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I have seen many posts refer to the calculation of the ICC by utilizing the variance of the random intercept and the residual variance from a multilevel model. Does anyone have this formula handy?

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You can find the formula on the Wikipedia page. It is:
$$ \text{ICC} = \frac{\sigma^2_{\text{intercepts}}}{\sigma^2_\text{intercepts} + \sigma^2_\text{residuals}} $$

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