(This article was originally published at The Analysis Factor, and syndicated at StatsBlogs.)
Like some of the other terms in our list–level and beta–GLM has two different meanings.
It’s a little different than the others, though, because it’s an abbreviation for two different terms:
General Linear Model and Generalized Linear Model.
It’s extra confusing because their names are so similar on top of having the same abbreviation.
And, oh yeah, Generalized Linear Models are an extension of General Linear Models.
And neither should be confused with Generalized Linear Mixed Models, abbreviated GLMM.
Naturally.
So what’s the difference? And does it really matter?
General Linear Models
You’re probably familiar with General Linear Models, though possibly through the names linear regression, OLS regression, least-squares regression, ordinary regression, ANOVA, ANCOVA.
In all of these models, there are two defining features:
1. The residuals (aka errors) are normally distributed.
2. The model parameters–regression coefficients, means, residual variance–are estimated using a technique called Ordinary Least Squares.
This latter feature is important, because many of the nice statistics we get from these models–R-squared, MSE, Eta-Squared–come directly from OLS methods.
And this is why you can run regressions and ANOVAs in the same General Linear Model software procedure.
Generalized Linear Models
But it turns out that not all dependent variables can result in residuals that are normally distributed.
Count variables and categorical variables are both good examples. But it turns out that as long as the errors follow a distribution within a certain family of distributions, we can still fit a model.
You’re probably familiar with these through one of its common examples–logistic regression, Poisson regression, probit regression, negative binomial regression.
In all of these models, there are a few more defining features:
1. The residuals come from a distribution in the exponential family. (And yes, you need to specify which one).
2. The mean of y has a linear form with model parameters only through a link function.
3. The model parameters are estimated using Maximum Likelihood Estimation. OLS doesn’t work.
Just like you can run a linear regression using either a linear regression or a General Linear Model procedure, you can run a logistic regression through either a logistic regression or a Generalized Linear Model procedure.
The logistic procedure is just making some default assumptions about the model–for example, that the link function is a logit. In the Generalized Linear Models procedure, you’d have to specify that.
If you’d like to learn more about some generalized linear models, download a recording of the webinar Poisson and Negative Binomial Regression for Count Data or Binary, Ordinal, and Multinomial Logistic Regression for Categorical Outcomes. They’re both free.
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