---
id: wiki-2026-0508-statistical-power
title: Statistical Power
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Power-Analysis, 1-beta]
duplicate_of: none
source_trust_level: A
confidence_score: 0.92
verification_status: applied
tags: [statistics, power, hypothesis-testing, ab-testing, sample-size]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: statsmodels-scipy
---

# Statistical Power

## 매 한 줄
> **"매 effect 가 있을 때 매 detect 의 확률 = 1 − β"**. Power 의 true positive rate — sample size, effect size, α, variance 의 함수. 1928 Neyman-Pearson 에서 등장 — 2026 A/B test, clinical trial, 모든 hypothesis test 의 design 의 핵심.

## 매 핵심

### 매 4 quantities (lock 3, solve 1)
- **n** — sample size.
- **α** — Type I error rate (false positive), 보통 0.05.
- **β** — Type II error rate (false negative); **power = 1 − β**, 보통 0.8.
- **effect size** — Cohen's d (means), Cohen's h (props), r (correlation), f² (regression).

### 매 effect size 의 convention (Cohen)
- d = 0.2 (small), 0.5 (medium), 0.8 (large) — for two-sample t-test.
- 매 domain-specific minimum detectable effect (MDE) 의 정의 의 우선 — Cohen 의 default 의 X.

### 매 sample size 공식 (two-sample t-test)
- n_per_group ≈ 2·(z_{1-α/2} + z_{1-β})² · σ² / Δ²
- α=0.05, power=0.8, Δ=0.5σ (medium d) → n ≈ 64/group.

### 매 응용
1. A/B test design — pre-experiment sample-size calc.
2. Clinical trial — FDA mandate.
3. Replication crisis — underpowered studies → false discovery.
4. Sequential testing — α-spending for early stopping.

## 💻 패턴

### 1. statsmodels — t-test power
```python
from statsmodels.stats.power import TTestIndPower
analysis = TTestIndPower()
n = analysis.solve_power(effect_size=0.5, alpha=0.05, power=0.8, ratio=1.0)
# n ≈ 63.77 per group
```

### 2. Proportion (A/B test) sample size
```python
from statsmodels.stats.proportion import proportion_effectsize
from statsmodels.stats.power import NormalIndPower
es = proportion_effectsize(0.10, 0.12)  # baseline 10%, MDE 12%
n = NormalIndPower().solve_power(effect_size=es, alpha=0.05, power=0.8)
# 매 conversion uplift 2pp → ~3800 / arm
```

### 3. Power curve (visualize tradeoff)
```python
import numpy as np
import matplotlib.pyplot as plt
ns = np.arange(20, 500, 10)
powers = [analysis.solve_power(effect_size=0.3, nobs1=n, alpha=0.05) for n in ns]
plt.plot(ns, powers); plt.axhline(0.8, color="r")
```

### 4. Simulation-based (custom test, complex design)
```python
import numpy as np
def simulate_power(n, effect, sigma, alpha=0.05, B=10000):
    rejects = 0
    for _ in range(B):
        a = np.random.normal(0, sigma, n)
        b = np.random.normal(effect, sigma, n)
        t = (b.mean() - a.mean()) / np.sqrt(2*sigma**2/n)
        if abs(t) > 1.96:
            rejects += 1
    return rejects / B
```

### 5. MDE given fixed n (post-hoc planning)
```python
mde = analysis.solve_power(nobs1=1000, alpha=0.05, power=0.8)
# 매 n=1000 에서 detect 가능한 minimum d
```

### 6. Bonferroni-corrected power (multiple tests)
```python
k = 10  # 매 10 hypotheses
adjusted_alpha = 0.05 / k
n_corrected = analysis.solve_power(effect_size=0.5, alpha=adjusted_alpha, power=0.8)
```

### 7. Sequential / group-sequential (early stopping)
```python
# 매 O'Brien-Fleming spending function 의 사용
# 매 이미 누적 α 를 나눠서 spend → 매 final n 의 약간 증가
from statsmodels.stats.proportion import proportions_ztest  # for fixed test
# Group-sequential: see `gsDesign` in R or `confseq` in Python (2026)
```

### 8. Cluster-randomized (intra-class correlation)
```python
# 매 design effect = 1 + (m-1)·ρ
# 매 effective sample = n / DE
icc = 0.05; m = 30  # cluster size
DE = 1 + (m-1)*icc  # ≈ 2.45
n_effective = 1000 / DE
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Mean comparison | TTestIndPower (Cohen's d) |
| Proportion (CTR, conversion) | NormalIndPower |
| Complex design / non-standard test | Simulation |
| Multiple testing | Bonferroni / FDR adjusted α |
| Early-stopping | Group-sequential / α-spending |
| Cluster trial | Inflate n by design effect |

**기본값**: α=0.05, power=0.8, MDE 의 domain-driven 정의 → statsmodels solve_power.

## 🔗 Graph
- 부모: [[Statistics]] · [[Probability Theory]]
- 변형: [[Sampling-Techniques]]
- 응용: [[Regression-Analysis-Foundations]] · [[Decision Theory]]
- Adjacent: [[Standard-Deviation-and-Variance]] · [[Multivariate-Analysis]]

## 🤖 LLM 활용
**언제**: explaining tradeoffs (n vs MDE vs power), interpreting power-analysis result, recommending design.
**언제 X**: 매 numerical solve — statsmodels / G*Power 의 사용.

## ❌ 안티패턴
- **Post-hoc power on observed effect**: 매 always ≈ 50% near significance — 매 meaningless. 매 MDE 의 reporting 의 사용.
- **Underpowered + p-hack**: small n + multiple comparisons → false-positive 의 폭증.
- **Ignoring variance assumption**: σ 의 estimate 의 잘못 → n calc 의 wrong.
- **Power=0.5 acceptable**: 매 0.8 minimum (medical 0.9) — 매 50% 의 coin flip.
- **One-sided test for convenience**: 매 effect direction 의 진짜 사전 hypothesis 면 OK, 매 fishing 면 X.

## 🧪 검증 / 중복
- Verified (Cohen 1988 *Statistical Power Analysis*, statsmodels docs, ICH E9 guideline).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — power formulas, statsmodels patterns, A/B test design. |