Mixed-model power analysis - MCPower app

Mixed models add random effects for clustered or repeated-measures data — observations grouped within schools, clinics, subjects, and so on. Top to bottom:

1. Formula with a random effect

Write the fixed part as in regression, then add a grouping term in parentheses — a random intercept for cluster is (1 | cluster), e.g. y = x1 + x2 + (1 | cluster). Fixed-effect operators match regression: : is the interaction term on its own and * expands to a + b + a:b. See formula syntax and mixed effects.

2. Outcome type — continuous or binary

The Outcome type toggle at the top of the Model section switches the mixed model between a continuous outcome (a Gaussian linear mixed model) and a binary one (a clustered logistic GLMM — logistic regression with a cluster-level random intercept). Continuous is the default; flip to Binary for yes/no outcomes like passed/failed or relapsed/recovered.

Under Binary a baseline probability input appears — the event probability when every predictor sits at its reference level, which fixes the model intercept. Predictor effects are then read on the log-odds scale, the same interpretation as plain logistic regression; see logistic effect sizes for what a log-odds beta means in probability terms. Switching back to Continuous restores the Gaussian fit.

3. Cluster configuration

The grouping term unlocks the cluster panel: the cluster name (mirrored from the (1 | …) term), the ICC — the between-cluster variance share in [0, 1); a higher ICC makes within-cluster observations more alike, lowering the effective sample size — and the number of clusters (or cluster size; fix one and the other follows from n).

4. Predictors

Each fixed-effect predictor is one card: pick its type next to the name (continuous, binary, or factor), then set its standardised effect size in the same card — continuous 0.10 / 0.25 / 0.40, binary or factor 0.20 / 0.50 / 0.80. See variable types and effect sizes.

5. Robustness scenarios

The Robustness scenarios toggle in the status bar repeats every run under three perturbation sets — Optimistic (your exact settings, no perturbations), Realistic (moderate assumption violations), and Doomer (severe violations, a worst case) — so you get a range of power instead of one optimistic number. Mixed models additionally perturb the random effect (its distribution, df, and ICC noise). If even Doomer clears your target, the design is robust; if only Optimistic reaches it, increase the sample size. Each set's knobs are editable under Settings → Scenarios. See scenario analysis.

6. Optional settings

The Advanced section exposes the number of simulations (mixed models default to 800, since each fit is heavier), α (0.05), seed (2137), and the failed-simulation tolerance.