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In simple terms, how would you explain (perhaps with simple examples) the difference between fixed effect, random effect and mixed effect models?

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I also find that sometimes is difficult to determine when an effect must be considered as fixed or as random effect. Althought there are some recommendations about this fact, not always is easy to take the right decision. –  Manuel Ramón Nov 19 '10 at 10:29
I think that this link may be helpful in clarifying the underlying principles of mixed models: <a href="support.sas.com/documentation/cdl/en/statug/63033/HTML/default/…, Random, and Mixed Models (SAS documentation)</a> –  pietrop Sep 4 '13 at 10:27
An extremely helpful answer can also be found here: What is a difference between random effects-, mixed effects- & marginal model? –  gung Nov 19 '14 at 20:21

4 Answers 4

up vote 29 down vote accepted

Blogger Andrew Gelman says that the terms 'fixed effect' and 'random effect' have variable meanings depending on who uses them (link). Perhaps you can pick out which one of the 5 definitions applies to your case. In general it may be better to either look for equations which describe the probability model the authors are using (when reading) or write out the full probability model you want to use (when writing).

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+1: very nice link! I guess the definition also varies depending on the field (e.g. #4 is very mathematical/statistical, but #1 and #2 are more "understandable" from a life science point of view) –  nico Nov 19 '10 at 6:39
Thanks - this is really helpful. –  Andrew Nov 19 '10 at 19:52
It is also informative to read the Discussion and Rejoinder to this paper. In the discussion, Peter McCullagh wrote that he disagrees with a substantial portion of what Gelman wrote. My point is not to favor one or the other, but to note that there is substantial disagreement among experts and not to put too much weight on one paper. –  julieth Jul 22 '12 at 1:19
Cool, I haven't seen that. Do you have a link to the paper(s) you're talking about? –  John Salvatier Jul 22 '12 at 6:06
The entire discussion is at link –  julieth Jul 22 '12 at 13:34

Fixed effect: Something the experimenter directly manipulates and is often repeatable, e.g., drug administration - one group gets drug, one group gets placebo.

Random effect: Source of random variation / experimental units e.g., individuals drawn (at random) from a population for a clinical trial. Random effects estimates the variability

Mixed effect: Includes both, the fixed effect in these cases are estimating the population level coefficients, while the random effects can account for individual differences in response to an effect, e.g., each person receives both the drug and placebo on different occasions, the fixed effect estimates the effect of drug, the random effects terms would allow for each person to respond to the drug differently.

General categories of mixed effects - repeated measures, longitudinal, hierarchical, split-plot.

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Your not wrong, but your definition of what a fixed effect is is not what I would think of when someone says fixed effect. Here is what I think of when someone says fixed effect en.wikipedia.org/wiki/Difference_in_differences , or this stata.com/support/faqs/stat/xtreg2.html (particularly equation 3 on the Stata page) –  Andy W Nov 19 '10 at 13:44

The distinction is only meaningful in the context of non-Bayesian statistics. In Bayesian statistics, all model parameters are "random".

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Interesting. But since fixed or random can be considered a condition of a given variable (a given column of data) rather than of a parameter associated with that variable,...does your answer fully apply? –  rolando2 Jan 27 '12 at 0:48

Not really a formal definition, but I like the following slides: Mixed models and why sociolinguists should use them (mirror), from Daniel Ezra Johnson. A brief recap' is offered on slide 4. Although it mostly focused on psycholinguistic studies, it is very useful as a first step.

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I think I'm going to need to see that presentation in person to get the full impact. –  Andy W Nov 19 '10 at 13:36

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