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Entropy, Statistical Evidence, and Scientific Inference: Evidence Functions in Theory and Applications

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Multidisciplinary Applications".

Deadline for manuscript submissions: closed (31 May 2024) | Viewed by 9006

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Special Issue Editors

Prof. Dr. Brian Dennis

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Guest Editor

1. Department of Mathematics and Statistical Science, University of Idaho, Moscow, ID 83844, USA
2. Professor Emeritus, Department of Fish and Wildlife Sciences, University of Idaho, Moscow, ID 83844, USA
Interests: statistical ecology; biometrics; mathematical modeling; theoretical ecology; conservation biology; population dynamics

Dr. Mark L. Taper

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Guest Editor

1. Department of Ecology, Montana State University, Bozeman, MT 59717, USA
2. Marine Science Institute, University of California, Santa Barbara, CA 94720, USA
Interests: theoretical ecology; ecological statistics; statistical inference; evolution; philosophy of science

Prof. Dr. Jose Miguel Ponciano

E-Mail Website
Guest Editor

1. Biology Department , University of Florida, Gainesville, FL 32611, USA
2. Mathematics Department, University of Florida, Gainesville, FL 32611, USA
Interests: statistical ecology; population dynamics; theoretical ecology; statistical phylogenetics; conservation biology; mathematical population genetics

Special Issue Information

Dear Colleagues,

Modern statistical evidence compares the relative support in scientific data for mathematical models. The fundamental tool of comparison is the evidence function, which is a contrast of generalized entropy discrepancies. The most commonly used evidence functions are the differences of information criterion values. Statistical evidence has many desirable properties, combining attractive features of both Bayesian and classical frequentist analysis while simultaneously avoiding many of their philosophical and practical issues. The goals of this Special Issue are to stimulate the further theoretical development of statistical evidence and present real-world examples where the use of statistical evidence clarifies scientific inference. While many of the applications featured here are ecological, reflecting the editors’ areas of expertise, we welcome and anticipate accounts or critiques of evidence functions applied in other scientific areas.

Prof. Dr. Brian Dennis
Dr. Mark L. L. Taper
Prof. Dr. Jose Miguel Ponciano
Guest Editors

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Keywords

entropy
evidential statistics
evidence
hypothesis testing
information theory
Kullback–Leibler discrepancy
model misspecification
model selection

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Published Papers (7 papers)

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Editorial

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8 pages, 251 KiB

Open AccessEditorial

Entropy, Statistical Evidence, and Scientific Inference: Evidence Functions in Theory and Applications

by Mark L. Taper, José Miguel Ponciano and Brian Dennis

Entropy 2022, 24(9), 1273; https://doi.org/10.3390/e24091273 - 9 Sep 2022

Viewed by 1621

Abstract

Scope and Goals of the Special Issue: There is a growing realization that despite being the essential tool of modern data-based scientific discovery and model testing, statistics has major problems [...] Full article

(This article belongs to the Special Issue Entropy, Statistical Evidence, and Scientific Inference: Evidence Functions in Theory and Applications)

Research

Jump to: Editorial

17 pages, 452 KiB

Open AccessArticle

Bootstrap Approximation of Model Selection Probabilities for Multimodel Inference Frameworks

by Andres Dajles and Joseph Cavanaugh

Entropy 2024, 26(7), 599; https://doi.org/10.3390/e26070599 - 15 Jul 2024

Viewed by 641

Abstract

Most statistical modeling applications involve the consideration of a candidate collection of models based on various sets of explanatory variables. The candidate models may also differ in terms of the structural formulations for the systematic component and the posited probability distributions for the random component. A common practice is to use an information criterion to select a model from the collection that provides an optimal balance between fidelity to the data and parsimony. The analyst then typically proceeds as if the chosen model was the only model ever considered. However, such a practice fails to account for the variability inherent in the model selection process, which can lead to inappropriate inferential results and conclusions. In recent years, inferential methods have been proposed for multimodel frameworks that attempt to provide an appropriate accounting of modeling uncertainty. In the frequentist paradigm, such methods should ideally involve model selection probabilities, i.e., the relative frequencies of selection for each candidate model based on repeated sampling. Model selection probabilities can be conveniently approximated through bootstrapping. When the Akaike information criterion is employed, Akaike weights are also commonly used as a surrogate for selection probabilities. In this work, we show that the conventional bootstrap approach for approximating model selection probabilities is impacted by bias. We propose a simple correction to adjust for this bias. We also argue that Akaike weights do not provide adequate approximations for selection probabilities, although they do provide a crude gauge of model plausibility. Full article

(This article belongs to the Special Issue Entropy, Statistical Evidence, and Scientific Inference: Evidence Functions in Theory and Applications)

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8 pages, 226 KiB

Open AccessArticle

Multimodel Approaches Are Not the Best Way to Understand Multifactorial Systems

by Benjamin M. Bolker

Entropy 2024, 26(6), 506; https://doi.org/10.3390/e26060506 - 11 Jun 2024

Cited by 2 | Viewed by 749

Abstract

Information-theoretic (IT) and multi-model averaging (MMA) statistical approaches are widely used but suboptimal tools for pursuing a multifactorial approach (also known as the method of multiple working hypotheses) in ecology. (1) Conceptually, IT encourages ecologists to perform tests on sets of artificially simplified models. (2) MMA improves on IT model selection by implementing a simple form of shrinkage estimation (a way to make accurate predictions from a model with many parameters relative to the amount of data, by “shrinking” parameter estimates toward zero). However, other shrinkage estimators such as penalized regression or Bayesian hierarchical models with regularizing priors are more computationally efficient and better supported theoretically. (3) In general, the procedures for extracting confidence intervals from MMA are overconfident, providing overly narrow intervals. If researchers want to use limited data sets to accurately estimate the strength of multiple competing ecological processes along with reliable confidence intervals, the current best approach is to use full (maximal) statistical models (possibly with Bayesian priors) after making principled, a priori decisions about model complexity. Full article

(This article belongs to the Special Issue Entropy, Statistical Evidence, and Scientific Inference: Evidence Functions in Theory and Applications)

15 pages, 1792 KiB

Open AccessArticle

Likelihood Ratio Test and the Evidential Approach for 2 × 2 Tables

by Peter M. B. Cahusac

Entropy 2024, 26(5), 375; https://doi.org/10.3390/e26050375 - 28 Apr 2024

Viewed by 1212

Abstract

Categorical data analysis of 2 × 2 contingency tables is extremely common, not least because they provide risk difference, risk ratio, odds ratio, and log odds statistics in medical research. A

χ^{2}

test analysis is most often used, although some researchers use [...] Read more.

Categorical data analysis of 2 × 2 contingency tables is extremely common, not least because they provide risk difference, risk ratio, odds ratio, and log odds statistics in medical research. A

χ^{2}

test analysis is most often used, although some researchers use likelihood ratio test (LRT) analysis. Does it matter which test is used? A review of the literature, examination of the theoretical foundations, and analyses of simulations and empirical data are used by this paper to argue that only the LRT should be used when we are interested in testing whether the binomial proportions are equal. This so-called test of independence is by far the most popular, meaning the

χ^{2}

test is widely misused. By contrast, the

χ^{2}

test should be reserved for where the data appear to match too closely a particular hypothesis (e.g., the null hypothesis), where the variance is of interest, and is less than expected. Low variance can be of interest in various scenarios, particularly in investigations of data integrity. Finally, it is argued that the evidential approach provides a consistent and coherent method that avoids the difficulties posed by significance testing. The approach facilitates the calculation of appropriate log likelihood ratios to suit our research aims, whether this is to test the proportions or to test the variance. The conclusions from this paper apply to larger contingency tables, including multi-way tables. Full article

(This article belongs to the Special Issue Entropy, Statistical Evidence, and Scientific Inference: Evidence Functions in Theory and Applications)

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23 pages, 501 KiB

Open AccessArticle

How the Post-Data Severity Converts Testing Results into Evidence for or against Pertinent Inferential Claims

by Aris Spanos

Entropy 2024, 26(1), 95; https://doi.org/10.3390/e26010095 - 22 Jan 2024

Viewed by 1025

Abstract

The paper makes a case that the current discussions on replicability and the abuse of significance testing have overlooked a more general contributor to the untrustworthiness of published empirical evidence, which is the uninformed and recipe-like implementation of statistical modeling and inference. It is argued that this contributes to the untrustworthiness problem in several different ways, including [a] statistical misspecification, [b] unwarranted evidential interpretations of frequentist inference results, and [c] questionable modeling strategies that rely on curve-fitting. What is more, the alternative proposals to replace or modify frequentist testing, including [i] replacing p-values with observed confidence intervals and effects sizes, and [ii] redefining statistical significance, will not address the untrustworthiness of evidence problem since they are equally vulnerable to [a]–[c]. The paper calls for distinguishing between unduly data-dependant ‘statistical results’, such as a point estimate, a p-value, and accept/reject

H_{0}

, from ‘evidence for or against inferential claims’. The post-data severity (SEV) evaluation of the accept/reject

H_{0}

results, converts them into evidence for or against germane inferential claims. These claims can be used to address/elucidate several foundational issues, including (i) statistical vs. substantive significance, (ii) the large n problem, and (iii) the replicability of evidence. Also, the SEV perspective sheds light on the impertinence of the proposed alternatives [i]–[iii], and oppugns [iii] the alleged arbitrariness of framing

H_{0}

and

H_{1}

which is often exploited to undermine the credibility of frequentist testing. Full article

(This article belongs to the Special Issue Entropy, Statistical Evidence, and Scientific Inference: Evidence Functions in Theory and Applications)

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14 pages, 334 KiB

Open AccessArticle

Profile Likelihood for Hierarchical Models Using Data Doubling

by Subhash R. Lele

Entropy 2023, 25(9), 1262; https://doi.org/10.3390/e25091262 - 25 Aug 2023

Viewed by 982

Abstract

In scientific problems, an appropriate statistical model often involves a large number of canonical parameters. Often times, the quantities of scientific interest are real-valued functions of these canonical parameters. Statistical inference for a specified function of the canonical parameters can be carried out via the Bayesian approach by simply using the posterior distribution of the specified function of the parameter of interest. Frequentist inference is usually based on the profile likelihood for the parameter of interest. When the likelihood function is analytical, computing the profile likelihood is simply a constrained optimization problem with many numerical algorithms available. However, for hierarchical models, computing the likelihood function and hence the profile likelihood function is difficult because of the high-dimensional integration involved. We describe a simple computational method to compute profile likelihood for any specified function of the parameters of a general hierarchical model using data doubling. We provide a mathematical proof for the validity of the method under regularity conditions that assure that the distribution of the maximum likelihood estimator of the canonical parameters is non-singular, multivariate, and Gaussian. Full article

(This article belongs to the Special Issue Entropy, Statistical Evidence, and Scientific Inference: Evidence Functions in Theory and Applications)

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16 pages, 924 KiB

Open AccessArticle

Evidence of an Absence of Inbreeding Depression in a Wild Population of Weddell Seals (Leptonychotes weddellii)

by John H. Powell, Steven T. Kalinowski, Mark L. Taper, Jay J. Rotella, Corey S. Davis and Robert A. Garrott

Entropy 2023, 25(3), 403; https://doi.org/10.3390/e25030403 - 22 Feb 2023

Cited by 1 | Viewed by 1724

Abstract

Inbreeding depression can reduce the viability of wild populations. Detecting inbreeding depression in the wild is difficult; developing accurate estimates of inbreeding can be time and labor intensive. In this study, we used a two-step modeling procedure to incorporate uncertainty inherent in estimating individual inbreeding coefficients from multilocus genotypes into estimates of inbreeding depression in a population of Weddell seals (Leptonychotes weddellii). The two-step modeling procedure presented in this paper provides a method for estimating the magnitude of a known source of error, which is assumed absent in classic regression models, and incorporating this error into inferences about inbreeding depression. The method is essentially an errors-in-variables regression with non-normal errors in both the dependent and independent variables. These models, therefore, allow for a better evaluation of the uncertainty surrounding the biological importance of inbreeding depression in non-pedigreed wild populations. For this study we genotyped 154 adult female seals from the population in Erebus Bay, Antarctica, at 29 microsatellite loci, 12 of which are novel. We used a statistical evidence approach to inference rather than hypothesis testing because the discovery of both low and high levels of inbreeding are of scientific interest. We found evidence for an absence of inbreeding depression in lifetime reproductive success, adult survival, age at maturity, and the reproductive interval of female seals in this population. Full article

(This article belongs to the Special Issue Entropy, Statistical Evidence, and Scientific Inference: Evidence Functions in Theory and Applications)

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