Repeated Measures, Clusters and Independence

Definition A cluster is a group of observations linked through a shared unit or setting, such as the readings taken on one device, the pupils in one classroom or the visits made by one patient. Repeated measures are several measurements of the same variable on the same unit, taken at different times, under different conditions or simply repeated under one protocol, and each unit together with its repeated measures forms a cluster. The sampling unit is the unit that the sampling procedure selects, and the experimental unit is the unit to which a treatment is independently assigned. Either one can be larger than the row that stores a single measurement.

Definition Pseudoreplication is analyzing observations as independent replicates of the unit an inference is about when they are not. Typical cases are treating repeated readings of a few devices as readings of many different devices, and treating pupils as independent replications of a teaching method that was assigned to whole classrooms. When observations within a cluster are positively correlated, the usual consequence is a standard error that is too small, an interval that is too narrow and a p-value that is too small.

Definition Let $Y_{ij}$ be observation $j$ in cluster $i$, in a model where every observation has the same variance $\sigma_Y^2 > 0$, observations in different clusters are independent, and every two different observations in the same cluster have the same correlation $\rho$, a structure called exchangeable or compound symmetry. Then $\rho$ is the intraclass correlation, and for $j \ne k$

In a cluster of $m \ge 2$ observations the correlation must satisfy $-1/(m - 1) \le \rho \le 1$. When the dependence has another pattern, such as visits whose correlation weakens as the time between them grows, a single intraclass correlation does not describe it.

Definition The design effect of an estimator is its variance under the actual design or dependence model divided by its variance under a stated reference, usually independent sampling of the same number of observations. Take $k$ independent clusters of $m$ observations each, $n = km$ in all, with common variance $\sigma_Y^2$ and intraclass correlation $\rho$. The sum of one cluster has $m$ variance terms and $m(m - 1)$ covariance terms, so the overall mean $\bar{Y}$ of all $n$ observations has

When $D > 0$ the variance-equivalent effective sample size is $n_{\text{eff}} = n/D$, the number of independent observations whose mean would have the same variance.

Definition Suppose $k$ clusters were sampled independently. A cluster bootstrap sample draws $k$ clusters with replacement from the observed clusters, each draw choosing every cluster with probability $1/k$, and keeps all observations of each drawn cluster together. The estimator is recomputed on every bootstrap sample exactly as on the original data, and the standard deviation of these bootstrap estimates is the cluster bootstrap standard error.

Definition A random-intercept model for clustered data writes observation $j$ in cluster $i$ as

where $\mu$ is a fixed overall mean, the random intercepts $b_i$ have mean $0$ and variance $\tau^2$, the errors $e_{ij}$ have mean $0$ and variance $\sigma^2$, and all the $b_i$ and $e_{ij}$ are mutually independent. It is a mixed model because it combines fixed effects, here $\mu$ and possibly regression terms for covariates, with the random effects $b_i$. The model implies, for $j \ne k$,

so the intraclass correlation is $\rho = \tau^2/(\tau^2 + \sigma^2)$ whenever $\tau^2 + \sigma^2 > 0$, and the mean of $m$ observations in one cluster has variance $\operatorname{Var}(\bar{Y}_i) = \tau^2 + \sigma^2/m$.