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Multilevel models (aka hierarchical linear models, mixed models, etc.) incorporating random slopes can entail unstructured variance-covariance matrices decomposing level-specific heterogeneity into portions attributable to variables with random slopes and to the joint distributions of those variables with one another, and with level-specific outcomes. For example, in the two-level random slopes model of individuals $i$ nested within populations $j$

$$\hat{y}_{ij} = \beta_{0ij} + \beta_{xij}x_{ij} + \beta_{zj}z_{j},$$

where: $$\beta_{0ij} = \beta_{0} + \varepsilon_{0ij} + \mu_{0j}\\ \beta_{xij} = \beta_{x} + \varepsilon_{xij}\\ \beta_{zj} = \beta_{z} + \mu_{zj},$$

and where:

$$\begin{eqnarray*} \left[\begin{array}{c} \varepsilon_{0ij}\\ \varepsilon_{xij}\end{array}\right] & \sim & \mathcal{N}\left(0,\Omega_{\varepsilon}\right):\Omega_{\varepsilon} = \left[\begin{array}{cc} \sigma^{2}_{\varepsilon 0} &\\ \sigma_{\varepsilon 0x} & \sigma^{2}_{\varepsilon x} \\ \end{array}\right]\\ \\ \left[\begin{array}{c} \mu_{0j}\\ \mu_{zj}\end{array}\right] & \sim & \mathcal{N}\left(0,\Omega_{\mu}\right):\Omega_{\mu} = \left[\begin{array}{cc} \sigma^{2}_{\mu 0} &\\ \sigma_{\mu 0z} & \sigma^{2}_{\mu z} \\ \end{array}\right] \end{eqnarray*} $$

heteroscedastic variance at the individual level is decomposed into individual intercept variance $\left(\sigma^{2}_{\varepsilon 0}\right)$, variance contributed by $x_{ij}$ $\left(\sigma^{2}_{\varepsilon x}\right)$, and a covariance between individual-level $y_{ij}$ and $x_{ij}$ $\left(\sigma_{\varepsilon 0 x}\right)$. Likewise, heteroscedastic variance at the population level is decomposed into population intercept variance $\left(\sigma^{2}_{\mu 0}\right)$, variance contributed by the population-level predictor $z_{j}$ $\left(\sigma^{2}_{\mu z}\right)$, and a covariance between population-level average $\hat{y}_{j}$ and $z_{j}$ $\left(\sigma_{\mu 0 x}\right)$.

A positive population-level covariance $\sigma_{\mu 0 x}$ means that populations with higher values of $z$ are more likely to also have higher population-level intercepts $\left(\beta_{0} + \sigma^{2}_{\mu 0}\right)$. A positive individual-level covariance $\sigma_{\varepsilon 0 x}$ means that individuals with higher values of $x$ are more likely to also have higher individual-level intercepts $\left(\beta_{0} + \sigma^{2}_{\varepsilon 0}\right)$.

Question: Does it make sense to think about 'cross-level covariance'?
For example, imagine a term $\sigma_{\varepsilon\mu xz}$ with an interpretation along the lines of 'A positive cross-level covariance $\sigma_{\varepsilon \mu x z}$ means that individuals with higher values of $x$ are more likely to be nested within populations that have higher values of $z$.' Is this crazy? Is it already somehow captured in the model above, and I am just not interpreting it? I'm probably crazy.



I am aware that lots of folks approach multilevel models without the conceptual framework in place that recognizes that individual-level heteroscedastic variance can likewise be decomposed in multilevel models. Lots of statistical software that is fine modeling complex variance at higher levels does not offer a means of doing so at the lowest level (i.e. level 1), but some statistical software can estimate such models (e.g., MLwiN, JAGS, etc.).

References

Goldstein, H. (2003). Multilevel statistical models. Oxford University Press, 3rd edition.

Duncan, C., Jones, K., and Moon, G. (1998). Context, composition and heterogeneity: Using multilevel models in health research. Social Science & Medicine, 46(1):97–117.

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  • $\begingroup$ Perhaps the intra-class correlation coefficient fits into your thoughts here, where $$\rho=\frac{e_{xij}}{e_{xij}+\mu_{zj}}$$ $\endgroup$ – Jay Schyler Raadt Dec 13 '17 at 1:02
  • $\begingroup$ @JaySchylerRaadt Thank you for the comment. However, no, I don't think so. Variance partition coefficients – of which the ICC is one kind—simply apportion variance in multilevel null models (aka intercepts-only models), though sometimes in random intercepts models. VPCs are notable for not estimating any covariances. See, for example, Goldstein, H., Browne, W., and Rasbash, J. (2002). Partitioning variation in multilevel models. Understanding Statistics, 1(4):223–232. $\endgroup$ – Alexis Dec 13 '17 at 2:27

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