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I am confused about the expression "fixed effect" in the context of mixed models. I am more familiar with the terms like "fixed effects" and "random effects" in context of econometrics and the analysis of panel data.

The understanding of "random effects" seems similar in both disciplines and the "random effects model" in econometrics is equivalent to a "mixed model with random intercept". See for example: How exactly does a "random effects model" in econometrics relate to mixed models outside of econometrics?

But what is about the "fixed effect". In econometrics, with the help of fixed effects e.g. all time invariant effects will be absorbed. Or in other words, a dummy variable for each individual is introduced.

But, what is "fixed effect" in context of mixed models? In simple words? Hereby, I mean more the intuition and not the mathematically way.

It is not the same as a fixed effect in econometrics. That is what I understand, but I have no clue what it is instead.

EDIT:

Meanwhile I found this explanation in context of mixed models:

"The fixed effects are analogous to standard regression coefficients and are estimated directly."

This means to me, that a fixed effect in mixed models is not the same as fixed effects in econometrics. Or in other words: Fixed effects are the variables that are not declared as random effects (--> standard regression coefficients as in linear regression).

Source (slide 2): http://fmwww.bc.edu/EC-C/S2013/823/EC823.S2013.nn07.slides.pdf

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  • $\begingroup$ I think this is fully answered in the thread you linked to (even in the question itself). $\endgroup$
    – amoeba
    Dec 6, 2017 at 22:34
  • $\begingroup$ @amoeba: Thank you for your comment. In the linked thread, there is a full explanation for the "random effects" part. That is the focus of the thread. They also say something about "fixed effects", but more in a mathematically way. So for me, the difference in case of fixed effects is not clear to me. It is especially not explained in simple word. $\endgroup$
    – Olaf_SQL
    Dec 6, 2017 at 23:04
  • $\begingroup$ But it's exactly the same as in econometrics: as you said, "dummy variable for each individual is introduced". Maybe this thread stats.stackexchange.com/questions/4700 can also help you (see the second answer). $\endgroup$
    – amoeba
    Dec 6, 2017 at 23:12
  • $\begingroup$ @amoeba: In my opinion, that cannot be true. I found examples for mixed models where the "fixed effect" was a continuous variable. That does not make sense to me to introduce dummy variables for this variable. Meanwhile I found this explanation: "The fixed effects are analogous to standard regression coefficients and are estimated directly." This would be more logic to me. What do think about this? $\endgroup$
    – Olaf_SQL
    Dec 7, 2017 at 8:44
  • $\begingroup$ In mixed models, fixed effect is what you have in usual regression. If it's continuous variable, then it's one regression coefficient. If it's a categorical variable, then it's a separate regression coefficient for each dummy variable. $\endgroup$
    – amoeba
    Dec 7, 2017 at 8:52

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Even within the disciplines, there is sadly no council that ensures consistent usage. However, in my experience the key distinction as that in early econometric panel models they tended to be referring to the intercept, whereas in other fields they are referring to anything.

Consider a model predicting how much a person weighs, where you have 10 observations taken over 10 years for 1,000 children. In econometrics the 'fixed effects' model is one where you have a separate intercept for each of the 1,000 children (such models are not typically estimated with dummy variables, but it is OK to think of them that way). The random effects model is one where you instead assume that the intercept is a random variable and estimate its mean and variance.

Now, let's add into the model a predictor which is the height of the child. If we estimate a single coefficient, we treat the effect as being fixed. If we instead treat it as a random variable, we estimate the average effect and the variance of the effect (if assuming normally distributed random variables).

If you have some predictors that are assumed to be random variables and others that are not, be they the intercept or anything else, you have a mixed model.

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  • $\begingroup$ Thank you! But it still remains a little unclear to me. In your example, i would choose child as fixed effect (in the sense of econometrics). but then you add height to your regression and you call it also fixed effect. but for me it would be just a standard regression coefficients. $\endgroup$
    – Olaf_SQL
    Dec 7, 2017 at 8:56
  • $\begingroup$ EDIT: I think my problem was solved above. Thank you. $\endgroup$
    – Olaf_SQL
    Dec 7, 2017 at 9:02
  • $\begingroup$ A 'standard regression coefficient' is a fixed effect. $\endgroup$
    – Tim
    Dec 7, 2017 at 23:50

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