Probability and Statistics with R

710 Pages·2008·9.363 MB·English

by  Maria Dolores Ugarte, Ana F. Militino, Alan T. Arnholt

PROBABILITY
and
STATISTICS WITH R
C8911_FM.indd 1 3/3/08 1:21:00 PM
C8911_FM.indd 2 3/3/08 1:21:00 PM
PROBABILITY
and
STATISTICS WITH R
María Dolores Ugarte
Ana F. Militino
Alan T. Arnholt
C8911_FM.indd 3 3/3/08 1:21:00 PM
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2008 by Taylor & Francis Group, LLC
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Version Date: 20140904
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Preface
The authors would like to thank their parents
Lola: Pedro and Loli
Ana: Carmelo and Juanita
Alan: Terry and Loretta
for their unflagging support and encouragement.
The Book
Probability and Statistics with R is a work born of the love of statistics and the
advancements that havebeen made in the field as more powerfulcomputers canbe used to
performcalculationsand simulationsthat were only dreamedofby those who came before.
TheSlanguageanditsderivative,R,havemadethepracticeofstatisticsavailabletoanyone
with the time and inclination to do so.
Teachers will enjoy the real-world examples and the thoroughly worked out derivations.
Those wanting to use this book as a reference work will appreciate the extensive treat-
ments on data analysis using appropriate techniques, both parametric and nonparametric.
Students who are visual learners will appreciate the detailed graphics and clear captions,
while the hands-on learners will be pleased with the abundant problems and solutions.
(A solutions manual should be available from Taylor & Francis.) It is our hope that
practitioners of statistics at every level will welcome the features of this book and that
it will become a valuable addition to their statistics libraries.
The Purpose
Our primary intention when we undertook this project was to introduce R as a teaching
statistical package, rather than just a program for researchers. As much as possible, we
have made a great effort to link the statistical contents with the procedures used by R to
show consistency to undergraduate students. The reader who uses S-PLUS will also find
this text useful, as S-PLUS commands are included with those for R in the vastmajority of
the examples.
This book is intended to be practical, readable, and clear. It gives the reader real-world
examples of how S can be used to solve problems in every topic covered including, but
not limited to, general probability in both the univariate and multivariate cases, sampling
distributions and point estimation, confidence intervals, hypothesis testing, experimental
design, and regression. Most of the problems are taken from genuine data sets rather than
created out of thin air. Next, it is unusually thorough in its treatment of virtually every
topic,coveringboththetraditionalmethodstosolveproblemsaswellasmanynonparamet-
ric techniques. Third, the figures used to explain difficult topics are exceptionally detailed.
v
vi
Finally, the derivations of difficult equations are worked out thoroughly rather than being
left as exercises. These features, and many others, will make this book beneficial to any
reader interested in applying the S language to the world of statistics.
The Program
TheSlanguageincludesbothRandS-PLUS.“Rcanberegardedasanimplementationof
the S language which was developed at Bell Laboratories by Rick Becker,John Chambers,
and Allan Wilks, and also forms the basis of the S-PLUS systems.”
(http://cran.r-project.org/doc/manuals/R-intro.html#Preface)
The current R is the result of a collaborative effort with contributions from all over
the world. R was initially written by Robert Gentleman and Ross Ihaka of the Statistics
Department of the University of Auckland. Since mid-1997 there has been a core group
with write access to the R source (http://www.r-project.org/—click “Contributors”on the
sidebar).
Not only is R an outstanding statistical package, but it is offered free of charge and can
be downloaded from http://www.r-project.org/. The authors are greatly indebted to the
giantsof statistics andprogrammingonwhose shoulderswe have stoodto see whatwe will
show the readers of this text.
The Content
The core of the material covered in this text has been used in undergraduate courses at
thePublicUniversityofNavarreforthelasttenyears. Ithasbeenusedtoteachengineering
(agricultural, industrial, and telecommunications) and economics majors. Some of the
material in this book has also been used to teach graduate students studying agriculture,
biology, engineering, and medicine.
The book starts with a brief introduction to S that includes syntax, structures, and
functions. It is designed to provide an overview of how to use both R and S-PLUS so that
even a neophyte will be able to solve the problems by the end of the chapter.
Chapter 2, entitled “Exploring Data,”covers important graphical and numerical descrip-
tive methods. This chapter could be used to teach a first course in statistics.
Thenextthreechaptersdealwithprobabilityandrandomvariablesinagenerallyclassical
presentationthatincludesmanyexamplesandanextensivecollectionofproblemstopractice
all that has been learned.
Chapter 6 presents some important statistics and their sampling distributions. Solving
the exerciseswillgive any readerconfidence that the difficult topics coveredin this chapter
are understood.
Thenextfourchaptersencompasspointestimation,confidenceintervals,hypothesistest-
ing,anda wide rangeof nonparametricmethods including goodness-of-fittests, categorical
data analysis, nonparametric bootstrapping, and permutation tests.
Chapter 11 provides an introduction to experimental design using fixed and random
effects models as well as the randomized block design and the two-factor factorial design.
Thebookendswithachapteronsimpleandmultipleregressionanalysis. Theprocedures
from this chapter are used to solve three interesting case studies based on real data.
The Fonts
Knowing several typographical conventions will help the reader in understanding the
materialpresentedinthistext. Rcodeisdisplayedinamonospacedfontwiththe>symbol
in front of commands that are entered at the R prompt.
vii
> x<-0.28354
> round(x,2)
[1] 0.28
The same font is used for data sets and functions, though functions are followed by ().
For example, the PASWR package but the round() function would be shown. Throughout
the text, a is found at the end of solutions to examples. In the index, page numbers
in BOLD are where the primary occurrences of topics are found, while those in ITALICS
indicate the pages where a problem about a topic or using a given data set can be located.
The Web
Thistextissupportedathttp://www1.appstate.edu/∼arnholta/PASWRontheInternet.
Thewebsite hasup-to-dateerrata,chapterscripts,andacopyofthe PASWRpackage(which
is also on CRAN) available for download.
Acknowledgments
WegratefullyacknowledgetheinvaluablehelpprovidedbySusieArnholt. Herwillingness
to apply her expertise in LATEX and knowledge of English grammar to the production of
this book is appreciated beyond words.
Several people were instrumental in improving the overall readability of this text. The
recommendationsmadebyPhilSpector,theApplicationsManagerandSoftwareConsultant
for the Statistical Computing Facility in the Department of Statistics at the University of
CaliforniaatBerkeley,whoreviewedthis textforTaylor& Francis,wereusedinimproving
much of the original R code as well as decreasing the inevitable typographical error rate.
Toma´sGoicoa,a member of the SpatialStatistics ResearchGroupatthe Public University
of Navarre, was of great help in preparing and checking exercises. Celes Alexander, an
Appalachian State University graduate student, graciously read the entire text and found
several typos. Any remaining typos or errors are entirely the fault of the authors.
Thanksto our editoratTaylor& Francis,DavidGrubbs, forembracingandencouraging
our project. Many thanks the Statistics and Operations Research Department at Public
University of Navarre and to the Department of Mathematical Sciences at Appalachian
State University for the support they gave us in our writing of this text.
The“Youchoose,youdecide”initiativesponsoredbyCajaNavarraalsoprovidedfunding
forin-personcollaborations. ThankstotheUniversidadNacionaldeEducaci´onaDistancia,
inparticulartheCentroAsociadodePamplona,forallowingustopresentthisprojectunder
their auspices.
Special thanks to Jos´e Luis Iriarte, the former Vicerector of International Relations of
the Public University of Navarre, and to T. Marvin Williamsen, the former Associate Vice
Chancellor for International Programs at Appalachian State University. These men were
instrumentalingainingfundingandsupportforseveralin-personcollaborationsincludinga
year-longvisit at the Public University of Navarrefor the third author and two multi-week
visits for the first two authors to Appalachian State University.
Finally, to the geniuses of this age who first conceived of the idea of an excellent open
source software for statistics and those who reared the idea to adulthood, our gratitude
is immeasurable. May the lighthouse of your brilliance guide travelers on the ocean of
statistics for decades to come. Thank you, R Core Team.

Contents
1 A Brief Introduction to S 1
1.1 The Basics of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Using S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Data Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.1 S Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.2 Mathematical Operations . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.3 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.4 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.5 Reading Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.5.1 Using scan() . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.5.2 Using read.table(). . . . . . . . . . . . . . . . . . . . 8
1.4.5.3 Using write() . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.5.4 Using dump() and source() . . . . . . . . . . . . . . . 9
1.4.6 Logical Operators and Missing Values . . . . . . . . . . . . . . . . 9
1.4.7 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.8 Vector and Matrix Operations . . . . . . . . . . . . . . . . . . . . 14
1.4.9 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.10 Lists. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.11 Data Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.12 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.13 Functions Operating on Factors and Lists . . . . . . . . . . . . . . 19
1.5 Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Creating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7 Programming Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.8 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Exploring Data 29
2.1 What Is Statistics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Displaying Qualitative Data . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.2 Barplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.3 Dot Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.4 Pie Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Displaying Quantitative Data . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.1 Stem-and-Leaf Plots . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.2 Strip Charts (R Only) . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.3 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5 Summary Measures of Location . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5.1 The Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5.2 The Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
ix