R语言进行多重比较分析

Multiple

using

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"Multiple comparison using R" 本书《Multiple Comparisons Using R》主要针对统计学教师和学生，无论是本科还是研究生阶段。它详细介绍了如何在R语言环境下进行多重比较分析，这是统计学中一个重要的概念，特别是在实验设计和数据分析中。多重比较（Multiple Comparison）是统计分析的一个关键部分，尤其是在对多个组间差异进行测试时。传统的单因素方差分析（ANOVA）或t检验可能无法直接提供所有组间的比较结果，多重比较方法则能解决这个问题，帮助我们确定哪些组间存在显著差异。R语言作为一个强大的统计分析工具，提供了丰富的库和函数支持这种复杂的分析。书中可能涵盖了以下内容： 1. **基础理论**：介绍多重比较的基本概念，包括控制错误率（如家族错误率Family-Wise Error Rate, FWER）和错误发现率False Discovery Rate, FDR）等。 2. **统计方法**：详细阐述各种多重比较方法，如Tukey's HSD（Honestly Significant Difference）测试、Dunnett's test、Scheffé's method、Bonferroni校正以及Benjamini-Hochberg过程等。 3. **R包应用**：介绍如何使用R中的`multcomp`, `lsmeans`, `agricolae`等包进行多重比较分析，包括数据导入、模型构建、结果解释和可视化。 4. **实例分析**：通过实际案例来演示如何在R中执行多重比较，可能涵盖农业、生物医学、社会科学等多个领域的研究。 5. **实验设计**：讨论如何设计有效的实验以优化多重比较分析，包括完全随机设计、随机区组设计、裂区设计等。 6. **错误控制**：讲解如何理解和控制多重比较中的类型I错误（误拒绝原假设）和类型II错误（未能拒绝虚假的原假设）。 7. **结果解释**：指导读者如何正确解读多重比较的结果，以及如何根据结果做出决策。 8. **代码示例**：提供详细的R代码示例，帮助读者掌握实际操作步骤。通过阅读这本书，学生和教师将能够掌握在R环境中进行多重比较分析的技巧，提升统计学研究和实践中的数据分析能力。同时，作者们提醒读者注意版权问题，并鼓励在发现任何未被承认的版权材料时及时告知，以便在后续重印中进行修正。

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PREFACE xv

the selected multiple comparison procedures in Chapter 5 can be read without

necessarily having read the previous chapters.

Who should read this book

This book is for statistics teachers and students, undergraduate or graduate,

who are learning about multiple comparison procedures, whether in a stand-

alone course on multiple testing, a course in analysis-of-variance, a course in

multivariate analysis, or a course in nonparameterics. Additionally, the book

is helpful for scientists who need to use multiple comparison procedures, in-

cluding biometricians, clinicians, medical doctors, molecular biologists, agri-

cultural analysts, etc. It is oriented towards users (i) who have only limited

knowledge of multiple comparison procedures but need to apply those tech-

niques in R, and (ii) who are already familiar with multiple comparison pro-

cedures but lack the related implementations in R. We assume that the reader

has a basic knowledge of R and can perform elementary data handling steps.

Otherwise, we recommend standard textbooks on R for a quick introduction;

see Dalgaard (2002) and Everitt and Hothorn (2009) among others.

Conventions used in this book

We use bold-face capital letters (e.g., C, R, X, . . .) to denote matrices and

bold-face small letters (e.g., c, x, y, . . .) to indicate vectors. If the distinction

between column and row vectors matters, we introduce vectors as column

vectors. A transpose of a vector or matrix is indicated by the superscript

;

for example, c

denotes the transpose of the (column) vector c. To simplify the

notation, we do not distinguish between random variables and their observed

values, unless noted explicitly.

We use many code examples in R throughout this book. Samples of code

that could be entered interactively at the R command line are formatted as

follows:

R> library("multcomp")

Here, R> denotes the prompt sign from the R command line and the user enters

everything else. In some instances the expressions to be entered will be longer

than a single line and it will appear as follows:

R> summary(glht(recovery.aov, linfct = mcp(blanket = contr),

+ alternative = "less"))

The symbol + indicates additional lines which are appropriately indented.

Finally, output produced by function calls is shown below the associated code

R> rnorm(10)

[1] 0.0310 1.3378 0.4158 0.0413 -1.4383 1.0761 -1.2687

[8] 0.8262 -0.8603 -0.7134

xvi PREFACE

Computational details and reproducibility

In this book, we use several R packages to access diﬀerent example datasets

(such as ISwR, MASS, etc.), standard functions for the general parametric

analyses (such as aov, lm, nlme, etc.) and the multcomp package to perform

the multiple comparsion procedures. All of the packages used in this book are

available at the Comprehensive R Archive Network (CRAN), which can be

accessed from http://CRAN.R-project.org.

The source code for the analyses presented in this book is available from

the multcomp package. A demo containing the R code to reproduce the

individual results is available for each chapter by invoking

R> library("multcomp")

R> demo("Ch_Intro")

R> demo("Ch_Theory")

R> demo("Ch_GLM")

R> demo("Ch_Appl")

R> demo("Ch_Misc")

The results presented in this book and obtained with the multcomp package

in general have been validated to the extent possible against the SAS macros

described in Westfall et al. (1999). The multcomp package also includes SAS

code for most of the examples presented in Chapters 1, 3 and 4. It can be

accessed in R via

R> file.show(system.file("MCMT", "multcomp.sas",

+ package = "multcomp"))

Readers should not expect to get exactly the same results as stated in

this book. For example, diﬀerences in random number generating seeds or

truncating the number of signiﬁcant digits in the results may result in slight

diﬀerences to the output shown here.

Acknowledgments

This book started as a presentation at the 2004 UseR! conference in Vi-

enna, Austria. Many individuals have inﬂuenced the writing of this book since

then. Our joint work over many years with Werner Brannath, Edgar Brunner,

Alan Genz, Ekkehard Glimm, Gerhard Hommel, Ludwig Hothorn, Jason Hsu,

Franz K

onig, Willi Maurer, Jos´e Pinheiro, Martin Posch, Stephen Senn, Klaus

Strassburger, Ajit Tamhane, Randy Tobias, and James Troendle had a direct

impact on this book and we are grateful for the many fruitful discussions.

We are indebted to Sylvia Davis and Kevin Henning for their close reading

and criticism of an earlier version of this book. Their comments and sugges-

tions have greatly improved the presentation of the material. Bj

orn Bornkamp,

Richard Heiberger, Wei Liu, Hans-Peter Piepho, and Gernot Wassmer read

sections of the manuscript and provided much needed help during the progress

of this book. We are especially indebted to Richard Heiberger who, over many

years, provided valuable feedback on new functionality in multcomp and do-

CHAPTER 1

Introduction

Many scientiﬁc experiments subject to rigorous statistical analyses involve the

simultaneous evaluation of more than one question. For example, in clinical

trials one may compare more than one treatment group with a control group,

assess several outcome variables, measure at various time points, analyze mul-

tiple subgroups or look at any combination of these and related questions; but

multiplicity problems occur if we want to make simultaneous inference across

multiple questions. Similar problems may arise in agricultural ﬁeld experi-

ments which simultaneously compare several irrigation systems, investigate

the dose response relationship of a fertilizer, involve repeated assessments of

growth curves for a particular culture, etc. Recently, high-dimensional screen-

ing studies have become widely available in molecular biology and its appli-

cations, such as gene expression experiments and high throughput screenings

in early drug discovery. Those screening studies have in common the problem

of identifying a small subset of relevant variables from a huge set of candi-

date variables (e.g., genes, compounds, proteins). Scientiﬁc research provides

many examples of well-designed experiments involving multiple investigational

questions. Multiplicity is likely to become important when strong evidence and

good decision making is required.

In hypotheses test problems involving a single null hypothesis H the sta-

tistical tests are often chosen to control the Type I error rate of incorrectly

rejecting H at a pre-speciﬁed signiﬁcance level α. If multiple hypotheses, m

say, are tested simultaneously and the ﬁnal inferences should be valid across

all experimental questions of interest, the probability of declaring non-existing

eﬀects signiﬁcant increases in m. Assume, for example, that m = 2 hypotheses

and H

are each tested at level α = 0.05 using independent test statistics.

For example, let H

, i = 1, 2, denote the null hypotheses that a drug does not

show a beneﬁcial eﬀect over placebo for two primary outcome variables. As-

sume further that both H

and H

are true. Then the probability of retaining

both hypotheses is (1−α)

= 0.9025 under the independence assumption. The

complementary probability of incorrectly rejecting at least one null hypothesis

is 1 − (1 −α)

= 2α − α

= 0.0975. This is substantially larger than the ini-

tial signiﬁcance level of α = 0.05. In general, when testing m null hypotheses

using independent test statistics, the probability of committing at least one

Type I error is 1 − (1 − α)

, which reduces to the previous expression for

m = 2. Figure 1.1 displays the probability of committing at least one Type I

error for m = 1, . . . , 100 and α = 0.01, 0.05, and 0.10. Clearly, the probability

quickly reaches 1 for suﬃciently large values of m. In other words, if there is

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