## STAT412 Generalised Linear Models

Second Semester |

In 1972, Nelder and Wedderburn published a paper entitled *Generalized linear models* in which they presented a unified approach to linear models that does not require an assumption of normal errors. This paper produced a dramatic change in the type of data that could be analyzed within a linear modeling framework.

### Paper details

We consider the theory and use of generalized linear models. This paper will include theory, but primarily emphasize application with a focus on parameter interpretation and hypothesis testing in real data analysis using R.

### Potential students

For any student who wishes to know more about this important framework for analyzing many types of data. A working knowledge of these models is extremely valuable for any practicing statistician.

### Main topics

- Introduction to generalized linear models
- Nominal and ordinal logistic models
- Poisson models
- Overdispersion
- Zero-inflated models
- Modelling rates using an offset
- Log-linear models
- Gamma and lognormal models
- Quasi-likelihood
- Generalised additive models
- Survival models

### Prerequisites

STAT 341

STAT 362

### Required text

None

### Useful references

Collett (1991) *Modelling binary data*, Chapman and Hall

Dobson and Barnett (2008) *An Introduction to generalized linear models*, Chapman and Hall

Faraway (2006) *Extending the linear model with R*, Chapman and Hall

Krzanowski (1998) *An introduction to statistical modelling*, Arnold

McCullagh and Nelder (1989) *Generalized linear models*, Chapman and Hall

### Lecturer

Dr Ting Wang, Room 518, Extension 7773

Dr Graham Wood, Room 516

### Lectures

Two hours of lecture per week. Time and location to be arranged.

### Tutorial

One per week at a time to be arranged

### Internal Assessment

The final internal assessment is based on 4 assignments and a project. The assignments are worth 75% in total and the project 25%. The project will involve a presentation to the rest of the class.

### Exam format

The final exam is three hours long.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = max(E, 0.6E + 0.3A + 0.1P)**

where:

- E is the Exam mark
- A is the Assignments mark
- P is the Project mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.

**Adrien-Marie Legendre**1752-1833.

In 1805 Legendre presented the method of least squares and applied it to the determination of the ellipticity of the earth, and hence the length of the metre (a metre was defined to be 1/10,000,000 of a meridian quadrant — the distance from the equator to the North Pole). Gauss published the method of least squares in 1809 but claimed that he had been using the technique since 1795. Even now — two centuries later — researchers are still trying to establish who was the first to discover the method of least squares. Even though Legendre derived the least squares estimator based on the minimization of the sum of squares error, he only treated the special case that there is a perfect fit. However, Gauss developed the method by using the method of maximum likelihood and showed that the least squares estimator was the solution when the random errors are normally distributed. Furthermore, Gauss also considered the case of unequal variances that leads to the weighted least squares.

### Sum of squares

Bartlett presented the above geometric representation in 1934 to illustrate the extra sum of squares principle. Here*y*is the observed value, is the predicted value from the full model and is the predicted value from the reduced model. Hence