1st Edition

Understanding Advanced Statistical Methods

By Peter Westfall, Kevin S. S. Henning Copyright 2013
    569 Pages 220 B/W Illustrations
    by Chapman & Hall

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    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.

    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.


    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.

    "This nicely written textbook fills the gap between elementary statistics courses and more advanced research methods courses. The book helps one to grasp the key assumptions and machinery behind advanced statistical topics … Each chapter ends with useful exercises."
    Mathematical Reviews, August 2014

    "… 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. Popović, 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.