17.3 Components of Variance

Statistics can be thought of as the study of uncertainty, and variance is a measure of uncertainty (and information). So yet again we see that we’re partitioning the variance. Recall that

• Measurement error: \(\sigma^{2}_{\epsilon}\)
• County level error: \(\sigma^{2}_{\alpha}\)

The intraclass correlation (ICC, \(\rho\)) is interpreted as

• the proportion of total variance that is explained by the clusters.
  
• the expected correlation between two individuals who are drawn from the same cluster.

\[ \rho = \frac{\sigma^{2}_{\alpha}}{\sigma^{2}_{\alpha} + \sigma^{2}_{\epsilon}} \]

• When \(\rho\) is large, a lot of the variance is at the macro level
  • units within each group are very similar
• If \(\rho\) is small enough, one may ask if fitting a multi-level model is worth the complexity.
• No hard and fast rule to say “is it large enough?”
  • rules of thumb include
    • under 10% (0.1) then a single level analysis may still be appropriate,
    • over 10% (0.1) then a multilevel model can be justified.