Statistical Modelling in R

by Aitkin, Murray; Francis, Brian; Hinde, John; Darnell, Ross

• ISBN13: 9780199219131
• ISBN10: 0199219133
• Format: Paperback
• Copyright: 2009-04-29
• Publisher: Oxford University Press

Summary

R is now the most widely used statistical package/language in university statistics departments and many research organizations. Its great disadvantages are that for many years, it has been the leading statistical package/language and that it can be freely downloaded from the R website. This text provides a comprehensive treatment of the theory of statistical modelling in R with an emphasis on applications to practical problems and an expanded discussion of statistical theory. A wide range of case studies is provided, using the normal, binomial, Poisson, multinomial, gamma, exponential and Weibull distributions, making this book ideal for graduates and research students in applied statistics and a wide range of qualitative disciplines.

Author Biography

Murray Aitkin is a Professorial Fellow at the Department of Mathematics and Statistics, University of Melbourne. In 1992 he was awarded an ARC Senior Research Fellowship, initially at the Australian National University and then at the University of Western Australia, where he worked on foundational issues in statistics. At the conclusion of the fellowship he was appointed to the Chair of Statistics at the University of Newcastle, UK, from which he took early retirement in 2004. In 2000-2002 he held a consulting position as Chief Statistician at the Education Statistics Services Institute, a division of the American Institutes for Research which provided consultancy to the National Center for Education Statistics of the US Department of Education. He continued to work as a consultant for NCES after 2002 at Newcastle, and this continues in Melbourne. John Hinde is Professor of Statistics at the National University of Ireland Galway having previously worked at the Universities of Exeter and Lancaster in the UK. It was while at Lancaster that he met his co-authors and wrote the original book Statistical Modelling in GLIM that was later revised and has now been translated to R. His interests are in all aspects of statistical modelling, including generalized linear models and their extensions, and statistical computing. Particular interests are in overdispersion modelling, mixture models, and random effect models. He was a joint founding editor of the journal Statistical Modelling and served as Chairman of the Statistical Modelling Society. He is currently President of the Irish Statistical Association.

Table of Contents

Introducing R	p. 1
Statistical packages and statistical modelling	p. 1
Getting started in R	p. 1
Reading data into R	p. 3
Assignment and data generation	p. 6
Displaying data	p. 8
Data structures and the workspace	p. 9
Transformations and data modification	p. 11
Functions and suffixing	p. 12
Structure functions	p. 13
Mathematical functions	p. 13
Logical operators	p. 14
Control functions	p. 14
Statistical functions	p. 14
Random numbers	p. 15
Suffixes in expressions	p. 16
Extracting subsets of data	p. 17
Recoding variates and factors into new factors	p. 17
Graphical facilities	p. 18
Text functions	p. 21
Writing your own functions	p. 22
Sorting and tabulation	p. 23
Editing R code	p. 26
Installing and using packages	p. 27
Statistical modelling and inference	p. 28
Statistical models	p. 28
Types of variables	p. 30
Population models	p. 31
Random sampling	p. 44
The likelihood function	p. 44
Inference for single parameter models	p. 46
Comparing two simple hypotheses	p. 47
Information about a single parameter	p. 49
Comparing a simple null hypothesis and a composite alternative	p. 54
Inference with nuisance parameters	p. 58
Profile likelihoods	p. 59
Marginal likelihood for the variance	p. 63
Likelihood normalizing transformations	p. 66
Alternative test procedures	p. 68
Bayes inference	p. 71
Binomial model	p. 74
Hypergeometric sampling from finite populations	p. 80
The effect of the sample design on inference	p. 81
The exponential family	p. 82
Mean and variance	p. 83
Generalized linear models	p. 83
Maximum likelihood fitting of the GLM	p. 84
Model comparisons through maximized likelihoods	p. 87
Likelihood inference without models	p. 89
Likelihoods for percentiles	p. 89
Empirical likelihood	p. 92
Regression and analysis of variance	p. 97
An example	p. 97
Strategies for model simplification	p. 107
Stratified, weighted and clustered samples	p. 111
Model criticism	p. 114
Mis-specification of the probability distribution	p. 116
Mis-specification of the link function	p. 119
The occurrence of aberrant and influential observations	p. 119
Mis-specification of the systematic part of the model	p. 123
The Box-Cox transformation family	p. 123
Modelling and background information	p. 126
Link functions and transformations	p. 136
Regression models for prediction	p. 138
Model choice and mean square prediction error	p. 140
Model selection through cross-validation	p. 141
Reduction of complex regression models	p. 144
Sensitivity of the Box-Cox transformation	p. 153
The use of regression models for calibration	p. 156
Measurement error in the explanatory variables	p. 159
Factorial designs	p. 161
Unbalanced cross-classifications	p. 168
The Bennett hostility data	p. 168
ANOVA of the cross-classification	p. 170
Regression analysis of the cross-classification	p. 174
Statistical package treatments of cross-classifications	p. 176
Missing data	p. 178
Approximate methods for missing data	p. 180
Modelling of variance heterogeneity	p. 180
Poison example	p. 184
Tree example	p. 191
Binary response data	p. 195
Binary responses	p. 195
Transformations and link functions	p. 197
Profile likelihoods for functions of parameters	p. 202
Model criticism	p. 207
Mis-specification of the probability distribution	p. 207
Mis-specification of the link function	p. 207
The occurrence of aberrant and influential observations	p. 207
Binary data with continuous covariates	p. 208
Contingency table construction from binary data	p. 223
The prediction of binary outcomes	p. 235
Profile and conditional likelihoods in 2 × 2 tables	p. 242
Three-dimensional contingency tables with a binary response	p. 246
Prenatal care and infant mortality	p. 246
Coronary heart disease	p. 248
Multidimensional contingency tables with a binary response	p. 255
Multinomial and Poisson response data	p. 269
The Poisson distribution	p. 269
Cross-classified counts	p. 271
Multicategory responses	p. 279
Multinomial logit model	p. 285
The Poisson-multinomial relation	p. 287
Fitting the multinomial logit model	p. 293
Ordered response categories	p. 298
Common slopes for the regressions	p. 299
Linear trend over response categories	p. 301
Proportional slopes	p. 304
The continuation ratio model	p. 304
Other models	p. 308
An Example	p. 310
Multinomial logit model	p. 313
Continuation ratio model	p. 320
Structured multinomial responses	p. 330
Independent outcomes	p. 331
Correlated outcomes	p. 339
Survival data	p. 347
Introduction	p. 347
The exponential distribution	p. 347
Fitting the exponential distribution	p. 349
Model criticism	p. 354
Comparison with the normal family	p. 361
Censoring	p. 364
Likelihood function for censored observations	p. 365
Probability plotting with censored data: the Kaplan-Meier estimator	p. 368
The gamma distribution	p. 377
Maximum likelihood with uncensored data	p. 379
Maximum likelihood with censored data	p. 382
Double modelling	p. 384
The Weibull distribution	p. 388
Maximum likelihood fitting of the Weibull distribution	p. 390
The extreme value distribution	p. 394
The reversed extreme value distribution	p. 397
Survivor function plotting for the Weibull and extreme value distributions	p. 398
The Cox proportional hazards model and the piecewise exponential distribution	p. 400
Maximum likelihood fitting of the piecewise exponential distribution	p. 403
Examples	p. 404
The logistic and log-logistic distributions	p. 407
The normal and lognormal distributions	p. 411
Evaluating the proportional hazard assumption	p. 414
Competing risks	p. 420
Time-dependent explanatory variables	p. 427
Discrete time models	p. 427
Finite mixture models	p. 433
Introduction	p. 433
Example - girl birthweights	p. 434
Finite mixtures of distributions	p. 434
Maximum likelihood in finite mixtures	p. 435
Standard errors	p. 437
Testing for the number of components	p. 440
Example	p. 443
Likelihood 'spikes'	p. 448
Galaxy data	p. 450
Kernel density estimates	p. 458
Random effect models	p. 461
Overdispersion	p. 461
Testing for overdispersion	p. 464
Conjugate random effects	p. 466
Normal kernel: the t-distribution	p. 466
Poisson kernel: the negative binomial distribution	p. 472
Binomial kernel: beta-binomial distribution	p. 477
Gamma kernel	p. 478
Difficulties with the conjugate approach	p. 478
Normal random effects	p. 479
Predicting from the normal random effect model	p. 481
Gaussian quadrature examples	p. 481
Overdispersion model fitting	p. 481
Poisson - the fabric fault data	p. 482
Binomial - the toxoplasmosis data	p. 484
Other specified random effect distributions	p. 487
Arbitrary random effects	p. 487
Examples	p. 489
The fabric fault data	p. 489
The toxoplasmosis data	p. 492
Leukaemia remission data	p. 493
The Brownlee stack-loss data	p. 493
Random coefficient regression models	p. 496
Example - the fabric fault data	p. 498
Algorithms for mixture fitting	p. 499
The trypanosome data	p. 499
Modelling the mixing probabilities	p. 503
Mixtures of mixtures	p. 504
Variance component models	p. 508
Models with shared random effects	p. 508
The normal/normal model	p. 508
Exponential family two-level models	p. 511
Other approaches	p. 513
NPML estimation of the masses and mass-points	p. 514
Random coefficient models	p. 514
Variance component model fitting	p. 515
Children's height development	p. 516
Multi-centre trial of beta-blockers	p. 524
Longitudinal study of obesity	p. 530
Autoregressive random effect models	p. 537
Latent variable models	p. 543
The normal factor model	p. 543
IRT models	p. 544
The Rasch model	p. 544
The two-parameter model	p. 545
The three-parameter logit (3PL) model	p. 547
Example - The Law School Aptitude Test (LSAT)	p. 547
Spatial dependence	p. 551
Multivariate correlated responses	p. 552
Discreteness of the NPML estimate	p. 552
Bibliography	p. 554
R function and constant index	p. 567
Dataset index	p. 570
Subject index	p. 571
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