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| association and causation math: An Introduction to Causal Inference Judea Pearl, 2015 This paper summarizes recent advances in causal inference and underscores the paradigmatic shifts that must be undertaken in moving from traditional statistical analysis to causal analysis of multivariate data. Special emphasis is placed on the assumptions that underly all causal inferences, the languages used in formulating those assumptions, the conditional nature of all causal and counterfactual claims, and the methods that have been developed for the assessment of such claims. These advances are illustrated using a general theory of causation based on the Structural Causal Model (SCM) described in Pearl (2000a), which subsumes and unifies other approaches to causation, and provides a coherent mathematical foundation for the analysis of causes and counterfactuals. In particular, the paper surveys the development of mathematical tools for inferring (from a combination of data and assumptions) answers to three types of causal queries: (1) queries about the effects of potential interventions, (also called causal effects or policy evaluation) (2) queries about probabilities of counterfactuals, (including assessment of regret, attribution or causes of effects) and (3) queries about direct and indirect effects (also known as mediation). Finally, the paper defines the formal and conceptual relationships between the structural and potential-outcome frameworks and presents tools for a symbiotic analysis that uses the strong features of both. The tools are demonstrated in the analyses of mediation, causes of effects, and probabilities of causation. -- p. 1. |
| association and causation math: Spurious Correlations Tyler Vigen, 2015-05-12 Spurious Correlations ... is the most fun you'll ever have with graphs. -- Bustle Military intelligence analyst and Harvard Law student Tyler Vigen illustrates the golden rule that correlation does not equal causation through hilarious graphs inspired by his viral website. Is there a correlation between Nic Cage films and swimming pool accidents? What about beef consumption and people getting struck by lightning? Absolutely not. But that hasn't stopped millions of people from going to tylervigen.com and asking, Wait, what? Vigen has designed software that scours enormous data sets to find unlikely statistical correlations. He began pulling the funniest ones for his website and has since gained millions of views, hundreds of thousands of likes, and tons of media coverage. Subversive and clever, Spurious Correlations is geek humor at its finest, nailing our obsession with data and conspiracy theory. |
| association and causation math: Causality Judea Pearl, 2009-09-14 Causality offers the first comprehensive coverage of causal analysis in many sciences, including recent advances using graphical methods. Pearl presents a unified account of the probabilistic, manipulative, counterfactual and structural approaches to causation, and devises simple mathematical tools for analyzing the relationships between causal connections, statistical associations, actions and observations. The book will open the way for including causal analysis in the standard curriculum of statistics, artificial intelligence ... |
| association and causation math: Causal Inference in Statistics Judea Pearl, Madelyn Glymour, Nicholas P. Jewell, 2016-01-25 CAUSAL INFERENCE IN STATISTICS A Primer Causality is central to the understanding and use of data. Without an understanding of cause–effect relationships, we cannot use data to answer questions as basic as Does this treatment harm or help patients? But though hundreds of introductory texts are available on statistical methods of data analysis, until now, no beginner-level book has been written about the exploding arsenal of methods that can tease causal information from data. Causal Inference in Statistics fills that gap. Using simple examples and plain language, the book lays out how to define causal parameters; the assumptions necessary to estimate causal parameters in a variety of situations; how to express those assumptions mathematically; whether those assumptions have testable implications; how to predict the effects of interventions; and how to reason counterfactually. These are the foundational tools that any student of statistics needs to acquire in order to use statistical methods to answer causal questions of interest. This book is accessible to anyone with an interest in interpreting data, from undergraduates, professors, researchers, or to the interested layperson. Examples are drawn from a wide variety of fields, including medicine, public policy, and law; a brief introduction to probability and statistics is provided for the uninitiated; and each chapter comes with study questions to reinforce the readers understanding. |
| association and causation math: The Book of Why Judea Pearl, Dana Mackenzie, 2018-05-15 A Turing Award-winning computer scientist and statistician shows how understanding causality has revolutionized science and will revolutionize artificial intelligence Correlation is not causation. This mantra, chanted by scientists for more than a century, has led to a virtual prohibition on causal talk. Today, that taboo is dead. The causal revolution, instigated by Judea Pearl and his colleagues, has cut through a century of confusion and established causality -- the study of cause and effect -- on a firm scientific basis. His work explains how we can know easy things, like whether it was rain or a sprinkler that made a sidewalk wet; and how to answer hard questions, like whether a drug cured an illness. Pearl's work enables us to know not just whether one thing causes another: it lets us explore the world that is and the worlds that could have been. It shows us the essence of human thought and key to artificial intelligence. Anyone who wants to understand either needs The Book of Why. |
| association and causation math: Causation, Prediction, and Search Peter Spirtes, Clark Glymour, Richard Scheines, 2012-12-06 This book is intended for anyone, regardless of discipline, who is interested in the use of statistical methods to help obtain scientific explanations or to predict the outcomes of actions, experiments or policies. Much of G. Udny Yule's work illustrates a vision of statistics whose goal is to investigate when and how causal influences may be reliably inferred, and their comparative strengths estimated, from statistical samples. Yule's enterprise has been largely replaced by Ronald Fisher's conception, in which there is a fundamental cleavage between experimental and non experimental inquiry, and statistics is largely unable to aid in causal inference without randomized experimental trials. Every now and then members of the statistical community express misgivings about this turn of events, and, in our view, rightly so. Our work represents a return to something like Yule's conception of the enterprise of theoretical statistics and its potential practical benefits. If intellectual history in the 20th century had gone otherwise, there might have been a discipline to which our work belongs. As it happens, there is not. We develop material that belongs to statistics, to computer science, and to philosophy; the combination may not be entirely satisfactory for specialists in any of these subjects. We hope it is nonetheless satisfactory for its purpose. |
| association and causation math: Lectures on Categorical Data Analysis Tamás Rudas, 2018-03-30 This book offers a relatively self-contained presentation of the fundamental results in categorical data analysis, which plays a central role among the statistical techniques applied in the social, political and behavioral sciences, as well as in marketing and medical and biological research. The methods applied are mainly aimed at understanding the structure of associations among variables and the effects of other variables on these interactions. A great advantage of studying categorical data analysis is that many concepts in statistics become transparent when discussed in a categorical data context, and, in many places, the book takes this opportunity to comment on general principles and methods in statistics, addressing not only the “how” but also the “why.” Assuming minimal background in calculus, linear algebra, probability theory and statistics, the book is designed to be used in upper-undergraduate and graduate-level courses in the field and in more general statistical methodology courses, as well as a self-study resource for researchers and professionals. The book covers such key issues as: higher order interactions among categorical variables; the use of the delta-method to correctly determine asymptotic standard errors for complex quantities reported in surveys; the fundamentals of the main theories of causal analysis based on observational data; the usefulness of the odds ratio as a measure of association; and a detailed discussion of log-linear models, including graphical models. The book contains over 200 problems, many of which may also be used as starting points for undergraduate research projects. The material can be used by students toward a variety of goals, depending on the degree of theory or application desired. |
| association and causation math: Actual Causality Joseph Y. Halpern, 2016-08-12 Explores actual causality, and such related notions as degree of responsibility, degree of blame, and causal explanation. The goal is to arrive at a definition of causality that matches our natural language usage and is helpful, for example, to a jury deciding a legal case, a programmer looking for the line of code that cause some software to fail, or an economist trying to determine whether austerity caused a subsequent depression. |
| association and causation math: Progress in Artificial Intelligence Goreti Marreiros, Francisco S. Melo, Nuno Lau, Henrique Lopes Cardoso, Luís Paulo Reis, 2021-09-07 This book constitutes the refereed proceedings of the 20th EPIA Conference on Artificial Intelligence, EPIA 2021, held virtually in September 2021. The 62 full papers and 6 short papers presented were carefully reviewed and selected from a total of 108 submissions. The papers are organized in the following topical sections: artificial intelligence and IoT in agriculture; artificial intelligence and law; artificial intelligence in medicine; artificial intelligence in power and energy systems; artificial intelligence in transportation systems; artificial life and evolutionary algorithms; ambient intelligence and affective environments; general AI; intelligent robotics; knowledge discovery and business intelligence; multi-agent systems: theory and applications; and text mining and applications. |
| association and causation math: Fundamentals of Causal Inference Babette A. Brumback, 2021-11-10 One of the primary motivations for clinical trials and observational studies of humans is to infer cause and effect. Disentangling causation from confounding is of utmost importance. Fundamentals of Causal Inference explains and relates different methods of confounding adjustment in terms of potential outcomes and graphical models, including standardization, difference-in-differences estimation, the front-door method, instrumental variables estimation, and propensity score methods. It also covers effect-measure modification, precision variables, mediation analyses, and time-dependent confounding. Several real data examples, simulation studies, and analyses using R motivate the methods throughout. The book assumes familiarity with basic statistics and probability, regression, and R and is suitable for seniors or graduate students in statistics, biostatistics, and data science as well as PhD students in a wide variety of other disciplines, including epidemiology, pharmacy, the health sciences, education, and the social, economic, and behavioral sciences. Beginning with a brief history and a review of essential elements of probability and statistics, a unique feature of the book is its focus on real and simulated datasets with all binary variables to reduce complex methods down to their fundamentals. Calculus is not required, but a willingness to tackle mathematical notation, difficult concepts, and intricate logical arguments is essential. While many real data examples are included, the book also features the Double What-If Study, based on simulated data with known causal mechanisms, in the belief that the methods are best understood in circumstances where they are known to either succeed or fail. Datasets, R code, and solutions to odd-numbered exercises are available at www.routledge.com. |
| association and causation math: The Way We're Working Isn't Working Tony Schwartz, Jean Gomes, Catherine McCarthy, 2010-05-18 This book was previously titled, Be Excellent at Anything. The Way We're Working Isn't Working is one of those rare books with the power to profoundly transform the way we work and live. Demand is exceeding our capacity. The ethic of more, bigger, faster exacts a series of silent but pernicious costs at work, undermining our energy, focus, creativity, and passion. Nearly 75 percent of employees around the world feel disengaged at work every day. The Way We're Working Isn't Working offers a groundbreaking approach to reenergizing our lives so we’re both more satisfied and more productive—on the job and off. By integrating multidisciplinary findings from the science of high performance, Tony Schwartz, coauthor of the #1 bestselling The Power of Full Engagement, makes a persuasive case that we’re neglecting the four core needs that energize great performance: sustainability (physical); security (emotional); self-expression (mental); and significance (spiritual). Rather than running like computers at high speeds for long periods, we’re at our best when we pulse rhythmically between expending and regularly renewing energy across each of our four needs. Organizations undermine sustainable high performance by forever seeking to get more out of their people. Instead they should seek systematically to meet their four core needs so they’re freed, fueled, and inspired to bring the best of themselves to work every day. Drawing on extensive work with an extra-ordinary range of organizations, among them Google, Ford, Sony, Ernst & Young, Shell, IBM, the Los Angeles Police Department, and the Cleveland Clinic, Schwartz creates a road map for a new way of working. At the individual level, he explains how we can build specific rituals into our daily schedules to balance intense effort with regular renewal; offset emotionally draining experiences with practices that fuel resilience; move between a narrow focus on urgent demands and more strategic, creative thinking; and balance a short-term focus on immediate results with a values-driven commitment to serving the greater good. At the organizational level, he outlines new policies, practices, and cultural messages that Schwartz’s client companies have adopted. The Way We're Working Isn't Working offers individuals, leaders, and organizations a highly practical, proven set of strategies to better manage the relentlessly rising demands we all face in an increasingly complex world. |
| association and causation math: The Art of Regression Modeling in Road Safety Ezra Hauer, 2014-12-10 This unique book explains how to fashion useful regression models from commonly available data to erect models essential for evidence-based road safety management and research. Composed from techniques and best practices presented over many years of lectures and workshops, The Art of Regression Modeling in Road Safety illustrates that fruitful modeling cannot be done without substantive knowledge about the modeled phenomenon. Class-tested in courses and workshops across North America, the book is ideal for professionals, researchers, university professors, and graduate students with an interest in, or responsibilities related to, road safety. This book also: · Presents for the first time a powerful analytical tool for road safety researchers and practitioners · Includes problems and solutions in each chapter as well as data and spreadsheets for running models and PowerPoint presentation slides · Features pedagogy well-suited for graduate courses and workshops including problems, solutions, and PowerPoint presentations · Equips readers to perform all analyses on a spreadsheet without requiring mastery of complex and costly software · Emphasizes understanding without esoteric mathematics · Makes assumptions visible and explains their role and consequences |
| association and causation math: Causality from the Point of View of Statistics José A. Ferreira, 2023-08-21 Most are familiar with the adage correlation does not imply causation. Since much of science is concerned with problems of causality and statistics is so widely used in research, one may wonder whether statistics possesses the tools to study such problems and contribute to their resolution. These were the questions posed over thirty years ago by Pearl, Robins, Rubin, Shafer, etc. when they set out to incorporate notions of causality into statistics theory and develop methods for estimating causal relationships. Since then, the schools of statistical causality they founded have produced interesting results and methods that help us think about causality and are potentially useful in real-life problems. Yet, despite its appeal, statistical causality is still disregarded by many mainstream statisticians, and its methods are not widely known. In part this is explained by the unorthodox and apparently disparate character of the various schools, in particular by the distinct languages they developed and that are not readily accessible. Thus, even some advanced researchers seemed startled by things like Rubin's counterfactuals that in one guise or another appear in all theories but that seem potentially incompatible with Kolmogorov's formalism, the very foundation of statistics. It turns out that statistical causality is firmly rooted in Kolmogorov's axiomatization of probability as the elements required by it are essentially those proposed a century ago by Steinhaus, and, perhaps surprisingly, that statistics has always engaged with causality. The present book makes this plain, providing a basis for statistical causality that subsumes and reconciles the theories of all other schools and that to a mainstream statistician will appear entirely familiar and natural. |
| association and causation math: The Chicago Guide to Writing about Multivariate Analysis, Second Edition Jane E. Miller, 2013-09-26 Many different people, from social scientists to government agencies to business professionals, depend on the results of multivariate models to inform their decisions. Researchers use these advanced statistical techniques to analyze relationships among multiple variables, such as how exercise and weight relate to the risk of heart disease, or how unemployment and interest rates affect economic growth. Yet, despite the widespread need to plainly and effectively explain the results of multivariate analyses to varied audiences, few are properly taught this critical skill. The Chicago Guide to Writing about Multivariate Analysis is the book researchers turn to when looking for guidance on how to clearly present statistical results and break through the jargon that often clouds writing about applications of statistical analysis. This new edition features even more topics and real-world examples, making it the must-have resource for anyone who needs to communicate complex research results. For this second edition, Jane E. Miller includes four new chapters that cover writing about interactions, writing about event history analysis, writing about multilevel models, and the “Goldilocks principle” for choosing the right size contrast for interpreting results for different variables. In addition, she has updated or added numerous examples, while retaining her clear voice and focus on writers thinking critically about their intended audience and objective. Online podcasts, templates, and an updated study guide will help readers apply skills from the book to their own projects and courses. This continues to be the only book that brings together all of the steps involved in communicating findings based on multivariate analysis—finding data, creating variables, estimating statistical models, calculating overall effects, organizing ideas, designing tables and charts, and writing prose—in a single volume. When aligned with Miller’s twelve fundamental principles for quantitative writing, this approach will empower readers—whether students or experienced researchers—to communicate their findings clearly and effectively. |
| association and causation math: Causal Inference Scott Cunningham, 2021-01-26 An accessible, contemporary introduction to the methods for determining cause and effect in the Social Sciences “Causation versus correlation has been the basis of arguments—economic and otherwise—since the beginning of time. Causal Inference: The Mixtape uses legit real-world examples that I found genuinely thought-provoking. It’s rare that a book prompts readers to expand their outlook; this one did for me.”—Marvin Young (Young MC) Causal inference encompasses the tools that allow social scientists to determine what causes what. In a messy world, causal inference is what helps establish the causes and effects of the actions being studied—for example, the impact (or lack thereof) of increases in the minimum wage on employment, the effects of early childhood education on incarceration later in life, or the influence on economic growth of introducing malaria nets in developing regions. Scott Cunningham introduces students and practitioners to the methods necessary to arrive at meaningful answers to the questions of causation, using a range of modeling techniques and coding instructions for both the R and the Stata programming languages. |
| association and causation math: Elements of Causal Inference Jonas Peters, Dominik Janzing, Bernhard Scholkopf, 2017-11-29 A concise and self-contained introduction to causal inference, increasingly important in data science and machine learning. The mathematization of causality is a relatively recent development, and has become increasingly important in data science and machine learning. This book offers a self-contained and concise introduction to causal models and how to learn them from data. After explaining the need for causal models and discussing some of the principles underlying causal inference, the book teaches readers how to use causal models: how to compute intervention distributions, how to infer causal models from observational and interventional data, and how causal ideas could be exploited for classical machine learning problems. All of these topics are discussed first in terms of two variables and then in the more general multivariate case. The bivariate case turns out to be a particularly hard problem for causal learning because there are no conditional independences as used by classical methods for solving multivariate cases. The authors consider analyzing statistical asymmetries between cause and effect to be highly instructive, and they report on their decade of intensive research into this problem. The book is accessible to readers with a background in machine learning or statistics, and can be used in graduate courses or as a reference for researchers. The text includes code snippets that can be copied and pasted, exercises, and an appendix with a summary of the most important technical concepts. |
| association and causation math: Research Methods in Education Joseph Check, Russell K. Schutt, 2011-10-27 Research Methods in Education introduces research methods as an integrated set of techniques for investigating questions about the educational world. This lively, innovative text helps students connect technique and substance, appreciate the value of both qualitative and quantitative methodologies, and make ethical research decisions. It weaves actual research stories into the presentation of research topics, and it emphasizes validity, authenticity, and practical significance as overarching research goals. The text is divided into three sections: Foundations of Research (5 chapters), Research Design and Data Collection (7 chapters), and Analyzing and Reporting Data (3 chapters). This tripartite conceptual framework honors traditional quantitative approaches while reflecting the growing popularity of qualitative studies, mixed method designs, and school-based techniques. This approach provides a comprehensive, conceptually unified, and well-written introduction to the exciting but complex field of educational research. |
| association and causation math: Statistical Models David Freedman, 2005-08-08 This lively and engaging textbook provides the knowledge required to read empirical papers in the social and health sciences, as well as the techniques needed to build statistical models. The author explains the basic ideas of association and regression, and describes the current models that link these ideas to causality. He focuses on applications of linear models, including generalized least squares and two-stage least squares. The bootstrap is developed as a technique for estimating bias and computing standard errors. Careful attention is paid to the principles of statistical inference. There is background material on study design, bivariate regression, and matrix algebra. To develop technique, there are computer labs, with sample computer programs. The book's discussion is organized around published studies, as are the numerous exercises - many of which have answers included. Relevant papers reprinted at the back of the book are thoroughly appraised by the author. |
| association and causation math: Statistics and Causality Wolfgang Wiedermann, Alexander von Eye, 2016-05-20 b”STATISTICS AND CAUSALITYA one-of-a-kind guide to identifying and dealing with modern statistical developments in causality Written by a group of well-known experts, Statistics and Causality: Methods for Applied Empirical Research focuses on the most up-to-date developments in statistical methods in respect to causality. Illustrating the properties of statistical methods to theories of causality, the book features a summary of the latest developments in methods for statistical analysis of causality hypotheses. The book is divided into five accessible and independent parts. The first part introduces the foundations of causal structures and discusses issues associated with standard mechanistic and difference-making theories of causality. The second part features novel generalizations of methods designed to make statements concerning the direction of effects. The third part illustrates advances in Granger-causality testing and related issues. The fourth part focuses on counterfactual approaches and propensity score analysis. Finally, the fifth part presents designs for causal inference with an overview of the research designs commonly used in epidemiology. Statistics and Causality: Methods for Applied Empirical Research also includes: New statistical methodologies and approaches to causal analysis in the context of the continuing development of philosophical theories End-of-chapter bibliographies that provide references for further discussions and additional research topics Discussions on the use and applicability of software when appropriate Statistics and Causality: Methods for Applied Empirical Research is an ideal reference for practicing statisticians, applied mathematicians, psychologists, sociologists, logicians, medical professionals, epidemiologists, and educators who want to learn more about new methodologies in causal analysis. The book is also an excellent textbook for graduate-level courses in causality and qualitative logic. |
| association and causation math: The Chicago Guide to Writing About Numbers Jane E. Miller, 2015-04-09 For students, scientists, journalists and others, a comprehensive guide to communicating data clearly and effectively. Acclaimed by scientists, journalists, faculty, and students, The Chicago Guide to Writing about Numbers has helped thousands communicate data clearly and effectively. It offers a much-needed bridge between good quantitative analysis and clear expository writing, using straightforward principles and efficient prose. With this new edition, Jane Miller draws on a decade of additional experience and research, expanding her advice on reaching everyday audiences and further integrating non-print formats. Miller, an experienced teacher of research methods, statistics, and research writing, opens by introducing a set of basic principles for writing about numbers, then presents a toolkit of techniques that can be applied to prose, tables, charts, and presentations. She emphasizes flexibility, showing how different approaches work for different kinds of data and different types of audiences. The second edition adds a chapter on writing about numbers for lay audiences, explaining how to avoid overwhelming readers with jargon and technical issues. Also new is an appendix comparing the contents and formats of speeches, research posters, and papers, to teach writers how to create all three types of communication without starting each from scratch. An expanded companion website includes new multimedia resources such as slide shows and podcasts that illustrate the concepts and techniques, along with an updated study guide of problem sets and suggested course extensions. This continues to be the only book that brings together all the tasks that go into writing about numbers, integrating advice on finding data, calculating statistics, organizing ideas, designing tables and charts, and writing prose all in one volume. Field-tested with students and professionals alike, this is the go-to guide for everyone who writes or speaks about numbers. |
| association and causation math: All of Statistics Larry Wasserman, 2013-12-11 Taken literally, the title All of Statistics is an exaggeration. But in spirit, the title is apt, as the book does cover a much broader range of topics than a typical introductory book on mathematical statistics. This book is for people who want to learn probability and statistics quickly. It is suitable for graduate or advanced undergraduate students in computer science, mathematics, statistics, and related disciplines. The book includes modern topics like non-parametric curve estimation, bootstrapping, and classification, topics that are usually relegated to follow-up courses. The reader is presumed to know calculus and a little linear algebra. No previous knowledge of probability and statistics is required. Statistics, data mining, and machine learning are all concerned with collecting and analysing data. |
| association and causation math: Because Without Cause Marc Lange, 2017 Not all scientific explanations work by describing causal connections between events or the world's overall causal structure. In addition, mathematicians regard some proofs as explaining why the theorems being proved do in fact hold. This book proposes new philosophical accounts of many kinds of non-causal explanations in science and mathematics. |
| association and causation math: Statistical Models in Epidemiology, the Environment, and Clinical Trials M.Elizabeth Halloran, Donald Berry, 1999-10-29 This IMA Volume in Mathematics and its Applications STATISTICAL MODELS IN EPIDEMIOLOGY, THE ENVIRONMENT,AND CLINICAL TRIALS is a combined proceedings on Design and Analysis of Clinical Trials and Statistics and Epidemiology: Environment and Health. This volume is the third series based on the proceedings of a very successful 1997 IMA Summer Program on Statistics in the Health Sciences. I would like to thank the organizers: M. Elizabeth Halloran of Emory University (Biostatistics) and Donald A. Berry of Duke University (Insti tute of Statistics and Decision Sciences and Cancer Center Biostatistics) for their excellent work as organizers of the meeting and for editing the proceedings. I am grateful to Seymour Geisser of University of Minnesota (Statistics), Patricia Grambsch, University of Minnesota (Biostatistics); Joel Greenhouse, Carnegie Mellon University (Statistics); Nicholas Lange, Harvard Medical School (Brain Imaging Center, McLean Hospital); Barry Margolin, University of North Carolina-Chapel Hill (Biostatistics); Sandy Weisberg, University of Minnesota (Statistics); Scott Zeger, Johns Hop kins University (Biostatistics); and Marvin Zelen, Harvard School of Public Health (Biostatistics) for organizing the six weeks summer program. I also take this opportunity to thank the National Science Foundation (NSF) and the Army Research Office (ARO), whose financial support made the workshop possible. Willard Miller, Jr. |
| association and causation math: Essential Math for AI Hala Nelson, 2023-01-04 Companies are scrambling to integrate AI into their systems and operations. But to build truly successful solutions, you need a firm grasp of the underlying mathematics. This accessible guide walks you through the math necessary to thrive in the AI field such as focusing on real-world applications rather than dense academic theory. Engineers, data scientists, and students alike will examine mathematical topics critical for AI--including regression, neural networks, optimization, backpropagation, convolution, Markov chains, and more--through popular applications such as computer vision, natural language processing, and automated systems. And supplementary Jupyter notebooks shed light on examples with Python code and visualizations. Whether you're just beginning your career or have years of experience, this book gives you the foundation necessary to dive deeper in the field. Understand the underlying mathematics powering AI systems, including generative adversarial networks, random graphs, large random matrices, mathematical logic, optimal control, and more Learn how to adapt mathematical methods to different applications from completely different fields Gain the mathematical fluency to interpret and explain how AI systems arrive at their decisions |
| association and causation math: Middle Math Mary B. Eron, Sidney L. Rachlin, 2015-10-01 (Orginally published in 2005) This monograph represents the work of many mathematics teacher educators explored the content knowledge and pedagogical knowledge that make up the middle grades learning experience. The middle grades remains a unique period of time in students' development and as such provides both challenges and promising opportunities for those who prepare teachers of middle grades mathematics. This work is the final product of an exciting NSF supported endeavor that gathered leaders in the field and explored curriculum, case studies of program models at several institutions, as well as issue papers on such key topics as assessment, technology, and preparing culturally responsive teachers. AMTE hopes this monograph will stimulate discussion and bring attention to this critical period of schooling. |
| association and causation math: Statistics Robin H. Lock, Patti Frazer Lock, Kari Lock Morgan, Eric F. Lock, Dennis F. Lock, 2020-10-13 Statistics: Unlocking the Power of Data, 3rd Edition is designed for an introductory statistics course focusing on data analysis with real-world applications. Students use simulation methods to effectively collect, analyze, and interpret data to draw conclusions. Randomization and bootstrap interval methods introduce the fundamentals of statistical inference, bringing concepts to life through authentically relevant examples. More traditional methods like t-tests, chi-square tests, etc. are introduced after students have developed a strong intuitive understanding of inference through randomization methods. While any popular statistical software package may be used, the authors have created StatKey to perform simulations using data sets and examples from the text. A variety of videos, activities, and a modular chapter on probability are adaptable to many classroom formats and approaches. |
| association and causation math: The Common Core Mathematics Companion: The Standards Decoded, High School Frederick L. Dillon, W. Gary Martin, Basil M. Conway IV, Marilyn E. Strutchens, 2017-09-12 Your User’s Guide to the Mathematics Standards When it comes to mathematics, standards aligned is achievement aligned... In the short time since The Common Core Mathematics Companions for grades K–2, 3–5 and 6–8 burst on the scene, they have been lauded as the best resources for making critical mathematics ideas easy to teach. With this brand-new volume, high school mathematics success is at your fingertips. Page by page, the authors lay out the pieces of an in-depth explanation, including The mathematical progression of each conceptual category, starting with modeling as a unifying theme, and moving through number & quantity, algebra, functions, geometry, and statistics and probability, building from the 8th grade standards The mathematics embedded in each conceptual category for a deeper understanding of the content How standards connect within and across domains, and to previous grade standards, so teachers can better appreciate how they relate How standards connect with the standards for mathematical practice, with a focus on modeling as a unifying theme Example tasks, progressions of tasks, and descriptions of what teachers and students should be doing to foster deep learning The Common Core Mathematics Companion: The Standards Decoded, High School has what every high school teacher needs to provide students with the foundation for the concepts and skills they will be expected to know . |
| association and causation math: Concepts of Mathematical Modeling Walter J. Meyer, 2012-10-23 Appropriate for undergraduate and graduate students, this text features independent sections that illustrate the most important principles of mathematical modeling, a variety of applications, and classic models. Students with a solid background in calculus and some knowledge of probability and matrix theory will find the material entirely accessible. The range of subjects includes topics from the physical, biological, and social sciences, as well as those of operations research. Discussions cover related mathematical tools and the historical eras from which the applications are drawn. Each section is preceded by an abstract and statement of prerequisites, and answers or hints are provided for selected exercises. 1984 edition. |
| association and causation math: Foundations of Epidemiology Marit L. Bovbjerg, 2020-10 Foundations of Epidemiology is an open access, introductory epidemiology text intended for students and practitioners in public or allied health fields. It covers epidemiologic thinking, causality, incidence and prevalence, public health surveillance, epidemiologic study designs and why we care about which one is used, measures of association, random error and bias, confounding and effect modification, and screening. Concepts are illustrated with numerous examples drawn from contemporary and historical public health issues. |
| association and causation math: Counterfactuals and Causal Inference Stephen L. Morgan, Christopher Winship, 2014-11-17 In this second edition of Counterfactuals and Causal Inference, completely revised and expanded, the essential features of the counterfactual approach to observational data analysis are presented with examples from the social, demographic, and health sciences. Alternative estimation techniques are first introduced using both the potential outcome model and causal graphs; after which, conditioning techniques, such as matching and regression, are presented from a potential outcomes perspective. For research scenarios in which important determinants of causal exposure are unobserved, alternative techniques, such as instrumental variable estimators, longitudinal methods, and estimation via causal mechanisms, are then presented. The importance of causal effect heterogeneity is stressed throughout the book, and the need for deep causal explanation via mechanisms is discussed. |
| association and causation math: Causal Inference Miquel A. Hernan, James M. Robins, 2019-07-07 The application of causal inference methods is growing exponentially in fields that deal with observational data. Written by pioneers in the field, this practical book presents an authoritative yet accessible overview of the methods and applications of causal inference. With a wide range of detailed, worked examples using real epidemiologic data as well as software for replicating the analyses, the text provides a thorough introduction to the basics of the theory for non-time-varying treatments and the generalization to complex longitudinal data. |
| association and causation math: Your Mathematics Standards Companion, High School Frederick L. Dillon, W. Gary Martin, Basil M. Conway IV, Marilyn E. Strutchens, 2018-02-06 Transforming the standards into learning outcomes just got a lot easier In this resource, you can see in an instant how teaching to your state standards should look and sound in the classroom. Under the premise that math is math, the authors provide a Cross-Referencing Index for states implementing their own specific mathematics standards, allowing you to see and understand which page number to turn to for standards-based teaching ideas. It’s all here, page by page: The mathematical progression of each conceptual category, starting with modeling as a unifying theme and moving through number and quantity, algebra, functions, geometry, and statistics and probability, building from eighth-grade standards The mathematics embedded in each conceptual category for a deeper understanding of the content How standards connect within and across domains and to previous grade standards, so teachers can better appreciate how they relate How content standards connect with the standards for mathematical practice, with a focus on modeling as a unifying theme Example tasks, progressions of tasks, and descriptions of what teachers and students should be doing to foster deep learning Your Mathematics Standards Companion is your one-stop guide for teaching, planning, assessing, collaborating, and designing powerful high school mathematics curriculum in any state or district. |
| association and causation math: Special Issue, Current and Future Challenges in the Applications of Mathematics , 1998 |
| association and causation math: Making Sense of Numbers Jane E. Miller, 2021-08-30 Making Sense of Numbers teaches students the skills they need to be both consumers and producers of quantitative research: able to read about, collect, calculate, and communicate numeric information for both everyday tasks and school or work assignments. The text teaches how to avoid making common errors of reasoning, calculation, or interpretation by introducing a systematic approach to working with numbers, showing students how to figure out what a particular number means. The text also demonstrates why it is important to apply a healthy dose of skepticism to the numbers we all encounter, so that we can understand how those numbers can (and cannot) be interpreted in their real-world context. Jane E. Miller uses annotated examples on a wide variety of topics to illustrate how to use new terms, concepts, and approaches to working with numbers. End-of-chapter engagement activities designed based on Miller’s three decades of teaching experience can be used in class or as homework assignments, with some for students to do individually and others intended for group discussion. The book is ideally suited for a range of courses, including quantitative reasoning, research methods, basic statistics, data analysis, and communicating quantitative information. An instructor website for the book includes a test bank, editable PowerPoint slides, and tables and figures from the book. |
| association and causation math: Methods in Social Epidemiology J. Michael Oakes, Jay S. Kaufman, 2006-05-11 Social epidemiology is the study of how social interactions—social norms, laws, institutions, conventia, social conditions and behavior—affect the health of populations. This practical, comprehensive introduction to methods in social epidemiology is written by experts in the field. It is perfectly timed for the growth in interest among those in public health, community health, preventive medicine, sociology, political science, social work, and other areas of social research. Topics covered are: Introduction: Advancing Methods in Social Epidemiology The History of Methods of Social Epidemilogy to 1965 Indicators of Socioeconomic Position Measuring and Analyzing 'Race' Racism and Racial Discrimination Measuring Poverty Measuring Health Inequalities A Conceptual Framework for Measuring Segregation and its Association with Population Outcomes Measures of Residential Community Contexts Using Census Data to Approximate Neighborhood Effects Community-based Participatory Research: Rationale and Relevance for Social Epidemiology Network Methods in Social Epidemiology Identifying Social Interactions: A Review, Multilevel Studies Experimental Social Epidemiology: Controlled Community Trials Propensity Score Matching Methods for Social Epidemiology Natural Experiments and Instrumental Variable Analyses in Social Epidemiology and Using Causal Diagrams to Understand Common Problems in Social Epidemiology. Publication of this highly informative textbook clearly reflects the coming of age of many social epidemiology methods, the importance of which rests on their potential contribution to significantly improving the effectiveness of the population-based approach to prevention. This book should be of great interest not only to more advanced epidemiology students but also to epidemiologists in general, particularly those concerned with health policy and the translation of epidemiologic findings into public health practice. The cause of achieving a ‘more complete’ epidemiology envisaged by the editors has been significantly advanced by this excellent textbook. —Moyses Szklo, professor of epidemiology and editor-in-chief, American Journal of Epidemiology, Johns Hopkins University Social epidemiology is a comparatively new field of inquiry that seeks to describe and explain the social and geographic distribution of health and of the determinants of health. This book considers the major methodological challenges facing this important field. Its chapters, written by experts in a variety of disciplines, are most often authoritative, typically provocative, and often debatable, but always worth reading. —Stephen W. Raudenbush, Lewis-Sebring Distinguished Service Professor, Department of Sociology, University of Chicago The roadmap for a new generation of social epidemiologists. The publication of this treatise is a significant event in the history of the discipline. —Ichiro Kawachi, professor of social epidemiology, Department of Society, Human Development, and Health, Harvard University Methods in Social Epidemiology not only illuminates the difficult questions that future generations of social epidemiologists must ask, it also identifies the paths they must boldly travel in the pursuit of answers, if this exciting interdisciplinary science is to realize its full potential. This beautifully edited volume appears at just the right moment to exert a profound influence on the field. —Sherman A. James, Susan B. King Professor of Public Policy Studies, professor of Community and Family Medicine, professor of African-American Studies, Duke University |
| association and causation math: The Psychology of Music Diana Deutsch, 2012-10-29 The Psychology of Music serves as an introduction to an interdisciplinary field in psychology, which focuses on the interpretation of music through mental function. This interpretation leads to the characterization of music through perceiving, remembering, creating, performing, and responding to music. In particular, the book provides an overview of the perception of musical tones by discussing different sound characteristics, like loudness, pitch and timbre, together with interaction between these attributes. It also discusses the effect of computer resources on the psychological study of music through computational modeling. In this way, models of pitch perception, grouping and voice separation, and harmonic analysis were developed. The book further discusses musical development in social and emotional contexts, and it presents ways that music training can enhance the singing ability of an individual. The book can be used as a reference source for perceptual and cognitive psychologists, neuroscientists, and musicians. It can also serve as a textbook for advanced courses in the psychological study of music. - Encompasses the way the brain perceives, remembers, creates, and performs music - Contributions from the top international researchers in perception and cognition of music - Designed for use as a textbook for advanced courses in psychology of music |
| association and causation math: An Invitation to Abstract Mathematics Béla Bajnok, 2020-10-27 This undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics. Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise. This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts. From reviews of the first edition: Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. ... The book can be used as a text for traditional transition or structure courses ... but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA Reviews The style of writing is careful, but joyously enthusiastic.... The author’s clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background ... and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATH |
| association and causation math: Regression and Machine Learning for Education Sciences Using R Cody Dingsen, 2024-11-01 This book provides a conceptual introduction to regression analysis and machine learning and their applications in education research. It discusses their diverse applications, including its role in predicting future events based on the current data or explaining why some phenomena occur. These identified important predictors provide data-based evidence for educational and psychological decision-making. Offering an applications-oriented approach while mapping out fundamental methodological developments, this book lays a sound foundation for understanding essential regression and machine learning concepts for data analytics. The first part of the book discusses regression analysis and provides a sturdy foundation to understand the logic of machine learning. With each chapter, the discussion and development of each statistical concept and data analytical technique is presented from an applied perspective, with the statistical results providing insights into decisions and solutions to problems using R. Based on practical examples, and written in a concise and accessible style, the book is learner-centric and does a remarkable job in breaking down complex concepts. Regression and Machine Learning for Education Sciences Using R is primarily for students or practitioners in education and psychology, although individuals from other related disciplines can also find the book beneficial. The dataset and examples used in the book are from an educational setting, and students will find that this text provides a good preparation ground for studying more statistical and data analytical materials. |
| association and causation math: Mendelian Randomization Stephen Burgess, Simon G. Thompson, 2015-03-06 Presents the Terminology and Methods of Mendelian Randomization for Epidemiological StudiesMendelian randomization uses genetic instrumental variables to make inferences about causal effects based on observational data. It, therefore, can be a reliable way of assessing the causal nature of risk factors, such as biomarkers, for a wide range of disea |
| association and causation math: Stats with Cats Charles Kufs, 2011 When you took statistics in school, your instructor gave you specially prepared datasets, told you what analyses to perform, and checked your work to see if it was correct. Once you left the class, though, you were on your own. Did you know how to create and prepare a dataset for analysis? Did you know how to select and generate appropriate graphics and statistics? Did you wonder why you were forced to take the class and when you would ever use what you learned? That's where Stats with Cats can help you out. The book will show you: How to decide what you should put in your dataset and how to arrange the data. How to decide what graphs and statistics to produce for your data. How you can create a statistical model to answer your data analysis questions. The book also provides enough feline support to minimize any stress you may experience. Charles Kufs has been crunching numbers for over thirty years, first as a hydrogeologist, and since the 1990s as a statistician. He is certified as a Six Sigma Green Belt by the American Society for Quality. He currently works as a statistician for the federal government and he is here to help you. |
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