Statistical Power

Statistical Power & Sample Size

Overview

Power analysis plans the probability of rejecting a specified null under an assumed alternative. It answers how many independent experimental units are needed to detect a scientifically important effect, or what effects a feasible sample could detect. Choose the inferential goal first: precision, equivalence, noninferiority, or sequential monitoring need their own calculations; the bundled superiority-test helpers do not cover them.

Four quantities are locked together for any given test: sample size (n), effect size, significance level (α), and power (1 − β). For a fixed design, analysis, and nuisance parameters, fixing three permits solving for the fourth when a solution exists. Every calculation in this skill is some rearrangement of that relationship.

This skill covers the two ways to do power analysis:

• Analytical power calculations (exact under some standard-test assumptions; asymptotic approximations for others) — see references/closed_form_recipes.md.
• Simulation / Monte Carlo (requires a credible data-generating process and a calibrated planned analysis) — see references/simulation_based_power.md.

For choosing and converting effect sizes — usually the hardest part — see references/effect_sizes.md.

When to Use This Skill

• Determining required sample size before collecting data (a priori power analysis)
• Finding the minimum detectable effect (MDE) for a fixed, already-determined sample size
• Producing power curves (power vs. n, or power vs. effect size) for a grant or protocol
• Justifying a sample size for an IRB submission, grant, or pre-registration
• Powering designs with unequal group sizes or non-1:1 allocation
• Planning complex mixed models, GLMs, clustering, survival, mediation, and interactions through simulation when simpler approximations do not fit
• Accounting for multiple comparisons, attrition/dropout, or clustering in the sample-size estimate

Installation

The local examples were checked on Python 3.13 with statsmodels 0.15.0, SciPy 1.18.1, NumPy 2.5.3, pandas 2.3.3, and Matplotlib 3.11.2. Use an environment separate from the repository's development environment:

uv venv --python 3.13 .venv-power
uv pip install --python .venv-power/bin/python "statsmodels==0.15.0" "scipy==1.18.1" "numpy==2.5.3" "pandas==2.3.3" "matplotlib==3.11.2"
# Optional comparison / survival methods (also checked for current API use):
uv pip install --python .venv-power/bin/python "pingouin==0.7.0" "lifelines==0.30.3"

On Windows use .venv-power/Scripts/python.exe. Lifelines 0.30.3 requires pandas<3; the tested pin above accommodates it. Mixed models and GLMs are included in statsmodels. Record versions and seeds with the output; numerical smoke tests do not establish a study's effect assumptions or Type I error control.

The one decision that drives everything: the effect size

Power calculations are only as trustworthy as the effect size you feed them. Do not invent a number. Use, in rough order of preference:

1. A minimally important effect — the smallest effect that would actually change a decision or matter scientifically/clinically (the "smallest effect size of interest", SESOI). This is the most defensible basis: you power to detect what matters, not what you hope to see.
2. A pilot or prior-study estimate, with uncertainty and selection bias considered. Small pilots are imprecise; publication or significance-based selection can inflate effects. Use a justified uncertainty model or sensitivity range instead of an arbitrary shrinkage factor.
3. A convention (Cohen's small/medium/large) only as a last resort, and say so explicitly.

Whatever you pick, run a sensitivity analysis: report how required n changes across a plausible range of effect sizes, not a single point. A power analysis presented as one number hides its biggest source of uncertainty. See references/effect_sizes.md for benchmarks and conversions between d, f, r, η², odds ratios, and Cohen's h/w.

Avoid post-hoc ("observed") power. Computing power from the effect size you just estimated is circular: for standard tests it largely restates the test statistic/p-value and adds no independent evidence of adequacy. If a study is already done and you want to know what it could have detected, report a sensitivity analysis (MDE at the achieved n) or, better, the confidence interval around the observed effect. This is a common reviewer complaint — do not produce observed power even if asked without flagging the issue.

Quick recipes (closed-form)

The bundled scripts/power.py wraps statsmodels and SciPy into one consistent interface so you don't have to remember which solver belongs to which test. Run from skills/statistical-power/scripts/ or add that directory to sys.path.

from power import sample_size, power, mde, power_curve

# 1. How many per group to detect Cohen's d = 0.5, two-sided, 80% power?
sample_size(test="t_ind", effect_size=0.5, power=0.80, alpha=0.05)
# -> 64 in sample 1; equal allocation gives 64 in sample 2

# 2. Two groups, 3:1 allocation (e.g. more controls than cases)
sample_size(test="t_ind", effect_size=0.5, power=0.80, ratio=3.0)

# 3. Fixed n=30/group — what's the minimum detectable d at 80% power?
mde(test="t_ind", nobs1=30, power=0.80, alpha=0.05)

# 4. One-way ANOVA, 4 groups, detect Cohen's f = 0.25
sample_size(test="anova", effect_size=0.25, k_groups=4, power=0.80)

# 5. Two proportions: 0.40 vs 0.55 (auto-converts to Cohen's h)
sample_size(test="two_proportions", prop1=0.40, prop2=0.55, power=0.80)

# 6. Correlation: detect r = 0.30
sample_size(test="correlation", effect_size=0.30, power=0.80)

# 7. Power curve for the grant figure
power_curve(test="t_ind", effect_size=0.5, n_range=range(10, 120, 5),
            save="power_curve.png")

For two-sample tests the return is n1, with n2 = ceil(ratio * n1); ratio=n2/n1. Recheck power using the realized integer ratio. ANOVA rounds total n to a multiple of k_groups; paired n counts pairs. One-sided alternatives use signed effects and "larger"/"smaller". Proportion MDEs return signed Cohen's h and need a baseline to convert to feasible probabilities.

Supported test= values: t_ind (two independent means), t_paired/t_one (paired or one-sample mean), anova (one-way), two_proportions, one_proportion, correlation, chi2 (goodness-of-fit / contingency via effect size w), linear_regression (R² increment / f²). Full argument tables and the underlying statsmodels calls are in references/closed_form_recipes.md.

When analytical assumptions do not fit: simulate

Use an analytical method when its design and assumptions match the planned analysis. For logistic/Poisson regression, mixed-effects / repeated-measures models, cluster-randomized trials, survival analysis, mediation, or multi-way interactions, simulation is often useful when available approximations omit material design features. The logic is always the same three steps:

1. Simulate a dataset from your assumed truth (the effect you want to detect, plus realistic noise, baseline rates, cluster structure, etc.).
2. Analyze it with the exact test/model you plan to use on the real data.
3. Repeat many times (≥1,000; 5,000–10,000 for a stable estimate near 80%). Power is the fraction of replicates in which the test is significant.

scripts/simulate_power.py provides a reusable harness plus worked examples (two-group difference, logistic regression, cluster-randomized trial with an ICC, and a linear mixed model). The core is just:

from simulate_power import simulate_power, example_two_group_difference

# Runnable software check: n is per group, effect is a raw mean difference.
gen_and_test = example_two_group_difference(effect=0.5, sd=1.0, alpha=0.05)
est = simulate_power(gen_and_test, n=64, n_sims=2000, alpha=0.05, seed=0)
print(est)  # power, 95% Monte Carlo CI, failure and warning counts

The callback must return a boolean rejection decision using the planned alpha internally. The harness's alpha argument does not threshold returned p-values or pass alpha into the callback; returning a raw p-value now raises TypeError instead of counting a nonzero float as rejection. The harness rejects nonpositive sample/replicate counts and invalid search bounds; an unmet target at the sample-size cap raises an explicit error. The examples check convergence and finite p-values. Expected failures raise SimulationFitError; the harness counts them as non-rejections and reports n_failures/failure_reasons, retaining every replicate in the denominator. It also reports warning counts; unexpected errors propagate. First check Type I error under the null. A noisy bisection search yields a candidate n: verify nearby sizes with more replicates and a fresh seed.

Report the Monte Carlo confidence interval on the estimate (the harness returns it) to quantify simulation sampling error; it does not cover uncertainty in the assumed effect, model, or adaptively chosen n. See references/simulation_based_power.md for the full patterns, including how to search for the n that hits target power and how to model dropout and clustering.