Confusing Statistical Term #7: GLM

by Karen Grace-Martin

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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.

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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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{ 7 comments… read them below or add one }

Jose

Hi, a good article.
Go on, we need people like you!!!
I am interested in special in factor analysis and clusters.

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Jose

Hi, a good article. Thanks.

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Ankit

Hi Karen,

Can you provide information on accessing fitness of a Generalised linear model using Maximum likelihood estimation. Though I have read few articles and pdfs on internet but none of them provide information as coherent and precise as you do.

Looking forward for this piece of knowledge.

Thanks 🙂

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sam

Hi Karen,

I have come to this page because I wasn’t sure which GLM you were referring to on another of your (very helpful) pages…I’m still not sure (though I suspect generalised linear model)! Ironic?

http://www.theanalysisfactor.com/when-dependent-variables-are-not-fit-for-glm-now-what/

Thanks,

Sam

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Karen

Hi Sam, haha! I’m using my own confusing terms. I meant General Linear Model. 🙂

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Karen Alexandre

Thanks, Karen – I am just starting to work through these similar yet varying concepts and this was a very helpful post!

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Karen

Hi Karen–glad it was helpful.

Karen

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Please note that Karen receives hundreds of comments at The Analysis Factor website each week. Since Karen is also busy teaching workshops, consulting with clients, and running a membership program, she seldom has time to respond to these comments anymore. If you have a question to which you need a timely response, please check out our low-cost monthly membership program, or sign-up for a quick question consultation.

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