1.LinearRegression........................................... 1
1.1Introduction ............................................. 1
1.2TheMethodofLeastSquares .............................. 3
1.2.1Correlationversion ................................. 7
1.2.2Large-samplelimit .................................. 8
1.3Theoriginsofregression ................................... 9
1.4Applicationsofregression.................................. 11
1.5TheBivariateNormalDistribution ......................... 14
1.6MaximumLikelihoodandLeastSquares ..................... 21
1.7SumsofSquares .......................................... 23
1.8Tworegressors ........................................... 26
Exercises .................................................... 28
2.TheAnalysisofVariance(ANOVA) ........................ 33
2.1TheChi-SquareDistribution ............................... 33
2.2ChangeofvariableformulaandJacobians ................... 36
2.3TheFisherF-distribution .................................. 37
2.4Orthogonality ............................................ 38
2.5Normalsamplemeanandsamplevariance ................... 39
2.6One-WayAnalysisofVariance ............................. 42
2.7Two-WayANOVA;NoReplications ......................... 49
2.8Two-WayANOVA:ReplicationsandInteraction .............. 52
Exercises .................................................... 56
3.MultipleRegression ........................................ 61
3.1TheNormalEquations .................................... 61
3.2SolutionoftheNormalEquations .......................... 64
3.3PropertiesofLeast-SquaresEstimators ...................... 70
3.4Sum-of-SquaresDecompositions ............................ 73
3.4.1Coefficientofdetermination.......................... 79
3.5Chi-SquareDecomposition ................................. 80
3.5.1Idempotence,TraceandRank........................ 81
3.5.2Quadraticformsinnormalvariates ................... 82
3.5.3SumsofProjections ................................ 82
3.6OrthogonalProjectionsandPythagoras’sTheorem ........... 85
3.7Workedexamples ......................................... 89
Exercises .................................................... 94
.FurtherMultilinearRegression ............................. 99
4.1PolynomialRegression .................................... 99
4.1.1ThePrincipleofParsimony .......................... 102
4.1.2Orthogonalpolynomials ............................. 103
4.1.3Packages .......................................... 103
4.2AnalysisofVariance ...................................... 104
4.3TheMultivariateNormalDistribution ....................... 105
4.4TheMultinormalDensity .................................. 111
4.4.1Estimationforthemultivariatenormal ................ 113
4.5ConditioningandRegression ............................... 115
4.6Mean-squareprediction ................................... 121
4.7Generalisedleastsquaresandweightedregression ............. 123
Exercises .................................................... 125
5.AddingadditionalcovariatesandtheAnalysis
ofCovariance ............................................... 129
5.1Introducingfurtherexplanatoryvariables .................... 129
5.1.1Orthogonalparameters .............................. 133
5.2ANCOVA ............................................... 135
5.2.1NestedModels ..................................... 139
5.3Examples ................................................ 140
Exercises .................................................... 145
6.LinearHypotheses .......................................... 149
6.1MinimisationUnderConstraints ............................ 149
6.2Sum-of-SquaresDecompositionandF-Test ................... 152
6.3Applications:SequentialMethods ........................... 157
6.3.1Forwardselection ................................... 157
6.3.2Backwardselection ................................. 158
6.3.3Stepwiseregression ................................. 159
Exercises .................................................... 160
7.ModelCheckingandTransformationofData ............... 163
7.1DeviationsfromStandardAssumptions ..................... 163
7.2TransformationofData ................................... 168
7.3Variance-StabilisingTransformations ........................ 171
7.4Multicollinearity ......................................... 174
Exercises .................................................... 177
8.GeneralisedLinearModels ................................. 181
8.1Introduction ............................................. 181
8.2Definitionsandexamples .................................. 183
8.2.1Statisticaltestingandmodelcomparisons ............. 185
8.2.2Analysisofresiduals ................................ 187
8.2.3Athleticstimes ..................................... 188
8.3Binarymodels ........................................... 190
8.4Countdata,contingencytablesandlog-linearmodels ......... 193
8.5Over-dispersionandtheNegativeBinomialDistribution ....... 197
8.5.1Practicalapplications:Analysisofover-dispersedmodels
inR ............................................. 199
Exercises .................................................... 200
9.Othertopics ................................................ 203
9.1Mixedmodels ............................................ 203
9.1.1MixedmodelsandGeneralisedLeastSquares .......... 206
9.2Non-parametricregression ................................. 211
9.2.1Kriging ........................................... 213
9.3ExperimentalDesign ...................................... 215
9.3.1Optimalitycriteria ................................. 215
9.3.2Incompletedesigns ................................. 216
9.4Timeseries .............................................. 219
9.4.1Cointegrationandspuriousregression ................. 220
9.5Survivalanalysis ......................................... 222
9.5.1Proportionalhazards ............................... 224
9.6 p>>n.................................................. 225
Solutions ....................................................... 227
DramatisPersonae:Whodidwhatwhen ....................... 269
Bibliography .................................................... 271
Index ........................................................... 279