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Generalized Linear Models

Generalized Linear Models
Spring 2013

Texts:
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


Syllabus:
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
sparseness
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
over-dispersion
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
over-dispersion
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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