Stat 473/873: Generalized Linear Models

Course Description: In real-life situations, binary, categorical or counting variables often arise as the outcomes of an experiment or observational study. Generalized linear models (GLM) provide useful statistical tools to study the association between the discrete outcome variable and other variables (factors, covariates) of interest, and is a generalization of ordinary least squares regression.  The GLM can be used, for example, in a study to examine the relationship between heart attack (binary outcome) and factors such as smoking status and alcohol consumption.

Binomial regression models with different link functions will be introduced for data with binary outcomes, and special attention will be given to the logistic regression in this class of models.

Poisson regression models will be introduced to handle data with counting response variables, and one major topic in this class is the log-linear model. The analysis of contingency tables and its connection to GLMs will be discussed for studying the association between categorical variables. The GLM will be presented in its full generality for responses from the exponential family, and the modeling and analysis will be based on likelihood methods. Random effects models will be introduced as away of handling the problem of over-dispersion, which occurs when there is more variability in the data than is explained by the model under the distributional assumption. Each topic will be developed in theory, discussed in terms of parameter estimation, hypothesis testing, model selection and goodness of fittests. The illustrative data examples of each topic arise from studies in medicine, biology, industry, etc.

The objectives of this course are to help students develop an understanding of advanced regression theory and techniques beyond linear regression, and equip them with the ability to fit GLMs for a wide range of problems.

References:

McCullagh, P. and Nelder, J. A., Generalized Linear Models, Chapman and Hall, 1989 (2nd edition).

Dobson,A. J., An Introduction to Generalized Linear Models, Chapman and Hall, 2008 (3rd edition).

Agresti, A., Categorical Data Analysis, John Wiley, 2002 (2nd  edition).

Evaluation:

The final grade for the course will be based on 4 assignments (25%), an in-person midterm exam (30%) and a take-home final exam with derivation and data analysis problems (45%).

The assignments and exams will include theoretical problems, and data analysis problems.

Midterm Exam: February 29, Thursday, 18:30-20:30, Jeffery 225.

Two (size 8.5”x11”) double-sided aid sheets are allowed on the midterm exam.

Calculator: CASIO 991 for the midterm exam.

Tentative Outline:

Review of Likelihood Based Inference

Basic Methods for the Analysis of Contingency Tables

Brief Introduction to Binomial Regression Models: logistic regression

Generalized Linear Models: the exponential family; link functions; goodness of fittests; estimation procedures; hypothesis testing

Binomial Regression Models Revisited: different link functions; parameter interpretation; model selection; analysis of residuals

Poisson Regression Models: log-linear model, analysis of contingency tables

Random Effects Models: over-dispersion, hierarchical log-linear models

Analysis of Multinomial Outcome Data

Acknowledgement: Prof. Richard Cook of University of Waterloo for his lectures, notes and course materials of Generalized Linear Models.

Important Messages:

1.   Assignments will be distributed at onQ for STAT 473/873. Assignments will be collected at Crowdmark. Brief solutions to the assignments will be posted after the due dates.

2.   Make-up midterm exam will only be offered in the case of illness, or participation in an intercollegiate event (sports or academic competitions, etc.), or a schedule conflict with another course that you are taking. You must contact the instructor about your absence prior to the midterm exam. You must provide proper documents to support your request for the make-up midterm exam. Your request will only be considered if you have submitted at least 1 assignment in time. There will only be ONE makeup midterm exam, it will be within 7 days of the original midterm exam. Problems on the makeup exam may be different from those on the original exam.

3. Academic integrity Academic integrity is constituted by the five core fundamental values of honesty, trust, fairness, respect and responsibility (see www.academicintegrity.org). These values are central to the building, nurturing and sustaining of an academic community in which all members of the community will thrive. Adherence to the values expressed through academic integrity forms a foundation for the "freedom of inquiry and exchange of ideas" essential to the intellectual life of the University (see the Senate Report on Principles and Priorities

http://www.queensu.ca/secretariat/policies/senateandtrustees/principlespriorities.html).

Students are responsible for familiarizing themselves with the regulations concerning academic integrity and for ensuring that their assignments conform to the principles of academic integrity. Information on academic integrity is available in the Arts and Science Calendar 2023 - 2024 (Academic Regulation 1), on the Arts and Science website (see

http://www.queensu.ca/artsci/academics/undergraduate/academic-integrity), and from the instructor of this course. Departures from academic integrity include plagiarism, use of unauthorized materials, facilitation, forgery and falsification, and are antithetical to the development of an academic community at Queen's. Given the seriousness of these matters, actions which contravene the regulation on academic integrity carry sanctions that can range from awarning or the loss of grades on an assignment to the failure of a course to a requirement to withdraw from the university.

4. Arts & Science Calculator Policy As noted in Academic Regulation 9.2, Calculators

acceptable for use during quizzes, tests and examinations are intended to support the basic calculating functions required by most Arts and Science courses. For this purpose, the use of the Casio 991 series calculator is permitted and is the only approved calculator for Arts and Science students. This calculator sells for around $25 at the Queen's Campus Bookstore, Staples and other popular suppliers of school and office supplies. If you attend an exam with an unapproved calculator (non- Casio 991),then it will be removed from you. An instructor will not be able to override this policy.