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Heck et al (2013) describe a process of building multilevel models that makes use of the concept of intraclass correlation (ICC).

They recommend calculating the ICC from a random-effects ANOVA

FIRST MODEL $$ y_{ij} = \gamma_{00} + u_j + e_{ij} $$

where $u_j$ are the level-2 residuals and $e_{ij}$ are the level-1 residuals. Then we obtain estimates, $\hat{\sigma}_u^2$ and $\hat{\sigma}_e^2$ for the variance of $u_j$ and $e_{ij}$ respectively, and plug them into the following equation:

$$ ρ = \frac{\hat{\sigma}_u^2}{\hat{\sigma}_u^2 +\hat{\sigma}_e^2} $$

Then, if $ρ$ (i.e. ICC) is relatively high they regard that as a sign that a multilevel model might be useful to explain the variability between groups. We might make a random intercept model with the Level 1 predictor $x_1$.

SECOND MODEL $$ y_{ij} = \gamma_{00} + \gamma_{10}x_1{_{ij}} + u_{0j} + e_{ij} $$

Then we might calculate the ICC again, and maybe we'll notice that it's lower - after controlling for $x_1$ the variability between groups has decreased.

Does it follow that ICC must decrease between the first model I've described an the second model? In Heck et al's example ICC decreases at this point, but I'm not sure if it necessarily follows that that must happen.

What if we made a third model with another Level 1 predictor $x_2$:

THIRD MODEL $$ y_{ij} = \gamma_{00} + \gamma_{10}x_1{_{ij}} + \gamma_{20}x_2{_{ij}} + u_{0j} + e_{ij} $$

Would ICC necessarily decrease between the second model and the third model? Why/Why not?

Heck, R. H., Thomas, S. L., & Tabata, L. N. (2013). Multilevel and longitudinal modeling with IBM SPSS. Routledge. Chicago

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I am pretty confident that the following answer explains at least one way in which ICC can increase from the first model to the second model (and similarly from the second to third model)

As shown in the equation below, the ICC is measuring the proportion of random variation between groups ($\hat{\sigma}_u^2$) relative to the total amount of unexplained variance (${\hat{\sigma}_u^2 +\hat{\sigma}_e^2}$). $$ ρ = \frac{\hat{\sigma}_u^2}{\hat{\sigma}_u^2 +\hat{\sigma}_e^2} $$

When we include a predictor variable it is likely going to account for some proportion of the total unexplained variance (i.e. ${\hat{\sigma}_u^2 +\hat{\sigma}_e^2}$ will be reduced). In this sense, including a predictor can potentially reduce level 1 residual variation ($\hat{\sigma}_e^2$) by some amount and can potentially reduce the level 2 random variation between groups ($\hat{\sigma}_u^2$) by some amount.

Based on this, we should expect the ICC to increase between the first and second model when the predictor variable accounts for a larger proportion of residual variation than variation between groups. This is because it would reduce the denominator (${\hat{\sigma}_u^2 +\hat{\sigma}_e^2}$) by proportionally more than it reduces the numerator ($\hat{\sigma}_u^2$).

The issue is more complicated than as described above though. For example, you can get seemingly strange situations in multi-level models where including a predictor actually increases the amount of unexplained variance (e.g. http://andrewgelman.com/2006/05/15/in_a_multilevel/).

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