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I'm seeking clarification regarding notation for Bayesian hierarchical models, specifically the mixed effects model. Consider the following hierarchical model for the outcome of unit $i \in N$ in group $j \in J$ from Gelman and Hill (p.265):

$$y_i \sim Normal(X_i\beta + \eta_{j[i]}, \sigma_y^2) \\ \eta_j \sim Normal(0, \sigma_\alpha^2)$$

  • $X_{(0)}$ is a constant.
  • $X_{(1)}$ is a unit/level-1 predictor.
  • $X_{(2)}$ is a group/level-2 predictor.

The authors call this the "regression with multiple error terms" notation but I find this confusing for several reasons.

  1. In this model isn't there only one error term ($\sigma_y^2$) that captures the total unexplained error in $y$? What then is the other error term in the name "multiple error terms?"

  2. If each group is supposed to have its own error, why is $\eta_j$ part of the model for the mean of $y_i$ rather than the error of $y_i$? I read this to imply a heteroskedastic model in which the error is decomposed into group-level terms and individual terms like $\sigma_y^2 = \sigma_\eta^2 + \sigma_i^2$.

  3. What, if anything, is multivariate here? Would it be more appropriate to write $\eta_j \sim MVN(0, \Sigma)$? If so, would $\Sigma$ have a single parameter $\sigma_\alpha^2$ on the diagonal and zeroes elsewhere or does each element of the diagonal vary (i.e. is its own parameter)?

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1 Answer 1

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Let's consider a simplification of your model with only a constant in the $X_i \beta$ part, i.e., $$\left\{ \begin{array}{l} y_{ji} \mid \eta_{j} \sim N(\eta_j, \sigma_y^2),\\ \eta_{j} \sim N(\beta, \sigma_\eta^2). \end{array} \right.$$ Note that I have explicitly conditioned on $\eta_j$ in the first line. That is, the first line represents the model at a given level $j$ $(j = 1, \ldots, n)$. The variance parameter $\sigma_y^2$ denotes the variance of the outcome $y_{ij}$ around the mean at the $j$-level denoted by $\beta + \eta_j$. The second line is the model for the means $\eta_1, \ldots, \eta_{n}$ that has a mean $\beta$ and variance $\sigma_{\eta}^2$.

From the above formulation, you can derive the marginal model for $y_j$, i.e., $$y_j \sim N(\beta, \Sigma),$$ where $\Sigma$ has a compound symmetry structure, with diagonal elements equal to $\sigma_{\eta}^2 + \sigma_y^2$, and off-diagonal elements equal to $\sigma_{\eta}^2$. This marginal model does not presume the existence of the latent variable $\eta_j$. That is, a hierarchical model implies a marginal model but not vice versa. For example, you could define a marginal model that does not correspond to a hierarchical one, or you could have that two different hierarchical models imply the same marginal model. You may have a look at Section 3.3 of my course notes for additional explanation.

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