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
Purely Probabilistic Statistical Models
Statistical Models with Both Deterministic and Probabilistic Components
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
Generating Random Numbers
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 q–q 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
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
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
Maximum Likelihood Estimates
Wald Standard Error
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 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
Frequentist Confidence Interval for s
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
Choosing a Sample Size for Prespecified Power
Post Hoc Power: A Useless Statistic
Robustness and Nonparametric Methods
Nonparametric Tests Based on the Rank Transformation
Level and Power Robustness
Bootstrap Percentile-t Confidence Interval
Vocabulary, Formula Summaries, and Exercises appear at the end of each chapter.
"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.