  In this article, we develop methods for sample size and power calculations in
four-level intervention studies when intervention assignment is carried out at
any level, with a particular focus on cluster randomized trials (CRTs). CRTs
involving four levels are becoming popular in health care research, where the
effects are measured, for example, from evaluations (level 1) within
participants (level 2) in divisions (level 3) that are nested in clusters
(level 4). In such multi-level CRTs, we consider three types of intraclass
correlations between different evaluations to account for such clustering: that
of the same participant, that of different participants from the same division,
and that of different participants from different divisions in the same
cluster. Assuming arbitrary link and variance functions, with the proposed
correlation structure as the true correlation structure, closed-form sample
size formulas for randomization carried out at any level (including
individually randomized trials within a four-level clustered structure) are
derived based on the generalized estimating equations approach using the
model-based variance and using the sandwich variance with an independence
working correlation matrix. We demonstrate that empirical power corresponds
well with that predicted by the proposed method for as few as 8 clusters, when
data are analyzed using the matrix-adjusted estimating equations for the
correlation parameters with a bias-corrected sandwich variance estimator, under
both balanced and unbalanced designs.
