Methods
Estimators and inference
Power is not a property of a design alone. It is a property of a design together with an analysis method. This page sets out what each estimator in the calculator computes and when it will mislead you.
The common core
Every estimator starts from a design matrix \(X\) with one row per cluster-period and a model-implied covariance matrix \(\Sigma\). The generalised least squares information matrix is
$$M = X^{\top}\Sigma^{-1}X$$
and the model-based variance of the treatment effect estimate is the diagonal element of \(M^{-1}\) corresponding to the treatment column:
$$\operatorname{Var}(\hat\theta) = \left[(X^{\top}\Sigma^{-1}X)^{-1}\right]_{kk}$$
Everything else is a variation on how \(\Sigma\) is built, whether a different working covariance is used for estimation than for the true variance, and what reference distribution the test statistic is compared against.
Model-based inference
Normal approximation
The default. Treats \(\hat\theta / \operatorname{SE}(\hat\theta)\) as standard normal under the null, giving
$$\text{power} = \Phi\!\left(\frac{|\theta|}{\operatorname{SE}} - z_{1-\alpha/2}\right)$$
Accurate when the number of clusters is large. With few clusters it is optimistic, because it ignores the fact that the variance components are estimated rather than known.
t-test
Same standard error, but referred to a t distribution with degrees of freedom based on the number of clusters minus the number of estimated fixed effects. A minimal correction, but it removes the worst of the anti-conservatism at essentially no cost.
Satterthwaite
Approximates the degrees of freedom by matching the first two moments of the estimated variance to a scaled chi-squared distribution. Accounts for the fact that different contrasts in an unbalanced design carry different effective sample sizes, so the degrees of freedom need not be an integer and can be far below the cluster count.
Kenward–Roger
Inflates the variance estimate to account for the downward bias induced by estimating the variance components, and adjusts the degrees of freedom. The most conservative of the model-based options and generally the best behaved with small numbers of clusters. If your analysis plan is a linear mixed model and you have fewer than about 30 clusters, this is usually the honest choice.
Robust and marginal methods
GEE estimators use a working covariance \(V\) that need not equal the truth. The variance is then estimated by a sandwich:
$$\operatorname{Var}(\hat\theta) = \underbrace{(X^{\top}V^{-1}X)^{-1}}_{\text{bread}}\; \underbrace{X^{\top}V^{-1}\Sigma V^{-1}X}_{\text{meat}}\; \underbrace{(X^{\top}V^{-1}X)^{-1}}_{\text{bread}}$$
Independence working correlation takes \(V = I\): the point estimate ignores clustering entirely, but the sandwich variance restores valid inference. Simple and robust to covariance misspecification, at some cost in efficiency.
Exchangeable working correlation uses a correctly shaped working covariance, recovering most of the efficiency while keeping the robustness of the sandwich.
The sandwich variance is consistent as the number of clusters grows, but it is biased downward when that number is small — the well known small-sample problem with GEE. A t-test variant is provided for this reason. Below roughly 30 clusters, treat an uncorrected robust GEE result with the same suspicion as an uncorrected model-based one.
Conditional and marginal effects
For continuous outcomes with an identity link, a mixed model and a GEE estimate the same quantity. For binary and count outcomes they do not.
A GLMM with a logit link estimates a conditional, cluster-specific log odds ratio: the effect of treatment for a given cluster. A GEE estimates a marginal, population-averaged log odds ratio: the effect averaged over the cluster distribution. Because the logit is non-linear, these differ, with the marginal effect attenuated toward the null relative to the conditional one. The gap widens as the between-cluster variance grows.
Decide which estimand your trial question is about, then match the estimator to it, then match the effect size you enter to the same scale. A conditional log odds ratio entered as if it were marginal will overstate power. The calculator labels the binary and count estimators explicitly as conditional or marginal for this reason.
Design effect
Included for comparison with textbook and older calculations. The individually randomised sample size is inflated by
$$\text{DE} = 1 + (\bar{m} - 1)\rho$$
for equal cluster sizes, or, allowing a coefficient of variation \(k\) in cluster size,
$$\text{DE} = 1 + \left\{\bar{m}(1 + k^{2}) - 1\right\}\rho$$
This is exact for a parallel design with a single period and no covariates. It is not correct for any longitudinal design, and applying it to a stepped wedge will overstate the required sample size substantially, because it ignores the within-cluster comparisons entirely. It is offered so you can see the size of that discrepancy rather than as a recommendation.
Choosing, in practice
| Situation | Reasonable choice |
|---|---|
| Continuous outcome, < 30 clusters | Kenward–Roger |
| Continuous outcome, many clusters | Model-based, Satterthwaite for reassurance |
| Binary outcome, conditional estimand | GLMM with Satterthwaite or Kenward–Roger |
| Binary outcome, marginal estimand | GEE exchangeable robust, with t-test if few clusters |
| Covariance structure genuinely uncertain | GEE robust, and compare against model-based |
Whichever you choose, look at the spread across the alternatives before you commit. If adequate power depends on which estimator you pick, the design is fragile, and that is worth knowing at the protocol stage rather than at the analysis.