"Magnitude-based inference": a statistical review

Purpose: We consider ‘‘magnitude-based inference’’ and its interpretation by examining in detail its use in the problem of comparing two means. Methods: We extract from the spreadsheets, which are provided to users of the analysis (http:// www.sportsci.org/), a precise description of how ‘‘magnitude-based inference’’ is implemented.We compare the implemented version of the method with general descriptions of it and interpret the method in familiar statistical terms. Results and Conclusions: We show that ‘‘magnitude-based inference’’ is not a progressive improvement on modern statistics. The additional probabilities introduced are not directly related to the confidence interval but, rather, are interpretable either as P values for two different nonstandard tests (for different null hypotheses) or as approximate Bayesian calculations, which also lead to a type of test. We also discuss sample size calculations associated with ‘‘magnitude-based inference’’ and show that the substantial reduction in sample sizes claimed for the method (30% of the sample size obtained from standard frequentist calculations) is not justifiable so the sample size calculations should not be used. Rather than using ‘‘magnitude-based inference,’’ a better solution is to be realistic about the limitations of the data and use either confidence intervals or a fully Bayesian analysis.Alan H. Welsh, Emma J. Knigh

Statistical Tests, Tests of Significance, and Tests of a Hypothesis Using Excel

Microsoft’s spreadsheet program Excel has many statistical functions and routines. Over the years there have been criticisms about the inaccuracies of these functions and routines (see McCullough 1998, 1999). This article reviews some of these statistical methods used to test for differences between two samples. In practice, the analysis is done by a software program and often with the actual method used unknown. The user has to select the method and variations to be used, without full knowledge of just what calculations are used. Usually there is no convenient trace back to textbook explanations. This article describes the Excel algorithm and gives textbook related explanations to bolster Microsoft’s Help explanations

Valid and efficient imprecise-probabilistic inference with partial priors, III. Marginalization

As Basu (1977) writes, "Eliminating nuisance parameters from a model is universally recognized as a major problem of statistics," but after more than 50 years since Basu wrote these words, the two mainstream schools of thought in statistics have yet to solve the problem. Fortunately, the two mainstream frameworks aren't the only options. This series of papers rigorously develops a new and very general inferential model (IM) framework for imprecise-probabilistic statistical inference that is provably valid and efficient, while simultaneously accommodating incomplete or partial prior information about the relevant unknowns when it's available. The present paper, Part III in the series, tackles the marginal inference problem. Part II showed that, for parametric models, the likelihood function naturally plays a central role and, here, when nuisance parameters are present, the same principles suggest that the profile likelihood is the key player. When the likelihood factors nicely, so that the interest and nuisance parameters are perfectly separated, the valid and efficient profile-based marginal IM solution is immediate. But even when the likelihood doesn't factor nicely, the same profile-based solution remains valid and leads to efficiency gains. This is demonstrated in several examples, including the famous Behrens--Fisher and gamma mean problems, where I claim the proposed IM solution is the best solution available. Remarkably, the same profiling-based construction offers validity guarantees in the prediction and non-parametric inference problems. Finally, I show how a broader view of this new IM construction can handle non-parametric inference on risk minimizers and makes a connection between non-parametric IMs and conformal prediction.Comment: Follow-up to arXiv:2211.14567. Feedback welcome at https://researchers.one/articles/23.09.0000

Exact Approaches for Bias Detection and Avoidance with Small, Sparse, or Correlated Categorical Data

Every day, traditional statistical methodology are used world wide to study a variety of topics and provides insight regarding countless subjects. Each technique is based on a distinct set of assumptions to ensure valid results. Additionally, many statistical approaches rely on large sample behavior and may collapse or degenerate in the presence of small, spare, or correlated data. This dissertation details several advancements to detect these conditions, avoid their consequences, and analyze data in a different way to yield trustworthy results. One of the most commonly used modeling techniques for outcomes with only two possible categorical values (eg. live/die, pass/fail, better/worse, ect.) is logistic regression. While some potential complications with this approach are widely known, many investigators are unaware that their particular data does not meet the foundational assumptions, since they are not easy to verify. We have developed a routine for determining if a researcher should be concerned about potential bias in logistic regression results, so they can take steps to mitigate the bias or use a different procedure altogether to model the data. Correlated data may arise from common situations such as multi-site medical studies, research on family units, or investigations on student achievement within classrooms. In these circumstance the associations between cluster members must be included in any statistical analysis testing the hypothesis of a connection be-tween two variables in order for results to be valid. Previously investigators had to choose between using a method intended for small or sparse data while assuming independence between observations or a method that allowed for correlation between observations, while requiring large samples to be reliable. We present a new method that allows for small, clustered samples to be assessed for a relationship between a two-level predictor (eg. treatment/control) and a categorical outcome (eg. low/medium/high)

On the Robustness of Robustness Checks of the Environmental Kuznets Curve

Since its first inception in the debate on the relationship between environment and growth in 1992, the Environmental Kuznets Curve has been subject of continuous and intense scrutiny. The literature can be roughly divided in two historical phases. Initially, after the seminal contributions, additional work aimed to extend the investigation to new pollutants and to verify the existence of an inverted-U shape as well as assessing the value of the turning point. The following phase focused instead on the robustness of the empirical relationship, particularly with respect to the omission of relevant explanatory variables other than GDP, alternative datasets, functional forms, and grouping of the countries examined. The most recent line of investigation criticizes the Environmental Kuznets Curve on more fundamental grounds, in that it stresses the lack of sufficient statistical testing of the empirical relationship and questions the very existence of the notion of Environmental Kuznets Curve. Attention is in particular drawn on the stationarity properties of the series involved – per capita emissions or concentrations and per capita GDP – and, in case of presence of unit roots, on the cointegration property that must be present for the Environmental Kuznets Curve to be a well-defined concept. Only at that point can the researcher ask whether the long-run relationship exhibits an inverted-U pattern. On the basis of panel integration and cointegration tests for sulphur, Stern (2002, 2003) and Perman and Stern (1999, 2003) have presented evidence and forcefully stated that the Environmental Kuznets Curve does not exist. In this paper we ask whether similar strong conclusions can be arrived at when carrying out tests of fractional panel integration and cointegration. As an example we use the controversial case of carbon dioxide emissions. The results show that more EKCs come back into life relative to traditional integration/cointegration tests. However, we confirm that the EKC remains a fragile concept.Environment, Growth, CO2 Emissions, Panel Data, Fractional Integration, Panel Cointegration Tests

Glosarium Matematika

273 p.; 24 cm

On the Robustness of Robustness Checks of the Environmental Kuznets Curve

Since its first inception in the debate on the relationship between environment and growth in 1992, the Environmental Kuznets Curve has been subject to continuous and intense scrutiny. The literature can be roughly divided in two historical phases. Initially, after the seminal contributions, additional work aimed to extend the investigation to new pollutants and to verify the existence of an inverted-U shape as well as assessing the value of the turning point. The following phase focused instead on the robustness of the empirical relationship, particularly with respect to the omission of relevant explanatory variables other than GDP, alternative datasets, functional forms, and grouping of the countries examined. The most recent line of investigation criticizes the Environmental Kuznets Curve on more fundamental grounds, in that it stresses the lack of sufficient statistical testing of the empirical relationship and questions the very existence of the notion of Environmental Kuznets Curve. Attention is drawn in particular on the stationarity properties of the series involved – per capita emissions or concentrations and per capita GDP – and, in case of unit roots, on the cointegration property that must be present for the Environmental Kuznets Curve to be a well-defined concept. Only at that point can the researcher ask whether the long-run relationship exhibits an inverted-U pattern. On the basis of panel integration and cointegration tests for sulphur, Stern (2002, 2003) and Perman and Stern (1999, 2003) have presented evidence and forcefully stated that the Environmental Kuznets Curve does not exist. In this paper we ask whether similar strong conclusions can be arrived at when carrying out tests of fractional panel integration and cointegration. As an example we use the controversial case of carbon dioxide emissions. The results show that more EKCs come back into life relative to traditional integration/cointegration tests. However, we confirm that the EKC remains a fragile concept.Environment, Growth, CO2 Emissions, Panel data, Fractional integration, Panel cointegration tests