為什麼必須報告效應量？
樣本量夠大時，連 0.1% 的差異也會「顯著」。效應量 (effect size) 才告訴你差異有多大、是否值得在生物學上關心。Cohen (1988) 在 Statistical Power Analysis for the Behavioral Sciences 提出 d=0.2/0.5/0.8 為 small/medium/large，但 2024 年 Giner-Sorolla 等 (Personality Soc Psych Rev) 建議改用 SESOI (Smallest Effect Size of Interest)——由領域決定何謂「值得」的效應。
Power analysis 由 Faul et al. 2007 (Behav Res Methods) 的 G*Power 3 普及；R 的 pwr 與 Python 的 statsmodels/pingouin 提供等效功能。Post hoc power（用觀察效應量回算功效）被 Hoenig & Heisey 2001 嚴正反對——它與 p 值一一對應，沒有提供新資訊。
With large n, even a 0.1% difference becomes "significant." Effect size tells you how big the difference is and whether it matters biologically. Cohen (1988) proposed d=0.2/0.5/0.8 as small/medium/large; Giner-Sorolla et al. 2024 (Personality Soc Psych Rev) advocate the SESOI (Smallest Effect Size of Interest) — let the field decide what counts as "meaningful."
Power analysis was popularized by G*Power 3 (Faul et al. 2007, Behav Res Methods); R's pwr and Python's statsmodels/pingouin are equivalent. Post hoc power (computed from observed effect size) is rejected by Hoenig & Heisey 2001 — it's one-to-one with the p-value and adds nothing.
一、常用效應量速查
| 效應量 | 適用 | 公式 | 慣例 |
|---|---|---|---|
| Cohen's d | 兩組均值差 | (M₁−M₂)/SD_pooled | 0.2 / 0.5 / 0.8 |
| Hedges' g | 小樣本 | d × (1 − 3/(4df−1)) | 同 d |
| η² / partial η² | ANOVA | SS_effect / SS_total | 0.01 / 0.06 / 0.14 |
| Pearson r | 連續相關 | cov(X,Y)/(SD_X·SD_Y) | 0.1 / 0.3 / 0.5 |
| OR / RR | 類別結果 | odds₁/odds₀ ; risk₁/risk₀ | 領域決定 |
| Cliff's δ | 無母數 | P(X>Y) − P(Y>X) | 0.147 / 0.33 / 0.474 |
a priori 樣本量計算器（兩組 t-test）
給定預期效應量 d 與目標功效 (1−β)，計算每組需要的樣本量（兩側 α=0.05, equal n）。
Given an expected effect size d and target power (1−β), compute required per-group sample size (two-sided α=0.05, equal n).
不同 n 對應的 power
二、效應量與 Power 計算
library(pwr); library(effectsize) # 1. 效應量計算（兩組） cohens_d(value ~ group, data=df) # Cohen's d hedges_g(value ~ group, data=df) # 小樣本校正 cliffs_delta(value ~ group, data=df) # 無母數 # 2. a priori 樣本量計算（兩組 t-test） pwr.t.test(d=0.5, power=0.8, sig.level=0.05, type="two.sample") # 3. ANOVA 樣本量 pwr.anova.test(k=4, f=0.25, power=0.8, sig.level=0.05) # 4. Sensitivity（最小可偵測效應；給定 n、α、power） pwr.t.test(n=20, power=0.8, sig.level=0.05, type="two.sample")$d # 5. Mixed-effect / repeated measure 用 simr 模擬 library(simr) powerSim(fit_lmer, nsim=100)
import pingouin as pg from statsmodels.stats.power import TTestIndPower, FTestAnovaPower # 1. 效應量 print(pg.compute_effsize(df.A, df.B, eftype="cohen")) # Cohen's d print(pg.compute_effsize(df.A, df.B, eftype="hedges")) # Hedges' g print(pg.compute_effsize(df.A, df.B, eftype="CLES")) # common language effect # 2. a priori 樣本量（兩組 t-test） analysis = TTestIndPower() n = analysis.solve_power(effect_size=0.5, power=0.8, alpha=0.05) print(f"每組需要 n ≈ {n:.0f}") # 3. ANOVA 樣本量 anova_pow = FTestAnovaPower() n_per = anova_pow.solve_power(effect_size=0.25, k_groups=4, power=0.8, alpha=0.05) # 4. Sensitivity d_min = analysis.solve_power(nobs1=20, power=0.8, alpha=0.05) print(f"n=20 時可偵測最小 d = {d_min:.2f}")
三、最常見錯誤
❌ Post hoc power 自慰
用觀察 effect size 算 power 與 p 一一對應，毫無新資訊。改做 sensitivity analysis 或 equivalence test。
Power computed from observed ES is 1-to-1 with p — uninformative. Do sensitivity analysis or equivalence testing instead.
❌ 效應量不附 CI
d = 0.5 (95% CI: 0.1 – 0.9) 與 d = 0.5 (95% CI: 0.48 – 0.52) 訊息天差地遠。永遠報告 CI。
d = 0.5 (95% CI 0.1–0.9) vs d = 0.5 (95% CI 0.48–0.52) carry very different information. Always report CIs.
❌ 盲套 Cohen 0.2/0.5/0.8
不同領域典型效應量差異極大。如老年學 Brydges 2019 重估 small/medium/large 應為 0.16/0.38/0.76。
Typical effects vary by field. Brydges 2019 re-estimated gerontology cutoffs as 0.16/0.38/0.76.
❌ 80% power 視為定律
80% 是慣例非定律。重大臨床試驗常用 ≥ 90%；探索性 pilot 可接受 60-70%。
80% is convention, not law. Pivotal trials often use ≥ 90%; exploratory pilots can accept 60–70%.
🎯 章末小測驗
1. n=10 t-test p=0.4 應做？
2. 小樣本兩組比較最佳 ES？
3. 效應量報告必附？