R语言实战：多元统计分析指南

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《应用多元统计分析：R语言实践指南》是一部由Daniel Zelterman编著的实用教材，专为生物与健康领域的统计分析而设计。本书是Statistics for Biology and Health系列的一部分，该系列由Mitchell Gail、Jonathan M. Samet、Anastasios Tsiatis和Wing Wong共同编辑，旨在提供针对特定学科领域深入且实用的统计学习资料。作者以其在耶鲁大学公共卫生学院的深厚背景，将复杂的多元统计理论融入到R语言的实际操作中，帮助读者掌握在实际问题中运用这一工具进行数据分析的能力。本书主要关注的是如何利用R语言进行多元统计分析，涵盖了诸如主成分分析（PCA）、因子分析、聚类分析、回归分析、判别分析、生存分析等多变量方法。它不仅介绍了统计原理，还提供了丰富的R代码示例，使得学习者能够在实践中理解和掌握各种技术。读者不仅能学到理论知识，还能通过实例了解到如何处理实际数据集中的复杂性，如变量间的关联性、多重共线性和缺失值等问题。本书适合对R语言有基本了解的统计学或应用科学专业人员，以及希望在生物医学研究、公共卫生、经济学、社会科学等领域应用多元统计分析的工作者。此外，对于研究生、教师和数据分析师而言，这本书也是一个不可或缺的参考资料，因为它提供了一个将理论知识转化为实战技能的桥梁。《应用多元统计分析：R语言实践指南》的ISBN号码为978-3-319-14092-6和978-3-319-14093-3，电子版可以通过DOI 10.1007/978-3-319-14093-3获取。版权方面，所有权利归Springer International Publishing所有，强调了对内容复制、分发和电子形式的保护。如果你是一位对R语言的多元统计分析感兴趣的读者，这本书将为你打开一扇深入探索这一领域的大门。

CONTENTS xv

7.5 Inference on Multivariate Normal Variances .......... 187

7.6 Fitting Patterned Covariance Matrices ............. 189

7.7 Tests for Multivariate Normality ................ 194

7.8 Exercises ............................. 202

8 Factor Methods 207

8.1 Principal Components Analysis ................. 208

8.2 Example 1: Investment Allocations ............... 210

8.3 Example 2: Kuiper Belt Objects ................ 214

8.4 Example 3: Health Outcomes in US Hospitals ......... 217

8.5 Factor Analysis .......................... 218

8.6 Exercises ............................. 223

9 Multivariable Linear Regression 231

9.1 Univariate Regression ...................... 232

9.2 Multivariable Regression in R .................. 238

9.3 A Large Health Survey ...................... 243

9.4 Exercises ............................. 250

10 Discrimination and Classiﬁcation 257

10.1 An Introductory Example .................... 257

10.2 Multinomial Logistic Regression ................. 261

10.3 Linear Discriminant Analysis .................. 265

10.4 Support Vector Machine ..................... 273

10.5 Regression Trees ......................... 278

10.6 Exercises ............................. 283

11 Clustering 287

11.1 Hierarchical Clustering ...................... 287

11.2 K-Means Clustering ....................... 295

11.3 Diagnostics, Validation, and Other Methods .......... 301

11.4 Exercises ............................. 308

12 Time Series Models 315

12.1 Introductory Examples and Simple Analyses .......... 315

12.2 Autoregressive Models ...................... 322

12.3 Spectral Decomposition ..................... 333

12.4 Exercises ............................. 336

13 Other Useful Methods 339

13.1 Ranking from Paired Comparisons ............... 339

13.2 Canonical Correlations ...................... 342

13.3 Methods for Extreme Order Statistics ............. 348

13.4 Big Data and Wide Data .................... 354

13.5 Exercises ............................. 356

1: INTRODUCTION 2

Table 1.1: Costs of living in each of the 50 states

Median Median Cost of 2009 Average

apartment home value living population gross income

State rent in $ in $1000 index in 1000s in $1000

AK 949 237.8 133.2 698.47 68.60

AL 631 121.5 93.3 4708.71 36.11

AR 606 105.7 90.4 2889.45 34.03

WV 528 95.9 95.0 1819.78 33.88

WY 636 188.2 99.6 544.27 64.88

Source: US Census, 2007 and 2009 data

As an example of a multivariate statistical analysis, let us create a 95 %

joint (simultaneous) conﬁdence interval of both the mean rent and housing

prices.

In Fig. 1.1 we present both the marginal conﬁdence intervals and the

joint, bivariate conﬁdence ellipsoid that is described more fully in Sect. 6.3.

The marginal conﬁdence intervals treat each variable individually, and

the resulting 95 % conﬁdence interval for the two means is pictured as a rect-

angle. The bivariate conﬁdence ellipsoid takes into account the correlation

between rents and housing prices resulting in an elongated elliptical shape

oriented to reﬂect the positive correlation between these two prices.

The elliptical area and the rectangle overlap. There are also areas in-

cluded in one ﬁgure but not the other. More importantly, notice the area of

the ellipse is smaller than that of the rectangle. This diﬀerence in area illus-

trates the beneﬁt of using multivariate methods over the marginal approach.

If we were using univariate methods and obtaining conﬁdence intervals for

each variable individually, then the resulting conﬁdence region is larger than

the region that takes the bivariate relationship of rents and housing costs

into account. This ﬁgure provides a graphical illustration of the beneﬁts of

using multivariate methods over the use of a series of univariate analyses.

The thoughtful reader might argue this exercise does not make sense because there

are only 50 states and this census represents the values of the complete population. In

other words, there is no uncertainty associated with this data because there is no larger

population of states for us to sample from.

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