View source for Multivariate Statistical Modelling Based on Generalized Linear Models

=Chapter 1: Book Outline and Notes=

Chapter 2: review of univariate generalized linear models and extensions

Chapter 3: models for multicategorical responses (i.e. multiple, unordered responses)

Chapter 4: selecting variables for models, variable reduction procedures, checking models, goodness-of-fit, residual analysis (outliers or consistent trend?)

Chapter 5: Semi- and non-parametric approaches

Chapter 6: Fixed-parameter models for time series (extends Ch. 2 and Ch. 3)

Chapter 7: Random effects models for non-normal data

Chapter 8: State space models for analyzing non-normal time series; relate time series observations y_t to unobserved states, like trend and seasonal components

Chapter 9: Survival models; determination of factors that determine survival/transition



=Chapter 2: Univariate Generalized Linear Models=

Cross-sectional regression analysis: univariate variable of primary interest (response variable) <math>y</math>

Explained by a vector <math>x = (x_1, x_2, \dots x_m)</math>

Data consist of observations on <math>(y,x)</math>:

<math>
(y_i , x_i ), i = 1, \dots, n
</math>

==Definition of Univariate Generalized Linear Models==

classical linear model for ungrouped normal responses and deterministic covariates is:

<math>
y_i = z_i^{\prime} \beta + \epsilon_i
</math>

where:

<math>z_i</math> = design vector, function of covariate vector <math>x_i</math>

<math>\beta</math> = vector of unknown parameters

<math>\epsilon_i</math> = errors, normally distributed and independent, <math>\epsilon_i \sim N(0, \sigma^2)</math>

The observations <math>y_i</math> are independent and normally distributed, 

<math>
y_i \sim N(\mu_i, \sigma^2), i=1, \dots, n
</math>

A specific generalized linear model is fully characterized by three components:
* type of exponential family
* response or link function
* design vector

Example:

Exponential family
* important members: normal, binomial, poisson, gamma, inverse Gaussian distributions

==Models for Continuous Responses==

Normal distribution

Gamma distribution 

Inverse Gaussian distribution

==Models for Binary and Binomial Responses==

Linear probability model

Probit model

Logit model

Complementary log-log model

==Models for Counted Data==

Log-linear Poisson model

Linear Poisson model


==Likelihood Inference==

Regression analysis with generalized linear models is based on likelihoods

This section contains inferential tools for:
* parameter estimation
* hypothesis testing
* good-ness-of-fit tests
* more detailed material on model choice/checking: see Chapter 4

Assumes that model is completely and correctly specified

Maximum likelihood estimator (MLE): MLE of unknown parameter vector obtained by maximizing the likelihood

===Goodness of fit Statistics===

two measures of adequacy of model (goodness of fit) are:

Pearson statistic <math>\chi^2 = \sum_{i=1}^{g} \frac{ \left( y_i - \hat{\mu}_i \right)^2 }{ v(\hat{\mu}_i) }</math>

Deviance <math>D = - 2 \phi \sum_{i=1}^{g} \left[ l_i ( \hat{\mu}_i ) - l_i (y_i) \right]</math>

where

<math>\hat{\mu}_i</math> = estimated mean function

<math>v(\hat{\mu}_i)</math> = estimated variance function

<math>l_i(y_i)</math> = individual log-likelihood





















=References=

Generalized linear models:
* McCullagh and Nelder 1989 - standard source of information about generalized linear models
* Santner and Duffy 1989 - consider cross-classified data and univariate discrete data


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