= Nested measurements
<Nested measurements> are responses grouped inside larger units, with each measured object belonging to just one such unit. Three <leaves> within each pot give <nested measurements>; repeatedly reading one <leaf> gives another level of repetition. Either can yield <clustered data>. A common <random effect> can describe the shared influence of the containing <experimental unit>.
For the constructed pot example, label <treatments> by $g\in\{A,B\}$, pots within each group by $i=1,\ldots,n$, and <leaves> within each pot by $j=1,\ldots,m$. Consider the <statistical model>
$$
Y_{gij}=\mu+\alpha_g+U_{gi}+\varepsilon_{gij},
\qquad
\mathbb E[U_{gi}]=\mathbb E[\varepsilon_{gij}]=0,
\qquad
\operatorname{Var}(U_{gi})=\tau^2,
\quad
\operatorname{Var}(\varepsilon_{gij})=\sigma^2.
$$
Here $Y_{gij}$ is a <leaf> <length>, $\mu+\alpha_g$ is its group <expected value>, $U_{gi}$ is a pot <random effect>, and $\varepsilon_{gij}$ describes remaining variation among <leaves> and <measurement error>. Assume that all pot <random effects> are mutually <independent>, all remaining errors are mutually <independent>, and the two collections are <independent> of one another and of the <randomization>. These are explicit modelling assumptions, conditional on the assigned <treatments>; no <normal distribution> assumption is needed for the following <variance> calculation.
Two distinct <leaf> measurements in one pot share $U_{gi}$. Their <covariance> is $\tau^2$, their individual <variance> is $\tau^2+\sigma^2$, and their <intraclass correlation coefficient> is
$$
\rho=\frac{\tau^2}{\tau^2+\sigma^2}
\qquad (\tau^2+\sigma^2>0).
$$
Measurements from different pots have zero <covariance> and are <independent> under the stated <statistical model>. Statistical <independence> therefore concerns the whole response structure, not just the fact that the measured <leaves> are different objects.
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