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I am having difficulty finding information on the assumptions for the intraclass correlation. Can someone please tell me what they are?

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There is no context to your request... here, I give an attempt to answer it in the context of regression models. More specifically, I shall refer to the usual linear mixed model.

Let $y_{ij}$ be the observation for subject $j \in \{ 1, \ldots{}, n_i\}$ from group $i \in \{1, \ldots{}, g\}$. The model I shall consider takes the form

$y_{ij} = \underbrace{\mu_{ij}}_{\textrm{fixed part}} + \underbrace{u_{i}}_{\textrm{random part}} + e_{ij}$,

under the assumption that $u_{i}$ is a realisation of a $N(0, \sigma^2_u)$ random variable, $e_{ij}$ is a realisation of a $N(0, \sigma^2)$ random variable, and under independence between these two random variables.

Observe that all subjects from the $i$th group share the same value for the random effect. Therefore, the random effect accounts for the association between observations from the same group. Mathematically, this can be seen by computing $\textrm{Corr}(Y_{ij}, Y_{i'j'})$ (see, e.g., here). When $ i = i'$, $\textrm{Corr}(Y_{ij}, Y_{i'j'}) = \textrm{Corr}(Y_{ij}, Y_{ij'}) > 0$ is known as the intraclass correlation.

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Adding to @Ocram's answer, the assumption for intraclass correlation coefficient is that the variance of observation of a subject before a fixed effect is same as after the fixed effect.

For example, in an experiment measuring the (fixed) effect 'j' of a medicine on the blood pressure (BP= response 'y') of different subjects, the BP of a subject before and after the medicinal treatment are assumed to have equal variance.

Let '$y_{i1}$' be the observation for ith subject i∈{1,…,ni} with fixed effect value j=1 representing before medicinal treatment and '$y_{i2}$' the observation for ith subject with fixed effect value j=2 after medicinal treatment, then

Variance $\sigma^2$ = $Var(Y_{i1})$ = V$ar(Y_{i2}$)

This assumption is not applicable for measuring Pearson correlation coefficient.

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