The only difference of note is that the random effects model could be non-linear (e g in analysing binary or count data). The mixed effects model includes a 'fixed part' - that is means, and a 'random part' (that variances and covariances).

Lot of resources here

https://www.bristol.ac.uk/cmm/software/mlwin/mlwin-resources.html

and the free online course

https://www.bristol.ac.uk/cmm/learning/online-course/

Hi Kelvyn Jones from what i understand, the random effects regression looks like this:

•Yit = + 𝛽1Xit,1+ 𝛽2Xit,2+…+ 𝛽kXit,k+ 𝛼𝑖 + e𝑖t

and also has a fixed part (𝛽1Xit,1+ 𝛽2Xit,2+…+ 𝛽kXit,k)

and random effects (𝛼𝑖)

e𝑖t is idiosyncratic error.

The difference between a random effects model and a linear mixed model is that they are estimated with different methods. Random effects model is estimated with the feasible generalized least squares (FGLS) estimator and Linear mixed model is estimated usually with restricted maximum likelihood (REML).

At random effects model we have the assumption:

- COV (Xit,𝛼𝑖)No time-constant unobserved heterogeneity

also the independent variables are not correlated to the unobserved heterogeneity.

My question is should this assumption be satisfied also for linear mixed models?

Your model specification is:

•Yit = + 𝛽1Xit,1+ 𝛽2Xit,2+…+ 𝛽kXit,k+ 𝛼𝑖 + e𝑖t

and also has a fixed part (𝛽1Xit,1+ 𝛽2Xit,2+…+ 𝛽kXit,k)

and random effects (𝛼𝑖)

e𝑖t is idiosyncratic error.

where i represents in a repeated measures model, the person (or subject), and t represents the occasion. In multilevel terms the repeated measures at level 1 are nested within subjects at level 2.

Unfortunately, it is not a random effects model. (As specified it is a fixed effects model - we will come to that) You have left out the most important bit!

A Random variable (𝛼𝑖) means that it is "allowed to vary according to a distribution" and you do not specify that. You need an extra line such as

𝛼𝑖 is distributed as a Normal Variable with a mean of zero and a variance to be estimated. It is this variance that is distinctive of a random-effects model.

It also means that you need a distinctive estimation procedure to estimate it. I use RIGLS which has REML properties or I use (in complex models) some Bayesian procedure such as MCMC, which is full uncertainty modelling (the 𝛽's are estimated in the belief that the 𝛼𝑖 are estimates and vice versa). These more complex procedure are needed as to estimate the 𝛽's, we need to know the variance and vice versa.

The random -effects model is very flexible and there are three important extensions

1) the current model is a random intercepts one, but you can have random slopes too, where there is a random term associated with the 𝛽's (one or several), this necessitates a set of variance/covariances between subjects at level 2;

2) you can also include a time invariant (fixed) term for a variable about the subject (eg whether they are male of female), that is a variable measured at level 2; for many problems this is very important.

3) you can include an additional random terms at level : e0it, e1𝑖t to model within-person heterogeneity.

There is an alternative! You could use a set of dummies for each and every subject and thereby dummy out the subject effects (𝛼𝑖), You can then use standard OLS for estimation. This is known (confusingly) as a 'fixed effects' model. That is what you have specified in your post. The 𝛽's are then estimated 'controlling' for subject characteristics without measuring them. But you cannot do any of the three extensions. You can also estimate your model by demeaning the response variable. But again you cannot do any of the extensions especially extension 2 as all the level 2 variation would have been dummied or demeaned out.

Have a look at these model specifications:

https://en.wikipedia.org/wiki/Kelvyn_Jones#/media/File:Two_level_panel_model_alternative_specifications.jpg

which also gives some models that are useful for tackling some forms of endogeneity.

For fuller discussion see the Wikipedia site:

https://en.wikipedia.org/wiki/Kelvyn_Jones

The section on the "gold standard of Fixed effects" is the pertinent bit, and I suggest this ordering of your reading:

Article Explaining Fixed Effects: Random Effects Modeling of Time-Se...

Article Fixed and Random effects models: making an informed choice

Article Understanding and misunderstanding Group mean centering: a c...

Article Do multilevel models ever give different results?.

In case of doubt, the random effects model is a linear mixed effects model ; it is defined by its specification and not by its mode of estimation. Thus such models could be estimated by the EM algorithm, RIGLS , MCMC amongst other. (see https://core.ac.uk/download/pdf/82406287.pdf). It is mixed because it has a both fixed part (averages) and a random part (summarised by variance covariances) in one model. It is linear in the parameters ; see https://datascience.stackexchange.com/questions/12274/what-does-linear-in-parameters-mean#:~:text=It%20simply%20means%20that%20Y,a%20linear%20function%20of%20X.&text=%22Linear%20in%20parameters%22%20in%20Linear,or%20divided%20by%20another%20parameter.

In relation to the assumption of COV (Xit,𝛼𝑖), the papers cited above discuss group-mean centering and the alternative Mundlak specification as a way getting around this assumption in estimating the 𝛽's of the fixed part.

Hi @Kelvyn Jones

you are right that 𝛼𝑖 is distributed as a Normal Variable with a mean of zero and a variance to be estimated. That means that the random effects model has a fixed part and a random part. You are also right that the current model is a random intercept one.

But is the difference between a random effects model and linear mixed effects model that Random effects model is estimated with the feasible generalized least squares (FGLS) estimator and Linear mixed model is estimated usually with restricted maximum likelihood (REML)?

And does this assumption or condition

-COV (Xit,𝛼𝑖) :No time-constant unobserved heterogeneity

that should be satisfied in order for the random effects model to be unbiased, should also be satisfied in order for the linear mixed model to be unbiased?

Or linear mixed models do not make that assumption?

Thanks for the suggestions by the way. I'm actually deciding whether to use fixed effects random effects or linear mixed effects for my model. The only reason. I am not 100% sure on using fixed effects is that i don't seem have enough within variation in the data.

I do not agree with anything after "random intercept one". I clearly cannot help you. Good luck.