Advanced Statistical Modelling
General Information
1
Introduction and Review
1.1
Introduction
1.2
Random Vectors and Random Matrices: A Review
1.3
Generalised Linear Models: A Review
1.4
Exercises
Question 1
Question 2
Question 3
Question 4
Question 5
Question 6
2
Estimation
2.1
Likelihood Function
2.2
Log-Likelihood Function
2.3
Score Function and Score Equation
2.3.1
Special Case: Natural Link
2.4
Fisher Information
2.5
Example: Poisson Regression
2.5.1
With Natural Link
2.5.2
With Identity Link
2.6
Properties of
\(\boldsymbol{S}(\boldsymbol{\beta})\)
and
\(\boldsymbol{F}(\boldsymbol{\beta})\)
2.6.1
Expectation of
\(\boldsymbol{S}(\boldsymbol{\beta})\)
2.6.2
Variance of
\(\boldsymbol{S}(\boldsymbol{\beta})\)
2.6.3
Property of
\(\boldsymbol{F}(\boldsymbol{\beta})\)
2.6.4
Special Case: Natural Link
2.7
Matrix Notation
2.7.1
Score Function and Fisher Information
2.7.2
Natural Link
2.8
Iterative Solution of
\(\boldsymbol{S}(\hat{\boldsymbol{\beta}}) = 0\)
2.8.1
Iteratively Reweighted Least Squares (IRLS)
2.8.2
IRLS Pseudo-Code
2.9
Practical Example: US Polio Data
2.10
Estimation of Dispersion Parameter
\(\phi\)
2.10.1
Special Cases
2.10.2
Practical Example: Hospital Stay Data
2.11
Asymptotic Properties of
\(\hat{\boldsymbol{\beta}}\)
2.11.1
Fisher Scoring
2.11.2
Expectation
2.11.3
Variance
2.11.4
Asymptotic Normality
2.11.5
Closing The Circle
2.11.6
Next Step
2.12
Exercises
Question 1
Question 2
Question 3
Question 4
Question 5
Question 6
Question 7
Question 8
Question 9
Question 10
3
Prediction and Inference
3.1
Prediction and Confidence Intervals
Example: Hospital Stay Data
3.2
Hypothesis Tests
3.2.1
Simple Tests
3.2.2
Generalisation to Nested Models
3.2.3
Example: Hospital Stay Data
3.3
Confidence Regions for
\(\hat{\boldsymbol{\beta}}\)
3.3.1
Hessian Confidence Region
3.3.2
Method of Support Confidence Region
3.4
Issues with GLMs and the Wald Test
3.4.1
Separation
3.4.2
Hauck-Donner Effect
3.4.3
Next Step
3.5
Exercises
Question 1
Question 2
Question 3
Question 4
4
Deviance
4.1
Goodness-of-Fit
4.1.1
The Saturated Model
4.1.2
Deviance
4.1.3
Examples for Special Cases
4.2
Asymptotic Properties
4.3
Pearson Statistic
4.3.1
Relation to Deviance
4.3.2
Pearson Residuals
4.3.3
Example: US Polio Data
4.4
Residuals and Diagnostics
4.4.1
Example: Hospital Stay Data
4.4.2
Example: US Polio Data
4.5
Analysis of Deviance
4.5.1
Interpretation and Testing
4.5.2
General Case
4.5.3
Example: Hospital Stay Data
4.6
Exercises
Question 1
Question 2
Question 3
Question 4
5
Quasi-likelihood Methods
5.1
More on Dispersion
5.1.1
Example: Hospital Stay Data
5.2
Overdispersion
5.2.1
Example: US Polio Data
5.2.2
Reasons and Impacts of Overdispersion
5.3
Quasi-likelihood Methods
5.3.1
Example: US Polio Data
5.4
Generalized Estimating Equations
5.4.1
Example: US Polio Data
5.5
Exercises
Question 1
Question 2
Question 3
Question 4
6
Marginal models
6.1
Repeated measures data
6.1.1
Example: Oxford boys data
6.1.2
Example: Mathematics achievement data
6.2
Marginal model for repeated measures
6.2.1
Some examples
6.3
Estimation
6.3.1
Example
7
Linear mixed models
8
Practical Sheets
8.1
Practical 1
8.1.1
Load and visualise the data
8.1.2
Gaussian GLM and linear model
8.1.3
Gamma GLMs
8.1.4
Fisher information matrices
8.2
Practical 2
8.2.1
Exploratory data analysis
8.2.2
Rescaled binomial logit model
8.2.3
Rescaled Poisson model
8.2.4
Compare the models
9
Practical Sheet Solutions
9.1
Practical 1
9.1.1
Load and visualise the data
9.1.2
Gaussian GLM and linear model
9.1.3
Gamma GLMs
9.1.4
Fisher information matrices
9.1.5
Models without the intercept
9.2
Practical 2
9.2.1
Exploratory data analysis
9.2.2
Rescaled binomial logit model
9.2.3
Rescaled Poisson model
9.2.4
Compare the models
9.2.5
Dispersion and confidence interval for binomial GLM
10
Solutions to Selected Exercises
10.1
Chapter 1
Question 5
Question 6
10.2
Chapter 2
Question 1b
Question 2
Question 3
Question 5
Question 6
Question 9
10.3
Chapter 3
Question 2a
Question 4
10.4
Chapter 4
Question 2
Question 3
Published with bookdown
Advanced Statistical Modelling III (Epiphany term)
Chapter 7
Linear mixed models
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