# Intra-class correlation coefficient interpretation

I am getting a hard time to go around this and I am not sure my interpretation is correct. I have several models that look like this:

change | trials ~ treatmentA + treatmentB + treatmentC + (1|person)

I would like to correctly interpret my group-level estimate (random effect) and its derived intraclass correlation coefficient (ICC). By definition, the ICC can be interpreted as “the proportion of the variance explained by the grouping structure in the population”. The ICC is calculated by dividing the random effect variance, σ2i, by the total variance, i.e. the sum of the random effect variance and the residual variance, σ2ε.

Would an ICC close to 1 indicate high consistency of response across treatments within the same person, while an ICC close to zero refer to an heterogeneous intra-patient response?

Could you recommend me some literature or provide me a bit more insight on this?

I am not interested on 'siblings studies' or 'sample replicability'... I would rather know if I could use the ICC measure for estimating how similar or divergent are patients in regards to the response in each of the models.

Interpreting the intraclass correlation coefficient is questionable, since your class is a single person.

Unless you have multiple change data per person, your per-person random effect cannot be resolved from the per-data noise.

I am new to R lmer notation, so it took a little bit to interpret your code, but this thread helped.

change | trials ~ treatmentA + treatmentB + treatmentC + (1|person)


As I understand it, this implies the following.

• $$Y_i$$: change on person $$i$$
• $$\beta_t$$: fixed effect of treatment $$t$$
• $$x_{i,t} \in \{0, 1\}$$: indicator of whether person $$i$$ received treatment $$t$$
• $$R_i$$: random effect on person $$i$$
• $$\epsilon$$: random noise

$$Y_i = \beta_0 + \beta_A x_{i,A} + \beta_B x_{i,B} + \beta_C x_{i,C} + R_i + \epsilon$$

Parameters $$\beta_0$$, $$\beta_A$$, $$\beta_B$$, and $$\beta_C$$ are fixed and have no dependence on person $$i$$ in this model. The only variable that depends on person $$i$$ is the random effect $$R_i$$. As you can see, if you only have one data $$Y_i$$ per person, then $$R_i$$ and $$\epsilon$$ are indistinguishable because they could be rewritten as $$e = R_i + \epsilon$$ where $$e$$ is also a noise.

If you have the multiple data per person to resolve $$R_i$$ from $$\epsilon$$, an intraperson correlation coefficient (ICC) close to one does suggest consistency of response across treatments. (It also suggests that the treatments might be irrelevant to the response.) An ICC close to zero would suggest that either noise $$\epsilon$$ or the treatment effects outweigh $$R_i$$.

If you only have one data per person, a more informative comparison of variances would be the ratio of fixed effect variance to total variance.