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I'm having a hard time understanding how to write a multi-level hierarchical linear regression model in mathematical language. Suppose, for example, that our independent variable is x, our dependent variable is y, and that we have a 3 level hierarchy: C is nested inside B which is nested inside A. Also, suppose that we have a random-slopes model. How do we write this model? Here's what I have so far:

$Y_{iabc} = \beta_0 + \beta_1 C_{abc} x_i + \epsilon_{iabc}$

$C_{abc} = \beta_{10} + \beta_{11} B_{ab} + \delta_{abc}$

$B_{ab} = \beta_{110} + \beta_{111} A_a + \gamma_{ab}$

$A_a \sim N(0, \xi^2)$

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I was wondering if it is the other way around. The higher levels are B and C. So x is at level-1 as the same level as Y. Just substituting (2) and (3) in 1 will give single line mathematical representation for the HLM model. Basically, it will have within group and between group interaction coefficients to be determined.

I found this document to be very helpful: here .It gives step by step, how to go about developing a HLM model.

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  • $\begingroup$ Thanks, that doc was very helpful! It seems like they only go in to detail with a 2-level model though, and I'm really like to find an example where they generalize the notation to 3 levels... $\endgroup$ – random_forest_fanatic May 18 '15 at 12:35

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