MATH68341 - 2012/2013

 

Specification

Aims

This course unit aims to familiarise students with the methodology and practical applications
of regression analysis, analysis of variance and analysis of covariance within the class of
linear and generalized linear models.

Brief Description of the unit

As an important modelling strategy, regression analysis is concerned with investigating whether, and how, one or more so-called explanatory variables, such as age, sex, blood pressure, etc., influence a response variable, such as a patient's diagnosis, by taking random variations of data into account. In Linear Models, the Normality assumption of the response and the linear relationship imposed between the response and the explanatory variables could sometimes be inappropriate to model certain data. An extension of Linear Models to Generalized Linear Models can help overcome this issue, by extending linearity to non-linearity and normality to a commonly encountered distribution family, called the exponential family of distributions. The course starts with an introduction to the concepts of Linear Models, Generalized Linear Models and the exponential family of distributions. It then continues on to study model fitting, assessing (model diagnostics, variable selection) and inference (hypothesis testing, confidence intervals) in the Linear Model setting. An extension of the aforementioned processes to Generalized Linear Models is also considered.

Learning Outcomes

Students should be able to:

• understand the methods for fitting linear and generalized linear model to data and the
  relevant statistical assumptions
• have the ability to interpret a statistical model
• fit data to a given linear or generalized linear model and comment on the adequacy of the fit
• produce confidence intervals for linear combinations of the model parameters
• formulate hypotheses in terms of the model parameters and construct test procedures for
  testing such hypotheses
• interpret the results of such analyses
• use the model to predict outcomes of new but related data sets
• perform statistical calculations using SPlus/R
• communicate and discuss the results of the statistical analysis of data

Future topics requiring this course unit

None

Syllabus

1. Linear Model and its specification, least squares estimators and maximum likelihood
   estimators and their statistical properties, residuals and residual sum of squares, their
   statistical properties and their use in assessing the goodness of fit of the model. Model
   comparison and model building. Multicollinearity.
2. Confidence intervals and t-tests involving a parametric function, prediction. The general
   linear hypothesis and its test using the likelihood ratio test, Ftests. Linear regression problems using dummy variables.
3. Generalized linear model, the exponential family of distributions, linear predictor, link
   function, canonical link, mean-variance relationship, maximum likelihood estimation,
   iteratively weighted least square (IWLS) estimation, asymptotic properties of the model
   estimators, Fisher information matrix, Wald statistic and its use in hypothesis testing and
   confidence interval construction, deviance function, general linear hypothesis and its test
   based on analysis of deviance.
4. Applications to Binary, Binomial, Multinomial data and Poisson data arising in medicine
   and public health.

Textbooks

• An introduction to generalized linear models - Dobson, Annette J., 2007
  Book | Recommended | 3rd ed
• Applied regression analysis - Draper, N. R., Smith, Harry, c1998
  Book | Recommended | 3rd ed
• Linear models with R - Faraway, Julian J., 2004
  Book | Recommended
• Extending the linear model with R: generalized linear, mixed effects and nonparametric regression models - Faraway, Julian J., 2005 (electronic resource)
  Book | Recommended | E-Book. Available via Dawsonera. Requires central username and password for access

Teaching and learning methods

Three lectures and one exercise class a week

Assessment

End of semester examination: two and a half hours, 80%

Coursework 20%

 

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