F 검정의 정규성 민감도와 로버스트 대안¶

import inspect
import numpy as np
from scipy.stats import levene, fligner, shapiro, f

print("levene 의 center 기본값 =", inspect.signature(levene).parameters["center"].default)

x1 = np.array([12, 15, 14, 10, 13, 14, 12, 11], dtype=float)
x2 = np.array([22, 25, 20, 18, 24, 23, 19, 21], dtype=float)

# 1단계: F 검정은 정규성에 매우 민감하므로 먼저 그것부터 확인한다.
# 다만 n=8 로 작아 검정력이 낮다 — 기각되지 않았다고 정규라는 뜻은 아니다.
W1, p1 = shapiro(x1)
W2, p2 = shapiro(x2)
print(f"Shapiro-Wilk x1: W={W1:.4f}, p={p1:.4f}")
print(f"Shapiro-Wilk x2: W={W2:.4f}, p={p2:.4f}")

# 2단계: F 검정과 로버스트 대안 셋을 나란히 둔다. 평균 중심 Levene,
# 중앙값 중심 Brown-Forsythe, 순위를 쓰는 Fligner-Killeen 순으로
# 더 로버스트해진다.
F = x1.var(ddof=1) / x2.var(ddof=1)
pF_test = 2 * min(f.cdf(F, 7, 7), f.sf(F, 7, 7))
print(f"F 검정:                           F={F:.4f}, p={pF_test:.6f}")

Wm, pm = levene(x1, x2, center='mean')
print(f"Levene (mean-centered):           W={Wm:.4f}, p={pm:.6f}")

Wmed, pmed = levene(x1, x2, center='median')
print(f"Brown-Forsythe (median-centered): W={Wmed:.4f}, p={pmed:.6f}")

Wdef, pdef = levene(x1, x2)
print(f"levene(x1, x2)  — 인자 없이:      W={Wdef:.4f}, p={pdef:.6f}")

X2, pFK = fligner(x1, x2)
print(f"Fligner-Killeen:                  X2={X2:.4f}, p={pFK:.6f}")

# n=8 에서 샤피로-윌크가 비정규를 잡아내는 비율
rng = np.random.default_rng(5)
B, n, alpha = 20_000, 8, 0.05
print(f"\n샤피로-윌크의 검정력 (n={n}, 반복 {B:,}회, alpha={alpha})")
for name, draw in [("정규", lambda: rng.normal(size=n)),
                   ("t_5", lambda: rng.standard_t(5, size=n)),
                   ("지수", lambda: rng.exponential(size=n))]:
    rej = sum(shapiro(draw()).pvalue < alpha for _ in range(B))
    print(f"  {name} 모집단: {rej / B:.3f}")

levene 의 center 기본값 = median
Shapiro-Wilk x1: W=0.9657, p=0.8619
Shapiro-Wilk x2: W=0.9749, p=0.9332
F 검정:                           F=0.4732, p=0.344772
Levene (mean-centered):           W=1.4831, p=0.243425
Brown-Forsythe (median-centered): W=1.4706, p=0.245320
levene(x1, x2)  — 인자 없이:      W=1.4706, p=0.245320
Fligner-Killeen:                  X2=1.6063, p=0.205015

샤피로-윌크의 검정력 (n=8, 반복 20,000회, alpha=0.05)
  정규 모집단: 0.052
  t_5 모집단: 0.094
  지수 모집단: 0.331

import numpy as np
from scipy.stats import shapiro

x1 = np.array([12, 15, 14, 10, 13, 14, 12, 11], dtype=float)
x2 = np.array([22, 25, 20, 18, 24, 23, 19, 21], dtype=float)

W1, p1 = shapiro(x1)
W2, p2 = shapiro(x2)
print(f"x1: W={W1:.4f}, p={p1:.4f}")
print(f"x2: W={W2:.4f}, p={p2:.4f}")

x1: W=0.9657, p=0.8619
x2: W=0.9749, p=0.9332

import numpy as np
from scipy.stats import levene, fligner, f as fdist

rng = np.random.default_rng(42)
x1 = rng.chisquare(df=3, size=50)
x2 = rng.chisquare(df=5, size=50)

print(f"sample variances: {np.var(x1, ddof=1):.3f}, "
      f"{np.var(x2, ddof=1):.3f} (true: 6, 10)")

# F 검정 — 정규성에 민감하다
F = np.var(x1, ddof=1) / np.var(x2, ddof=1)
p_f = 2 * min(fdist(49, 49).cdf(F), fdist(49, 49).sf(F))
print(f"F-test:          F={F:.3f}, p={p_f:.4g}")

# Levene / Brown-Forsythe / Fligner-Killeen — 로버스트한 대안들
W, p_l = levene(x1, x2, center='mean')
print(f"Levene (mean):   W={W:.3f}, p={p_l:.4g}")
W, p_bf = levene(x1, x2, center='median')
print(f"Brown-Forsythe:  W={W:.3f}, p={p_bf:.4g}")
X2, p_fk = fligner(x1, x2)
print(f"Fligner-Killeen: X2={X2:.3f}, p={p_fk:.4g}")

sample variances: 2.207, 10.648 (true: 6, 10)
F-test:          F=0.207, p=1.681e-07
Levene (mean):   W=17.140, p=7.358e-05
Brown-Forsythe:  W=12.596, p=0.0005955
Fligner-Killeen: X2=11.175, p=0.0008291

import numpy as np
from scipy.stats import shapiro, levene, fligner

def check_equal_variances(*groups, alpha=0.05):
    """
    Workflow for checking equality of variances.
    Returns a dict with test results and recommendation.
    """
    # 1단계: 집단마다 정규성 확인
    normality_ok = True
    shapiro_results = []
    for g in groups:
        W, p = shapiro(g)
        shapiro_results.append((round(W, 4), round(p, 4)))
        if p < alpha:
            normality_ok = False

    # 2단계: 그에 맞는 검정 고르기
    if normality_ok:
        test_name = "Levene (mean-centered)"
        stat, pval = levene(*groups, center='mean')
    else:
        test_name = "Brown-Forsythe (median-centered)"
        stat, pval = levene(*groups, center='median')

    # 3단계: 확인 삼아 Fligner-Killeen 도 돌려 본다
    fk_stat, fk_p = fligner(*groups)

    return {
        "normality_assumed": normality_ok,
        "shapiro_results": shapiro_results,
        "primary_test": test_name,
        "statistic": round(stat, 4),
        "p_value": pval,
        "fligner_killeen_p": fk_p,
        "equal_variances": pval >= alpha,
    }

# 사용 예
rng = np.random.default_rng(0)
g1 = rng.normal(0, 1, 30)
g2 = rng.normal(0, 1.5, 30)
g3 = rng.normal(0, 2, 30)
result = check_equal_variances(g1, g2, g3)
for k, v in result.items():
    print(f"{k}: {v}")

normality_assumed: True
shapiro_results: [(0.9752, 0.6879), (0.9578, 0.2725), (0.9695, 0.5253)]
primary_test: Levene (mean-centered)
statistic: 14.4774
p_value: 3.736985145892101e-06
fligner_killeen_p: 1.5741483418264177e-05
equal_variances: False