Power Analysis

/power-analysis — Power / MDE for study design

Compute the three interlocking quantities of an ex-ante design calculation — power, required N, and minimum detectable effect (MDE) — and emit a power section the user can paste straight into a preregistration. Analytical for standard designs; simulation-based (reusing the /simulation-study harness pattern) for non-standard ones.

Core principle: a power calculation is a design-time commitment made before the data exist. Fix any two of {effect size, N, power} and solve for the third; never back out a "power" number from a realised estimate (that is post-hoc power, and it is uninformative — see "What this skill does NOT do").

When to use

• Before launching an RCT / field / survey experiment — to choose N (or clusters) for a target MDE at 80–90% power.
• Invoked by [`/preregister`](../preregister/SKILL.md) for RCTs — the AEA RCT Registry and most IRBs require a power/MDE justification; /preregister's aea-rct style calls this skill to fill that section.
• During R&R — when a referee asks "was this study adequately powered to detect the effect you claim?"
• Designing a Monte Carlo — to set R and sample sizes before handing off to /simulation-study.

Inputs

$ARGUMENTS may carry flags; missing pieces are elicited in Phase 0.

• --mode mde|n|power — solve for MDE given N+power, N given MDE+power, or power given N+MDE. Default mde.
• --design rct|cluster|multiarm|sim — two-arm RCT, clustered RCT (ICC), multiple arms, or simulation-based. Default inferred from the elicited design.
• --input <path> — a spec from `/interview-me` (under quality_reports/specs/) to pull the RQ, outcome, and design from.

Workflow

Phase 0 — Elicit the design

Gather the design parameters; ask once for anything missing rather than fabricating. Required:

• Estimand & test: primary outcome, one- vs two-sided test, alpha (default 0.05), and whether the target is a difference in means, a proportion, or a regression coefficient.
• Two of {effect size, N, power}: the effect as a raw difference and in standardized units (Cohen's d = effect / SD) — record both; power default 0.80.
• Baseline mean and SD (or baseline proportion for a binary outcome) — needed to translate raw ↔ standardized effects.
• Allocation: treated:control ratio (default 1:1; unequal allocation costs power — note it).
• Clustering: if randomization is at a group level (village, school, clinic), the ICC (ρ), the average cluster size (m), and number of clusters. Compute the design effect DEFF = 1 + (m − 1)·ρ and the effective N.
• Multiplicity: number of arms / primary outcomes; the correction (Bonferroni, Holm, or none) and whether power is per-comparison or familywise.

Echo a Pre-Flight Report (design, the two fixed quantities, the one being solved for, alpha, power, allocation, ICC/clusters, multiplicity) before computing. If the estimand or the SD source is ambiguous, stop and ask.

Phase 1 — Analytical power (standard designs)

For two-arm RCTs, clustered RCTs, and multi-arm comparisons, compute analytically. Prefer R pwr / WebPower (or a closed-form power.t.test / power.prop.test); for clustered designs inflate variance by DEFF, or use pwr on the effective N. Stata users: power twomeans / power twoproportions / power, cluster; Python: statsmodels.stats.power. Emit a short script to scripts/R/power_<slug>.R (or .do / .py) so the calc is reproducible, not a one-off console number.

• MDE mode: MDE = (z_{1−α/2} + z_{1−β}) · SE(effect), where SE is built from the SD, N, allocation, and DEFF. Report MDE in raw and standardized units.
• N mode: invert the above for total N (and #clusters when clustered) given the target MDE.
• Power mode: given N and a hypothesized effect, return achieved power.
• Multi-arm: divide alpha by the number of comparisons in the family m (Bonferroni alpha/m): m = K−1 for all-vs-control, m = K(K−1)/2 for all-pairwise. Report per-comparison and familywise power.

Sweep a grid (N or #clusters × effect size) so Phase 3 can draw a power curve and an MDE-vs-N curve.

Phase 2 — Simulation-based power (non-standard designs)

When the design is not a clean two-arm comparison — DiD / staggered event-study, IV / 2SLS (weak-instrument-aware), panel with serial correlation, a non-normal or censored outcome, or any estimator with no closed-form SE — switch to simulation. Reuse the `/simulation-study` harness exactly (see `simulation-study` and `.claude/rules/simulation-conventions.md`):

1. Seeded, parameterized DGP that embeds the hypothesized effect (and the null DGP for size). set.seed(YYYYMMDD) once; L'Ecuyer streams if parallel. 2. Estimator = the one you will actually use on the real data (e.g. fixest::feols two-way FE, did::att_gt, AER::ivreg), returning est, se, ci, p, reject. 3. Power = share of reps rejecting H0 at alpha; size = rejection rate under the null DGP (verify it is near nominal before trusting power). Report each with its Monte Carlo SE = sqrt(p(1−p)/R). 4. Sweep N (or #clusters / #periods) to trace the power curve; save the raw per-rep tibble via saveRDS() to scripts/R/_outputs/.

A simulated power number without an MCSE, or without a verified size check, is not yet an answer.

Phase 3 — Write the power section

Produce the deliverables under quality_reports/power/:

• `power_<slug>.md` — a table and a methods paragraph (below).
• `power_curve_<slug>.png` — power vs N (and/or MDE vs N), with reference lines at the target power and the design's planned N.
• The reproducible script under scripts/R/ (or .do / .py).

# Power Analysis: <study title>
**Date:** YYYY-MM-DD · **Design:** <rct|cluster|multiarm|sim> · **Method:** <analytical|simulation, R/Stata/Python>

| Quantity | Value |
|---|---|
| alpha (sided) | 0.05 (two-sided) |
| Target power | 0.80 |
| Baseline mean (SD) | <m0> (<sd>) |
| Allocation (T:C) | 1:1 |
| ICC / cluster size / #clusters | <ρ> / <m> / <J>  (DEFF = <…>) |
| Total N (analysis sample) | <N> |
| **MDE (raw / standardized)** | **<Δ> / <d>** |
| Achieved power at planned N | <…>  (± MCSE <…> if simulated) |

## Methods paragraph (paste into preregistration)
> Assuming a baseline outcome mean of <m0> (SD <sd>), 1:1 allocation, and a two-sided
> test at α = 0.05, a total sample of <N> [<J> clusters of <m>, ICC = <ρ>] yields 80%
> power to detect a minimum effect of <Δ> (<d> SD). [Simulation: under the hypothesized
> DGP, <P>% of <R> replications rejected H0 (MCSE <…>); size under the null was <…>.]

Phase 4 — Handoff

If invoked by /preregister, return the methods paragraph + MDE row for the preregistration's power section. If standalone, print the save paths and remind the user the MDE is a design commitment to record before data collection.

Exit behavior

• Computation succeeds: exit 0; print the MDE / N / power result, the save paths, and (for simulation mode) the size-check value next to power.
• Under-identified design (only one of {effect, N, power} supplied) or ambiguous SD source: halt in Phase 0 with a single specific question — never guess the SD or the ICC.
• Simulation size check fails (empirical size far from nominal under the null DGP): report power as UNRELIABLE and surface the size value; the estimator/DGP must be fixed before the power number is trustworthy.

Flags

• --mode <mde|n|power> — What to solve for: minimum detectable effect, required N, or achieved power.
• --design <rct|cluster|multiarm|sim> — Design family — two-arm RCT, clustered/ICC, multi-arm with corrections, or simulation-based for non-standard designs.
• --input <spec> — Path to an /interview-me spec or preregistration draft to read design parameters from.

Cross-references

• `.claude/skills/preregister/SKILL.md` — invokes this skill to fill the power/MDE section of an aea-rct (and OSF) preregistration; this skill returns the methods paragraph.
• `.claude/skills/simulation-study/SKILL.md` — the Monte Carlo harness Phase 2 reuses (seeded DGP, estimator grid, % rejecting H0).
• `.claude/rules/simulation-conventions.md` — the simulation contract (truth from DGP, MCSE, size-under-the-null) that Phase 2 must honor.
• `.claude/skills/data-analysis/SKILL.md` · `.claude/skills/stata-replication/SKILL.md` — where the realised analysis (and its actual estimator/SE) lives; the power calc should use the same estimator.
• `.claude/rules/confidential-data.md` — when baseline mean/SD/ICC are taken from restricted-access data, disclosure-avoidance limits apply; cite published or pilot moments rather than embedding raw confidential statistics in the (externally-uploaded) preregistration.

What this skill does NOT do

• Post-hoc / observed power. It refuses to compute "the power we had to detect our estimate" from a realised result — that is a deterministic function of the p-value and tells you nothing. Power is ex-ante only.
• Pick your effect size for you. The MDE is your design commitment; the skill computes consequences of an assumed effect (from theory, a pilot, or a meta-analysis), it does not invent a plausible one.
• Submit to a registry. Like /preregister, it writes a document; the user uploads it.
• Replace `/simulation-study`. Phase 2 borrows the harness for a single power question; a full bias/RMSE/coverage study is /simulation-study's job.