, in which the null hypothesis about the properties of a population is almost always known a priori to be false, is contrasted with scientific hypothesis testing, which examines a credible null hypothesis about phenomena in nature. More meaningful alternatives are briefly outlined, including estimation and confidence intervals for determining the importance of factors, decision theory for guiding actions in the face of uncertainty, and Bayesian approaches to hypothesis testing and other statistical practices.

We consider the problem of uncertainty assessment for low dimensional components in high dimensional models. Specifically, we propose a novel decorrelated score function to handle the impact of high dimensional nuisance parameters. We consider both hypothesis tests and confidence regions for generic penalized M-estimators. Unlike most existing inferential methods which are tailored for individual models, our method provides a general framework for high dimensional inference and is applicable to a wide variety of applications. In particular, we apply this general framework to study five illustrative examples: linear regression, logistic regression, Poisson regression, Gaussian graphical model and additive hazards model. For hypothesis testing, we develop general theorems to characterize the limiting distributions of the decorrelated score test statistic under both null hypothesis and local alternatives. These results provide asymptotic guarantees on the type I errors and local powers. For confidence region construction, we show that the decorrelated score function can be used to construct point estimators that are asymptotically normal and semiparametrically efficient. We further generalize this framework to handle the settings of misspecified models. Thorough numerical results are provided to back up the developed theory.

Statistical inference often fails to replicate. One reason is that many results may be selected for drawing inference because some threshold of a statistic like the P-value was crossed, leading to biased reported effect sizes. Nonetheless, considerable non-replication is to be expected even without selective reporting, and generalizations from single studies are rarely if ever warranted. Honestly reported results must vary from replication to replication because of varying assumption violations and random variation; excessive agreement itself would suggest deeper problems, such as failure to publish results in conflict with group expectations or desires. A general perception of a \"replication crisis\" may thus reflect failure to recognize that statistical tests not only test hypotheses, but countless assumptions and the entire environment in which research takes place. Because of all the uncertain and unknown assumptions that underpin statistical inferences, we should treat inferential statistics as highly unstable local descriptions of relations between assumptions and data, rather than as providing generalizable inferences about hypotheses or models. And that means we should treat statistical results as being much more incomplete and uncertain than is currently the norm. Acknowledging this uncertainty could help reduce the allure of selective reporting: Since a small P-value could be large in a replication study, and a large P-value could be small, there is simply no need to selectively report studies based on statistical results. Rather than focusing our study reports on uncertain conclusions, we should thus focus on describing accurately how the study was conducted, what problems occurred, what data were obtained, what analysis methods were used and why, and what output those methods produced.

Bayesian parameter estimation and Bayesian hypothesis testing present attractive alternatives to classical inference using confidence intervals and p values. In part I of this series we outline ten prominent advantages of the Bayesian approach. Many of these advantages translate to concrete opportunities for pragmatic researchers. For instance, Bayesian hypothesis testing allows researchers to quantify evidence and monitor its progression as data come in, without needing to know the intention with which the data were collected. We end by countering several objections to Bayesian hypothesis testing. Part II of this series discusses JASP, a free and open source software program that makes it easy to conduct Bayesian estimation and testing for a range of popular statistical scenarios (Wagenmakers et al. this issue ).

Many high-dimensional hypothesis tests aim to globally examine marginal or low-dimensional features of a high-dimensional joint distribution, such as testing of mean vectors, covariance matrices and regression coefficients. This paper constructs a family of U-statistics as unbiased estimators of the ℓp -norms of those features.We show that under the null hypothesis, the U-statistics of different finite orders are asymptotically independent and normally distributed. Moreover, they are also asymptotically independent with the maximum-type test statistic, whose limiting distribution is an extreme value distribution. Based on the asymptotic independence property, we propose an adaptive testing procedure which combines p-values computed from the U-statistics of different orders.We further establish power analysis results and show that the proposed adaptive procedure maintains high power against various alternatives.

We develop a general approach to valid inference after model selection. At the core of our framework is a result that characterizes the distribution of a post-selection estimator conditioned on the selection event. We specialize the approach to model selection by the lasso to form valid confidence intervals for the selected coefficients and test whether all relevant variables have been included in the model.

Uncertain hypothesis test is a statistical tool that uses uncertainty theory to determine whether some hypotheses are correct or not based on observed data. As an application of uncertain hypothesis test, this paper proposes a method to test whether an uncertain differential equation fits the observed data or not. In order to demonstrate the test method, some numerical examples are provided. Finally, both uncertain currency model and stochastic currency model are used to model US Dollar to Chinese Yuan (USD–CNY) exchange rates. As a result, it is shown that the uncertain currency model fits the exchange rates well, but the stochastic currency model does not.