Understanding Advanced Statistical Methods

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ISBN 9781466512108
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    • Shows students how a statistical model is a recipe for producing random data
    • Provides a self-contained treatment of mathematical statistics, requiring no prerequisite of calculus and assuming familiarity with algebra, functions, graphs, and spreadsheet software
    • Defines and uses the "process" terminology
    • Helps students understand how logical conclusions follow from the assumptions—always encouraging them to ask why
    • Teaches Bayesian methods before classical (frequentist) methods, which provides a seamless transition from probability to likelihood to Bayes as well as enables a well-rounded and thoughtful discussion on the frequentist-based confidence interval and hypothesis testing concepts

    Pedagogical Features

    • Provides vocabulary terms in bold, with definitions summarized at the end of each chapter
    • Presents important formulas at the end of every chapter and gives reasons for each step of the derivations
    • Includes end-of-chapter exercises essential to understanding the material
    • Offers computer code, sample quizzes, exams, and other supplements on the book’s website

    Solutions manual available upon qualifying course adoption


      Providing a much-needed bridge between elementary statistics courses and advanced research methods courses, Understanding Advanced Statistical Methods helps students grasp the fundamental assumptions and machinery behind sophisticated statistical topics, such as logistic regression, maximum likelihood, bootstrapping, nonparametrics, and Bayesian methods. The book teaches students how to properly model, think critically, and design their own studies to avoid common errors. It leads them to think differently not only about math and statistics but also about general research and the scientific method.

      With a focus on statistical models as producers of data, the book enables students to more easily understand the machinery of advanced statistics. It also downplays the "population" interpretation of statistical models and presents Bayesian methods before frequentist ones. Requiring no prior calculus experience, the text employs a "just-in-time" approach that introduces mathematical topics, including calculus, where needed. Formulas throughout the text are used to explain why calculus and probability are essential in statistical modeling. The authors also intuitively explain the theory and logic behind real data analysis, incorporating a range of application examples from the social, economic, biological, medical, physical, and engineering sciences.

      Enabling your students to answer the why behind statistical methods, this text teaches them how to successfully draw conclusions when the premises are flawed. It empowers them to use advanced statistical methods with confidence and develop their own statistical recipes. Ancillary materials are available on the book’s website.

      Table of Contents

      Introduction: Probability, Statistics, and Science
      Reality, Nature, Science, and Models
      Statistical Processes: Nature, Design and Measurement, and Data
      Deterministic Models
      Purely Probabilistic Statistical Models
      Statistical Models with Both Deterministic and Probabilistic Components
      Statistical Inference
      Good and Bad Models
      Uses of Probability Models

      Random Variables and Their Probability Distributions
      Types of Random Variables: Nominal, Ordinal, and Continuous
      Discrete Probability Distribution Functions
      Continuous Probability Distribution Functions
      Some Calculus–Derivatives and Least Squares
      More Calculus–Integrals and Cumulative Distribution Functions

      Probability Calculation and Simulation
      Analytic Calculations, Discrete and Continuous Cases
      Simulation-Based Approximation
      Generating Random Numbers

      Identifying Distributions
      Identifying Distributions from Theory Alone
      Using Data: Estimating Distributions via the Histogram
      Quantiles: Theoretical and Data-Based Estimates
      Using Data: Comparing Distributions via the Quantile–Quantile Plot
      Effect of Randomness on Histograms and qq Plots

      Conditional Distributions and Independence
      Conditional Discrete Distributions
      Estimating Conditional Discrete Distributions
      Conditional Continuous Distributions
      Estimating Conditional Continuous Distributions

      Marginal Distributions, Joint Distributions, Independence, and Bayes’ Theorem
      Joint and Marginal Distributions
      Estimating and Visualizing Joint Distributions
      Conditional Distributions from Joint Distributions
      Joint Distributions When Variables Are Independent
      Bayes’ Theorem

      Sampling from Populations and Processes
      Sampling from Populations
      Critique of the Population Interpretation of Probability Models
      The Process Model versus the Population Model
      Independent and Identically Distributed Random Variables and Other Models
      Checking the iid Assumption

      Expected Value and the Law of Large Numbers
      Discrete Case
      Continuous Case
      Law of Large Numbers
      Law of Large Numbers for the Bernoulli Distribution
      Keeping the Terminology Straight: Mean, Average, Sample Mean, Sample Average, and Expected Value
      Bootstrap Distribution and the Plug-In Principle

      Functions of Random Variables: Their Distributions and Expected Values
      Distributions of Functions: The Discrete Case
      Distributions of Functions: The Continuous Case
      Expected Values of Functions and the Law of the Unconscious Statistician
      Linearity and Additivity Properties
      Nonlinear Functions and Jensen’s Inequality
      Standard Deviation, Mean Absolute Deviation, and Chebyshev’s Inequality
      Linearity Property of Variance
      Skewness and Kurtosis

      Distributions of Totals
      Additivity Property of Variance
      Covariance and Correlation
      Central Limit Theorem

      Estimation: Unbiasedness, Consistency, and Efficiency
      Biased and Unbiased Estimators
      Bias of the Plug-In Estimator of Variance
      Removing the Bias of the Plug-In Estimator of Variance
      The Joke Is on Us: The Standard Deviation Estimator Is Biased after All
      Consistency of Estimators
      Efficiency of Estimators

      Likelihood Function and Maximum Likelihood Estimates
      Likelihood Function
      Maximum Likelihood Estimates
      Wald Standard Error

      Bayesian Statistics
      Introduction: Play a Game with Hans!
      Prior Information and Posterior Knowledge
      Case of the Unknown Survey
      Bayesian Statistics: The Overview
      Bayesian Analysis of the Bernoulli Parameter
      Bayesian Analysis Using Simulation
      What Good Is Bayes?

      Frequentist Statistical Methods
      Large-Sample Approximate Frequentist Confidence Interval for the Process Mean
      What Does Approximate Really Mean for an Interval Range?
      Comparing the Bayesian and Frequentist Paradigms

      Are Your Results Explainable by Chance Alone?
      What Does by Chance Alone Mean?
      The p-Value
      The Extremely Ugly "pv ≤ 0.05" Rule of Thumb

      Chi-Squared, Student’s t, and F-Distributions, with Applications
      Linearity and Additivity Properties of the Normal Distribution
      Effect of Using an Estimate of s
      Chi-Squared Distribution
      Frequentist Confidence Interval for s
      Student’s t-Distribution
      Comparing Two Independent Samples Using a Confidence Interval
      Comparing Two Independent Homoscedastic Normal Samples via Hypothesis Testing
      F-Distribution and ANOVA Test
      F-Distribution and Comparing Variances of Two Independent Groups

      Likelihood Ratio Tests
      Likelihood Ratio Method for Constructing Test Statistics
      Evaluating the Statistical Significance of Likelihood Ratio Test Statistics
      Likelihood Ratio Goodness-of-Fit Tests
      Cross-Classification Frequency Tables and Tests of Independence
      Comparing Non-Nested Models via the AIC Statistic

      Sample Size and Power
      Choosing a Sample Size for a Prespecified Accuracy Margin
      Noncentral Distributions
      Choosing a Sample Size for Prespecified Power
      Post Hoc Power: A Useless Statistic

      Robustness and Nonparametric Methods
      Nonparametric Tests Based on the Rank Transformation
      Randomization Tests
      Level and Power Robustness
      Bootstrap Percentile-t Confidence Interval

      Final Words


      Vocabulary, Formula Summaries, and Exercises appear at the end of each chapter.

      Author Bio(s)

      Peter H. Westfall is the Paul Whitfield Horn Professor of Statistics and James Niver Professor of Information Systems and Quantitative Sciences at Texas Tech University. A Fellow of the ASA and the AAAS, Dr. Westfall has published several books and over 100 papers on statistical theory and methods. He also has won several teaching awards and is the former editor of The American Statistician. He earned a PhD in statistics from the University of California, Davis.

      Kevin S.S. Henning is a clinical assistant professor of business analysis in the Department of Economics and International Business at Sam Houston State University, where he teaches business statistics and forecasting. He earned a PhD in business statistics from Texas Tech University.

      Editorial Reviews

      "… full of interesting insights and excellent examples and explanations for essential basic statistical concepts. The use of thought experiments; the detailed algebraic developments of proofs; and the explanations of frequentist and Bayesian statistics, confidence intervals, hypothesis testing, and so on, are all first rate. … a solid teaching resource."
      Australian & New Zealand Journal of Statistics, 2014

      "… useful as a prerequisite for advanced study of statistical analysis, such as regression, experimental design, survival analysis, and categorical data analysis … examples in this book seem very useful and may help expand the view of newcomers to statistics."
      Biometrics, June 2014

      "This book contains just as many formulas as other statistics texts, but with intuitive, engaging, insightful, and irreverent explanations … the authors strive mightily to part the curtain that hides the fundamentals of statistical thinking from most students. … The book has 20 chapters that cover the usual topics, and more, in an undergraduate/graduate math stat text; it is suitable for a fast-paced semester course offered to serious students. The ‘and more’ refers to the strong emphasis throughout the book on thoughtful applications in a wide variety of disciplines. … The coverage of mathematical statistics is extensive and benefits from a substantial effort by the authors to explain the intuition motivating the procedures and the correct interpretation of specific results. … A companion Web site has a wealth of material useful for the instructor and students. … the text represents a successful effort by the authors to advance and improve the statistics education paradigm for courses offered to upper-level undergraduate and graduate students."
      The American Statistician, May 2014

      "There is a gap between elementary statistics courses and advanced research techniques. This gap is reflected by difficulties in linking statistical theory with its application in the real world. This book is an ideal way to overcome this problem. …
      The main advantage of this book is the possibility to achieve advanced research skills. The theory behind data analysis is well explained, using plenty of real examples from social, economic, medical, physical and engineering sciences. The theory and application are well balanced and very well linked. All examples are illustrated in MS Excel.
      This book helps to teach students to explore statistics more deeply, avoiding the typical trap of students learning little about the applications of what they are studying and why they are doing it. I think this book will be very useful in the sense that students will be forced to think differently about things, not only about math and statistics, but also about research and the scientific method.
      The reviewer enjoyed reading the book and it is worth emphasising its usefulness for teachers, students and researchers."
      —Božidar V. Popovic, Journal of Applied Statistics, 2014

      "The book covers the content of a typical undergraduate math stat text, but with much more thought to application than a typical text. It appears to be close to Rice’s text (Mathematical Statistics and Data Analysis) in spirit and level, but perhaps comes closer to that spirit than Rice’s. It would be worth considering for a course using Rice. I also recommend it as a reference for anyone teaching applied statistics."
      —Martha K. Smith, Professor Emerita of Mathematics, University of Texas at Austin

      "I work with scientists who are pioneers in their fields and their ignorance of statistical concepts never ceases to amaze me. I believe most of this can be traced to the way we teach statistics to non-statisticians: as a bag of tools rather than a systematic way to think about data collection and analysis. This book is unique in the way it approaches this topic. It does not subscribe to the cookbook template of teaching statistics but focuses instead on understanding the distinction between the observed data and the mechanisms that generated it. This focus allows a better distinction between models, parameters, and estimates and should help pave a way to instill statistical thinking to undergraduate students."
      —Mithat Gönen, Memorial Sloan-Kettering Cancer Center

      "Understanding Advanced Statistical Methods is an excellent source for the curious student. The book introduces a novel approach to learning statistics by providing comprehensive coverage of concepts in a captivating framework. Students are not only encouraged to understand the intuition and structure behind the concepts, but also motivated to think seriously about the pertinent questions before they ask. Therefore, the book strives to build a solid background in fundamental concepts and to equip students with the necessary skills so that they can expand their toolbox in their future endeavors. The book will no doubt be the standard reference in advanced statistics courses and bring about profound changes in how statistics should be taught."
      —Ozzy Akay, Assistant Professor, Texas Tech University

      "Don't let the authors' exuberant and iconoclastic style fool you into thinking that this book is not a serious text. It definitely is. The style has a purpose—to romp around the field's sacred cows and show the reader as quickly as possible the real working principles behind how statistical methods are developed and some of the methods’ most important applications. In that sense, the subject of the book truly is theoretical statistics, but both the motivation and the presentation are so thoroughly grounded in practice that many readers will see it as a practical guide. But the authors don’t intend for it to be a statistical cheat sheet: each of their many engaging and illuminating examples points forward to more that could be studied, and invites readers to pursue those studies. This isn’t the last statistics textbook students will ever need, but it should be the first."
      —Randy Tobias, Director, Linear Models R&D, SAS Institute Inc.

      Downloads / Updates

      Resource OS Platform Updated Description Instructions
      Cross Platform April 09, 2013 Author web site click on http://www.understandingadvancedstatisticalmethods.com/