Biostatistics with R: An Introduction to Statistics Through Biological Data

354 Pages·2012·4.86 MB·English

by  Babak Shahbaba (auth.)

Use R!
SeriesEditors:
RobertGentleman KurtHornik GiovanniParmigiani
Forfurthervolumes:
http://www.springer.com/series/6991
Babak Shahbaba
Biostatistics with R
An Introduction to Statistics
Through Biological Data
Prof.BabakShahbaba
DepartmentofStatistics
UniversityofCalifornia,Irvine
Irvine,CA92697-1250
USA
[email protected]
SeriesEditors:
RobertGentleman KurtHornik
PrograminComputationalBiology DepartmentofStatistikandMathematik
DivisionofPublicHealthSciences WirtschaftsuniversitätWien
FredHutchinsonCancerResearchCenter Augasse2-6
1100FairviewAvenue,N.M2-B876 1090Wien
Seattle,WA98109 Austria
USA
GiovanniParmigiani
TheSidneyKimmelComprehensive
CancerCenteratJohnsHopkinsUniversity
550NorthBroadway
Baltimore,MD21205-2011
USA
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Preface
This book will discuss basic statistical analysis methods through a series of bio-
logical examples using R and R-Commander as computational tools. The book is
intended for a wide range of readers, from people with relatively strong analyti-
calbackgroundwhowanttolearnaboutstatisticsanditsapplicationinbiology,to
nonstatisticianscientistswhousestatisticalmethodsintheirresearch.
While the theoretical aspects of statistics are intriguing and interesting on their
own,webelievethatwhatseparatesstatisticsfromotherbranchesofmathematicsis
itsintimaterelationshipwithotherfields,suchasbiology,economics,andsocialsci-
ences,anditswidespreadapplicationintheseareas.Instatistics,atheoreticalwork
is usually inspired by applied problems, and new theories usually find immediate
applications in real-world problems. This interweaving of theory and application
hasputstatisticsinaspecialplaceinthescientificworld.
In this book, most topics are motivated by real examples first. We believe that
learning a new topic becomes easier if it is motivated by interesting and engaging
appliedproblems.Wealsohopethatthisapproachhelpsstudentstoimprovetheir
criticalthinkingandproblem-solvingskillsforsituationswheretheyarepresented
withnewproblems.Tothisend,wemotivateeachnewtopicwitharelevantproblem
from biology. We then try to reach the solution intuitively before discussing the
relatedstatisticalmethods.Forexample,whendiscussingBayes’theorem,wefirst
present a biological problem (finding the probability of lung cancer for smokers)
and find the answer to that problem intuitively based on what we already know.
Then,weintroduceBayes’theoremasageneralformofoursolutionforthistype
ofproblem.
While discussing statistical methods and their applications, our goal is to keep
abalancebetweenmathematicalrigorandreadability.Toaccomplishthis,wehave
movedconceptsthattendtobemorecomplexwithlimitedapplicationsineveryday
analysistotheendofeachchapterin“Advanced”sections.Forthemostpart,these
sectionscouldbeskippedinthefirstreadingofthisbook.
Throughoutthebook,weuseR-Commander,afreeandpubliclyavailablecom-
puterprogram,toshowhowstatisticalmethodscanbeusedinpractice.Webelieve
thatusingthesemethodswhilelearningthemcouldhelpwiththelearningprocess.
vii
viii Preface
Most ofthe examplesdiscussedinthis bookare basedonscientificstudies whose
dataarepubliclyavailable.Foreachexample,weprovidethestep-by-stepapplica-
tionofR-Commander.Readersareencouragedtofollowthesestepswhilereading
thebooksothattheycanlearnstatisticsthroughhands-onexperience.
For some examples, the data are available through R and R-Commander. For
these examples, we provide the steps required to obtain the data. For some
other examples, the data are available online and can be downloaded from
http://extras.springer.com. Appendix A shows the steps for installing and using
R-Commander. Before reading the chapters, readers should follow these steps to
installR-Commanderontheircomputer.
The chapters are arranged according to what a typical statistical analysis in-
volves. We usually start with some specific scientific questions in mind. Then, we
designascientificstudytoanswerthosequestions.InChap.1,weverybrieflydis-
cussdifferenttypesofstudiesandtheirobjectives.Wealsopresentanoverviewof
typicalstepswetakefromraisingascientificquestiontoansweringitthroughsta-
tisticalmethods.Thesestepsalwaysinvolveidentifyingatargetpopulation,which
is the group of individuals we want to study (e.g., population of humans, orange
trees,cells).
Because the target populations are usually very large, we conduct our studies
onarelativelysmallnumberofindividualsrandomlysampled (i.e.,selected)from
thepopulation.Fromtheseindividualswecollectinformationintheformofmea-
surements of some specific characteristics such as age, size, and counts. We refer
totheinformationobtainedfromtheseindividualsasdatacollectively.InChaps.2
and3,wediscussseveraldataexplorationtechniques,whichinvolvesummarizing
andvisualizingdatatoobtainahigh-levelunderstandingofthedataandthetarget
population.
We want to generalize what we learn from the individuals participating in our
study(i.e.,therandomlyselectedindividuals)tothewholepopulation.Thisgener-
alization should always be accompanied by our acknowledgement that we are not
completelycertainaboutourfindingssinceourknowledgeofthepopulationisbased
onarelativelysmallsampleofindividualsfromthatpopulation.Specifically,weal-
wayspresentourfindingsalongwithsomemeasurementsthatreflecttheextentof
ouruncertainty.Tothisend,weuseprobabilityasapowerfulmathematicaltoolto
measureuncertainty.WediscussprobabilityinChaps.4and5.
Theprocessofanalyzingdatatolearnaboutthewholepopulationisreferredtoas
statisticalinference.Thisusuallyinvolvesguessingsomeunknownvalues,drawing
conclusions,andmakingdecisions.Chapters6,7,and8discusssomebasicmeth-
odsofinferentialstatistics.Chapters9,10,and11provideslightlymoreadvanced
statistical inference methods, which for the most part could be considered as the
generalizationoftopicscoveredinChap.8.
Finally,Chap.12discussesclusteringmethods,andChap.13discussesBayesian
analysis very briefly. These topics are not traditionally included in introductory
booksonstatistics.Wedecidedtoincludethesetopicsduetotheirimmenseimpor-
tanceinscientificstudies.Whilethisbookdoesnotdojusticetothesetwotopics,
wehopethatitservesasanintroductionforinterestedreaders.
Preface ix
As mentioned above, we use R-Commander to show how statistical methods
canbeusedforrealproblems.UsingR-Commanderdoesnotrequireanycomputer
programming. For readers who are comfortable with learning a programming lan-
guage,wediscusstheequivalentRprogramsattheendofeachchapterinAdvanced
sections.ThesereadersshouldstartfromAppendixB,whereweprovideabriefin-
troductiontoRprogramming.
The methods discussed in this book have been developed by many researchers
overmanyyears.Toavoidoverburdeningthereader,weprovideonlyasmallnum-
ber of references, mainly for related books that go beyond what we have covered
here,andalsoforrealproblemsthatwehaveusedasexamples.
Writing this book has helped me to improve my teaching, and the feedback I
havereceivedfrommystudentshashelpedmetoimprovethebookduringthepast
several years. I would like to thank all my students who challenged me with their
questions;theyhavebeenmytoughestcritics.
I would like to thank John Fox for developing R-Commander. This is an ex-
tremely useful tool for teaching basic statistics to students without programming
background.
IwouldalsoliketothankJessicaUtts,MichaelPhelan,SamBehseta,andWesley
Johnsonforreviewingthebookandprovidingthoughtfulcommentsandconstruc-
tivecriticismstoimprovethequalityofthisbook.Iamverygratefultohavesuch
goodfriendsandsupportivecolleagues.
AveryspecialthanksgoestoLauraBalzer,whoiscurrentlyagraduatestudentat
UCBerkeley.Shehasbeenextremelyhelpfulintheprocessofpreparingtheinitial
draftandeditingthebook.
Finally,Iwouldliketothankmyfamilyforbeingpatientandsupportivethrough-
outtheprocessofwritingthisbook;itwouldnothavebeenpossiblewithouttheir
loveandsupport.
Irvine,CA BabakShahbaba
November11,2011
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 StatisticalMethodsintheContextofScientificStudies . . . . . . . 1
1.2 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 ObservationalStudiesandExperiments . . . . . . . . . . . . . . . 4
1.4 DataExplorationandAnalysis . . . . . . . . . . . . . . . . . . . 5
1.5 StatisticalInference . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6.1 UsingR-Commander . . . . . . . . . . . . . . . . . . . . 6
1.6.2 UsingR . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7.1 MoreonSampling . . . . . . . . . . . . . . . . . . . . . . 11
1.7.2 MoreonObservationalStudies . . . . . . . . . . . . . . . 12
1.7.3 MoreonExperiments . . . . . . . . . . . . . . . . . . . . 13
1.7.4 Cross-Sectional,Longitudinal,andTimeSeriesData . . . . 14
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 DataExploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 DataVisualizationandSummaryStatistics . . . . . . . . . . . . . 17
2.2 VariableTypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 ExploringCategoricalVariables . . . . . . . . . . . . . . . . . . . 20
2.3.1 RelativeFrequencyandPercentage . . . . . . . . . . . . . 21
2.3.2 BarGraph . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.3 PieChart . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 ExploringNumericalVariables . . . . . . . . . . . . . . . . . . . 25
2.4.1 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.2 MeanandMedian . . . . . . . . . . . . . . . . . . . . . . 32
2.4.3 VarianceandStandardDeviation . . . . . . . . . . . . . . 34
2.4.4 Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.5 Boxplots . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 DataPreprocessing. . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.1 MissingData . . . . . . . . . . . . . . . . . . . . . . . . . 40
xi
xii Contents
2.5.2 Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5.3 DataTransformation . . . . . . . . . . . . . . . . . . . . . 43
2.5.4 CreatingNewVariableBasedonTwoorMoreExisting
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5.5 CreatingCategoriesforNumericalVariables . . . . . . . . 45
2.6 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.6.1 CoefficientofVariation . . . . . . . . . . . . . . . . . . . 46
2.6.2 ScalingandShiftingVariables . . . . . . . . . . . . . . . . 48
2.6.3 VariableStandardization . . . . . . . . . . . . . . . . . . . 50
2.6.4 DataExplorationwithRProgramming . . . . . . . . . . . 51
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 ExploringRelationships . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1 VisualizingandSummarizingRelationshipsBetweenVariables . . 61
3.2 RelationshipsBetweenTwoNumericalRandomVariables . . . . . 61
3.3 RelationshipsBetweenCategoricalVariables . . . . . . . . . . . . 69
3.4 RelationshipsBetweenNumericalandCategoricalVariables . . . . 73
3.5 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1 ProbabilityasaMeasureofUncertainty. . . . . . . . . . . . . . . 83
4.2 SomeCommonlyUsedGeneticTerms . . . . . . . . . . . . . . . 83
4.3 TheSampleSpace . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 ProbabilityMeasure . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.5 Complement,Union,andIntersection . . . . . . . . . . . . . . . . 87
4.5.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5.2 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5.3 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5.4 Jointvs.MarginalProbability . . . . . . . . . . . . . . . . 91
4.6 DisjointEvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.7 ConditionalProbabilities . . . . . . . . . . . . . . . . . . . . . . 93
4.8 TheLawofTotalProbability . . . . . . . . . . . . . . . . . . . . 95
4.9 IndependentEvents . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.10 Bayes’Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.10.1 ApplicationofBayes’TheoreminMedicalDiagnosis . . . 99
4.10.2 BayesianStatistics . . . . . . . . . . . . . . . . . . . . . . 101
4.11 InterpretationofProbabilityastheRelativeFrequency . . . . . . . 101
4.12 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.12.1 UsingTreeDiagramstoObtainJointProbabilities . . . . . 103
4.12.2 MakingDecisionsunderUncertainty . . . . . . . . . . . . 105
4.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5 RandomVariablesandProbabilityDistributions . . . . . . . . . . . 109
5.1 RandomVariables . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 Discretevs.Continuous . . . . . . . . . . . . . . . . . . . . . . . 110
5.3 ProbabilityDistributions . . . . . . . . . . . . . . . . . . . . . . . 111
Contents xiii
5.4 DiscreteProbabilityDistributions . . . . . . . . . . . . . . . . . . 112
5.4.1 BernoulliDistribution . . . . . . . . . . . . . . . . . . . . 113
5.4.2 BinomialDistribution . . . . . . . . . . . . . . . . . . . . 115
5.4.3 PoissonDistribution . . . . . . . . . . . . . . . . . . . . . 119
5.5 ContinuousProbabilityDistributions . . . . . . . . . . . . . . . . 121
5.5.1 ProbabilityDensityCurvesandDensityHistograms . . . . 124
5.5.2 NormalDistribution . . . . . . . . . . . . . . . . . . . . . 125
5.5.3 Student’st-distribution. . . . . . . . . . . . . . . . . . . . 130
5.6 CumulativeDistributionFunctionandQuantiles . . . . . . . . . . 131
5.7 ScalingandShiftingRandomVariables . . . . . . . . . . . . . . . 134
5.8 SumofTwoRandomVariables . . . . . . . . . . . . . . . . . . . 135
5.9 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.9.1 MoreonProbabilityDistributions . . . . . . . . . . . . . . 137
5.9.2 SomeOtherCommonlyUsedProbabilityDistributions . . 138
5.9.3 Quantile–QuantilePlots . . . . . . . . . . . . . . . . . . . 142
5.9.4 ProbabilityDistributionswithRProgramming . . . . . . . 143
5.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.1 ParameterEstimation . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2 PointEstimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2.1 PopulationMean . . . . . . . . . . . . . . . . . . . . . . . 152
6.2.2 PopulationVariance . . . . . . . . . . . . . . . . . . . . . 154
6.3 SamplingDistribution . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4 ConfidenceIntervalsforthePopulationMean . . . . . . . . . . . 158
6.5 ConfidenceIntervalWhenthePopulationVarianceIsUnknown . . 162
6.6 UsingCentralLimitTheoremforConfidenceInterval . . . . . . . 163
6.7 ConfidenceIntervalsforthePopulationProportion . . . . . . . . . 166
6.8 MarginofError . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.9 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.9.1 DerivingConfidenceIntervals . . . . . . . . . . . . . . . . 168
6.9.2 SampleSizeEstimation . . . . . . . . . . . . . . . . . . . 169
6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7 HypothesisTesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.2 HypothesisTestingforthePopulationMean . . . . . . . . . . . . 174
7.3 StatisticalSignificance . . . . . . . . . . . . . . . . . . . . . . . . 176
7.3.1 z-TestsofthePopulationMean . . . . . . . . . . . . . . . 177
7.3.2 Interpretationofp-value. . . . . . . . . . . . . . . . . . . 178
7.3.3 One-SidedHypothesisTesting . . . . . . . . . . . . . . . 181
7.3.4 Two-SidedHypothesisTesting . . . . . . . . . . . . . . . 183
7.4 HypothesisTestingUsingt-tests . . . . . . . . . . . . . . . . . . 184
7.5 HypothesisTestingforPopulationProportion. . . . . . . . . . . . 186
7.6 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.6.1 TestofNormality . . . . . . . . . . . . . . . . . . . . . . 188