Relationship between mixed-effects models and fixed-effects models

Occasionally I encounter discussions of what are called fixed-effects
models in econometrics but I haven't seen descriptions of the underlying
statistical model.  Can anyone point me to a description of these models,
in particular a description in terms of a probability distribution of the
response? I would be particularly interested in a discussion of how they
relate to mixed-effects models as we think of them in lme4 and nlme.

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Occasionally I encounter discussions of what are called fixed-effects
models in econometrics but I haven't seen descriptions of the underlying
statistical model.  Can anyone point me to a description of these models,
in particular a description in terms of a probability distribution of the
response? I would be particularly interested in a discussion of how they
relate to mixed-effects models as we think of them in lme4 and nlme.

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If I understand correctly, "fixed effects" in econometrics are simply
categorical variables, especially ones with a large number of levels.
There are "fixed" in the sense that they are observed at fixed
(discrete) levels instead of as continuously.

I don't have access to my copy at the moment, but this is discussed in
Gelman & Hill (2006).

Phillip

On 07/06/2021 10:09, Douglas Bates wrote:

Occasionally I encounter discussions of what are called fixed-effects
models in econometrics but I haven't seen descriptions of the underlying
statistical model.  Can anyone point me to a description of these models,
in particular a description in terms of a probability distribution of the
response? I would be particularly interested in a discussion of how they
relate to mixed-effects models as we think of them in lme4 and nlme.

	[[alternative HTML version deleted]]

Somewhat related to this and what James wrote, in the world of fMRI and
other two-stage analyses in psychology and neuroscience, the "fixed
effect" vs "random effect" distinction is used in the same sense as in
meta-analysis, which lines up more closely with the use in mixed models,
i.e. whether or not the individual estimates are treated as observed
draws from a random variable in the group-level analysis.

On 07/06/2021 10:27, Phillip Alday wrote:

If I understand correctly, "fixed effects" in econometrics are simply
categorical variables, especially ones with a large number of levels.
There are "fixed" in the sense that they are observed at fixed
(discrete) levels instead of as continuously.

I don't have access to my copy at the moment, but this is discussed in
Gelman & Hill (2006).

Phillip

On 07/06/2021 10:09, Douglas Bates wrote:

Occasionally I encounter discussions of what are called fixed-effects
models in econometrics but I haven't seen descriptions of the underlying
statistical model.  Can anyone point me to a description of these models,
in particular a description in terms of a probability distribution of the
response? I would be particularly interested in a discussion of how they
relate to mixed-effects models as we think of them in lme4 and nlme.

	[[alternative HTML version deleted]]

I think FE (fixed effects) models are used in particular in panel data or
longitudinal data analysis, when time varying predictors are included, e.g.
"income". Income has a between-subject effect (we have higher- and
lower-income groups) and a within-subject effect (income of person A can
increase over time, while it can decrease for person B - no matter, if A or
B belong to low- or high-income groups!).

The arguments from a FE perspective against mixed models is that you cannot
include "income" as predictor, because income has an effect on both
individual level (within) and higher levels (between), i.e. it would
introduce correlated error terms between the fixed effects and random
effects, which violates model assumptions. The solution is now to "demean"
the "income" variable and only include the within-effect, i.e. the time
varying component in the model. All between effects, and in general all
predictors that could be seen as "between" effects (gender, education, ...)
have to be omitted from the model. The group-level variation (e.g.
"subject", or whatever would be the group factor in mixed models) is
included as normal predictor.

So, a FE model is a classical linear model, where
- Intercept is removed
- time-invariant predictors are not allowed to be included
- the group-level factor is included as predictor
- time-varying predictors are de-meaned (?person-mean centered?, indicating
the ?within-subject? effect)

However, in particular Bell et al. [1, 2] have shown that the "demeaning"
trick also applies to mixed models, so that essentially, mixed models are
probably much better for panel data / longitudinal data analysis. You may
be
interested in this vignette, describing the issue and comparing FE to mixed
models: https://easystats.github.io/parameters/articles/demean.html

There are some newer developments, like fixed effects individual slope
models (package feisr), or the panelr package (fun fact: which uses lme4 to
fit flexible models for panel data, so these models are actually mixed
models, no classical FE models).

Best
Daniel

1) Bell, Andrew, Malcolm Fairbrother, and Kelvyn Jones. 2019. ?Fixed and
Random Effects Models: Making an Informed Choice.? Quality & Quantity 53:
1051?74. https://doi.org/10.1007/s11135-018-0802-x.

2) Bell, Andrew, and Kelvyn Jones. 2015. ?Explaining Fixed Effects: Random
Effects Modeling of Time-Series Cross-Sectional and Panel Data.? Political
Science Research and Methods 3 (1): 133?53.
https://doi.org/10.1017/psrm.2014.7.

-----Urspr?ngliche Nachricht-----
Von: R-sig-mixed-models <r-sig-mixed-models-bounces at r-project.org> Im
Auftrag von Douglas Bates
Gesendet: Montag, 7. Juni 2021 17:10
An: R-mixed models mailing list <r-sig-mixed-models at r-project.org>
Betreff: [R-sig-ME] Relationship between mixed-effects models and
fixed-effects models

Occasionally I encounter discussions of what are called fixed-effects
models in econometrics but I haven't seen descriptions of the underlying
statistical model.  Can anyone point me to a description of these models,
in particular a description in terms of a probability distribution of the
response? I would be particularly interested in a discussion of how they
relate to mixed-effects models as we think of them in lme4 and nlme.

        [[alternative HTML version deleted]]