We will prefer to use GLM to mean "generalized" linear model in this course.
Unfortunately, different authors and texts may use GLM to mean either "general" or "generalized" linear model, so it's best to rely on context to determine which is meant. Today, GLIMs are fit by many packages, including SAS's Genmod procedure and R's glm() function. Because of this program, "GLIM" became a well-accepted abbreviation for generalized linear models, as opposed to "GLM" which often is used for general linear models.
#Glims on the stuff software#
The first widely used software package for fitting these models was called GLIM. Some would call these “nonlinear” because \(\mu_i\) is often a nonlinear function of the covariates, but McCullagh and Nelder consider them to be linear because the covariates affect the distribution of \(y_i\) only through the linear combination \(x_i^T\beta\). In these models, the response variable \(y_i\) is assumed to follow an exponential family distribution with mean \(\mu_i\), which is assumed to be some (often nonlinear) function of \(x_i^T\beta\). The term "generalized" linear model (GLIM or GLM) refers to a larger class of models popularized by McCullagh and Nelder (1982, 2nd edition 1989). These models are fit by least squares and weighted least squares using, for example, SAS's GLM procedure or R's lm() function. The form is \(y_i\sim N(x_i^T\beta, \sigma^2),\) where \(x_i\) contains known covariates and \(\beta\) contains the coefficients to be estimated.
It includes multiple linear regression, as well as ANOVA and ANCOVA (with fixed effects only). The term "general" linear model (GLM) usually refers to conventional linear regression models for a continuous response variable given continuous and/or categorical predictors. As we introduce the class of models known as the generalized linear model, we should clear up some potential misunderstandings about terminology.
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