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A/B Testing Power Analysis: A Comprehensive Guide
Author: Dr. Emily Carter, PhD. in Statistics, 10+ years experience in data science with a focus on A/B testing and experimental design for major e-commerce companies.
Publisher: DataScienceCentral.com – A leading online resource providing insightful articles, tutorials, and webinars on data science, machine learning, and analytics, with a dedicated section on statistical methodologies for business applications.
Editor: John Smith, MSc in Data Analytics, 5 years experience editing technical articles and publications in the data science field.
Summary: This guide provides a comprehensive overview of A/B testing power analysis, a crucial step in designing effective experiments. We'll explore the key concepts, demonstrate how to perform the analysis using various methods, outline best practices, and highlight common pitfalls to avoid. Understanding A/B testing power analysis is essential for ensuring your experiments have the statistical power to detect meaningful differences, saving time and resources.
Keywords: A/B testing power analysis, statistical power, sample size calculation, type I error, type II error, effect size, A/B testing, experiment design, online experimentation, conversion rate optimization, hypothesis testing.
1. Introduction to A/B Testing Power Analysis
A/B testing, a cornerstone of online experimentation, involves comparing two versions (A and B) of a website, app, or marketing campaign to determine which performs better. A crucial but often overlooked aspect of A/B testing is power analysis. A/B testing power analysis determines the sample size needed to detect a statistically significant difference between the two versions, given a specific effect size and significance level. Without proper power analysis, your A/B test may fail to detect a real difference (Type II error) or falsely claim a difference where none exists (Type I error).
2. Key Concepts in A/B Testing Power Analysis
Significance Level (α): The probability of rejecting the null hypothesis when it is true (Type I error). Typically set at 0.05 (5%).
Power (1-β): The probability of rejecting the null hypothesis when it is false (correctly identifying a true difference). Generally, a power of 0.8 (80%) is considered acceptable.
Effect Size: The magnitude of the difference you want to detect between version A and version B. A larger effect size requires a smaller sample size. This is often expressed as a difference in proportions (for conversion rates) or means (for other metrics).
Sample Size: The number of users or observations needed in each group (A and B) to achieve the desired power.
Null Hypothesis: The assumption that there is no difference between the two versions.
Alternative Hypothesis: The assumption that there is a difference between the two versions.
3. Performing A/B Testing Power Analysis
There are several methods for performing A/B testing power analysis. These include:
Using statistical software: Packages like R, Python (with libraries like `statsmodels` and `pwr`), and specialized A/B testing platforms offer functions for calculating sample size and power.
Online calculators: Numerous online calculators are available that simplify the process by requiring only the input of key parameters (significance level, power, effect size).
Manual calculation: While more complex, manual calculation provides a deeper understanding of the underlying statistical principles. This typically involves using formulas based on the chosen statistical test (e.g., z-test for proportions).
4. Choosing the Right Effect Size
Determining the appropriate effect size is critical. It's based on prior knowledge, industry benchmarks, or a Minimum Detectable Effect (MDE) – the smallest difference considered practically meaningful. Underestimating the effect size leads to underpowered tests, while overestimating leads to unnecessarily large sample sizes.
5. Best Practices for A/B Testing Power Analysis
Clearly define your goals and metrics: Before performing any A/B testing power analysis, clearly define your objectives and the key metrics you'll use to measure success.
Consider the variability of your data: The variability in your data impacts the required sample size. Higher variability requires larger samples.
Account for potential dropouts: If you anticipate a significant number of dropouts or incomplete observations, increase the planned sample size accordingly.
Iterative approach: Power analysis isn't a one-time event. Refine your estimates based on early results or new insights.
Document your methodology: Maintain a detailed record of your A/B testing power analysis, including the chosen parameters and calculations.
6. Common Pitfalls in A/B Testing Power Analysis
Ignoring power analysis altogether: This is the most significant pitfall, leading to inconclusive or misleading results.
Using an unrealistic effect size: An overly optimistic or pessimistic effect size can severely skew the results.
Failing to consider variability: Ignoring data variability results in inaccurate sample size estimations.
Not accounting for dropouts: This leads to underpowered tests, potentially masking true differences.
Misinterpreting the results: A lack of understanding of statistical concepts can lead to incorrect conclusions.
7. Interpreting the Results of Your A/B Testing Power Analysis
The results of your A/B testing power analysis will provide the required sample size for each variation. This is the minimum number of participants or observations needed to achieve the desired statistical power. If the calculated sample size is too large or impractical to obtain, you may need to reconsider your effect size, significance level, or power level.
8. Advanced Considerations in A/B Testing Power Analysis
For more complex scenarios, you might need to consider factors like:
Multiple comparisons: If you are testing multiple variations simultaneously, you need to adjust for multiple comparisons to avoid inflating the Type I error rate.
Sequential testing: This involves continuously monitoring the results of your A/B test and stopping early if a significant difference is detected.
Bayesian methods: Bayesian approaches offer an alternative to frequentist methods for power analysis, incorporating prior knowledge and allowing for updating beliefs as data accumulates.
Conclusion
A/B testing power analysis is an essential step in ensuring the success of your online experiments. By carefully considering the key concepts, performing the analysis correctly, and avoiding common pitfalls, you can significantly increase the likelihood of obtaining reliable and meaningful results, ultimately leading to more effective optimization and better business decisions. Investing the time upfront in a thorough power analysis saves valuable resources in the long run.
FAQs
1. What is the difference between Type I and Type II error in A/B testing? Type I error is concluding there's a difference when there isn't (false positive), while Type II error is concluding there's no difference when there is (false negative).
2. How do I choose the appropriate effect size for my A/B test? Consider past data, industry benchmarks, and the minimum detectable effect that's practically significant for your business.
3. What happens if my A/B test is underpowered? An underpowered test is likely to fail to detect a real difference between variations, leading to incorrect conclusions and wasted resources.
4. Can I use A/B testing power analysis for metrics other than conversion rates? Yes, you can apply power analysis to various metrics, adjusting the calculations based on the data distribution (e.g., t-test for continuous data).
5. How do I account for seasonality in my A/B testing power analysis? Consider historical data to understand seasonal patterns and incorporate this into your sample size calculations or use a control group unaffected by seasonality.
6. What software or tools can I use for A/B testing power analysis? R, Python (with `statsmodels` and `pwr`), GPower, and various online calculators are available.
7. What is the significance of choosing the right statistical test in power analysis? The choice of test (e.g., z-test, t-test) depends on your data type and assumptions. Using an inappropriate test can lead to incorrect results.
8. How often should I perform power analysis for my A/B tests? Before each A/B test to ensure sufficient power to detect meaningful differences.
9. What's the role of sample size in the accuracy of my A/B testing results? Larger sample sizes generally lead to more accurate and reliable results, reducing the chance of Type I and Type II errors.
Related Articles:
1. Optimizing A/B Testing Sample Size for Maximum ROI: This article focuses on balancing sample size with the cost and time constraints of A/B testing.
2. Understanding and Avoiding Common A/B Testing Pitfalls: This article delves into common mistakes made during A/B testing and offers solutions to avoid them.
3. Bayesian A/B Testing: A Powerful Alternative to Frequentist Methods: This article explores the advantages of using Bayesian methods for A/B testing and power analysis.
4. Advanced A/B Testing Techniques for Complex Experiments: This article covers techniques for handling scenarios like multiple variations and sequential testing.
5. A/B Testing with Multiple Metrics: How to Handle Multiple Comparisons: This article discusses how to adjust for multiple comparisons when testing several metrics simultaneously.
6. Using R for A/B Testing Power Analysis: A Practical Guide: This article provides a step-by-step guide to performing A/B testing power analysis using the R programming language.
7. Python for A/B Testing Power Analysis: Code Examples and Tutorials: This article offers practical examples and tutorials on how to use Python for A/B testing power analysis.
8. The Impact of Effect Size on A/B Testing Power and Sample Size: This article explores the relationship between effect size and the resources required for a successful A/B test.
9. Case Studies in A/B Testing Power Analysis: Real-World Examples and Lessons Learned: This article presents real-world case studies illustrating the importance and application of power analysis in A/B testing.
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| a b testing power analysis: Statistical Methods in Online A/B Testing Georgi Zdravkov Georgiev, 2019-09-28 Statistical Methods in Online A/B Testing is a comprehensive guide to statistics in online controlled experiments, a.k.a. A/B tests, that tackles the difficult matter of statistical inference in a way accessible to readers with little to no prior experience with it. Each concept is built from the ground up, explained thoroughly, and illustrated with practical examples from website testing. The presentation is straight to the point and practically oriented so you can apply the takeaways in your daily work.It is a must-read for anyone looking for a deep understanding of how to make data-driven business decisions through experimentation: conversion rate optimizers, product managers, growth experts, data analysts, marketing managers, experts in user experience and design. The new research presented and the fresh perspective on how to apply statistics and experimentation to achieve business goals make for an interesting read even for experienced statisticians.The book deals with scientific methods, but their introductions and explanations are grounded in the business goals they help achieve, such as innovating under controlled risk, and estimating the effect of proposed business actions before committing to them. While the book doesn't shy away from math and formulas, it is to the extent to which these are essential for understanding and applying the underlying concepts. The presentation is friendly to readers with little to no prior knowledge in statistics. Artificial and impractical examples like dice rolling and betting are absent, instead statistical concepts are illustrated through scenarios which might well be mistaken with the last couple of A/B tests you managed.This book also doesn't shy away from the fact that much of the current statistical theory and practice in online A/B testing is misguided, misinterpreted, or misapplied. It also addresses the issue of blind copying of scientific applications without due consideration of the unique features of online business, which is widespread. The book will help you avoid these malpractices by explicitly pointing out frequent mistakes, while also helping you align your usage of statistics and experimentation with any business goals you might want to pursue. |
| a b testing power analysis: Practical Statistical Power Analysis Zhiyong Zhang, 2018-05 The importance of conducting a statistical power analysis at the beginning of a study is universally accepted. Without careful planning, a study can easily fail to detect an existing effect by chance. This book teaches how to conduct power analysis for a range of models from correlation and t-test to structural equation models and multilevel models. Both a graphical Web software WebPower and an R package are introduced to practically conduct the analysis. |
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| a b testing power analysis: Statistical Inference as Severe Testing Deborah G. Mayo, 2018-09-20 Mounting failures of replication in social and biological sciences give a new urgency to critically appraising proposed reforms. This book pulls back the cover on disagreements between experts charged with restoring integrity to science. It denies two pervasive views of the role of probability in inference: to assign degrees of belief, and to control error rates in a long run. If statistical consumers are unaware of assumptions behind rival evidence reforms, they can't scrutinize the consequences that affect them (in personalized medicine, psychology, etc.). The book sets sail with a simple tool: if little has been done to rule out flaws in inferring a claim, then it has not passed a severe test. Many methods advocated by data experts do not stand up to severe scrutiny and are in tension with successful strategies for blocking or accounting for cherry picking and selective reporting. Through a series of excursions and exhibits, the philosophy and history of inductive inference come alive. Philosophical tools are put to work to solve problems about science and pseudoscience, induction and falsification. |
| a b testing power analysis: Smart Persuasion Philippe AIMÉ, Jochen GRÜNBECK, 2019-03-01 Conversions begin in the brain. Every purchase starts with a decision, and every decision is shaped by consumer psychology. This book explains how mental shortcuts (cognitive biases) affect your customers' decision making and shows you how to be more persuasive online. Philippe Aimé and Jochen Grünbeck are optimisation addicts and have been at the forefront of digital marketing since the beginning. Inspired by behavioural economists like Daniel Kahneman, Dan Ariely and Richard Thaler, the techniques described in Smart Persuasion leverage powerful decision-making biases to make marketing more effective. Alongside these behavioural insights, Smart Persuasion incorporates research from marketing experts such as Jonah Berger, Robert Cialdini and Roger Dooley. Principles relating to attention and perception, as well as the cognitive effects that make consumers predictably irrational, are distilled into concrete website optimisation strategies. Drawing from hundreds of unique studies, Smart Persuasion lists proven effects such as Anchoring and Framing. Each one is illustrated with case-studies, examples and ideas that you can apply immediately. Using the persuasive strategies outlined in this book will allow you to influence consumers more effectively, unlocking your website's potential. All profits from the sale of this book help provide educational resources for children in Africa. |
| a b testing power analysis: Applied Power Analysis for the Behavioral Sciences Christopher L. Aberson, 2019-01-24 Applied Power Analysis for the Behavioral Sciences is a practical how-to guide to conducting statistical power analyses for psychology and related fields. The book provides a guide to conducting analyses that is appropriate for researchers and students, including those with limited quantitative backgrounds. With practical use in mind, the text provides detailed coverage of topics such as how to estimate expected effect sizes and power analyses for complex designs. The topical coverage of the text, an applied approach, in-depth coverage of popular statistical procedures, and a focus on conducting analyses using R make the text a unique contribution to the power literature. To facilitate application and usability, the text includes ready-to-use R code developed for the text. An accompanying R package called pwr2ppl (available at https://github.com/chrisaberson/pwr2ppl) provides tools for conducting power analyses across each topic covered in the text. |
| a b testing power analysis: Statistical Power Analysis with Missing Data Adam Davey, Jyoti "Tina" Savla, 2009-08-20 Statistical power analysis has revolutionized the ways in which we conduct and evaluate research. Similar developments in the statistical analysis of incomplete (missing) data are gaining more widespread applications. This volume brings statistical power and incomplete data together under a common framework, in a way that is readily accessible to those with only an introductory familiarity with structural equation modeling. It answers many practical questions such as: How missing data affects the statistical power in a study How much power is likely with different amounts and types of missing data How to increase the power of a design in the presence of missing data, and How to identify the most powerful design in the presence of missing data. Points of Reflection encourage readers to stop and test their understanding of the material. Try Me sections test one’s ability to apply the material. Troubleshooting Tips help to prevent commonly encountered problems. Exercises reinforce content and Additional Readings provide sources for delving more deeply into selected topics. Numerous examples demonstrate the book’s application to a variety of disciplines. Each issue is accompanied by its potential strengths and shortcomings and examples using a variety of software packages (SAS, SPSS, Stata, LISREL, AMOS, and MPlus). Syntax is provided using a single software program to promote continuity but in each case, parallel syntax using the other packages is presented in appendixes. Routines, data sets, syntax files, and links to student versions of software packages are found at www.psypress.com/davey. The worked examples in Part 2 also provide results from a wider set of estimated models. These tables, and accompanying syntax, can be used to estimate statistical power or required sample size for similar problems under a wide range of conditions. Class-tested at Temple, Virginia Tech, and Miami University of Ohio, this brief text is an ideal supplement for graduate courses in applied statistics, statistics II, intermediate or advanced statistics, experimental design, structural equation modeling, power analysis, and research methods taught in departments of psychology, human development, education, sociology, nursing, social work, gerontology and other social and health sciences. The book’s applied approach will also appeal to researchers in these areas. Sections covering Fundamentals, Applications, and Extensions are designed to take readers from first steps to mastery. |
| a b testing power analysis: Statistical Power Analysis Kevin R. Murphy, Brett Myors, Kevin Murphy, Allen Wolach, 2003-08-01 This book presents a simple and general method for conducting statistical power analysis based on the widely used F statistic. The book illustrates how these analyses work and how they can be applied to problems of studying design, to evaluate others' research, and to choose the appropriate criterion for defining statistically significant outcomes. Statistical Power Analysis examines the four major applications of power analysis, concentrating on how to determine: *the sample size needed to achieve desired levels of power; *the level of power that is needed in a study; *the size of effect that can be reliably detected by a study; and *sensible criteria for statistical significance. Highlights of the second edition include: a CD with an easy-to-use statistical power analysis program; a new chapter on power analysis in multi-factor ANOVA, including repeated-measures designs; and a new One-Stop PV Table to serve as a quick reference guide. The book discusses the application of power analysis to both traditional null hypothesis tests and to minimum-effect testing. It demonstrates how the same basic model applies to both types of testing and explains how some relatively simple procedures allow researchers to ask a series of important questions about their research. Drawing from the behavioral and social sciences, the authors present the material in a nontechnical way so that readers with little expertise in statistical analysis can quickly obtain the values needed to carry out the power analysis. Ideal for students and researchers of statistical and research methodology in the social, behavioral, and health sciences who want to know how to apply methods of power analysis to their research. |
| a b testing power analysis: Applied Analysis of Variance in Behavioral Science Lynne Edwards, 1993-06-16 A reference devoted to the discussion of analysis of variance (ANOVA) techniques. It presents ANOVA as a research design, a collection of statistical models, an analysis model, and an arithmetic summary of data. Discussion focuses primarily on univariate data, but multivariate generalizations are to |
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| a b testing power analysis: Introduction to Bayesian Statistics William M. Bolstad, James M. Curran, 2016-09-02 ...this edition is useful and effective in teaching Bayesian inference at both elementary and intermediate levels. It is a well-written book on elementary Bayesian inference, and the material is easily accessible. It is both concise and timely, and provides a good collection of overviews and reviews of important tools used in Bayesian statistical methods. There is a strong upsurge in the use of Bayesian methods in applied statistical analysis, yet most introductory statistics texts only present frequentist methods. Bayesian statistics has many important advantages that students should learn about if they are going into fields where statistics will be used. In this third Edition, four newly-added chapters address topics that reflect the rapid advances in the field of Bayesian statistics. The authors continue to provide a Bayesian treatment of introductory statistical topics, such as scientific data gathering, discrete random variables, robust Bayesian methods, and Bayesian approaches to inference for discrete random variables, binomial proportions, Poisson, and normal means, and simple linear regression. In addition, more advanced topics in the field are presented in four new chapters: Bayesian inference for a normal with unknown mean and variance; Bayesian inference for a Multivariate Normal mean vector; Bayesian inference for the Multiple Linear Regression Model; and Computational Bayesian Statistics including Markov Chain Monte Carlo. The inclusion of these topics will facilitate readers' ability to advance from a minimal understanding of Statistics to the ability to tackle topics in more applied, advanced level books. Minitab macros and R functions are available on the book's related website to assist with chapter exercises. Introduction to Bayesian Statistics, Third Edition also features: Topics including the Joint Likelihood function and inference using independent Jeffreys priors and join conjugate prior The cutting-edge topic of computational Bayesian Statistics in a new chapter, with a unique focus on Markov Chain Monte Carlo methods Exercises throughout the book that have been updated to reflect new applications and the latest software applications Detailed appendices that guide readers through the use of R and Minitab software for Bayesian analysis and Monte Carlo simulations, with all related macros available on the book's website Introduction to Bayesian Statistics, Third Edition is a textbook for upper-undergraduate or first-year graduate level courses on introductory statistics course with a Bayesian emphasis. It can also be used as a reference work for statisticians who require a working knowledge of Bayesian statistics. |
| a b testing power analysis: Statistical Methods for Machine Learning Jason Brownlee, 2018-05-30 Statistics is a pillar of machine learning. You cannot develop a deep understanding and application of machine learning without it. Cut through the equations, Greek letters, and confusion, and discover the topics in statistics that you need to know. Using clear explanations, standard Python libraries, and step-by-step tutorial lessons, you will discover the importance of statistical methods to machine learning, summary stats, hypothesis testing, nonparametric stats, resampling methods, and much more. |
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| a b testing power analysis: Introductory Business Statistics 2e Alexander Holmes, Barbara Illowsky, Susan Dean, 2023-12-13 Introductory Business Statistics 2e aligns with the topics and objectives of the typical one-semester statistics course for business, economics, and related majors. The text provides detailed and supportive explanations and extensive step-by-step walkthroughs. The author places a significant emphasis on the development and practical application of formulas so that students have a deeper understanding of their interpretation and application of data. Problems and exercises are largely centered on business topics, though other applications are provided in order to increase relevance and showcase the critical role of statistics in a number of fields and real-world contexts. The second edition retains the organization of the original text. Based on extensive feedback from adopters and students, the revision focused on improving currency and relevance, particularly in examples and problems. This is an adaptation of Introductory Business Statistics 2e by OpenStax. You can access the textbook as pdf for free at openstax.org. Minor editorial changes were made to ensure a better ebook reading experience. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution 4.0 International License. |
| a b testing power analysis: Developing a Protocol for Observational Comparative Effectiveness Research: A User's Guide Agency for Health Care Research and Quality (U.S.), 2013-02-21 This User’s Guide is a resource for investigators and stakeholders who develop and review observational comparative effectiveness research protocols. It explains how to (1) identify key considerations and best practices for research design; (2) build a protocol based on these standards and best practices; and (3) judge the adequacy and completeness of a protocol. Eleven chapters cover all aspects of research design, including: developing study objectives, defining and refining study questions, addressing the heterogeneity of treatment effect, characterizing exposure, selecting a comparator, defining and measuring outcomes, and identifying optimal data sources. Checklists of guidance and key considerations for protocols are provided at the end of each chapter. The User’s Guide was created by researchers affiliated with AHRQ’s Effective Health Care Program, particularly those who participated in AHRQ’s DEcIDE (Developing Evidence to Inform Decisions About Effectiveness) program. Chapters were subject to multiple internal and external independent reviews. More more information, please consult the Agency website: www.effectivehealthcare.ahrq.gov) |
| a b testing power analysis: The Love Hypothesis Ali Hazelwood, 2021-09-14 The Instant New York Times Bestseller and TikTok Sensation! As seen on THE VIEW! A BuzzFeed Best Summer Read of 2021 When a fake relationship between scientists meets the irresistible force of attraction, it throws one woman's carefully calculated theories on love into chaos. As a third-year Ph.D. candidate, Olive Smith doesn't believe in lasting romantic relationships--but her best friend does, and that's what got her into this situation. Convincing Anh that Olive is dating and well on her way to a happily ever after was always going to take more than hand-wavy Jedi mind tricks: Scientists require proof. So, like any self-respecting biologist, Olive panics and kisses the first man she sees. That man is none other than Adam Carlsen, a young hotshot professor--and well-known ass. Which is why Olive is positively floored when Stanford's reigning lab tyrant agrees to keep her charade a secret and be her fake boyfriend. But when a big science conference goes haywire, putting Olive's career on the Bunsen burner, Adam surprises her again with his unyielding support and even more unyielding...six-pack abs. Suddenly their little experiment feels dangerously close to combustion. And Olive discovers that the only thing more complicated than a hypothesis on love is putting her own heart under the microscope. |
| a b testing power analysis: 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. |
| a b testing power analysis: Introduction to Meta-Analysis Michael Borenstein, Larry V. Hedges, Julian P. T. Higgins, Hannah R. Rothstein, 2011-08-24 This book provides a clear and thorough introduction to meta-analysis, the process of synthesizing data from a series of separate studies. Meta-analysis has become a critically important tool in fields as diverse as medicine, pharmacology, epidemiology, education, psychology, business, and ecology. Introduction to Meta-Analysis: Outlines the role of meta-analysis in the research process Shows how to compute effects sizes and treatment effects Explains the fixed-effect and random-effects models for synthesizing data Demonstrates how to assess and interpret variation in effect size across studies Clarifies concepts using text and figures, followed by formulas and examples Explains how to avoid common mistakes in meta-analysis Discusses controversies in meta-analysis Features a web site with additional material and exercises A superb combination of lucid prose and informative graphics, written by four of the world’s leading experts on all aspects of meta-analysis. Borenstein, Hedges, Higgins, and Rothstein provide a refreshing departure from cookbook approaches with their clear explanations of the what and why of meta-analysis. The book is ideal as a course textbook or for self-study. My students, who used pre-publication versions of some of the chapters, raved about the clarity of the explanations and examples. David Rindskopf, Distinguished Professor of Educational Psychology, City University of New York, Graduate School and University Center, & Editor of the Journal of Educational and Behavioral Statistics. The approach taken by Introduction to Meta-analysis is intended to be primarily conceptual, and it is amazingly successful at achieving that goal. The reader can comfortably skip the formulas and still understand their application and underlying motivation. For the more statistically sophisticated reader, the relevant formulas and worked examples provide a superb practical guide to performing a meta-analysis. The book provides an eclectic mix of examples from education, social science, biomedical studies, and even ecology. For anyone considering leading a course in meta-analysis, or pursuing self-directed study, Introduction to Meta-analysis would be a clear first choice. Jesse A. Berlin, ScD Introduction to Meta-Analysis is an excellent resource for novices and experts alike. The book provides a clear and comprehensive presentation of all basic and most advanced approaches to meta-analysis. This book will be referenced for decades. Michael A. McDaniel, Professor of Human Resources and Organizational Behavior, Virginia Commonwealth University |
| a b testing power analysis: Bayesian Methods for Hackers Cameron Davidson-Pilon, 2015-09-30 Master Bayesian Inference through Practical Examples and Computation–Without Advanced Mathematical Analysis Bayesian methods of inference are deeply natural and extremely powerful. However, most discussions of Bayesian inference rely on intensely complex mathematical analyses and artificial examples, making it inaccessible to anyone without a strong mathematical background. Now, though, Cameron Davidson-Pilon introduces Bayesian inference from a computational perspective, bridging theory to practice–freeing you to get results using computing power. Bayesian Methods for Hackers illuminates Bayesian inference through probabilistic programming with the powerful PyMC language and the closely related Python tools NumPy, SciPy, and Matplotlib. Using this approach, you can reach effective solutions in small increments, without extensive mathematical intervention. Davidson-Pilon begins by introducing the concepts underlying Bayesian inference, comparing it with other techniques and guiding you through building and training your first Bayesian model. Next, he introduces PyMC through a series of detailed examples and intuitive explanations that have been refined after extensive user feedback. You’ll learn how to use the Markov Chain Monte Carlo algorithm, choose appropriate sample sizes and priors, work with loss functions, and apply Bayesian inference in domains ranging from finance to marketing. Once you’ve mastered these techniques, you’ll constantly turn to this guide for the working PyMC code you need to jumpstart future projects. Coverage includes • Learning the Bayesian “state of mind” and its practical implications • Understanding how computers perform Bayesian inference • Using the PyMC Python library to program Bayesian analyses • Building and debugging models with PyMC • Testing your model’s “goodness of fit” • Opening the “black box” of the Markov Chain Monte Carlo algorithm to see how and why it works • Leveraging the power of the “Law of Large Numbers” • Mastering key concepts, such as clustering, convergence, autocorrelation, and thinning • Using loss functions to measure an estimate’s weaknesses based on your goals and desired outcomes • Selecting appropriate priors and understanding how their influence changes with dataset size • Overcoming the “exploration versus exploitation” dilemma: deciding when “pretty good” is good enough • Using Bayesian inference to improve A/B testing • Solving data science problems when only small amounts of data are available Cameron Davidson-Pilon has worked in many areas of applied mathematics, from the evolutionary dynamics of genes and diseases to stochastic modeling of financial prices. His contributions to the open source community include lifelines, an implementation of survival analysis in Python. Educated at the University of Waterloo and at the Independent University of Moscow, he currently works with the online commerce leader Shopify. |
| a b testing power analysis: Mindset Carol S. Dweck, 2007-12-26 From the renowned psychologist who introduced the world to “growth mindset” comes this updated edition of the million-copy bestseller—featuring transformative insights into redefining success, building lifelong resilience, and supercharging self-improvement. “Through clever research studies and engaging writing, Dweck illuminates how our beliefs about our capabilities exert tremendous influence on how we learn and which paths we take in life.”—Bill Gates, GatesNotes “It’s not always the people who start out the smartest who end up the smartest.” After decades of research, world-renowned Stanford University psychologist Carol S. Dweck, Ph.D., discovered a simple but groundbreaking idea: the power of mindset. In this brilliant book, she shows how success in school, work, sports, the arts, and almost every area of human endeavor can be dramatically influenced by how we think about our talents and abilities. People with a fixed mindset—those who believe that abilities are fixed—are less likely to flourish than those with a growth mindset—those who believe that abilities can be developed. Mindset reveals how great parents, teachers, managers, and athletes can put this idea to use to foster outstanding accomplishment. In this edition, Dweck offers new insights into her now famous and broadly embraced concept. She introduces a phenomenon she calls false growth mindset and guides people toward adopting a deeper, truer growth mindset. She also expands the mindset concept beyond the individual, applying it to the cultures of groups and organizations. With the right mindset, you can motivate those you lead, teach, and love—to transform their lives and your own. |
| a b testing power analysis: The Frailty Model Luc Duchateau, Paul Janssen, 2007-10-23 Readers will find in the pages of this book a treatment of the statistical analysis of clustered survival data. Such data are encountered in many scientific disciplines including human and veterinary medicine, biology, epidemiology, public health and demography. A typical example is the time to death in cancer patients, with patients clustered in hospitals. Frailty models provide a powerful tool to analyze clustered survival data. In this book different methods based on the frailty model are described and it is demonstrated how they can be used to analyze clustered survival data. All programs used for these examples are available on the Springer website. |
| a b testing power analysis: Pain Management and the Opioid Epidemic National Academies of Sciences, Engineering, and Medicine, Health and Medicine Division, Board on Health Sciences Policy, Committee on Pain Management and Regulatory Strategies to Address Prescription Opioid Abuse, 2017-09-28 Drug overdose, driven largely by overdose related to the use of opioids, is now the leading cause of unintentional injury death in the United States. The ongoing opioid crisis lies at the intersection of two public health challenges: reducing the burden of suffering from pain and containing the rising toll of the harms that can arise from the use of opioid medications. Chronic pain and opioid use disorder both represent complex human conditions affecting millions of Americans and causing untold disability and loss of function. In the context of the growing opioid problem, the U.S. Food and Drug Administration (FDA) launched an Opioids Action Plan in early 2016. As part of this plan, the FDA asked the National Academies of Sciences, Engineering, and Medicine to convene a committee to update the state of the science on pain research, care, and education and to identify actions the FDA and others can take to respond to the opioid epidemic, with a particular focus on informing FDA's development of a formal method for incorporating individual and societal considerations into its risk-benefit framework for opioid approval and monitoring. |
| a b testing power analysis: Statistical Power Analysis Brett Myors, Kevin R. Murphy, 2023-03-03 Statistical Power Analysis explains the key concepts in statistical power analysis and illustrates their application in both tests of traditional null hypotheses (that treatments or interventions have no effect in the population) and in tests of the minimum-effect hypotheses (that the population effects of treatments or interventions are so small that they can be safely treated as unimportant). It provides readers with the tools to understand and perform power analyses for virtually all the statistical methods used in the social and behavioral sciences. Brett Myors and Kevin Murphy apply the latest approaches of power analysis to both null hypothesis and minimum-effect testing using the same basic unified model. This book starts with a review of the key concepts that underly statistical power. It goes on to show how to perform and interpret power analyses, and the ways to use them to diagnose and plan research. We discuss the uses of power analysis in correlation and regression, in the analysis of experimental data, and in multilevel studies. This edition includes new material and new power software. The programs used for power analysis in this book have been re-written in R, a language that is widely used and freely available. The authors include R codes for all programs, and we have also provided a web-based app that allows users who are not comfortable with R to perform a wide range of analyses using any computer or device that provides access to the web. Statistical Power Analysis helps readers design studies, diagnose existing studies, and understand why hypothesis tests come out the way they do. The fifth edition includes updates to all chapters to accommodate the most current scholarship, as well as recalculations of all examples. This book is intended for graduate students and faculty in the behavioral and social sciences; researchers in other fields will find the concepts and methods laid out here valuable and applicable to studies in many domains. |
| a b testing power analysis: Experimentation Works Stefan H. Thomke, 2020-02-18 Don't fly blind. See how the power of experiments works for you. When it comes to improving customer experiences, trying out new business models, or developing new products, even the most experienced managers often get it wrong. They discover that intuition, experience, and big data alone don't work. What does? Running disciplined business experiments. And what if companies roll out new products or introduce new customer experiences without running these experiments? They fly blind. That's what Harvard Business School professor Stefan Thomke shows in this rigorously researched and eye-opening book. It guides you through best practices in business experimentation, illustrates how these practices work at leading companies, and answers some fundamental questions: What makes a good experiment? How do you test in online and brick-and-mortar businesses? In B2B and B2C? How do you build an experimentation culture? Also, best practice means running many experiments. Indeed, some hugely successful companies, such as Amazon, Booking.com, and Microsoft, run tens of thousands of controlled experiments annually, engaging millions of users. Thomke shows us how these and many other organizations prove that experimentation provides significant competitive advantage. How can managers create this capability at their own companies? Essential is developing an experimentation organization that prizes the science of testing and puts the discipline of experimentation at the center of its innovation process. While it once took companies years to develop the tools for such large-scale experiments, advances in technology have put these tools at the fingertips of almost any business professional. By combining the power of software and the rigor of controlled experiments, today's managers can make better decisions, create magical customer experiences, and generate big financial returns. Experimentation Works is your guidebook to a truly new way of thinking and innovating. |
| a b testing power analysis: Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences Patricia Cohen, Stephen G. West, Leona S. Aiken, 2014-04-04 This classic text on multiple regression is noted for its nonmathematical, applied, and data-analytic approach. Readers profit from its verbal-conceptual exposition and frequent use of examples. The applied emphasis provides clear illustrations of the principles and provides worked examples of the types of applications that are possible. Researchers learn how to specify regression models that directly address their research questions. An overview of the fundamental ideas of multiple regression and a review of bivariate correlation and regression and other elementary statistical concepts provide a strong foundation for understanding the rest of the text. The third edition features an increased emphasis on graphics and the use of confidence intervals and effect size measures, and an accompanying CD with data for most of the numerical examples along with the computer code for SPSS, SAS, and SYSTAT. Applied Multiple Regression serves as both a textbook for graduate students and as a reference tool for researchers in psychology, education, health sciences, communications, business, sociology, political science, anthropology, and economics. An introductory knowledge of statistics is required. Self-standing chapters minimize the need for researchers to refer to previous chapters. |
| a b testing power analysis: The Design of Experiments Sir Ronald Aylmer Fisher, 1974 |
| a b testing power analysis: E-Commerce Website Optimization Dan Croxen-John, Johann van Tonder, 2020-12-03 Conversion rate optimization (CRO) is about understanding people and behaviour, not simply website visits. This book is an all-encompassing guide to the how, the why and the tools and techniques. Grounded in best-practice theory and research, E-commerce Website Optimization brings together usability, analytics and persuasion to offer a detailed, step-by-step guide to increasing the percentage of visitors who buy from your site and subsequently the amount that these visitors spend when they do. In a complex and evolving field, it will help you improve conversion rates, increase ROI from online marketing campaigns, generate higher levels of repeat business and increase the e-commerce value of websites. In the fast-moving world of e-commerce, this fully revised second edition includes updates on test metrics, prioritization and personalization, alongside updated case studies and newly recommended tools. E-commerce Website Optimization is an invaluable book for those seeking to implement a data-driven ethos for their organization's e-commerce programme, for everyone from chief digital officers and heads of online sales, to entrepreneurs and small business owners. |
| a b testing power analysis: Design and Analysis of Experiments Douglas C. Montgomery, 2005 This bestselling professional reference has helped over 100,000 engineers and scientists with the success of their experiments. The new edition includes more software examples taken from the three most dominant programs in the field: Minitab, JMP, and SAS. Additional material has also been added in several chapters, including new developments in robust design and factorial designs. New examples and exercises are also presented to illustrate the use of designed experiments in service and transactional organizations. Engineers will be able to apply this information to improve the quality and efficiency of working systems. |
| a b testing power analysis: Benchmarking the User Experience Jeff Sauro, 2018-06-25 This is a practical book about how to measure the user experience of websites, software, mobile apps, products, or just anything people use. This book is for UX researchers, designers, product owners, or anyone that has a vested interest in improving experience of websites and products--Introduction. |
| a b testing power analysis: Conversion Optimization Khalid Saleh, Ayat Shukairy, 2010-11-01 How do you turn website visitors into customers? Conversion Optimization offers practical advice on how to persuade visitors to make a buying decision -- without driving them away through data overload or tedious navigation. You'll learn how to use marketing principles, design, usability, and analytics on your site to increase your buyer-to-visitor ratio, whether you're involved with marketing or designing a large ecommerce site, or managing a modest online operation. Based on the authors' broad experience in helping businesses attract online customers, this book addresses every aspect of the process, from landing visitors to finalizing the sale. You'll learn several techniques for blending successful sales approaches with the particular needs of the people you want to attract. Are you ready to do what it takes to get a double-digit conversion rate? Explore case studies involving significant conversion rate improvements Walk through different stages of a sale and understand the value of each Understand your website visitors through persona creation Connect with potential customers and guide them toward a conversion Learn how to deal with FUDs -- customer fears, uncertainties, and doubts Examine the path that visitors take from landing page to checkout Test any change you make against your original design The Web is unique in its ability to deliver this almost improbable win-win: You can increase revenue AND make your customers happy. Yet most websites stink. Worry not, Khalid and Ayat to the rescue! Buy this book to follow their practical advice on how to create high converting websites that your visitors love.--Avinash Kaushik, author of Web Analytics 2.0 and Web Analytics: An Hour A Day (both Sybex) |
| a b testing power analysis: Doing Bayesian Data Analysis John Kruschke, 2010-11-25 There is an explosion of interest in Bayesian statistics, primarily because recently created computational methods have finally made Bayesian analysis tractable and accessible to a wide audience. Doing Bayesian Data Analysis, A Tutorial Introduction with R and BUGS, is for first year graduate students or advanced undergraduates and provides an accessible approach, as all mathematics is explained intuitively and with concrete examples. It assumes only algebra and 'rusty' calculus. Unlike other textbooks, this book begins with the basics, including essential concepts of probability and random sampling. The book gradually climbs all the way to advanced hierarchical modeling methods for realistic data. The text provides complete examples with the R programming language and BUGS software (both freeware), and begins with basic programming examples, working up gradually to complete programs for complex analyses and presentation graphics. These templates can be easily adapted for a large variety of students and their own research needs.The textbook bridges the students from their undergraduate training into modern Bayesian methods. - Accessible, including the basics of essential concepts of probability and random sampling - Examples with R programming language and BUGS software - Comprehensive coverage of all scenarios addressed by non-bayesian textbooks- t-tests, analysis of variance (ANOVA) and comparisons in ANOVA, multiple regression, and chi-square (contingency table analysis). - Coverage of experiment planning - R and BUGS computer programming code on website - Exercises have explicit purposes and guidelines for accomplishment |
| a b testing power analysis: Introduction to Robust Estimation and Hypothesis Testing Rand R. Wilcox, 2012-01-12 This book focuses on the practical aspects of modern and robust statistical methods. The increased accuracy and power of modern methods, versus conventional approaches to the analysis of variance (ANOVA) and regression, is remarkable. Through a combination of theoretical developments, improved and more flexible statistical methods, and the power of the computer, it is now possible to address problems with standard methods that seemed insurmountable only a few years ago-- |
| a b testing power analysis: Statistical Power Analysis Kevin R. Murphy, Brett Myors, Allen Wolach, 2011-04-27 First Published in 2009. Routledge is an imprint of Taylor & Francis, an informa company. |
| a b testing power analysis: Linear Model Theory Keith E. Muller, Paul W. Stewart, 2006-10-06 A precise and accessible presentation of linear model theory, illustrated with data examples Statisticians often use linear models for data analysis and for developing new statistical methods. Most books on the subject have historically discussed univariate, multivariate, and mixed linear models separately, whereas Linear Model Theory: Univariate, Multivariate, and Mixed Models presents a unified treatment in order to make clear the distinctions among the three classes of models. Linear Model Theory: Univariate, Multivariate, and Mixed Models begins with six chapters devoted to providing brief and clear mathematical statements of models, procedures, and notation. Data examples motivate and illustrate the models. Chapters 7-10 address distribution theory of multivariate Gaussian variables and quadratic forms. Chapters 11-19 detail methods for estimation, hypothesis testing, and confidence intervals. The final chapters, 20-23, concentrate on choosing a sample size. Substantial sets of excercises of varying difficulty serve instructors for their classes, as well as help students to test their own knowledge. The reader needs a basic knowledge of statistics, probability, and inference, as well as a solid background in matrix theory and applied univariate linear models from a matrix perspective. Topics covered include: A review of matrix algebra for linear models The general linear univariate model The general linear multivariate model Generalizations of the multivariate linear model The linear mixed model Multivariate distribution theory Estimation in linear models Tests in Gaussian linear models Choosing a sample size in Gaussian linear models Filling the need for a text that provides the necessary theoretical foundations for applying a wide range of methods in real situations, Linear Model Theory: Univariate, Multivariate, and Mixed Models centers on linear models of interval scale responses with finite second moments. Models with complex predictors, complex responses, or both, motivate the presentation. |
| a b testing power analysis: A First Course in Design and Analysis of Experiments Gary W. Oehlert, 2000-01-19 Oehlert's text is suitable for either a service course for non-statistics graduate students or for statistics majors. Unlike most texts for the one-term grad/upper level course on experimental design, Oehlert's new book offers a superb balance of both analysis and design, presenting three practical themes to students: • when to use various designs • how to analyze the results • how to recognize various design options Also, unlike other older texts, the book is fully oriented toward the use of statistical software in analyzing experiments. |
| a b testing power analysis: Psychometric Tests (the Ultimate Guide) Richard McMunn, 2010-11 |
| a b testing power analysis: Power Analysis for Experimental Research R. Barker Bausell, Yu-Fang Li, 2002-09-19 Power analysis is an essential tool for determining whether a statistically significant result can be expected in a scientific experiment prior to the experiment being performed. Many funding agencies and institutional review boards now require power analyses to be carried out before they will approve experiments, particularly where they involve the use of human subjects. This comprehensive, yet accessible, book provides practising researchers with step-by-step instructions for conducting power/sample size analyses, assuming only basic prior knowledge of summary statistics and the normal distribution. It contains a unified approach to statistical power analysis, with numerous easy-to-use tables to guide the reader without the need for further calculations or statistical expertise. This will be an indispensable text for researchers and graduates in the medical and biological sciences needing to apply power analysis in the design of their experiments. |
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知乎，中文互联网高质量的问答社区和创作者聚集的原创内容平台，于 2011 年 1 月正式上线，以「让人们更好的分享知识、经验和见解，找到自己的解答」为品牌使命。知乎凭借认真、专业 …
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