Intraclass correlation coefficient in Bayesian statistics

I need some references about intraclass correlation coefficient in Bayesian statistics and hypothesis testing.

I already take a look in A. Gelman, J.B. Carlin, H.S. Stern and D.B. Rubin, Bayesian Data Analysis. London: Chapman & Hall/CRC, 2003

but it has very little information there.

It doesn't surprise me that Bayesian data analysis by Gelman et al. does not cover ICC since it is a general handbook about Bayesian methods, so it cannot cover everything. For more focused book you can check handbook by Lee and Wagenmakers (2014) that describes similar kind of models (but as far as I remember does not describe ICC per se).

Recently ICC is defined in terms of random effects model (Gelman & Hill, 2007):

$$y_{ij} = \mu + \alpha_j + \varepsilon_{ij}$$

where $y_{ij}$ is $i$-th observation for $j$-th group, $\mu$ is overall mean, $\alpha_i$ is a random effect, and $\varepsilon_{ij}$ is error term. So we have random variables

$$\alpha_i \sim \mathcal{N}(0, \sigma_\alpha^2) \\ \varepsilon_{ij} \sim \mathcal{N}(0, \sigma_\varepsilon^2)$$

and ICC is defined as

$$\frac{\sigma_\alpha^2}{\sigma_\alpha^2+\sigma_\varepsilon^2}$$

If $\sigma_\alpha^2$ is variance explained by $\alpha_i$ grouping factor and $\sigma_\varepsilon^2$ is variance unexplained by it, then ICC is a fraction of total variance explained by grouping. This means that you are interested in Bayesian estimation of hierarchical/random effects models that are nicely described by Gelman and Hill (2007). Since you will be mostly focused on variance components, you can check paper by Gelman (2006) who discusses choosing priors for them.

Finally, you can easily find multiple papers focusing directly on Bayesian estimation of ICC, e.g. Burch and Harris (1999), Ahmed and Shoukri (2010), Jelenkowska (1998), or Chung and Dey (1998), to name some. But honestly, I never studied this topic in detail since ICC can be misleading and interpreting variances directly seems to be more straightforward approach.

Gelman, A. & Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.

Lee, M. D., & Wagenmakers, E. J. (2014). Bayesian cognitive modeling: A practical course. Cambridge University Press.

Burch, B. D., & Harris, I. R. (1999). Bayesian estimators of the intraclass correlation coefficient in the one-way random effects model. Communications in Statistics-Theory and Methods, 28(6), 1247-1272.

Ahmed, M., & Shoukri, M. (2010). A Bayesian estimator of the intracluster correlation coefficient from correlated binary responses. Journal of Data Science, 8(1), 127-37.

Jelenkowska, T. H. (1998). Bayesian estimation of the intraclass correlation coefficients in the mixed linear model. Applications of Mathematics, 43(2), 103-110.

Chung, Y., & Dey, D. K. (1998). Bayesian approach to estimation of intraclass correlation using reference prior. Communications in Statistics-Theory and Methods, 27(9), 2241-2255.

Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian analysis, 1(3), 515-534.

• The ICC is always lower than 1 when you have more than one replication right? I mean $\rho=\frac{\sigma_a^2}{\sigma_a^2+\sigma_e^2}=1$ if and only if you have one replication? – user72621 Jun 1 '16 at 16:35
• @PRAGAKHAM calculating it on only one case does not make sens -- you wouldn't be able to estimate underlying model in such case. So in general it should be 0 < ICC < 1. – Tim Jun 1 '16 at 17:03
• I ask it because in the article that I read they say "$\rho<1$ when $m\geq 2$ where $m$ is the number of replications". – user72621 Jun 1 '16 at 17:06
• @PRAGAKHAM if you have only one observation that error variance is 0 so ICC needs to be 1 -- but it's a rather pathological case. – Tim Jun 1 '16 at 17:08