When I fit a multilevel varying slope model, it is easy to summarize the variation in slope. However, I have not yet seen any materials that discusses how to explain such variation (i.e. what about this group that allows it to have a greater effect than the rest?)

For example, we want to test whether meal voucher helps student's performance. We fit a multilevel model of students within schools, allowing for the effect of meal voucher to vary across schools.

$$ \begin{align} performance_i &\sim N(\alpha_{j[i]} + \beta_{j[i]} * voucher_i \\ \alpha_j &\sim N(\text{school level covariates}, \sigma_\alpha) \\ \beta_j &\sim N(\text{school level covariates}, \sigma_\beta) \end{align} $$

Now we need to explain why the variation in $\beta_j$ occurs. Is this causal inference goal possible with multi-level model?

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    $\begingroup$ For clarification: what do you mean by "this group" and also "greater effect"? Does group = an aspect of grouping or clustering in your data (eg hospitals, schools, blocks, individuals with repeated measures) and for "greater effect" do you mean "Why does some aspect of clustering have a random slope but not others" in a model, with say, repeated measurements on students clustered within classrooms clustered within schools. $\endgroup$ – N Brouwer Nov 17 '14 at 3:42
  • $\begingroup$ By group I mean clusters (school is a cluster of students). By "greater effect" I mean that, for example, meal voucher helps students in one school a lot more than in another school. In other words, the slope of "meal voucher" varies across school - how do we explain this? $\endgroup$ – Heisenberg Nov 17 '14 at 3:48
  • $\begingroup$ I don't quite understand. If you have other measured school-level covariates that might explain some of the among-school variation (e.g. average income level -- ideally you would have this at the student level, but maybe it wouldn't be available), then you could include them as fixed effects and see how much of the among-school variation was explained/reduced in the model that incorporated them. If you don't have any other information, then I don't see how you can make any inferences to explain the variation ... $\endgroup$ – Ben Bolker Nov 17 '14 at 23:34
  • $\begingroup$ @BenBolker I'm interested in what you just said. If I do have measured school-level covariates, where do I include them in the model to explain the varying effect of meal voucher? What kind of assessment can I make after running that model? (I'll edit my questions with a defined model). $\endgroup$ – Heisenberg Nov 17 '14 at 23:37

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