Table Of ContentOUPCORRECTEDPROOF–FINAL,16/5/2019,SPi
Statistical Thinking
from Scratch
OUPCORRECTEDPROOF–FINAL,16/5/2019,SPi
OUPCORRECTEDPROOF–FINAL,16/5/2019,SPi
Statistical Thinking
from Scratch
A Primer for Scientists
M. D. EDGE
1
OUPCORRECTEDPROOF–FINAL,16/5/2019,SPi
3
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OUPCORRECTEDPROOF–FINAL,16/5/2019,SPi
Contents
Acknowledgments xi
Prelude 1
1 Encountering data 5
1.1 Thingstocome 7
2 R and exploratory data analysis 11
2.1 InteractingwithR 13
2.2 Tutorial:theIrisdata 15
2.3 Chaptersummary 24
2.4 Furtherreading 25
3 The line of best fit 26
Box3-1: Mathematicsrequiredforthischapter 27
3.1 Definingthe“best”fit 28
3.2 Derivation:findingtheleast-squaresline 31
3.3 Conclusions 34
3.4 Chaptersummary 37
3.5 Furtherreading 37
4 Probability and random variables 38
Box4-1: Setsandsetnotation 40
4.1 [Optionalsection]Theaxiomsofprobability 41
4.2 Relationshipsbetweenevents:conditionalprobabilityandindependence 43
4.3 Bayes’theorem 45
Box4-2: Bayes’theoreminstatistics 47
4.4 Discreterandomvariablesanddistributions 48
4.5 Continuousrandomvariablesanddistributions 51
Box4-3: Integration 52
4.6 Probabilitydensityfunctions 53
4.7 Familiesofdistributions 55
4.8 Chaptersummary 58
4.9 Furtherreading 58
OUPCORRECTEDPROOF–FINAL,16/5/2019,SPi
viii CONTENTS
5 Properties of random variables 60
5.1 Expectedvaluesandthelawoflargenumbers 60
Box5-1: Measuresoflocation 61
5.2 Varianceandstandarddeviation 65
5.3 Jointdistributions,covariance,andcorrelation 68
5.4 [Optionalsection]Conditionaldistribution,expectation,andvariance 72
5.5 Thecentrallimittheorem 74
5.6 Aprobabilisticmodelforsimplelinearregression 79
Box5-2: Assumptionsaboutthelinearmodelandclaimsthatfollow 80
5.7 Chaptersummary 86
5.8 Furtherreading 86
Interlude 88
6 Properties of point estimators 91
Box6-1: Processes,populations,andsamples 92
6.1 Bias 94
6.2 Variance 96
6.3 Meansquarederror 96
6.4 Consistency 97
6.5 Efficiency 99
6.6 [Optionalsection]Statisticaldecisiontheoryandrisk 100
6.7 Robustness 103
6.8 Estimatorsforsimplelinearregression 105
6.9 Conclusion 109
6.10 Chaptersummary 109
6.11 Furtherreading 109
7 Interval estimation and inference 111
7.1 Standarderror 111
7.2 Confidenceintervals 112
Box7-1: Athoughtexperimentaboutconfidenceintervals 114
7.3 FrequentistinferenceI:nullhypotheses,teststatistics,andpvalues 116
7.4 FrequentistinferenceII:alternativehypothesesandtherejection
framework 121
7.5 [Optionalsection]Connectinghypothesistestsandconfidenceintervals 124
7.6 NHSTandtheabuseoftests 125
7.6.1 Lackofreplication 125
7.6.2 Ossificationofα=0.05 125
7.6.3 α=0.05asagatekeeper 126
7.6.4 Identificationofscientifichypothesiswithastatisticalhypothesis 126
7.6.5 Neglectofothergoals,suchasestimationandprediction 126
7.6.6 Adegradedintellectualculture 127
7.6.7 EvaluatingsignificancetestsinlightofNHST 129
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CONTENTS ix
7.7 FrequentistinferenceIII:power 131
7.8 Puttingittogether:Whathappenswhenthesamplesizeincreases? 135
7.9 Chaptersummary 137
7.10 Furtherreading 137
8 Semiparametric estimation and inference 139
8.1 Semiparametricpointestimationusingthemethodofmoments 142
8.1.1 Plug-inestimators 143
8.1.2 Themethodofmoments 145
8.2 Semiparametricintervalestimationusingthebootstrap 149
Box8-1: Bootstrappivotalintervals 154
8.3 Semiparametrichypothesistestingusingpermutationtests 157
8.4 Conclusion 162
8.5 Chaptersummary 163
8.6 Furtherreading 163
9 Parametric estimation and inference 165
Box9-1: Logarithms 166
9.1 Parametricestimationusingmaximumlikelihood 168
9.1.1 Maximum-likelihoodestimationforsimplelinearregression 172
9.2 Parametricintervalestimation:thedirectapproachandFisher
information 175
9.2.1 Thedirectapproach 175
9.2.2 [Optionalsubsection]TheFisherinformationapproach 177
9.3 ParametrichypothesistestingusingtheWaldtest 180
9.3.1 TheWaldtest 180
9.4 [Optionalsection]Parametrichypothesistestingusingthe
likelihood-ratiotest 181
9.5 Chaptersummary 185
9.6 Furtherreading 185
10 Bayesian estimation and inference 186
10.1 Howtochooseapriordistribution? 187
10.2 Theunscaledposterior,conjugacy,andsamplingfromtheposterior 188
10.3 BayesianpointestimationusingBayesestimators 193
10.4 Bayesianintervalestimationusingcredibleintervals 196
10.5 [Optionalsection]Bayesian“hypothesistesting”usingBayesfactors 198
10.6 Conclusion:Bayesianvs.frequentistmethods 201
10.7 Chaptersummary 202
10.8 Furtherreading 202
