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arXiv2022

A Journey from Wild to Textbook Data to Reproducibly Refresh the Wages Data from the National Longitudinal Survey of Youth Database

Textbook data is essential for teaching statistics and data science methods because they are clean, allowing the instructor to focus on methodology. Ideally textbook data sets are refreshed regularly, especially when they are subsets taken from an on-going data collection. It is also important to use contemporary data for teaching, to imbue the sense that the methodology is relevant today. This paper describes the trials and tribulations of refreshing a textbook data set on wages, extracted from the National Longitudinal Survey of Youth (NLSY79) in the early 1990s. The data is useful for teaching modeling and exploratory analysis of longitudinal data. Subsets of NLSY79, including the wages data, can be found in supplementary files from numerous textbooks and research articles. The NLSY79 database has been continuously updated through to 2018, so new records are available. Here we describe our journey to refresh the wages data, and document the process so that the data can be regularly updated into the future. Our journey was difficult because the steps and decisions taken to get from the raw data to the wages textbook subset have not been clearly articulated. We have been diligent to provide a reproducible workflow for others to follow, which also hopefully inspires more attempts at refreshing data for teaching. Three new data sets and the code to produce them are provided in the open source R package called `yowie`.

stat.OT↗

arXiv2023

On the use of a local $\hat{R}$ to improve MCMC convergence diagnostic

Diagnosing convergence of Markov chain Monte Carlo is crucial and remains an essentially unsolved problem. Among the most popular methods, the potential scale reduction factor, commonly named $\hat{R}$, is an indicator that monitors the convergence of output chains to a target distribution, based on a comparison of the between- and within-variances. Several improvements have been suggested since its introduction in the 90s. Here, we aim at better understanding the $\hat{R}$ behavior by proposing a localized version that focuses on quantiles of the target distribution. This new version relies on key theoretical properties of the associated population value. It naturally leads to proposing a new indicator $\hat{R}_\infty$, which is shown to allow both for localizing the Markov chain Monte Carlo convergence in different quantiles of the target distribution, and at the same time for handling some convergence issues not detected by other $\hat{R}$ versions.

math.ST↗

arXiv2022

BayesMix: Bayesian Mixture Models in C++

We describe BayesMix, a C++ library for MCMC posterior simulation for general Bayesian mixture models. The goal of BayesMix is to provide a self-contained ecosystem to perform inference for mixture models to computer scientists, statisticians and practitioners. The key idea of this library is extensibility, as we wish the users to easily adapt our software to their specific Bayesian mixture models. In addition to the several models and MCMC algorithms for posterior inference included in the library, new users with little familiarity on mixture models and the related MCMC algorithms can extend our library with minimal coding effort. Our library is computationally very efficient when compared to competitor software. Examples show that the typical code runtimes are from two to 25 times faster than competitors for data dimension from one to ten. Our library is publicly available on Github at https://github.com/bayesmix-dev/bayesmix/.

stat.CO↗

arXiv2022

Visualising Multilevel Regression and Poststratification: Alternatives to the Current Practice

Surveys provide important evidence for policymaking, decision-making, and understanding of society. However, conducting the large surveys required to provide subpopulation level estimates is expensive and time-consuming. Multilevel Regression and Poststratification (MRP) is a promising method to provide reliable estimates for subpopulations from surveys without the amount of data needed for reliable direct estimates. Graphical displays have been widely used to communicate and diagnose MRP estimates. However, there have been few studies on how visualisation should be performed in this field. Accordingly, this study examines the current practice of MRP visualisation using a systematic literature review. This study also applies MRP to estimate the Trump vote share in the U.S. 2016 presidential election using the Cooperative Congressional Election Study (CCES) data to illustrate the implication of current visualisation practices and explore alternatives for improvement. We find that uncertainty is not often displayed in the current practice, despite its importance for survey inference. The choropleth map is the most frequently used to display MRP estimates even though it only shows point estimates and could hinder the information conveyed. Using various graphical representations, we show that visualisation with uncertainty can illustrate the effect of different model specifications on the estimation result. In addition, this study also proposes a visualisation strategy to also take the bias-variance trade-off into account when evaluating MRP models.

stat.AP↗

arXiv2022

The paradoxical nature of easily improvable evidence

Established frameworks to understand problems with reproducibility in science begin with the relationship between our understanding of the prior probability of a claim and the statistical certainty that should be demanded of it, and explore the ways in which independent investigations, biases in study design and publication bias interact with these considerations. We propose a complementary perspective; namely, that to improve reproducibility in science, our interpretation of the persuasiveness of evidence (e.g., statistical significance thresholds) should be responsive to our understanding of the effort that would be required to improve that evidence. We will quantify this notion in some formal settings. Indeed, we will demonstrate that even simplistic models of evidence publication can exhibit an improvable evidence paradox, where the publication of easily improvable evidence in favor of a claim can best seen as evidence the claim is false.

stat.OT↗

arXiv2022

A discrete analogue of Terrell's characterization of rectangular distributions

George R. Terrell (1983, {Ann. Probab., vol. 11(3), pp. 823--826) showed that the Pearson coefficient of correlation of an ordered pair from a random sample of size two is at most one-half, and the equality is attained only for rectangular (uniform over some interval) distributions. In the present note it is proved that the same is true for the discrete case, in the sense that the correlation coefficient attains its maximal value only for discrete rectangular (uniform over some finite lattice) distributions. MSC: Primary 60E15; 62E10; Secondary 62G30. Key words and phrases: discrete rectangular distribution; order statistics; Hahn polynomials; Pearson coefficient of correlation.

math.PR↗

arXiv2022

Interactive Exploration of Large Dendrograms with Prototypes

Hierarchical clustering is one of the standard methods taught for identifying and exploring the underlying structures that may be present within a data set. Students are shown examples in which the dendrogram, a visual representation of the hierarchical clustering, reveals a clear clustering structure. However, in practice, data analysts today frequently encounter data sets whose large scale undermines the usefulness of the dendrogram as a visualization tool. Densely packed branches obscure structure, and overlapping labels are impossible to read. In this paper we present a new workflow for performing hierarchical clustering via the R package called protoshiny that aims to restore hierarchical clustering to its former role of being an effective and versatile visualization tool. Our proposal leverages interactivity combined with the ability to label internal nodes in a dendrogram with a representative data point (called a prototype). After presenting the workflow, we provide three case studies to demonstrate its utility.

stat.OT↗

arXiv2022

Making Sense of Dependence: Efficient Black-box Explanations Using Dependence Measure

This paper presents a new efficient black-box attribution method based on Hilbert-Schmidt Independence Criterion (HSIC), a dependence measure based on Reproducing Kernel Hilbert Spaces (RKHS). HSIC measures the dependence between regions of an input image and the output of a model based on kernel embeddings of distributions. It thus provides explanations enriched by RKHS representation capabilities. HSIC can be estimated very efficiently, significantly reducing the computational cost compared to other black-box attribution methods. Our experiments show that HSIC is up to 8 times faster than the previous best black-box attribution methods while being as faithful. Indeed, we improve or match the state-of-the-art of both black-box and white-box attribution methods for several fidelity metrics on Imagenet with various recent model architectures. Importantly, we show that these advances can be transposed to efficiently and faithfully explain object detection models such as YOLOv4. Finally, we extend the traditional attribution methods by proposing a new kernel enabling an ANOVA-like orthogonal decomposition of importance scores based on HSIC, allowing us to evaluate not only the importance of each image patch but also the importance of their pairwise interactions. Our implementation is available at https://github.com/paulnovello/HSIC-Attribution-Method.

cs.CV↗

arXiv2022

Fast Computation of Highly G-optimal Exact Designs via Particle Swarm Optimization

Computing proposed exact $G$-optimal designs for response surface models is a difficult computation that has received incremental improvements via algorithm development in the last two-decades. These optimal designs have not been considered widely in applications in part due to the difficulty and cost involved with computing them. Three primary algorithms for constructing exact $G$-optimal designs are presented in the literature: the coordinate exchange (CEXCH), a genetic algorithm (GA), and the relatively new $G$-optimal via $I_λ$-optimality algorithm ($G(I_λ)$-CEXCH) which was developed in part to address large computational cost. Particle swarm optimization (PSO) has achieved widespread use in many applications, but to date, its broad-scale success notwithstanding, has seen relatively few applications in optimal design problems. In this paper we develop an extension of PSO to adapt it to the optimal design problem. We then employ PSO to generate optimal designs for several scenarios covering $K = 1, 2, 3, 4, 5$ design factors, which are common experimental sizes in industrial experiments. We compare these results to all $G$-optimal designs published in last two decades of literature. Published $G$-optimal designs generated by GA for $K=1, 2, 3$ factors have stood unchallenged for 14 years. We demonstrate that PSO has found improved $G$-optimal designs for these scenarios, and it does this with comparable computational cost to the state-of-the-art algorithm $G(I_λ)$-CEXCH. Further, we show that PSO is able to produce equal or better $G$-optimal designs for $K= 4, 5$ factors than those currently known. These results suggest that PSO is superior to existing approaches for efficiently generating highly $G$-optimal designs.

stat.CO↗

arXiv2022

Current state and prospects of R-packages for the design of experiments

Re-running an experiment is generally costly and, in some cases, impossible due to limited resources; therefore, the design of an experiment plays a critical role in increasing the quality of experimental data. In this paper, we describe the current state of R-packages for the design of experiments through an exploratory data analysis of package downloads, package metadata, and a comparison of characteristics with other topics. We observed that experimental designs in practice appear to be sufficiently manufactured by a small number of packages, and the development of experimental designs often occurs in silos. We also discuss the interface designs of widely utilized R packages in the field of experimental design and discuss their future prospects for advancing the field in practice.

stat.OT↗

arXiv2022

Process, Population, and Sample: the Researcher's Interest

A case is made that researchers are interested in studying processes. Often the inferences they are interested in making are about the process and its associated population. On other occasions, a researcher may be interested in making an inference about the collection of individuals the process has generated. We will call the statistical methods employed by the researcher to make such inferences about the process/population ``estimation methods.'' The statistical methods used in making an inference about the collection of individuals generated we call ``prediction methods.'' Methods for obtaining interval estimates of a parameter and prediction intervals for a statistic are given. The analytical and enumerative methods discussed in Deming (1953) are simply estimation and prediction methods, respectively.

stat.OT↗

arXiv2022

On the probability of invalidating a causal inference due to limited external validity

External validity is often questionable in empirical research, especially in randomized experiments due to the trade-off between internal validity and external validity. To quantify the robustness of external validity, one must first conceptualize the gap between a sample that is fully representative of the target population (i.e., the ideal sample) and the observed sample. Drawing on Frank & Min (2007) and Frank et al. (2013), I define such gap as the unobserved sample and intend to quantify its relationship with the null hypothesis statistical testing (NHST) in this study. The probability of invalidating a causal inference due to limited external validity, i.e., the PEV, is the probability of failing to reject the null hypothesis based on the ideal sample provided the null hypothesis has been rejected based on the observed sample. This study illustrates the guideline and the procedure of evaluating external validity with the PEV through an empirical example (i.e., Borman et al. (2008)). Specifically, one would be able to locate the threshold of the unobserved sample statistic that would make the PEV higher than a desired value and use this information to characterize the unobserved sample that would render external validity of the research in question less robust. The PEV is shown to be linked to statistical power when the NHST is thought to be based on the ideal sample.

stat.ME↗

arXiv2022

Approximate Bayesian Inference for the Interaction Types 1, 2, 3 and 4 with Application in Disease Mapping

We address in this paper a new approach for fitting spatiotemporal models with application in disease mapping using the interaction types 1,2,3, and 4. When we account for the spatiotemporal interactions in disease-mapping models, inference becomes more useful in revealing unknown patterns in the data. However, when the number of locations and/or the number of time points is large, the inference gets computationally challenging due to the high number of required constraints necessary for inference, and this holds for various inference architectures including Markov chain Monte Carlo (MCMC) and Integrated Nested Laplace Approximations (INLA). We re-formulate INLA approach based on dense matrices to fit the intrinsic spatiotemporal models with the four interaction types and account for the sum-to-zero constraints, and discuss how the new approach can be implemented in a high-performance computing framework. The computing time using the new approach does not depend on the number of constraints and can reach a 40-fold faster speed compared to INLA in realistic scenarios. This approach is verified by a simulation study and a real data application, and it is implemented in the R package INLAPLUS and the Python header function: inla1234().

stat.ME↗

arXiv2022

Analysis of Hydrogen Production Costs across the United States and over the next 30 years

Hydrogen can play an important role for decarbonization. While hydrogen is usually produced through SMR, it can also be produced through water electrolysis which is cleaner. The relative cost and carbon intensity of hydrogen production through SMR and electrolysis vary throughout the United States because of differences in the grid. While many hydrogen cost models exist, no regional hydrogen study has been conducted across the US. We studied how the Levelized Cost of Hydrogen (LCOH) and carbon intensity for producing hydrogen vary across the US. We looked at electrolysis technologies (Alkaline, PEM, and SOEC) and compared them to SMR. In 2020, SMR with 90 percent CCUS has a lower average LCOH and carbon intensity for hydrogen production than electrolysis by SOEC. For states with cleaner grids, hydrogen produced through SOEC has a lower carbon intensity than hydrogen produced using SMR with 90 percent CCUS. Washington has one of the lowest carbon footprints and the lowest LCOH to produce hydrogen through electrolysis (alkaline). We predict that the LCOH for hydrogen production will be 3.2 USD per kg for Alkaline, 3.1 USD per kg for PEM, and 2.6 USD per kg for SOEC by 2050 with constant electricity prices. These projected LCOHs are still higher than the LCOH for hydrogen produced through SMR with 90 percent CCUS. If electricity costs decrease to 2c per kWh, we expect to reach cost-parity with SMR with 90 percent CCUS. The results suggest that significant investment in decarbonizing the grid and lowering the cost of electricity needs to be made to make electrolysis more competitive compared to SMR.

stat.AP↗

arXiv2023

Run and Frequency quotas for q-binary trials

We study the distributions of waiting times in variations of the $q$-sooner and later waiting time problem. One variation imposes length and frequency quotas on the runs of successes and failures. Another case considers binary trials for which the probability of ones is geometrically varying. We also study the distributions of Longest run under the same variations. The main theorems are sooner and later waiting time problem and the joint distribution of the length of the longest success and longest failure runs when a run and frequency quotas imposed on runs of successes and failure. In the present work, we consider a sequence of independent binary $(0-1)$ trials with not necessarily identical distributed with probability of ones varying according to a geometric rule. Exact formulae for the distributions obtained by means of enumerative combinatorics.

math.PR↗

arXiv2022

On General Weighted Extropy of Ranked Set Sampling

In the past six years, a considerable attention has been given to the extropy measure proposed by Lad et al. (2015). Weighted Extropy of Ranked Set Sampling was studied and compared with simple random sampling by Qiu et al. (2022). The general weighted extropy and some results related to it are introduced in this paper. We provide general weighted extropy of ranked set sampling. We also studied characterization results, stochastic comparison and monotone properties of general weighted extropy.

math.ST↗

arXiv2022

The Inference Framework

The following three sections and appendices are taken from my thesis "The Foundations of Inference and its Application to Fundamental Physics" from 2021, in which I construct a theory of entropic inference from first principles. The majority of these chapters are not original, but are a collection of various sources through the history of the subject. The first section deals with deductive reasoning, which is inference in the presence of complete information. The second section expands on the deductive system by constructing a theory of inductive inference, a theory of probabilities, which is inference in the presence of incomplete information. Finally, section three develops a means of updating these probabilities in the presence of new information that comes in the form of constraints.

stat.OT↗

arXiv2023

Sharp hypotheses and organic fiducial inference

A fundamental class of inferential problems are those characterised by there having been a substantial degree of pre-data (or prior) belief that the value of a model parameter $θ_j$ was equal or lay close to a specified value $θ^{*}_j$, which may, for example, be the value that indicates the absence of a treatment effect or the lack of correlation between two variables. This paper puts forward a generally applicable 'push-button' solution to problems of this type that circumvents the severe difficulties that arise when attempting to apply standard methods of inference, including the Bayesian method, to such problems. Usually the only input of major note that is required from the user in implementing this solution is the assignment of a pre-data or prior probability to the hypothesis that the parameter $θ_j$ lies in a narrow interval $[θ_{j0},θ_{j1}]$ that is assumed to contain the value of interest $θ^{*}_j$. On the other hand, the end result that is achieved by applying this method is, conveniently, a joint post-data distribution over all the parameters $θ_1,θ_2,\ldots,θ_k$ of the model concerned. The proposed method is constructed by naturally combining a simple Bayesian argument with an approach to inference called organic fiducial inference that was developed in a number of earlier papers. To begin with, the main theoretical arguments underlying this combined Bayesian and fiducial method are presented and discussed in detail. Various applications and useful extensions of this methodology are then outlined in the latter part of the paper. The examples that are considered are made relevant to the analysis of clinical trial data where appropriate.

stat.ME↗