General description
This module builds on the work in MA36510 by focusing on some of the many and varied applications of the Linear Model and considers techniques and modifications that have been motivated by them. Modern developments in the area are also considered.

Aims
To make the student aware of some of the applications of Linear Models and to consider new developments.

Learning outcomes
On completion of this module, a student should be able to:

• explain and use the reduction in sum of squares principle;
• formulate and carry out a test of a linear hypothesis;
• explain the importance of good design matrix structures;
• compare different models suggested for the same data sets;
• construct and fit models involving coincident or parallel straight lines as arise in biological and pharmaceutical assays;
• analyse a design matrix with respect to leverage;
• fit models of less than full rank;
• explain the idea of estimability;
• describe the concept of a generalized linear model and, in some appropriate situations, construct suitable models.

Syllabus
1. THE GENERAL LINEAR HYPOTHESIS: Definition and rank of linear hypothesis. The reduction in sum of squares principle. Testing linear hypotheses. Examples including applications in biological and pharmaceutical assays. Slope ratio and parallel line assays.
2. COMPARISON OF MODELS: Orthogonality. Orthogonal polynomials. Weighing designs. Brief treatment of design optimality.
3. GENERALIZED LINEAR MODELS: Basic ideas. The exponential family. Link functions and canonical links. Deviance. Examples including models for exponential, binomial and Poisson data.
4. DIAGNOSTICS: Ordinary, standardized and studentized residuals. Leverages. Deletion statistics.