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510314e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | """Statistica appaiata: McNemar esatto, IC Newcombe (metodo 10) e Tango, Wilson, bootstrap a grappoli."""
import math
import random
Z90 = 1.6448536269514722
Z95 = 1.959963984540054
def binom_two_sided(b, c):
"""McNemar esatto a due code: binomiale(n=b+c, 1/2), p = min(1, 2*P(X<=min))."""
n = b + c
if n == 0:
return 1.0
k = min(b, c)
s = sum(math.comb(n, i) for i in range(k + 1)) / 2 ** n
return min(1.0, 2 * s)
def wilson(x, n, z=Z95):
if n == 0:
return (0.0, 1.0)
p = x / n
den = 1 + z * z / n
c = (p + z * z / (2 * n)) / den
h = z * math.sqrt(p * (1 - p) / n + z * z / (4 * n * n)) / den
return (max(0.0, c - h), min(1.0, c + h))
def paired_counts(x, y):
"""x, y: liste 0/1 appaiate. a = entrambe giuste, b = x giusta y sbagliata, c = x sbagliata y giusta, d."""
a = sum(1 for i, j in zip(x, y) if i and j)
b = sum(1 for i, j in zip(x, y) if i and not j)
c = sum(1 for i, j in zip(x, y) if not i and j)
d = sum(1 for i, j in zip(x, y) if not i and not j)
return a, b, c, d
def newcombe10(a, b, c, d, z=Z90):
"""Newcombe (1998) metodo 10, differenza p1 - p2 con p1 = (a+b)/n, p2 = (a+c)/n."""
n = a + b + c + d
p1, p2 = (a + b) / n, (a + c) / n
l1, u1 = wilson(a + b, n, z)
l2, u2 = wilson(a + c, n, z)
num = a * d - b * c
if num > 0:
num = max(num - n / 2, 0)
den = (a + b) * (c + d) * (a + c) * (b + d)
phi = num / math.sqrt(den) if den > 0 else 0.0
th = p1 - p2
dl = math.sqrt(max(0.0, (p1 - l1) ** 2 - 2 * phi * (p1 - l1) * (u2 - p2) + (u2 - p2) ** 2))
du = math.sqrt(max(0.0, (u1 - p1) ** 2 - 2 * phi * (u1 - p1) * (p2 - l2) + (p2 - l2) ** 2))
return th, th - dl, th + du
def _tango_z(b, c, n, delta):
"""Statistica score di Tango (1998) per Δ = (b - c)/n; p21 vincolata stimata per massima verosimiglianza."""
# log-verosimiglianza in q = p21 con p12 = q + Δ, resto 1 - 2q - Δ (a+d raggruppati)
lo = max(0.0, -delta) + 1e-12
hi = (1 - delta) / 2 - 1e-12
if hi <= lo:
return float("inf") if (b - c - n * delta) > 0 else float("-inf")
rest = n - b - c
def ll(q):
v = 0.0
if b:
v += b * math.log(q + delta)
if c:
v += c * math.log(q)
if rest:
v += rest * math.log(max(1e-300, 1 - 2 * q - delta))
return v
# massimo di una funzione concava: sezione aurea
g = (math.sqrt(5) - 1) / 2
x1, x2 = hi - g * (hi - lo), lo + g * (hi - lo)
f1, f2 = ll(x1), ll(x2)
for _ in range(200):
if f1 < f2:
lo, x1, f1 = x1, x2, f2
x2 = lo + g * (hi - lo)
f2 = ll(x2)
else:
hi, x2, f2 = x2, x1, f1
x1 = hi - g * (hi - lo)
f1 = ll(x1)
q = (lo + hi) / 2
var = n * (2 * q + delta - delta * delta)
if var <= 0:
var = 1e-12
return (b - c - n * delta) / math.sqrt(var)
def tango(a, b, c, d, z=Z90):
n = a + b + c + d
th = (b - c) / n
def root(lo, hi, target):
# Z(Δ) è decrescente in Δ
for _ in range(100):
m = (lo + hi) / 2
if _tango_z(b, c, n, m) > target:
lo = m
else:
hi = m
return (lo + hi) / 2
low = root(-0.999999, th, z) if th > -0.999999 else -1.0
up = root(th, 0.999999, -z) if th < 0.999999 else 1.0
return th, low, up
def ni_verdict(lower, margin=0.05):
return "dimostrata" if lower > -margin else "non dimostrata"
def cluster_bootstrap(pairs_by_cluster, B=10000, seed=20261017):
"""pairs_by_cluster: {cluster: [(x, y), ...]}. Restituisce la distribuzione della differenza p_x - p_y."""
rng = random.Random(seed)
keys = list(pairs_by_cluster)
out = []
for _ in range(B):
sx = sy = n = 0
for _k in range(len(keys)):
for x, y in pairs_by_cluster[keys[rng.randrange(len(keys))]]:
sx += x
sy += y
n += 1
out.append((sx - sy) / n)
out.sort()
return out
def pct(sorted_vals, q):
i = q * (len(sorted_vals) - 1)
lo, hi = math.floor(i), math.ceil(i)
return sorted_vals[lo] + (sorted_vals[hi] - sorted_vals[lo]) * (i - lo)
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