# Intraclass Correlation Coefficients (ICC) with Multiple Variables

Suppose I have measured some variable in siblings, which are nested within families. The data structure looks like this:

family sibling value
------ ------- -----
1      1       y_11
1      2       y_12
2      1       y_21
2      2       y_22
2      3       y_23
...    ...     ...


I want to know the correlation between measurements taken on siblings within the same family. The usual way of doing that is to calculate the ICC based on a random-intercept model:

res <- lme(yij ~ 1, random = ~ 1 | family, data=dat)
getVarCov(res)[[1]] / (getVarCov(res)[[1]] + res$s^2)  If I only had pairs of siblings within families, this would be equivalent to: res <- gls(yij ~ 1, correlation = corCompSymm(form = ~ 1 | family), data=dat)  except that the latter approach also allows for a negative ICC. Now suppose I have measured three items in siblings nested within families. So, the data structure looks like this: family sibling item value ------ ------- ---- ----- 1 1 1 y_111 1 1 2 y_112 1 1 3 y_113 1 2 1 y_121 1 2 2 y_122 1 2 3 y_123 2 1 1 y_211 2 1 2 y_212 2 1 3 y_213 2 2 1 y_221 2 2 2 y_222 2 2 3 y_223 2 3 1 y_231 2 3 2 y_232 2 3 3 y_233 ... ... ... ...  Now, I want to find out about: 1. the correlation between measurements taken on siblings within the same family for the same item 2. the correlation between measurements taken on siblings within the same family for different items If I only had pairs of siblings within families, I would just do: res <- gls(yijk ~ item, correlation = corSymm(form = ~ 1 | family), weights = varIdent(form = ~ 1 | item), data=dat)  which gives me a$6 \times 6$var-cov matrix on the residuals of the form:$\left[\begin{array}{ccc|ccc} \sigma^2_1 & \rho_{12} \sigma_1 \sigma_2 & \rho_{13} \sigma_1 \sigma_3 & \phi_{11} \sigma^2_1 & \phi_{12} \sigma_1 \sigma_2 & \phi_{13} \sigma_1 \sigma_3 \\ & \sigma^2_2 & \rho_{23} \sigma_2 \sigma_3 & & \phi_{22} \sigma^2_2 & \phi_{23} \sigma_2 \sigma_3 \\ & & \sigma^2_3 & & & \phi_{33} \sigma^2_3 \\ \hline & & & \sigma^2_1 & \rho_{12} \sigma_1 \sigma_2 & \rho_{13} \sigma_1 \sigma_3 \\ & & & & \sigma^2_2 & \rho_{23} \sigma_2 \sigma_3 \\ & & & & & \sigma^2_3 \\ \end{array}\right]$based on which I could easily estimate those cross-sibling correlations (the$\phi_{jj}$values are the ICCs for the same item; the$\phi_{jj'}\$ values are the ICCs for different items). However, as shown above, for some families, I have only two siblings, but for other families more than two. So, that makes me think that I need to get back to a variance-components type of model. However, the correlation between items may be negative, so I do not want to use a model that constraints the correlations to be positive.

Any ideas/suggestions of how I could approach this? Thanks in advance for any help!

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