Introduction to Generalized Linear Models (GLM) We introduce the notion of GLM as an extension of the traditional normal-theory-based linear regression models.

This will be very helpful in order to gain a general insight into all discussions till the end of this course since the speci?c models that will be discussed in details from now all, will turn out to be speci?c GLM. We already mentioned in the introductory lecture that when dealing with categorical data as output, it is not wise to model it (or for that matter, the probabilities for its particular categories) by using linear models. This is why one has tried to extend the Linear Models theory to make it suitable for such situations. There are at least two important aspects of the extension of the traditional normal- theory based linear regression model.

• It allows the mean of a population to depend on a linear predictor through a nonlinear link function

• The response probability distribution to be not necessarily normal but a member of the exponential family of distributions.

The set of generalized linear models is indeed quite large. These include: classical linear models with normal errors, logistic and probit models for binary categorical data, and log- linear models for multinomial data. Many other statistical models can also be shown to be
a particular GLM after choosing suitably the link function and the response probability distribution.