Statistical Power & Sample Size

Overview

Power analysis answers one of the most consequential questions in study planning: how large a sample do you need to reliably detect an effect of a given size, and what could you detect with the sample you can afford? An underpowered study wastes resources and produces inconclusive or irreproducible results; an overpowered one wastes participants, money, and (in clinical work) exposes more people to risk than necessary. Getting this right before data collection is the single highest-leverage statistical decision in a project.

Four quantities are locked together for any given test: sample size (n), effect size, significance level (α), and power (1 − β). Fix any three and the fourth is determined. Every calculation in this skill is some rearrangement of that relationship.

This skill covers the two ways to do power analysis:

• Closed-form formulas (fast, exact for standard tests) — see references/closed_form_recipes.md.
• Simulation / Monte Carlo (works for any design or model you can simulate and analyze) — 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
• Powering anything without a textbook formula (mixed models, logistic/Poisson regression, cluster-randomized trials, survival analysis, mediation, interactions) via simulation
• Accounting for multiple comparisons, attrition/dropout, or clustering in the sample-size estimate

Installation

Use uv. Pin versions in production; unpinned is fine for exploration.

uv pip install "statsmodels>=0.14.6" "scipy>=1.11" "pingouin>=0.6" "numpy>=1.26" matplotlib pandas
# For simulation-based power of advanced models (optional, add as needed):
uv pip install lifelines            # survival
# mixed models and GLMs come with statsmodels

Compatibility note: use statsmodels>=0.14.6 with scipy>=1.11 to avoid _lazywhere import errors on SciPy 1.16+. Pingouin 0.5+ renamed power-function arguments to match the names used below.

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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, but shrink it — published and pilot effects are inflated by publication bias and the winner's curse. Powering on a raw pilot estimate routinely underpowers the real study.
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: it is a deterministic function of the p-value and tells you nothing new. 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.

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Quick recipes (closed-form)

The bundled scripts/power.py wraps statsmodels into one consistent interface so you don't have to remember which solver belongs to which test. Run from the skill directory or add scripts/ 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)
# -> required n per group

# 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")

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.

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When there is no formula: simulate

Closed-form power exists only for a handful of simple tests. For logistic/Poisson regression, mixed-effects / repeated-measures models, cluster-randomized trials, survival analysis, mediation, multi-way interactions, or any non-standard analysis, the right tool is simulation. 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

def gen_and_test(n, rng):
    # build a dataset of size n under the assumed effect, run the planned test,
    # return True if the result is significant
    ...

est = simulate_power(gen_and_test, n=200, n_sims=2000, alpha=0.05)
print(f"Power at n=200: {est.power:.3f} (95% CI {est.ci_low:.3f}-{est.ci_high:.3f})")

Report the Monte Carlo confidence interval on the estimate (the harness returns it) so the reader knows whether 0.81 vs. 0.79 is signal or simulation noise. 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.

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Adjustments people forget

These routinely make the difference between an adequately powered study and an underpowered one. Apply them explicitly and state that you did.

• Multiple comparisons. If the analysis tests m hypotheses with a Bonferroni-style correction, power each test at the corrected α (e.g. α/m), which raises n. Better: power on the family-wise or FDR-controlled procedure directly via simulation. Ignoring this silently underpowers every secondary endpoint.
• Attrition / dropout / unusable samples. Power gives the n you need *analy