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The wikipedia page on generalized mixed models describes them as an "extension of" generalized linear models but doesn't mention regression. The latter Wikipedia page describes GLM as "a flexible generalization of ordinary linear regression". This helpful "INTRODUCTION TO GENERALIZED LINEAR MIXED MODELS" starts from a standard linear regression and then adds complications to get to GLMM. It sure makes GLMM sound like a sophisticated kind of regression. Would it be incorrect to describe it in that way? I'm worried that despite similarities, GLMM is different enough that describing it as a kind of regression would be incorrect.

(Answers to this question, which @Tim helpfully mentioned, are very useful in providing a broader context that is quite relevant. However, those answers don't specifically answer my question about GLMM per se. It's possible an the answer to my question is implied by what's said in those answers. If I knew enough to derive that implication on my own, I probably wouldn't need to ask my question!)

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    $\begingroup$ Possible duplicate of Definition and delimitation of regression model $\endgroup$
    – Tim ♦
    Jul 29, 2017 at 22:18
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    $\begingroup$ There are two things: linear regression per se and a broad family of [regression models]( stats.stackexchange.com/q/173660/35989). While GLMMs are not regression, they are regression models since the general idea (estimating conditional mean) is the same. Linear regression is a good (simple) introductory example for GLMs and GLMMs. $\endgroup$
    – Tim ♦
    Jul 29, 2017 at 22:21
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    $\begingroup$ Thanks @Tim. That sounds like most of an answer. I interpret you as saying that (a) "regression" often/usually/always means "linear regression", while "regression model" is a broader term that includes GLMM. This is consistent with what kjetil b halvorsen says in an answer to the question you linked. Halvorsen also suggests that there might be some variation in usage or vagueness in "regression model". This wouldn't be surprising to me. Often authors define terms slightly differently for the sake of pedagogy, conciseness, etc. $\endgroup$
    – Mars
    Jul 29, 2017 at 22:55

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Clearly, it depends on your exact definition of "regression". Here is what Wikipedia says:

In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one or more independent variables (often called 'predictors', 'covariates', 'explanatory variables' or 'features').

by this definition, GLMM are a form of regression, but multivariate regression is not, since it involves multiple dependent variables. A nearly identical definition is in Britanica, Investopedia, and other sources. OTOH, Mathworld says:

A method for fitting a curve (not necessarily a straight line) through a set of points using some goodness-of-fit criterion. The most common type of regression is linear regression.

Which would also include MLMM

The Oxford Dictionary of Statistics ed. by Upton and Cook, adds the notion of "expected value":

The expected value of one variable Y is presumed to be dependent on one or more other variables ...

which would also include MLMM.

I don't have access to the Encyclopedia of Statistical Sciences, maybe one of the academics on this site does, and would add that definition.

But, so far, I didn't find any definition which would exclude MLMM.

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  • $\begingroup$ I actually think we can go much farther than the Wiki definition. Recall it was Galton who referred to "regression to the mean". Indeed, regression may be taken to refer to modeling means. This would include the no-regressor case lm(y ~ 1). If "y" is a paired difference, this is recognizable as a paired t-test - a form of regression. Similarly, your mixed model may be lm(y ~ 1 + (1|sampleid)) tested against lm(y ~ 1) - which technically models variances, not means. However, the variance is modeled using latent means, so I would agree this is still (broad sense) regression. $\endgroup$
    – AdamO
    2 days ago
  • $\begingroup$ @AdamO OK, but writing an actual definition is a little tricky, as any lexicographer will tell you. Do you have one to propose? $\endgroup$
    – Peter Flom
    2 days ago
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    $\begingroup$ More precisely, we could caveat that independent variable or independent variable(s) are possibly singular. $\endgroup$
    – AdamO
    2 days ago

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