p値関数

軸は

縦軸 p値

見方

p値は次のいずれでもない

× "効果の大きさ"

× "結果の重要性"

信頼区間との関係

p値関数から信頼区間を作ることができる

信頼区間はp値関数の要約の一つといえる

S値関数との関係

$ s(\theta_0) = -\log_2 p(\theta_0)

p値関数・信頼区間が依存するもの

観察されたデータ

モデルを適用するための背後仮定

観測が独立であること

二項モデルが妥当であること

標本サイズが十分に大きいこと

Wald型のp値関数（compatibility curve）

ただしこれは近似的な手法なので、標本サイズが小さい場合などには精度が悪くなることがある

https://gyazo.com/66835885a4dcabe151c3c848ef313d0c

code:wald.js

let n = 20;

let marginL = 70, marginR = 50, marginT = 50, marginB = 65;

function setup() {

createCanvas(720, 480);

}

function draw() {

background(255);

let plotW = width - marginL - marginR;

let plotH = height - marginT - marginB;

let pHat = 0.5;

let X = round(n * pHat);

// カーソルYからalpha

let alphaRaw = map(mouseY, marginT + plotH, marginT, 0.0, 1.0, true);

let alpha = max(0.01, round(alphaRaw / 0.01) * 0.01);

// --- 軸 ---

stroke(200);

strokeWeight(1);

line(marginL, marginT, marginL, marginT + plotH);

line(marginL, marginT + plotH, marginL + plotW, marginT + plotH);

// 目盛り

fill(100);

noStroke();

textFont('sans-serif');

textSize(10);

textAlign(CENTER, TOP);

for (let i = 0; i <= 10; i++) {

let x = marginL + (i / 10) * plotW;

stroke(230);

line(x, marginT, x, marginT + plotH);

noStroke();

text(nf(i / 10, 1, 1), x, marginT + plotH + 8);

}

textAlign(RIGHT, CENTER);

for (let i = 0; i <= 5; i++) {

let y = marginT + plotH - (i / 5) * plotH;

stroke(230);

line(marginL, y, marginL + plotW, y);

noStroke();

text(nf(i / 5, 1, 1), marginL - 8, y);

}

// --- 軸ラベル ---

fill(60);

noStroke();

textSize(13);

textAlign(CENTER, TOP);

text('θ₀', marginL + plotW / 2, marginT + plotH + 24);

push();

translate(18, marginT + plotH / 2);

rotate(-HALF_PI);

textAlign(CENTER, CENTER);

text('p-value', 0, 0);

pop();

// --- p値関数を計算 ---

let steps = 500;

let pValues = [];

for (let i = 0; i <= steps; i++) {

let p0 = i / steps;

pValues.push(waldPValue(p0, pHat, n));

}

// --- 信頼区間を解析的に計算 ---

let se = sqrt(pHat * (1 - pHat) / n);

let zCrit = qnorm(1 - alpha / 2);

let ciLow = pHat - zCrit * se;

let ciHigh = pHat + zCrit * se;

// --- alpha水平線(red, full width) ---

let yAlpha = marginT + plotH - alpha * plotH;

stroke(220, 70, 70);

strokeWeight(1);

line(marginL, yAlpha, marginL + plotW, yAlpha);

// --- p値関数の曲線 ---

noFill();

stroke(40, 100, 200);

strokeWeight(2);

beginShape();

for (let i = 0; i <= steps; i++) {

let sx = marginL + (i / steps) * plotW;

vertex(sx, sy);

}

endShape();

// --- CI as green line on alpha level + dots (drawn on top of curve) ---

{

let cl = max(0, ciLow);

let ch = min(1, ciHigh);

let x1 = marginL + cl * plotW;

let x2 = marginL + ch * plotW;

// green line segment on alpha level

stroke(40, 160, 80);

strokeWeight(3);

line(x1, yAlpha, x2, yAlpha);

// dots at intersection points

noStroke();

fill(40, 160, 80);

circle(x1, yAlpha, 8);

circle(x2, yAlpha, 8);

}

// --- 点推定値の縦線 ---

let xPhat = marginL + pHat * plotW;

stroke(0, 0, 0, 40);

strokeWeight(1);

line(xPhat, marginT, xPhat, marginT + plotH);

drawingContext.setLineDash([]);

// 頂上の点

noStroke();

fill(40, 100, 200);

circle(xPhat, marginT, 6);

// --- alpha表示 ---

fill(220, 70, 70);

noStroke();

textFont('sans-serif');

textSize(11);

textAlign(LEFT, CENTER);

// --- CI表示 ---

fill(40, 160, 80);

textSize(12);

textAlign(LEFT, TOP);

let pct = ((1 - alpha) * 100).toFixed(0);

// --- タイトル・情報 ---

noStroke();

fill(40);

textAlign(LEFT, TOP);

textSize(14);

textStyle(BOLD);

textStyle(NORMAL);

textSize(11);

fill(100);

// 点推定値ラベル

fill(0, 0, 0, 120);

textSize(10);

textAlign(CENTER, BOTTOM);

// 操作説明

fill(160);

textSize(10);

textAlign(CENTER, BOTTOM);

text('Cursor: Change α-level ／ ↑↓: Change sample size', width / 2, height - 4);

}

function keyPressed() {

if (keyCode === UP_ARROW) {

if (n < 50) n += 2;

else if (n < 200) n += 10;

else if (n < 1000) n += 50;

else n += 200;

n = min(n, 10000);

}

if (keyCode === DOWN_ARROW) {

if (n <= 50) n -= 2;

else if (n <= 200) n -= 10;

else if (n <= 1000) n -= 50;

else n -= 200;

n = max(n, 2);

if (n % 2 !== 0) n++;

}

return false;

}

// --- 正規分布CDF ---

function normalCDF(x) {

if (x < -8) return 0;

if (x > 8) return 1;

let a1 = 0.254829592, a2 = -0.284496736, a3 = 1.421413741;

let a4 = -1.453152027, a5 = 1.061405429, p = 0.3275911;

let sign = x < 0 ? -1 : 1;

let u = abs(x) / sqrt(2);

let t = 1.0 / (1.0 + p * u);

let y = 1.0 - (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * exp(-x * x / 2);

return 0.5 * (1.0 + sign * y);

}

// --- Wald p値 ---

function waldPValue(p0, pHat, n) {

let se = sqrt(pHat * (1 - pHat) / n);

if (se === 0) return 0;

let z = abs((pHat - p0) / se);

return 2 * (1 - normalCDF(z));

}

// --- 逆正規分布CDF (Acklam's approximation) ---

function qnorm(p) {

if (p <= 0) return -Infinity;

if (p >= 1) return Infinity;

if (p === 0.5) return 0;

let a1 = -3.969683028665376e+01, a2 = 2.209460984245205e+02;

let a3 = -2.759285104469687e+02, a4 = 1.383577518672690e+02;

let a5 = -3.066479806614716e+01, a6 = 2.506628277459239e+00;

let b1 = -5.447609879822406e+01, b2 = 1.615858368580409e+02;

let b3 = -1.556989798598866e+02, b4 = 6.680131188771972e+01;

let b5 = -1.328068155288572e+01;

let c1 = -7.784894002430293e-03, c2 = -3.223964580411365e-01;

let c3 = -2.400758277161838e+00, c4 = -2.549732539343734e+00;

let c5 = 4.374664141464968e+00, c6 = 2.938163982698783e+00;

let d1 = 7.784695709041462e-03, d2 = 3.224671290700398e-01;

let d3 = 2.445134137142996e+00, d4 = 3.754408661907416e+00;

let pLow = 0.02425, pHigh = 1 - pLow;

let q, r;

if (p < pLow) {

q = sqrt(-2 * log(p));

return (((((c1*q+c2)*q+c3)*q+c4)*q+c5)*q+c6) / ((((d1*q+d2)*q+d3)*q+d4)*q+1);

} else if (p <= pHigh) {

q = p - 0.5; r = q * q;

return (((((a1*r+a2)*r+a3)*r+a4)*r+a5)*r+a6)*q / (((((b1*r+b2)*r+b3)*r+b4)*r+b5)*r+1);

} else {

q = sqrt(-2 * log(1 - p));

return -(((((c1*q+c2)*q+c3)*q+c4)*q+c5)*q+c6) / ((((d1*q+d2)*q+d3)*q+d4)*q+1);

}

}