Thread by @WomenInStat

I work primarily with nested data. One example is in experiments, with students nested in schools. Another is meta-analysis, with effect sizes nested in studies. In this thread, I’ll focus on students nested in schools, but this applies more generally.

Question 1: Do you need to take nesting into account in your analysis? Our world is naturally nested – students in classrooms in teachers in schools in districts and so on. Does this mean we need to take all of these levels into account? No.

Nesting only needs to be accounted for if it is part of how our sample of data is generated – either how the data is selected (sampled) or the who gets an intervention being studied (assignment).

Question 2: What does nesting do? It creates dependence in our data. Treating them as independent can lead to an inflated Type I error.

Question 3: How should you take this nesting into account? This is a big one and where things can get complicated. I’ll talk about two approaches here.

Option 1. Use a multi-level model: Y_ij = [fixed part] + [random effects], where the [fixed part] includes a regression model and the [random part] includes both student and school level residuals. This model assumes that schools are a random sample from some population.

Extension 1. This model can accommodate interactions between covariates and random effects. For example, this model allows us to interact the treatment indicator with the school random effect. We can now the distribution of treatment effects.

Extension 2. By centering covariates, this model can allow you to separate the effects of covariates at the student level and the school level – and if they differ – the contextual effect. (See Raudenbush & Bryk for some nice examples).

Option 2. Use cluster-robust variance estimation (CRVE): Y_i = [fixed part] + [generic residual] and then empirically estimating the standard errors using a sandwich estimator, called something like Huber-White or Eickler or so on, depending upon your field.

Extensions. There aren’t really extensions. This model really allows you to focus on estimation of the effects of covariates at the student level on outcomes. The nesting is more noise than signal.

Question 4: Why one or the other? I hear this question a lot. The simplest answer is that your choice is determined by your field. Economics adjacent fields use CRVE, while sociology adjacent fields use multi-level models. So, norms are at play.

But I think the root of this difference is in regard to what is considered signal and noise in these fields. Economics – like psychology – historically has focused on the behavior of individuals.

Thus, the fact that the data is nested is not interesting – it is simply error or noise. Sociology, however, focuses on the role of context and groups – in which case, the nesting is interesting – it is signal.

Another answer though, may have to do with assumptions. The multi-level model makes several modeling assumptions – e.g., normality and that the correlation structure is correctly specified.

The CRVE approach doesn’t require these assumptions – it depends upon asymptotics, so if you get your structure wrong, no problem.

Question 5: Do I have to choose? Ok, no one really ever asks me this. But @jepusto and I have been working on this for some time and the short answer is: use both‼ That is, first use a multi-level model with an assumed structure, then use CRVE to get your standard errors.

@jepusto Result 1. For those used to using a multi-level model - you get all the benefits of the multi-level model PLUS the safety of your inferences being robust to misspecification of the error structure.

@jepusto Result 2. For those used to using CRVE – you get better small-sample performance. Here I mean the number of clusters and the types of covariates you’re studying. The multi-level model becomes a ‘working model’, and this working model improves performance, even when it’s wrong.

@jepusto If you’re interested in learning more, see these papers: doi.org/10.1080/073500…
and
osf.io/preprints/meta…