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Regarding this basic model with an individual-level covariate $x_{ij}$, group effects $U_{0j}$, and individual effects $R_{ij}$:

$Y_{ij}=\gamma_{00}+\gamma_{10}x_{ij}+U_{0j}+R_{ij}$

the multilevel book I'm reading says that:

"[Modelling $U_0j$ as fixed parameters] is relevant if the groups $j$ refer to categories each with their own distinct interpretation -for example, a classification according to gender or religious denomination."

"[Modelling $U_0j$ as random variables] is relevant if the effects of the groups $j$ (which may be neighborhoods, schools, companies, etc.), controlling for the explanatory variables, can be considered exchangeable."

What are some examples of exchangeable group characteristics and what exactly makes them exchangeable (as opposed to the gender and religious denomination cases)?

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exchangeable here is in the sense of https://en.wikipedia.org/wiki/Exchangeable_random_variables exchangeable random variables. For the random variables $(X_1, X_2, \dotsc, X_n)$ to be exchangeable, all permutations of the variables should have the same distribution.

Some examples where this maybe holds (to decice in each applied case one would need more context):

  1. Schools in a study of educational outcomes. At least as far as the schools are not to different (in different countries, very different size, ...). Or classes within schools.
  2. In some studies, countries
  3. Often what you really want, is partial exchangeability, that is, exchageability given some predictor variables included in a model. That will extend applicability of exchangeability ideas.
  4. Individuals (person) selected randomly for some experimental study. This is the archetypical case, which is behind the iid assumption. So in modelling, exchangeability serves as extension of iid.

Some examples where we maybe shouldn't assume exchangeability:

  1. Groups selected for ability would not be exchangeable.
  2. Observations in a time series will not be exchangeable. Permutation of the variables destroys the time order.
  3. geolocalized variables in a spatial study. Permutation of the variables destroys spatial structure.

Some elaboration: Lets say you have a sample of high schools, some are ordinary high schools, other are selective "science high schools". Even after controlling for control variables, they should probably not be considered exchangeable. They would be too different in ways the control variables do not capture. If all are "ordinary" high schools, it could be safe to assume exchangeability. Order would be unimportant. One book with deep discussion of exchangeability in modelling is Bernardo & Smith.

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    $\begingroup$ I am familiar with exchangeable random variables and with the common examples that you cite, but something just isn't clicking. With the Wikipedia definition and three schools, what does it mean that the probability of observing $(U_{01}=A, U_{02}=A, U_{03}=B)$ is the same as the probability of $(U_{01} =B, U_{02} =A, U_{03}=A)$? That there is no "natural order" to our sample of schools, in the sense that time series or spatial data have a natural order? Ok, but then what kind of groups DO have a natural order? You mentioned groups selected for ability, but I'm not sure what you mean. Thanks. $\endgroup$ – suckrates Jun 29 '18 at 12:04

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