Nested or Crossed Effects with Nationalities

Question

I am working with mixed effects models and I am still a bit confused.

While I have read multiple explanations of what the differences between nested and crossed random effects are, I am not sure how to apply them to my data. I have read the following explanation already: Crossed vs nested random effects: how do they differ and how are they specified correctly in lme4?

My dataset is about people living in different cities. Thus, I have multiple nationalities as one variable (nationality of the person living in a city) and cities as another variable (the city the person lives in). What I want to see with my model is whether nationalities differ overall and whether they also differ between each city (e.g. whether someone with the nationality "Japan" living in San Francisco is different in terms of my dependent variable when compared to other Japanese that live somewhere else).

To answer this question, I thought about using a nested model, but I am not sure whether this is possible in my scenario. What is confusing to me is the example about class rooms and schools as described in the link above. While I understand that one class is part of only one school (nested), I am not sure whether this can also be said for nationalities. Especially in regards to the following: In my dataset, one and the same individual can only be observed in one city but the overall nationality factor can be observed in multiple cities. In other words: Person A134 lives in San Francisco and is Japanese. However, he is not the only Japanese person and I have Japanese people living in Tokyo, but also living in, London and other cities)

Would it still be possible to use a nested model or is it an issue that the nationality "Japan" appears across all cities? If not, I am not sure how else to answer my question.

The nested random effect I thought of would look like:

lmer(dependent_variable ~ variable1 + variable2 + (1|nationality/city), data=data)

Furthermore, what would the difference in interpretation be if the following model was used? What would change in terms of interpretation?

lmer(dependent_variable ~ variable1 + variable2 + (1|nationality) * (1|city), data=data)

EDIT: I am not sure, but maybe the following is what I am looking for? How does it differ from the two above?:

lmer(dependent_variable ~ variable1 + variable2 + (1|nationality:city), data=data)

Answer 1

Each person is measured (observed) once.

Person's belong to only one city - that is they are nested in city.

Person's belong to only one nationality - that is, they are nested in nationality.

There is no nesting of nationality in city or vice versa. Hence City and Nationality are crossed factors.

So in a mixed model setting you could fit:

lmer(dependent_variable ~ variable1 + variable2 + (1|nationality) + (1|city), data=data)

However, this will not answer your research question:

What I want to see with my model is whether nationalities differ overall and whether they also differ between each city (e.g. whether someone with the nationality "Japan" living in San Francisco is different in terms of my dependent variable when compared to other Japanese that live somewhere else).

To answer this the most obvious approach is to fit interactions for city and nationality as fixed effects

lm(dependent_variable ~ variable1 + variable2 + nationality*city, data=data)

and this would not be a mixed model. The problem with this is that for many cities and nationalities you are going to have a lot of interation terms.

Finally there is a bit of confusion in your question. You also posit these models:

> lmer(dependent_variable ~ variable1 + variable2 + (1|nationality/city), data=data)

This model says that city is nested in nationality and the software will fit random intercepts for nationality and the nationality:city interaction.

> lmer(dependent_variable ~ variable1 + variable2 + (1|nationality:city), data=data)

This model says that you are fitting random intercepts for the nationality:city only and that is rarely what is warranted.