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Generalized Linear Models

Generalized Linear Models
Spring 2013

Cox, D.R. and Snell, E.J. (1981) Applied Statistics. Chapman & Hall.
Davison, A.C. (2003) Statistical Models. Cambridge University Press.
McCullagh, P. and Nelder, J.A. (1989) Generalized Linear Models, 2nd edn. C&H.
Cox, D.R. and Snell, E.J. (1989) Binary data 2nd Edn. Chapman & Hall.
Chambers, J. and Hastie, T. Statistical models in S. Wadsworth.
Venables, W.N. and Ripley, B.D. Modern Applied Statistics with S. Springer

Other relevant books:
McCulloch, C.E. and Searle, S.R. (2001) Generalized, Linear, and Mixed Models, Wiley
Pawitan, Y. (2001) In All Likelihood. Oxford.
Myers, Montgomery and Vining (2002) Generalized Linear Models

Linear regression and weighted linear regression
general matrix form; projection matrix.
precision of estimates
t- tests and F-tests: p-values; AS section 4.7
reduction in residual sum of squares principle
adding a new variable to a regression model
Transformation: Tukey's 1DOFNA; Box-Cox method
interpretation of regression coeffients: AS section 4.3, 4.11
special case with replication in linear regression
decomposition of sum of squares between/within
simple variance components problems
reading: Applied Statistics, pages 1-50.

Model Formulae
factors and variates
intrinsic and extrinsic aliasing
constraints and generalized inverses
effect of choice of constraint
`+', `.' and `*': interactions
constructing an analysis of variance table in GLIM
homologous factors
reading: GLM sections 3.1-3.6

Examples with non-Normal response variable or non-linear effects
dilution assay GLM section 1.2.4
2x2 table
vitamin C decay rate: GLM p.96
water hardness and CNS disorders: GLM p.186
Yields of barley and sinapis in competition: GLM p.318
survival times of leukemia patients: AS example U
smoking and perinatal mortality: GLM p.190
food preference data: BD p.160; GLM p.175
mixtures of drugs: relative potency; synergy: GLM p.387
reading: Binary Data, Chapter 1

Definition of a generalized linear model
natural exponential families
cumulants and cumulant function
link function
model formula
problems connected with aggregation
fitting generalized linear models
deviance function
reading: GLM Chapter 2.

Some results concerning maximum likelihood estimation
score statistic, parameter estimate
observed and expected Fisher information
likelihood ratio statistics
nuisance parameters and composite hypotheses
reading: Binary Data Appendix 1; GLM Appendix 1.

Binary data
covariate classes and contingency tables
binomial distribution
link functions and parameter interpretation
retrospective sampling; BD section 4.2, 4.3; GLM section 4.3
parameter estimation
deviance function
extrapolation (point estimation vs interval estimation)
over-dispersion: BD section 3.2; GLM section 4.5
reading: BD chapter 2; GLM chapter4

Polytomous data
measurement scales nominal, ordinal, interval, nested
multinomial distribution
quadratic forms and generalized inverses
multinomial log likelihood
parameter estimation
paired preferences and the Bradley-Terry model formula
reading: BD sections 5.1-5.4; GLM chapter 5

Log-linear models
Poisson distribution
relative potency and Fieller's method
Contingency tables; independence and conditional independence
multiple responses
reading: BD sections 5.5-5.6; GLM chapter 6

Conditional and marginal likelihoods
Gaussian marginal likelihood: REML
2x2 tables
matched pairs and quasi-symmetry
odds-ratio regression and hypergeometric distribution
reading: BD sections 2.3-2.5, A1.8; GLM chapter 7

Quasi-likelihood and estimating functions
reading: GLM chapter 9

Distance-matrix methods
Wishart matrix and Wishart distance matrix
Trees rooted and unrooted
Projection methods

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