# Difference between random effect and random intercept model

I am looking at clustered data and because I was trained in economics I tend to look at fixed effects and random effects as solutions. An alternative would clearly be multi-level modelling. However, for me the clustering is just a nuisance, so I only care about "controlling it out".

However, looking at the equation of a random intercept model, I cannot see the difference to my random effect model. So I was wondering whether there just might be different names for the same thing, due to differences between disciplines? But then why can you (in Stata) estimate them differently?

• RIM: xtreg y x, i(id) mle
• RE: xtreg y x, re

However, the equation for both seems to be:

$$y_{ij}= b_{0} + b_{1}x1_{ij}+ u_{j} + e_{ij}$$

(Don't know how to format it nicely, but Google gives examples of RE and RIM equations, which look the same to me easily:

• Since an intercept is sort of an effect, a random intercept model is a special case of a random effect model. But you are right, there are many different terms for the same methodology. – Michael M Jul 16 '14 at 14:30
• Could you elaborate on how it is a special case or give me a reference? When I google I only find descriptions of either, but so far no comparison. All I read on RIM could (in my opinion) also be said about RE and vice versa. For example in RIM the group intercepts are the overall intercept+ the cluster error, but RE also has an overall intercept and cluster error? – John Gonway Jul 16 '14 at 14:51
• E.g. I wouldn't call a random effects model with random slopes a random intercept model. I don't know a reference of my first comment though. – Michael M Jul 16 '14 at 15:04
• It is worth explicitly pointing out here that these terms vary importantly & confusingly across disciplines (eg econometrics vs biostatistics). See here: What is a difference between random effects-, fixed effects-, and marginal model?, & here: Concepts behind fixed/random effects models. – gung Jul 17 '14 at 3:18

A random intercept model estimates separate intercepts for each unit of each level at which the intercept is permitted to vary. This is one kind of random effect model. Another kind of random effect model also includes random slopes, and estimates separate slopes (i.e. coefficients, betas, effects, etc. depending on your discipline) for each variable for each unit of each level at which that slope is permitted to vary.

It's a citation from epidemiology, not economics, but it is well written as an introduction to these kinds of models (including the "why would we care" bits): Duncan, C., Jones, K., and Moon, G. (1998). Context, composition and heterogeneity: Using multilevel models in health research. Social Science & Medicine, 46(1):97–117.

• Ok, so it seems to be a linguistic thing then, because both you and M.Mayer seem to argue that random effect is a family of estimator including random intercept and random slope models. Whereas in economics random effects implies random intercept. The only difference between the two stata commands I wrote up seems to be that once uses MLE and the other some least squares version. – John Gonway Jul 16 '14 at 15:26
• oh and thanks for the reference! – John Gonway Jul 16 '14 at 15:28
• Your second specification is generalized least squares (GLS). – Dimitriy V. Masterov Jul 16 '14 at 23:21
• I'd have said exactly as Alexis and Michael Mayer did - that a random intercept model is a special case of a random effects model. – Glen_b Jul 17 '14 at 5:32
• @DimitriyV.Masterov This is a late to the party response, but my answer was about the model, not the estimator: could be estimated using GLS, IGLS, MCMC/Bayesian, etc. Right? Or is differentiating model from estimator not the way folks should use their statistical concepts? – Alexis Oct 29 '16 at 19:24