85160 - Statistical Models and Applications

Learning outcomes

By the end of the course the student learns the basic notions to define statistical models. In particular, the student is able to estimate parameters, test hypothesis about them and build confidence intervals for generalized linear models and for linear mixed models, and to choose the most suitable model for the specific problem at hand.

Course contents

Statistical models: introduction.

Revision of linear regression models.

Linear mixed models:

• Fixed and random effects;
• Variance-covariance matrix structures;
• Maximum likelihood and restricted maximum likelihood estimators;
• Residual analysis;
• Inference about the parameters: confidence intervals and hypothesis testing;

Generalized linear models:

• Exponential families, linear predictor, link functions;
• Maximum likelihood estimators;
• Goodness of fit: the deviance of a model;
• Residual analysis;
• Inference on the parameters: likelihood ratio and Wald statistics;
• Poisson regression for count data;
• Logistic regression for categorical data;
• Other examples.
• Generalized linear mixed models: basic concepts.

Readings/Bibliography

Azzalini, A. (1996) Statistical inference based on the likelihood. Chapman & Hall/CRC.

Agresti, A. (2015) Foundations of linear and generalized linear models. Wiley.

Dobson, A. J. (2002) An Introduction to Generalized Linear Models. Second Edition. Chapman & Hall/CRC.

Everitt, B. S., Hothorn, T. (2009) A handbook of statistical analysis using R. Second edition. Chapman & Hall/CRC.

West, B. T., Welch, K. B. and Galecki, A. T. (2014) Linear mixed models. A practical guide using statistical software. Second edition. Chapman & Hall/CRC.

Handhouts provided by the teacher

Teaching methods

Lectures

Tutorial sessions in computer laboratory

Assessment methods

The exam will test the qualifications of each student on both a theoretical and a practical level.

The exam is composed of two parts: the first part is mandatory, the second part is optional. Both parts have to be taken during the same exam sitting.

The first mandatory part is a written exam. It consists of a set of questions concerning the models presented during the course. These questions focus both on theoretical properties and on output produced using the software R.

As far as the first sitting is concerned, students have the option of splitting the written exam into two partial exams. The first partial written exam takes place after the first 5 weeks and is focused on linear regression models and linear mixed models. The second partial written exam is scheluled after the end of the course, and it covers generalized linear models. Students must take both partial written exams. In particular, in order to register for the second partial exam, a student must have taken the first partial exam. In case of failure or rejection of the total mark obtained at the end of the second partial exam, students must repeat the whole written exam in one of the following sittings.

The second part is an optional oral exam. Only students with a mark for the written exam equal or larger than 18 can take this optional part. The oral exam consists of additional questions concerning the theoretical properties of linear and generalized linear models.

The overall mark is given by:

• The mark for the written exam, if a student does not take the oral exam;
• The average of the marks of the mandatory and of the optional parts, if a student takes also the oral exam. Non-integer average marks are rounded down to the next small integer.

Office hours

See the website of Giuliano Galimberti