Preface to First Edition - lib

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122 LOGISTIC REGRESSION AND GENERALISED LINEAR MODELS1. An error distribution giving the distribution of the response around itsmean. For analysis of variance and multiple regression this will be the normal;for logistic regression it is the binomial. Each of these (and othersused in other situations to be described later) come from the same, exponentialfamily of probability distributions, and it is this family that is usedin generalised linear modelling (see Everitt and Pickles, 2000).2. A link function, g, that shows how the linear function of the explanatoryvariables is related to the expected value of the response:g(µ) = β 0 + β 1 x 1 + · · · + β q x q .For analysis of variance and multiple regression the link function is simplythe identity function; in logistic regression it is the logit function.3. The variance function that captures how the variance of the response variabledepends on the mean. We will return to this aspect of GLMs later inthe chapter.Estimation of the parameters in a GLM is usually achieved through a maximumlikelihood approach – see McCullagh and Nelder (1989) for details.Having estimated a GLM for a data set, the question of the quality of its fitarises. Clearly the investigator needs to be satisfied that the chosen model describesthe data adequately, before drawing conclusions about the parameterestimates themselves. In practise, most interest will lie in comparing the fit ofcompeting models, particularly in the context of selecting subsets of explanatoryvariables that describe the data in a parsimonious manner. In GLMs ameasure of fit is provided by a quantity known as the deviance which measureshow closely the model-based fitted values of the response approximate the observedvalue. Comparing the deviance values for two models gives a likelihoodratio test of the two models that can be compared by using a statistic having aχ 2 -distribution with degrees of freedom equal to the difference in the numberof parameters estimated under each model. More details are given in Cook(1998).7.3 Analysis Using R7.3.1 ESR and Plasma ProteinsWe begin by looking at the ESR data from Table 7.1. As always it is good practiseto begin with some simple graphical examination of the data before undertakingany formal modelling. Here we will look at conditional density plots ofthe response variable given the two explanatory variables; such plots describehow the conditional distribution of the categorical variable ESR changes asthe numerical variables fibrinogen and gamma globulin change. The requiredR code to construct these plots is shown with Figure 7.1. It appears that higherlevels of each protein are associated with ESR values above 20 mm/hr.We can now fit a logistic regression model to the data using the glm func-© 2010 by Taylor and Francis Group, LLC
ANALYSIS USING R 123R> data("plasma", package = "HSAUR2")R> layout(matrix(1:2, ncol = 2))R> cdplot(ESR ~ fibrinogen, data = plasma)R> cdplot(ESR ~ globulin, data = plasma)ESRESR < 20 ESR > 200.0 0.2 0.4 0.6 0.8 1.0ESRESR < 20 ESR > 200.0 0.2 0.4 0.6 0.8 1.02.5 3.5 4.5fibrinogen30 35 40 45globulinFigure 7.1Conditional density plots of the erythrocyte sedimentation rate (ESR)given fibrinogen and globulin.tion. We start with a model that includes only a single explanatory variable,fibrinogen. The code to fit the model isR> plasma_glm_1 confint(plasma_glm_1, parm = "fibrinogen")2.5 % 97.5 %0.3387619 3.9984921© 2010 by Taylor and Francis Group, LLC
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A Handbook ofStatisticalAnalysesUsi
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DedicationTo our wives, Mary-Elizab
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Preface to First EditionThis book i
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List of Figures1.1 Histograms of th
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6.8 Normal probability plot of resi
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12.2 R output of the linear mixed-e
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18.7 Within-cluster sum of squares
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4.6 Lanza data. Misoprostol randomi
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15.2 BCG data. Meta-analysis on BCG
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6 Simple and Multiple Linear Regres
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CHAPTER 1An Introduction to R1.1 Wh
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INSTALLING R 3One can change the ap
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DATA OBJECTS IN R 5http://CRAN.R-pr
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DATA OBJECTS IN R 7R> help("Forbes2
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DATA IMPORT AND EXPORT 9As a simple
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BASIC DATA MANIPULATION 11The funct
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BASIC DATA MANIPULATION 13name sale
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COMPUTING WITH DATA 15Max. : 20.960
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COMPUTING WITH DATA 17Error in quan
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COMPUTING WITH DATA 19R> layout(mat
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© 2010 by Taylor and Francis Group
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SUMMARY 23examples of these functio
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26 DATA ANALYSIS USING GRAPHICAL DI
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Table 2.2:CHFLS data. Chinese Healt
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Table 2.5:USstates data. Socio-demo
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CHAPTER 3Simple Inference: Guessing
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INTRODUCTION 47table. Here there ar
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STATISTICAL TESTS 49Table 3.4:pisto
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STATISTICAL TESTS 51assumed to have
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ANALYSIS USING R 53procedure is McN
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ANALYSIS USING R 551 R> layout(matr
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ANALYSIS USING R 57R> wilcox.test(I
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ANALYSIS USING R 59R> t.test(moorin
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ANALYSIS USING R 61R> cor.test(~ mo
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SUMMARY 63R> mcnemar.test(rearrests
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CHAPTER 4Conditional Inference: Gue
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INTRODUCTION 67Table 4.4:Lanza data
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CONDITIONAL TEST PROCEDURES 69and x
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Page 101 and 102: INTRODUCTION 81The data in Table 5.
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Page 115 and 116: SUMMARY 95Table 5.4:schooldays data
Page 117 and 118: CHAPTER 6Simple and Multiple Linear
Page 119 and 120: SIMPLE LINEAR REGRESSION 99hours be
Page 121 and 122: MULTIPLE LINEAR REGRESSION 101The e
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Page 135 and 136: SUMMARY 115R> plot(clouds_lm)Cook's
Page 137 and 138: CHAPTER 7Logistic Regression and Ge
Page 139 and 140: INTRODUCTION 119Table 7.2:womensrol
Page 141: LOGISTIC REGRESSION AND GENERALISED
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Page 157 and 158: SUMMARY 137Table 7.5:bladdercancer
Page 159 and 160: CHAPTER 8Density Estimation: Erupti
Page 161 and 162: DENSITY ESTIMATION 141The Hertzspru
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Page 165 and 166: DENSITY ESTIMATION 1451 R> plot(xgr
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Page 171 and 172: ANALYSIS USING R 1510.360891 54.612
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Page 177 and 178: SUMMARY 157Table 8.4: birthdeathrat
Page 179 and 180: SUMMARY 159Table 8.5:schizophrenia
Page 181 and 182: 162 RECURSIVE PARTITIONINGTable 9.1
Page 183 and 184: 164 RECURSIVE PARTITIONINGBoth sets
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174 RECURSIVE PARTITIONINGR> plot(g
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CHAPTER 10Scatterplot Smoothers and
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INTRODUCTION 179Table 10.2:USairpol
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SMOOTHERS AND GENERALISED ADDITIVE
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ANALYSIS USING R 187R> plot(time ~
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ANALYSIS USING R 189R> x y men150
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ANALYSIS USING R 191R> USair_gam l
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ANALYSIS USING R 193R> layout(matri
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ANALYSIS USING R 195ter 9) where th
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198 SURVIVAL ANALYSISTable 11.1:gli
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200 SURVIVAL ANALYSISspect to time
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202 SURVIVAL ANALYSISHazard0.00 0.0
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204 SURVIVAL ANALYSISIn the Cox mod
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206 SURVIVAL ANALYSISExact Logrank
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208 SURVIVAL ANALYSISR> summary(GBS
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210 SURVIVAL ANALYSISR> layout(matr
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212 SURVIVAL ANALYSISmodels includi
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214 ANALYSING LONGITUDINAL DATA Iti
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216 ANALYSING LONGITUDINAL DATA ITa
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218 ANALYSING LONGITUDINAL DATA Iot
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220 ANALYSING LONGITUDINAL DATA IR>
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226 ANALYSING LONGITUDINAL DATA IKe
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228 ANALYSING LONGITUDINAL DATA Ido
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CHAPTER 13Analysing Longitudinal Da
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METHODS FOR NON-NORMAL DISTRIBUTION
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ANALYSIS USING R: GEE 239R> summary
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ANALYSIS USING R: GEE 245R> layout(
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ANALYSIS USING R: RANDOM EFFECTS 24
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ANALYSIS USING R: RANDOM EFFECTS 24
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SUMMARY 251Table 13.3:schizophrenia
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254 SIMULTANEOUS INFERENCE AND MULT
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CHAPTER 15Meta-Analysis: Nicotine G
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SYSTEMATIC REVIEWS AND META-ANALYSI
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STATISTICS OF META-ANALYSIS 271Sele
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ANALYSIS USING R 273the parameters
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ANALYSIS USING R 275R> plot(smoking
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PUBLICATION BIAS 277R> summary(BCG_
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SUMMARY 279R> summary(BCG_mod)Call:
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SUMMARY 2811 / standard error2 4 6
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Table 15.4:aspirin data. Meta-analy
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CHAPTER 16Principal Component Analy
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PRINCIPAL COMPONENT ANALYSIS 287nie
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ANALYSIS USING R 289R> score plot(
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ANALYSIS USING R 291R> score plot(
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ANALYSIS USING R 293Yuping (CHN) Ha
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SUMMARY 295R> biplot(heptathlon_pca
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SUMMARY 297rainJuly: rainfall in Ju
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Table 17.1:watervoles data. Water v
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302 MULTIDIMENSIONAL SCALINGn in nu
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304 MULTIDIMENSIONAL SCALINGwill be
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306 MULTIDIMENSIONAL SCALINGR> x y
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308 MULTIDIMENSIONAL SCALINGR> libr
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310 MULTIDIMENSIONAL SCALINGR> plot
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Table 17.3:eurodist data (package d
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CHAPTER 18Cluster Analysis: Classif
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INTRODUCTION 317mass, mass), the pe
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CLUSTER ANALYSIS 319x 20 2 4 6 8 10
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CLUSTER ANALYSIS 32141031678925Figu
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CLUSTER ANALYSIS 3231. Find some in
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ANALYSIS USING R 32518.3 Analysis U
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ANALYSIS USING R 327R> pottery_sing
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ANALYSIS USING R 329R> rge planet.
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ANALYSIS USING R 331R> plot(planet_
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ANALYSIS USING R 333R> scatterplot3
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BibliographyAdler, D. and Murdoch,
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BIBLIOGRAPHY 337Chambers, J. M. and
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BIBLIOGRAPHY 339Everitt, B. S. and
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BIBLIOGRAPHY 341Heitjan, D. F. (199
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BIBLIOGRAPHY 343Leisch, F. and Ross
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BIBLIOGRAPHY 345Proudfoot, J., Gold
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BIBLIOGRAPHY 347Stevens, J. (2001),
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