{"id":254,"date":"2022-09-04T20:24:51","date_gmt":"2022-09-04T11:24:51","guid":{"rendered":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/linear-regression-step-by-step\/"},"modified":"2026-01-01T09:41:51","modified_gmt":"2026-01-01T00:41:51","slug":"linear-regression-step-by-step","status":"publish","type":"post","link":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/linear-regression-step-by-step\/","title":{"rendered":"\u91cd\u56de\u5e30\u5206\u6790\u306e\u8a08\u7b97\u65b9\u6cd5"},"content":{"rendered":"\n

\u91cd\u56de\u5e30\u5206\u6790\u306f\u504f\u56de\u5e30\u4fc2\u6570\u3092\u6c42\u3081\u3066\u4e88\u6e2c\u5f0f\uff08\u56de\u5e30\u5f0f\uff09\u3092\u4f5c\u308a\u3001\u6a19\u6e96\u5316\u504f\u56de\u5e30\u4fc2\u6570\u306e\u8a08\u7b97\u3001\u56de\u5e30\u5f0f\u306e\u6709\u610f\u6027\u306e\u5206\u6563\u5206\u6790\u3001\u6c7a\u5b9a\u4fc2\u6570\u306e\u7b97\u51fa\u3001\u504f\u56de\u5e30\u4fc2\u6570\u306e\u691c\u5b9a\u3001\u306a\u3069\u884c\u3046\u304c\u3001\u5b9f\u969b\u3069\u3093\u306a\u8a08\u7b97\u3092\u3057\u3066\u3044\u308b\u306e\u304b\uff1f<\/p>\n\n\n\n

\u5b9f\u969b\u3069\u3093\u306a\u8a08\u7b97\u3092\u3057\u3066\u3044\u308b\u304b\u308f\u304b\u308b\u3068\u3001\u504f\u56de\u5e30\u4fc2\u6570\u3084\u6a19\u6e96\u5316\u504f\u56de\u5e30\u4fc2\u6570\u306e\u610f\u5473\u5408\u3044\u3001\u691c\u5b9a\u306e\u610f\u5473\u5408\u3044\u304c\u3088\u308a\u3088\u304f\u7406\u89e3\u3067\u304d\u308b\u3068\u601d\u3046\u3002<\/p>\n\n\n\n\n\n\n\n

\u91cd\u56de\u5e30\u5206\u6790\u306e\u7b2c\u4e00\u76ee\u6a19\u306f\u91cd\u56de\u5e30\u5f0f\u3092\u63a8\u5b9a\u3059\u308b\u3053\u3068\u3067\u3042\u308b<\/h2>\n\n\n\n

\u91cd\u56de\u5e30\u5206\u6790\u306e\u4e2d\u5fc3\u7684\u306a\u76ee\u6a19\u306f\u3001\u91cd\u56de\u5e30\u5f0f\u3092\u63a8\u5b9a\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u91cd\u56de\u5e30\u5f0f\u306f\u3001\u5358\u306b\u56de\u5e30\u5f0f\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u3082\u3042\u308b\u3002<\/p>\n\n\n\n

\u5358\u5909\u91cf\u3067\u3082\u591a\u5909\u91cf\u3067\u3082\u3042\u307e\u308a\u533a\u5225\u3057\u306a\u3044\u3068\u3044\u3046\u3053\u3068\u3060\u3002<\/p>\n\n\n\n

\u91cd\u304c\u4ed8\u304f\u5834\u5408\u306f\u3001\u591a\u5909\u91cf\u3092\u610f\u8b58\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

\u4ee5\u964d\u306f\u3001\u91cd\u56de\u5e30\u5f0f\u3092\u3001\u5358\u306b\u56de\u5e30\u5f0f\u3068\u8868\u73fe\u3059\u308b\u3002<\/p>\n\n\n\n

\u56de\u5e30\u5f0f\u306f\u3001\u5207\u7247\u3068\u547c\u3070\u308c\u308b\u5b9a\u6570\u3068\u3001\u5404\u8aac\u660e\u5909\u6570\u306b\u639b\u3051\u5408\u308f\u305b\u308b\u504f\u56de\u5e30\u4fc2\u6570\u3068\u547c\u3070\u308c\u308b\u4fc2\u6570\u304b\u3089\u69cb\u6210\u3055\u308c\u308b\u3002<\/p>\n\n\n\n

\u5207\u7247\u3092 $ \\beta_0 $\u3001\u8aac\u660e\u5909\u6570\u304c2\u3064\u3042\u308b\u3068\u3059\u308b\u3068\u3001\u504f\u56de\u5e30\u4fc2\u6570\u306f $ \\beta_1 $, $ \\beta_2 $\u3068\u8868\u3059\u3053\u3068\u304c\u591a\u3044\u3002<\/p>\n\n\n\n

\u56de\u5e30\u5f0f\u3067\u4e88\u6e2c\u3055\u308c\u308b\u76ee\u7684\u5909\u6570\u3092 $ \\hat{Y} $, \u8aac\u660e\u5909\u6570\u306f\u305d\u308c\u305e\u308c $ X_1 $, $ X_2 $ \u3068\u8868\u73fe\u3059\u308b\u3068\u3001\u56de\u5e30\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n\n\n\n

$$ \\hat{Y} = \\beta_0 + \\beta_1 X_1 + \\beta_2 X2 $$<\/p>\n\n\n\n\n\n\n\n

$ Y $ \u3068 $ X $ \u306f\u5c0f\u6587\u5b57\u3067\u66f8\u304b\u308c\u308b\u3053\u3068\u3082\u3042\u308b\u3002<\/p>\n\n\n\n

\u307e\u305f $ \\beta $ \u306f $ b $ \u3068\u66f8\u304b\u308c\u308b\u3053\u3068\u3082\u3042\u308b\u3002<\/p>\n\n\n\n

\u6a19\u6e96\u5316\u504f\u56de\u5e30\u4fc2\u6570\u3092 $ \\beta $\u3001\uff08\u975e\u6a19\u6e96\u5316\uff09\u504f\u56de\u5e30\u4fc2\u6570\u3092 $ b $ \u3068\u304b $ B $ \u3068\u8868\u73fe\u3059\u308b\u5834\u5408\u3084\u3001\u7d71\u8a08\u30bd\u30d5\u30c8\u30a6\u30a7\u30a2\u3082\u3042\u308b\u3002<\/p>\n\n\n\n

SPSS\u3067\u306f\u3001\u975e\u6a19\u6e96\u5316\u504f\u56de\u5e30\u4fc2\u6570\u3092 B\u3001\u6a19\u6e96\u5316\u504f\u56de\u5e30\u4fc2\u6570\u3092\u30d9\u30fc\u30bf\u3068\u8868\u793a\u3057\u5206\u3051\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

\u307e\u305f\u3001\u53b3\u5bc6\u306b\u306f\u3001\u30e2\u30c7\u30eb\u306e\u8a18\u8ff0\u3067\u3001\u6bcd\u96c6\u56e3\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u610f\u5473\u3057\u3066\u3044\u308b\u3068\u304d\u306f\u30ae\u30ea\u30b7\u30e3\u6587\u5b57\u3001\u63a8\u5b9a\u5024\u306f\u30a2\u30eb\u30d5\u30a1\u30d9\u30c3\u30c8\u306e\u3088\u3046\u306b\u304b\u304d\u5206\u3051\u305f\u308a\u3082\u3059\u308b\u3002<\/p>\n\n\n\n

\u3053\u306e\u8a18\u4e8b\u3067\u306f\u305d\u3053\u307e\u3067\u53b3\u5bc6\u306b\u306f\u304b\u304d\u5206\u3051\u305a\u306b\u9032\u3081\u308b\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n\n\n\n

\u91cd\u56de\u5e30\u5206\u6790\u306e\u8a08\u7b97\u65b9\u6cd5\u3000\u504f\u56de\u5e30\u4fc2\u6570<\/h2>\n\n\n\n

\u56de\u5e30\u5f0f\u306e\u3046\u3061\u3001$ X $ \u306f\u624b\u5143\u306b\u3042\u308b\u30c7\u30fc\u30bf\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u76ee\u7684\u5909\u6570 $ Y $ \u3082\u53d6\u5f97\u3055\u308c\u3066\u3044\u308b\u624b\u5143\u306b\u3042\u308b\u30c7\u30fc\u30bf\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u4e0a\u8a18\u3067\u767b\u5834\u3057\u305f\u4e88\u6e2c\u3055\u308c\u305f\u76ee\u7684\u5909\u6570\uff08\u4e88\u6e2c\u5024\uff09$ \\hat{Y} $ \u306f\u3001\u56de\u5e30\u5f0f\u3067\u63a8\u5b9a\u3055\u308c\u308b\u3001\u8a08\u7b97\u3067\u6c42\u307e\u308b\u5024\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u504f\u56de\u5e30\u4fc2\u6570 $ \\beta $ \u3082\u3001\u8a08\u7b97\u306b\u3088\u3063\u3066\u6c42\u307e\u308b\u6570\u5024\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u56de\u5e30\u5f0f\u306f\u3001\u5b9f\u6e2c\u306e\u76ee\u7684\u5909\u6570 $ Y $ \u306b\u306a\u308b\u3079\u304f\u8fd1\u3044\u4e88\u6e2c\u5024 $ \\hat{Y} $ \u3092\u7b97\u51fa\u3059\u308b\u3082\u306e\u304c\u826f\u3044\u3002<\/p>\n\n\n\n

\u3053\u306e\u300c\u306a\u308b\u3079\u304f\u8fd1\u3044\u300d\u3068\u3044\u3046\u306e\u304c\u3001\u5b9f\u6e2c\u5024\u3068\u4e88\u6e2c\u5024\u306e\u5dee\uff08\u3053\u308c\u3092\u6b8b\u5dee\u3068\u547c\u3076\uff09$ Y – \\hat{Y} $ \u304c\u306a\u308b\u3079\u304f\u5c0f\u3055\u304f\u3059\u308b\u3053\u3068\u3092\u76ee\u6a19\u306b\u3059\u308b\u3068\u3044\u3046\u610f\u5473\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

\u306a\u308b\u3079\u304f\u5c0f\u3055\u304f\u3059\u308b\u3068\u304d\u306b\u306f\u3001\u5408\u8a08\u3057\u3066\u5408\u8a08\u5024\u304c\u4e00\u756a\u5c0f\u3055\u3044\u306a\u3069\u3068\u3057\u305f\u3044\u308f\u3051\u3060\u304c\u3001\u6b8b\u5dee\u306f\u30d7\u30e9\u30b9\u3082\u30de\u30a4\u30ca\u30b9\u3082\u3042\u3063\u3066\u3001\u300c\u5c0f\u3055\u3044\u300d\u3068\u3044\u3046\u306e\u304c\u308f\u304b\u308a\u306b\u304f\u3044\u3002<\/p>\n\n\n\n

\u5168\u90e8\u304c\u30d7\u30e9\u30b9\uff08\u6b63\uff09\u3067\u3042\u308b\u3068\u5927\u304d\u3044\u5c0f\u3055\u3044\u304c\u308f\u304b\u308a\u3084\u3059\u3044\u3002<\/p>\n\n\n\n

\u305d\u3053\u3067\u3001\u5dee\u3092\uff12\u4e57\u3057\u3066\u3001\u5168\u90e8\u3092\u6b63\u306b\u3057\u3066\u3057\u307e\u3063\u3066\u304b\u3089\u3001\u5408\u8a08\u3059\u308b\u3002<\/p>\n\n\n\n

\u3053\u3046\u3059\u308b\u3068\u6b8b\u5dee\u3092\u6700\u5c0f\u306b\u3059\u308b\u6761\u4ef6\u3092\u898b\u51fa\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n\n\n\n

\u3053\u308c\u304c\u6700\u5c0f\uff12\u4e57\u6cd5\u3068\u547c\u3070\u308c\u3066\u3044\u308b\u65b9\u6cd5\u3060\u3002<\/p>\n\n\n\n

\uff1e\uff1e\u3082\u3046\u7d71\u8a08\u3067\u60a9\u3080\u306e\u306f\u7d42\u308f\u308a\u306b\u3057\u307e\u305b\u3093\u304b\uff1f\u00a0<\/a><\/strong><\/span><\/p>\r\n\"\"<\/a>\r\n

\u21911\u4e07\u4eba\u4ee5\u4e0a\u306e\u533b\u7642\u5f93\u4e8b\u8005\u304c\u8cfc\u8aad\u4e2d<\/span><\/strong><\/span><\/p><\/div>

\u91cd\u56de\u5e30\u5206\u6790\u306e\u8a08\u7b97\u65b9\u6cd5\u3000\u6700\u5c0f\uff12\u4e57\u6cd5<\/h2>\n\n\n\n

\u307e\u305a\u6b8b\u5dee\u3092\uff12\u4e57\u3057\u3066\u5408\u8a08\u3059\u308b\u3002<\/p>\n\n\n\n

$$ Q = \\sum (Y – \\hat{Y})^2 $$<\/p>\n\n\n\n\n\n\n\n

\u30a4\u30e1\u30fc\u30b8\u3092\u3064\u304b\u3080\u305f\u3081\u306e\u5f0f\u3068\u5272\u308a\u5207\u308a\u3001\u5358\u7d14\u5316\u306e\u305f\u3081\u3001\u6dfb\u3048\u5b57\u306f\u6975\u529b\u7701\u7565\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

\u4e88\u6e2c\u5024 $ \\hat{Y} $ \u306b\u306f\u4e0b\u8a18\u306e\u5f0f\u3092\u4ee3\u5165\u3059\u308b\u3002<\/p>\n\n\n\n

$$ \\hat{Y} = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2 $$<\/p>\n\n\n\n\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/span><\/p>\n\n\n\n

\u3053\u306e\u5f0f\u3092 $ \\beta_0 $, $ \\beta_1 $, $ \\beta_2 $ \u306b\u95a2\u3057\u3066\u3001\u504f\u5fae\u5206\u3057\u3066\uff1d\uff10\u3068\u304a\u304f\u3002<\/p>\n\n\n\n

\uff12\u6b21\u5f0f\u306e\uff11\u6b21\u5fae\u5206\u304c\uff10\u3068\u306a\u308b\u3068\u304d\u304c\u6700\u5927\u5024\u3068\u3044\u3046\u610f\u5473\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/span><\/p>\n\n\n\n

\u5909\u6570 $ Y $, $ X_1 $, $ X_2 $ \u306e\u5e73\u5747\u5024\u3092\u3001$ \\bar{Y} $, $ \\bar{X_1} $, $ \\bar{X_2} $ \u3068\u3057\u305f\u3068\u304d\u306e $ \\beta_0 $ \u3068\u3001$ X_1 $, $ X_2 $ \u9593\u306e\u5909\u52d5\u30fb\u5171\u5909\u52d5 $ S_{11} $, $ S_{12} $, $ S_{21} $, $ S_{22} $\u3001$ X_1 $, $ X_2 $ \u3068 $ Y $ \u306e\u5171\u5909\u52d5 $ S_{1y} $, $ S_{2y} $ \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u73fe\u3059\u308b\u3068\u3001\u6700\u7d42\u7684\u306b\u4e0a\u8a18\u306e\u504f\u5fae\u5206\u306e\u5f0f\u304c\u6574\u7406\u3067\u304d\u308b\u3002<\/p>\n\n\n\n

$$
\\begin{array} {rcl}
\\beta_0 &=& \\bar{Y} – \\beta_1 \\bar{X_1} – \\beta_2 \\bar{X_2} \\\\
S_{11} &=& \\sum (X_1 – \\bar{X_1})^2 \\\\
S_{12} &=& \\sum (X_1 – \\bar{X_1}) (X_2 – \\bar{X_2}) \\\\
S_{21} &=& \\sum (X_2 – \\bar{X_2}) (X_1 – \\bar{X_1}) \\\\
S_{22} &=& \\sum (X_2 – \\bar{X_2})^2 \\\\
S_{1y} &=& \\sum (X_1 – \\bar{X_1}) (Y – \\bar{Y}) \\\\
S_{2y} &=& \\sum (X_2 – \\bar{X_2}) (Y – \\bar{Y})
\\end{array}
$$<\/p>\n\n\n\n\n\n\n\n

\u305d\u308c\u305e\u308c\u3001\u5f0f\u5909\u5f62\u3057\u3066\u304a\u304f\u3002<\/p>\n\n\n\n

$ S_{11} $ \u306f\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u308b\u3002<\/p>\n\n\n\n

$$
\\begin{array}{rcl}
\\sum (X_1 – \\bar{X_1})^2
&=& \\sum (X_1^2 – 2 X_1 \\bar{X_1} + \\bar{X_1}^2) \\\\
&=& \\sum X_1^2 – 2 n \\bar{X_1}^2 + n \\bar{X_1}^2 \\\\
&=& \\sum X_1^2 – n \\bar{X_1}^2
\\end{array}
$$<\/p>\n\n\n\n\n\n\n\n

\u3042\u308b\u5909\u6570\u3001\u4f8b\u3048\u3070 $ X_1 $\u3001\u306e\u5168\u30c7\u30fc\u30bf\u6570 $\uff4e$ \u500b\u306e\u5408\u8a08\u3082\u3001\u305d\u306e\u5909\u6570\u306e\u5e73\u5747\u5024 $ \\bar{X_1} $ \u306e $\uff4e$ \u500d\u3082\u3069\u3061\u3089\u3082\u540c\u3058\u5408\u8a08\u5024\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

\u306a\u305c\u306a\u3089\u3001\u305d\u308c\u304c\u5e73\u5747\u5024\u306e\u7279\u5fb4\u3060\u304b\u3089\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u3053\u306e\u5e73\u5747\u5024\u306e\u7279\u5fb4\u3092\u5229\u7528\u3057\u305f\u5f0f\u5909\u5f62\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u540c\u69d8\u306b $ S_{22} $ \u306f\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n\n\n\n

$$ \\sum X_2^2 – n \\bar{X_2}^2 $$<\/p>\n\n\n\n\n\n\n\n

$ S_{12} $ \u306f\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u308b\u3002<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/span><\/p>\n\n\n\n

$ S_{21} $ \u306f\u3001\uff11\u3068\uff12\u3092\u9006\u306b\u3059\u308c\u3070\u3088\u3044\u306e\u3067\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

$$ \\sum X_2 X_1 – n \\bar{X_2} \\bar{X_1} $$<\/p>\n\n\n\n\n\n\n\n

$ S_{1y} $ \u306f\u3069\u3046\u306a\u308b\u304b\u3068\u3044\u3046\u3068\u3001<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/span><\/p>\n\n\n\n

\u3068\u3044\u3046\u3075\u3046\u306b\u5909\u5f62\u3067\u304d\u308b\u3002<\/p>\n\n\n\n

$ S_{2y} $ \u306f\u3001\u540c\u69d8\u306b<\/p>\n\n\n\n

$$ \\sum X_2 Y – n \\bar{X_2} \\bar{Y} $$<\/p>\n\n\n\n\n\n\n\n

\u3068\u66f8\u3051\u308b\u3002<\/p>\n\n\n\n

\u3053\u3053\u307e\u3067\u6e96\u5099\u3057\u3066\u304b\u3089\u3001\u4e0a\u8a18\u3067\u504f\u5fae\u5206\u3057\u305f\uff13\u3064\u306e\u5f0f\u3092\u6574\u7406\u3057\u3066\u3044\u304f\u3002<\/p>\n\n\n\n

$ \\beta_0 $ \u306b\u3064\u3044\u3066\u504f\u5fae\u5206\u3057\u305f\u5f0f\u306b $ \\beta_0 = \\bar{Y} – \\beta_1 \\bar{X_1} – \\beta_2 \\bar{X_2} $ \u3092\u6295\u5165\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/span><\/p>\n\n\n\n

\u306a\u306e\u3067\u3001$ \\beta_0 $ \u306e\u5f0f\u306f\u3001\u5f53\u7136\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

$ \\beta_1 $, $\\beta_2 $ \u3067\u5fae\u5206\u3057\u305f\u5f0f\u306f\u3069\u3046\u306a\u308b\u304b\uff1f<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/span><\/p>\n\n\n\n

\u4e0a\u8a18\u3067\u8a08\u7b97\u3057\u305f\u3001\u5909\u52d5\u30fb\u5171\u5909\u52d5\u3092\u4f7f\u3063\u3066\u6574\u7406\u3059\u308b\u3068\u3001<\/p>\n\n\n\n

$$ \\beta_1 S_{11} + \\beta_2 S_{12} = S_{1y} $$<\/p>\n\n\n\n\n\n\n\n

\u3068\u306a\u308b\u3002<\/p>\n\n\n\n

\u540c\u69d8\u306b\u3001<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/span><\/p>\n\n\n\n

\u3068\u306a\u3063\u305f\u3042\u3068\u3001\u4e0a\u8a18\u3067\u8a08\u7b97\u3057\u3066\u304a\u3044\u305f\u5909\u52d5\u30fb\u5171\u5909\u52d5\u3092\u4ee3\u5165\u3057\u3066\u6574\u7406\u3059\u308b\u3068\u3001<\/p>\n\n\n\n

$$ \\beta_1 S_{21} + \\beta_2 S_{22} = S_{2y} $$<\/p>\n\n\n\n\n\n\n\n

\u3068\u306a\u308b\u3002<\/p>\n\n\n\n

\u3053\u308c\u3089\u306e\uff12\u3064\u306e\u5f0f\u3092\u9023\u7acb\u65b9\u7a0b\u5f0f\u3068\u3057\u3066\u89e3\u304f\u3053\u3068\u306b\u3088\u308a\u3001\u504f\u56de\u5e30\u4fc2\u6570 $ \\beta_1 $, $ \\beta_2 $ \u304c\u6c42\u307e\u308b\u3002<\/p>\n\n\n\n

$$
\\begin{array}{rcl}
\\beta_1 S_{11} + \\beta_2 S_{12} &=& S_{1y} \\\\
\\beta_1 S_{21} + \\beta_2 S_{22} &=& S_{2y}
\\end{array}
$$<\/p>\n\n\n\n\n\n\n\n

\u3088\u308a\u4e00\u822c\u7684\u306a\u8868\u8a18\u3068\u3057\u3066\u306f\u3001\u884c\u5217\u3067\u8868\u73fe\u3059\u308b\u3068\u7c21\u5358\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

$$
\\begin{pmatrix}
S_{11} & S_{12} \\\\
S_{21} & S_{22}
\\end{pmatrix}
\\begin{pmatrix}
\\beta_1 \\\\
\\beta_2
\\end{pmatrix}
=
\\begin{pmatrix}
S_{1y} \\\\
S_{2y}
\\end{pmatrix}
$$<\/p>\n\n\n\n

\u884c\u5217\u3092\u6587\u5b57\u3067\u3042\u3089\u308f\u3059\u3088\u3046\u306b\u5909\u66f4\u3059\u308b\u3002<\/p>\n\n\n\n

$$ \\boldsymbol{S} = \\begin{pmatrix} S_{11} & S_{12} \\\\ S_{21} & S_{22} \\end{pmatrix} $$<\/p>\n\n\n\n

$$ \\boldsymbol \\beta =
\\begin{pmatrix} \\beta_1 \\\\
\\beta_2
\\end{pmatrix} $$<\/p>\n\n\n\n

$$ \\boldsymbol C =
\\begin{pmatrix} S_{1y} \\\\
S_{2y}
\\end{pmatrix} $$<\/p>\n\n\n\n

\u3068\u3044\u3046\u3075\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n\n\n\n

$$ \\boldsymbol S \\boldsymbol \\beta = \\boldsymbol C $$<\/p>\n\n\n\n\n\n\n\n

$ \\boldsymbol S $ \u306e\u9006\u884c\u5217\u3092 $ \\boldsymbol S^{-1} $ \u3068\u3059\u308b\u3068\u3001\u504f\u56de\u5e30\u4fc2\u6570 $ \\boldsymbol \\beta $ \u306f\u4ee5\u4e0b\u306e\u5f0f\u3067\u6c42\u3081\u3089\u308c\u308b\u3002<\/p>\n\n\n\n

$$ \\boldsymbol \\beta = \\boldsymbol S^{-1} \\boldsymbol C $$<\/p>\n\n\n\n\n\n\n\n

\u5b9a\u6570\u9805 $ \\beta_0 $ \u306f\u3001$ \\beta_0 = \\bar{Y} – \\beta_1 \\bar{X_1} – \\beta_2 \\bar{X_2} $ \u304b\u3089\u6c42\u3081\u3089\u308c\u308b\u3002<\/p>\n\n\n\n

\u3053\u3053\u307e\u3067\u3067\u3001\u56de\u5e30\u5f0f $ \\hat{Y} = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2 $ \u306e $ \\beta_0 $, $ \\beta_1 $, $ \\beta_2 $ \u306e 3\u3064\u304c\u6c42\u307e\u308a\u3001\u56de\u5e30\u5f0f\u304c\u5b8c\u6210\u3057\u305f\u3002<\/p>\n\n\n\n

\u91cd\u56de\u5e30\u5206\u6790\u306e\u8a08\u7b97\u65b9\u6cd5\u3000\u6a19\u6e96\u5316\u504f\u56de\u5e30\u4fc2\u6570<\/h2>\n\n\n\n

\u6b21\u306b\u51fa\u3066\u304f\u308b\u7591\u554f\u3068\u3057\u3066\u3001$ \\beta_1 $ \u3068$ \\beta_2 $ \u306f\u3069\u3061\u3089\u306e\u307b\u3046\u304c\u5927\u304d\u306a\u5f71\u97ff\u529b\u3092\u6301\u3064\u306e\u304b\u3067\u3042\u308d\u3046\u3002<\/p>\n\n\n\n

\u3060\u304c\u3001\u504f\u56de\u5e30\u4fc2\u6570\u306f\u3001\u305d\u308c\u305e\u308c\u306e\u9023\u7d9a\u30c7\u30fc\u30bf $ X_1 $, $ X_2 $ \u306e\u5358\u4f4d\uff08cm \u304b mm \u304b\u306a\u3069\uff09\u3084\u3070\u3089\u3064\u304d\u306b\u5de6\u53f3\u3055\u308c\u3066\u6c7a\u307e\u3063\u3066\u3044\u3066\u3001\u504f\u56de\u5e30\u4fc2\u6570\u81ea\u4f53\u306e\u6570\u5024\u3067\u6bd4\u3079\u308b\u3053\u3068\u306f\u3067\u304d\u306a\u3044\u3002<\/p>\n\n\n\n

\u305d\u3053\u3067\u3001\u5e73\u5747\uff10\u5206\u6563\uff11\u306b\u6a19\u6e96\u5316\u3057\u3066\u3001\u3069\u306e\u5909\u6570\u3082\u5e73\u5747\u3068\u3070\u3089\u3064\u304d\u3092\u540c\u3058\u306b\u3057\u3066\u63c3\u3048\u305f\u3089\u3001\u6bd4\u8f03\u3067\u304d\u308b\u3060\u308d\u3046\u3068\u8003\u3048\u308b\u308f\u3051\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u3053\u308c\u304c\u6a19\u6e96\u5316\u504f\u56de\u5e30\u4fc2\u6570\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u6a19\u6e96\u5316\u504f\u56de\u5e30\u4fc2\u6570\u306f\u3001\u6700\u521d\u306b\u8aac\u660e\u5909\u6570\u53ca\u3073\u76ee\u7684\u5909\u6570\u3092\u6a19\u6e96\u5316\u3057\u3066\u3001\u8a08\u7b97\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u308b\u3002<\/p>\n\n\n\n

\u4e00\u65b9\u3067\u3001\u504f\u56de\u5e30\u4fc2\u6570\u304b\u3089\u8a08\u7b97\u3067\u6a19\u6e96\u5316\u504f\u56de\u5e30\u4fc2\u6570\u3092\u6c42\u3081\u308b\u3053\u3068\u3082\u3067\u304d\u308b\u3002<\/p>\n\n\n\n

$$
\\begin{array} {rcl}
\\beta_1′ &=& \\beta_1 \\sqrt{\\frac{S_{11}}{S_{yy}}} \\\\
\\beta_2′ &=& \\beta_2 \\sqrt{\\frac{S_{22}}{S_{yy}}}
\\end{array}
$$<\/p>\n\n\n\n\n\n\n\n

\u3053\u3053\u3067 $ S_{yy} $ \u306f $ Y $ \u306e\u5909\u52d5\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u4ee5\u4e0b\u306e\u30b9\u30af\u30b7\u30e7\u306f\u3001\u504f\u56de\u5e30\u4fc2\u6570\u3092\u8a08\u7b97\u3057\u305f\u306e\u3061\u306b\u6a19\u6e96\u5316\u504f\u56de\u5e30\u4fc2\u6570\u3092\u6c42\u3081\u305f age, WBC.entry \u3068\u3001\u6a19\u6e96\u5316\u3057\u305f\uff08Z.age, Z.WBC.entry\uff09\u5f8c\u306b\u91cd\u56de\u5e30\u5206\u6790\u3092\u884c\u3063\u3066\u504f\u56de\u5e30\u4fc2\u6570\u3092\u6c42\u3081\u305f\u7d50\u679c\u304c\u4e00\u81f4\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/span><\/p>\n\n\n\n

\u91cd\u56de\u5e30\u5206\u6790\u306e\u8a08\u7b97\u65b9\u6cd5\u3000\u56de\u5e30\u30e2\u30c7\u30eb\u306e\u6709\u610f\u6027\u691c\u5b9a<\/h2>\n\n\n\n

\u504f\u56de\u5e30\u4fc2\u6570\u306e\u6709\u610f\u6027\u304c\u6c17\u306b\u306a\u308b\u3068\u3053\u308d\u3067\u3042\u308b\u304c\u3001\u305d\u306e\u524d\u306b\u3001\u56de\u5e30\u30e2\u30c7\u30eb\u81ea\u4f53\u306e\u7d71\u8a08\u5b66\u7684\u6709\u610f\u6027\u306e\u691c\u5b9a\u3092\u8aac\u660e\u3059\u308b\u3002<\/p>\n\n\n\n

\u6700\u7d42\u7684\u306b F \u5206\u5e03\u306b\u5f93\u3046 F \u5024\u3092\u8a08\u7b97\u3059\u308b\u70b9\u3001\u305d\u3053\u306b\u81f3\u308b\u904e\u7a0b\u3067\u3001\u5909\u52d5\u5e73\u65b9\u548c\u3001\u5e73\u5747\u5e73\u65b9\u548c\u3092\u8a08\u7b97\u3059\u308b\u70b9\u3067\u3001\u4e00\u5143\u914d\u7f6e\u5206\u6563\u5206\u6790\u3068\u540c\u69d8\u306e\u8a08\u7b97\u3092\u884c\u3046\u3002<\/p>\n\n\n\n

\u305d\u306e\u305f\u3081\u306b\u3001\u56de\u5e30\u306e\u5206\u6563\u5206\u6790\u3068\u547c\u3070\u308c\u308b\u3002<\/p>\n\n\n\n

\u3044\u307e\u307e\u3067\u306f\u3001\u3042\u3048\u3066\u56de\u5e30\u5f0f\u306b\u8aa4\u5dee $ \\varepsilon $\uff08\u30a4\u30d7\u30b7\u30ed\u30f3\uff09\u3092\u66f8\u304b\u306a\u304b\u3063\u305f\u304c\u3001\u691c\u5b9a\u306b\u306a\u308b\u3068\u8aa4\u5dee\u304c\u5fc5\u8981\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

\u3061\u306a\u307f\u306b\u3001\u8aa4\u5dee\u306e\u63a8\u5b9a\u5024 $\\hat{\\varepsilon} $ \u304c\u6b8b\u5dee\uff08\u76ee\u7684\u5909\u6570\u306e\u5b9f\u6e2c\u5024\u3068\u4e88\u6e2c\u5024\u306e\u5dee\uff09$ Y – \\hat{Y} $ \u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u3064\u307e\u308a\u91cd\u56de\u5e30\u30e2\u30c7\u30eb\uff08\u8aac\u660e\u5909\u6570\uff12\u3064\u7248\uff09\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n\n\n\n

$$ Y = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2 + \\varepsilon $$ <\/p>\n\n\n\n\n\n\n\n

\u4e88\u6e2c\u5024 $ \\hat{Y} $ \u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306e\u91cd\u56de\u5e30\u5f0f\u306f\u3001$ \\varepsilon $ \u306f\u3064\u304b\u306a\u3044\u3002<\/p>\n\n\n\n

$$ \\hat{Y} = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2 $$<\/p>\n\n\n\n\n\n\n\n

\u3064\u307e\u308a\u3001$ Y $ \u3068 $ \\hat{Y} $ \u306e\u5dee\u304c\u3001$ \\hat{\\varepsilon} $\uff08\u6b8b\u5dee\uff09\u3068\u3044\u3046\u3053\u3068\u3060\u3002<\/p>\n\n\n\n

$$ \\hat{\\varepsilon} = Y – \\hat{Y} $$<\/p>\n\n\n\n\n\n\n\n

\u5206\u6563\u5206\u6790\u3067\u306f\u3001\u307e\u305a\u3001\u3053\u306e\u6b8b\u5dee\u306e\u5e73\u65b9\u548c $ S_e $ \u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n

$$ S_e = \\sum (Y – \\hat{Y})^2 $$<\/p>\n\n\n\n\n\n\n\n

\u6b21\u306b\u3001\u5168\u4f53\u306e\u5909\u52d5\u8981\u56e0\u3068\u3057\u3066\u3001\u5e73\u65b9\u548c $ S_t $ \u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n

\u3053\u308c\u306f\u3001\u5b9f\u6e2c\u5024 $ Y $ \u3068\u5b9f\u6e2c\u5024\u306e\u5e73\u5747 $ \\bar{Y} $ \u306e\u5dee\u306e\u5e73\u65b9\u548c\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

$$ S_t = \\sum (Y – \\bar{Y})^2 $$<\/p>\n\n\n\n\n\n\n\n

\u3053\u306e\u5168\u4f53\u306e\u5909\u52d5\u8981\u56e0\u304b\u3089\u3001\u56de\u5e30\u3067\u8aac\u660e\u3067\u304d\u305f\u5909\u52d5\u8981\u56e0 $ S_r $ \u3092\u5f15\u304f\u3068\u3001\u6b8b\u308a\u304c\u6b8b\u5dee\u5909\u52d5\u8981\u56e0\u306a\u306e\u3067\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u3042\u308b\u3002<\/p>\n\n\n\n

$$ S_e = S_t – S_r $$<\/p>\n\n\n\n\n\n\n\n

$ S_e $ \u3092\u6c42\u3081\u308b\u306b\u306f\u3001\u4e0a\u8a18\u306e $ \\hat{Y} $ \u3092\u4f7f\u3063\u305f\u5f0f\u3067\u3082\u6c42\u3081\u3089\u308c\u308b\u304c\u3001\u624b\u8a08\u7b97\u3067\u8a08\u7b97\u3059\u308b\u5834\u5408\u3084\u30bd\u30d5\u30c8\u30a6\u30a7\u30a2\u3092\u4f7f\u3046\u306b\u3057\u3066\u3082\u624b\u9806\u3092\u8ffd\u3063\u3066\u8a08\u7b97\u3057\u3066\u307f\u308b\u5834\u5408\u306f\u3001$ S_r $ \u3092\u4ee5\u4e0b\u306e\u5f0f\u3092\u7528\u3044\u3066\u8a08\u7b97\u3057\u3066\u3001$ S_t $ \u304b\u3089\u5f15\u304f\u3068\u3044\u3046\u65b9\u6cd5\u3067\u8a08\u7b97\u3059\u308b\u3068\u3088\u3044\u3002<\/p>\n\n\n\n

\u4f8b\u3048\u3070\u8aac\u660e\u5909\u6570\u304c2\u3064\u306e\u5834\u5408\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u8a08\u7b97\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

$$
\\begin{array}{rcl}
S_r &=& \\sum \\beta_i S_{iy} \\\\
&=& \\beta_1 S_{1y} + \\beta_2 S_{2y}
\\end{array}
$$<\/p>\n\n\n\n\n\n\n\n

\u56de\u5e30\u306e\u5e73\u65b9\u548c $ S_r $ \u3082\u3001\u6b8b\u5dee\u306e\u5e73\u65b9\u548c $ S_e $ \u3082\u3001\u305d\u308c\u305e\u308c\u306e\u81ea\u7531\u5ea6 $ p $, $ n-p-1 $ \u3067\u5272\u308b\u3068\u5e73\u5747\u5e73\u65b9 $ MS_r $, $ MS_e $ \u3068\u306a\u308b\u3002<\/p>\n\n\n\n

\u3053\u3053\u3067 $ p $ \u306f\u504f\u56de\u5e30\u4fc2\u6570\u306e\u6570\u3001$ n $ \u306f\u5168\u4f53\u306e\u30b5\u30f3\u30d7\u30eb\u30b5\u30a4\u30ba\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u305d\u3057\u3066\u6700\u7d42\u7684\u306b $ MS_r $ \u3068 $ MS_e $ \u306e\u6bd4\u3092\u8a08\u7b97\u3059\u308b\u3068\u3001\u5206\u5b50\u306e\u81ea\u7531\u5ea6\u304c $ p $\u3001\u5206\u6bcd\u306e\u81ea\u7531\u5ea6\u304c $ n-p-1 $ \u306e F \u5206\u5e03\u306b\u5f93\u3046 F \u5024\u304c\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n

\u6709\u610f\u78ba\u7387\u304c\u8a08\u7b97\u3055\u308c\u3066\u3001\u6709\u610f\u6c34\u6e96\u4ee5\u4e0b\u3067\u3042\u308c\u3070\u3001\u7d71\u8a08\u5b66\u7684\u6709\u610f\u306b\u56de\u5e30\u30e2\u30c7\u30eb\u304c\u610f\u5473\u304c\u3042\u308b\u3068\u8a00\u3048\u308b\u3002<\/p>\n\n\n\n

\u8868\u306b\u307e\u3068\u3081\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

\u5909\u52d5\u8981\u56e0<\/th>\u5e73\u65b9\u548c<\/th>\u81ea\u7531\u5ea6<\/th>\u5e73\u5747\u5e73\u65b9<\/th>F \u5024<\/th><\/tr><\/thead>
\u56de\u5e30<\/td>$ S_r $<\/td>$ p $<\/td>$ MS_r = \\frac{S_r}{p} $<\/td>$ \\frac{MS_r}{MS_e} $<\/td><\/tr>
\u6b8b\u5dee<\/td>$ S_e $<\/td>$ n-p-1 $<\/td>$ MS_e = \\frac{S_e}{n-p-1} $<\/td><\/td><\/tr>
\u5168\u4f53<\/td>$ S_t $<\/td>$ n-1 $<\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

\u91cd\u56de\u5e30\u5206\u6790\u306e\u8a08\u7b97\u65b9\u6cd5\u3000\u6c7a\u5b9a\u4fc2\u6570<\/h2>\n\n\n\n

\u56de\u5e30\u30e2\u30c7\u30eb\u306e\u6709\u610f\u6027\u3092\u898b\u305f\u3064\u3044\u3067\u306b\u3001\u56de\u5e30\u30e2\u30c7\u30eb\u306b\u3088\u308b\u76ee\u7684\u5909\u6570\u306e\u8aac\u660e\u306e\u7a0b\u5ea6\u3092\u8868\u3057\u3066\u3044\u3066\u3001\u5f53\u3066\u306f\u307e\u308a\u306e\u6307\u6a19\u3067\u3082\u3042\u308b\u3001\u6c7a\u5b9a\u4fc2\u6570\u306e\u8a08\u7b97\u65b9\u6cd5\u3092\u898b\u3066\u304a\u304f\u3002<\/p>\n\n\n\n

\u6c7a\u5b9a\u4fc2\u6570\u306f\uff10\u304b\u3089\uff11\u306e\u5024\u3092\u53d6\u308a\u3001\uff11\u306b\u8fd1\u3044\u307b\u3069\u56de\u5e30\u30e2\u30c7\u30eb\u304c\u76ee\u7684\u5909\u6570\u306e\u5909\u52d5\u3092\u8aac\u660e\u3067\u304d\u3066\u3044\u308b\u3068\u7406\u89e3\u3057\u3066\u3001\u30e2\u30c7\u30eb\u304c\u30c7\u30fc\u30bf\u306b\u5f53\u3066\u306f\u307e\u3063\u3066\u3044\u308b\u3068\u8a55\u4fa1\u3059\u308b\u3002<\/p>\n\n\n\n

\u6163\u4f8b\u3068\u3057\u30660.5\u4ee5\u4e0a\u304c\u671b\u307e\u3057\u3044\u3068\u3055\u308c\u308b\u304c\u3001\u7814\u7a76\u5206\u91ce\u306b\u3088\u3063\u3066\u306f\u3082\u3063\u3068\u4f4e\u304f\u3066\u3082\u3088\u3044\u3068\u3055\u308c\u308b\u3053\u3068\u3082\u3042\u308b\u3002<\/p>\n\n\n\n

\u6c7a\u5b9a\u4fc2\u6570\u306e\u76ee\u5b89\u306b\u3064\u3044\u3066\u306f\u3001\u3053\u3061\u3089\u3082\u53c2\u7167\u3002<\/p>\n\n\n

\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\u3042\u308f\u305b\u3066\u8aad\u307f\u305f\u3044<\/span>\n\t\t\t\t\t
\"\"<\/figure><\/div>\t\t\t\t\t
\n\t\t\t\t\t\t\u6c7a\u5b9a\u4fc2\u6570\u306e\u76ee\u5b89 \u2015 \u6c7a\u5b9a\u4fc2\u6570 R 2 \u4e57\u5024\u306f\u3044\u304f\u3064\u306a\u3089\u3088\u3044\u304b\uff1f<\/a>\n\t\t\t\t\t\t\u91cd\u56de\u5e30\u5206\u6790\u306e\u5f53\u3066\u306f\u307e\u308a\u306e\u826f\u3055\u3092\u793a\u3059\u6c7a\u5b9a\u4fc2\u6570\u3002 \u6c7a\u5b9a\u4fc2\u6570\u306fR2\u4e57\u5024\u3068\u3082\u3044\u3046\u3002 \u6c7a\u5b9a\u4fc2\u6570\u306e\u76ee\u5b89\u306f\u3042\u308b\u306e\u3060\u308d\u3046\u304b\uff1f \u30bc\u30ed\u304b\u3089\uff11\u306e\u7bc4\u56f2\u3092\u3068\u308b\u308f\u3051\u3060\u304c\u3001\u3044\u304f\u3064\u306a\u3089\u826f\u3044\u306e\u304b\uff1f…<\/span>\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\n\n\n\n\n\n

\u6c7a\u5b9a\u4fc2\u6570 $ R^2 $ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n

$$ R^2 = \\frac{S_r}{S_t} = \\frac{S_t – S_e}{S_t} = 1 – \\frac{S_e}{S_t} $$<\/p>\n\n\n\n\n\n\n\n

\u56de\u5e30\u306e\u5206\u6563\u5206\u6790\u3067\u8a08\u7b97\u3057\u305f\u5e73\u65b9\u548c\u3092\u7528\u3044\u3066\u8a08\u7b97\u3055\u308c\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

\u8aac\u660e\u5909\u6570\u306e\u6570\u3092\u5897\u3084\u3057\u3066\u3044\u304f\u3068\u3001\u6c7a\u5b9a\u4fc2\u6570\u306f\u3060\u3093\u3060\u3093\u3068\uff11\u306b\u8fd1\u3065\u304f\u6027\u8cea\u304c\u3042\u308b\u3002<\/p>\n\n\n\n

\u3053\u308c\u3060\u3068\u8ffd\u52a0\u3057\u305f\u8aac\u660e\u5909\u6570\u306e\u305f\u3081\u306b\u8aac\u660e\u3067\u304d\u308b\u90e8\u5206\u304c\u5897\u3048\u305f\u306e\u304b\u3001\u5358\u306b\u8aac\u660e\u5909\u6570\u304c\u5897\u3048\u305f\u304b\u3089\u4e0a\u6607\u3057\u305f\u306e\u304b\u304c\u308f\u304b\u3089\u306a\u3044\u3002<\/p>\n\n\n\n

\u8aac\u660e\u5909\u6570\u306e\u6570\u3092\u5897\u3084\u3057\u305f\u3068\u304d\u306b\u3001\u8aac\u660e\u5909\u6570\u306e\u6570\u304c\u5897\u3048\u305f\u304b\u3089\u3060\u3051\u3067\u306f\u306a\u304f\u5b9f\u8cea\u7684\u306b\u6c7a\u5b9a\u4fc2\u6570\u304c\u4e0a\u6607\u3057\u305f\u3053\u3068\u3092\u793a\u3059\u306e\u304c\u3001\u81ea\u7531\u5ea6\u8abf\u6574\u6e08\u307f\u6c7a\u5b9a\u4fc2\u6570 $ R^{2*} $ \u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u5e73\u65b9\u548c\u3092\u81ea\u7531\u5ea6\u3067\u5272\u308a\u3001\u5e73\u5747\u5e73\u65b9\u306b\u3057\u3066\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n

$$ R^{2*} = 1 – \\frac{\\frac{S_e}{n-p-1}}{\\frac{S_t}{n-1}} = 1 – \\frac{MS_e}{MS_t} $$<\/p>\n\n\n\n\n\n\n\n

\u30e2\u30c7\u30eb\u306e\u5f53\u3066\u306f\u307e\u308a\u3084\u6709\u7528\u6027\u3092\u5224\u65ad\u3059\u308b\u306b\u306f\u81ea\u7531\u5ea6\u8abf\u6574\u6e08\u307f\u6c7a\u5b9a\u4fc2\u6570\u3092\u7528\u3044\u308b\u307b\u3046\u304c\u826f\u3044\u3002<\/p>\n\n\n\n

\u306a\u304a\u3001\u6c7a\u5b9a\u4fc2\u6570\u306f\u3001\u5bc4\u4e0e\u7387\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u3082\u3042\u308b\u3002<\/p>\n\n\n\n

\u91cd\u56de\u5e30\u5206\u6790\u306e\u8a08\u7b97\u65b9\u6cd5\u3000\u504f\u56de\u5e30\u4fc2\u6570\u306e\u691c\u5b9a<\/h2>\n\n\n\n

\u3055\u3066\u3001\u3084\u306f\u308a\u4e00\u756a\u6c17\u306b\u306a\u308b\u306e\u306f\u3001\u504f\u56de\u5e30\u4fc2\u6570\u306f\u7d71\u8a08\u5b66\u7684\u306b\u610f\u5473\u304c\u3042\u308b\u306e\u3060\u308d\u3046\u304b\uff1f\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u504f\u56de\u5e30\u4fc2\u6570\u304c\u6bcd\u96c6\u56e3\u3067\u30bc\u30ed\u3067\u306a\u3044\u3053\u3068\u304c\u8a00\u3048\u308c\u3070\u3001\u610f\u5473\u304c\u3042\u308b\u3060\u308d\u3046\u3002<\/p>\n\n\n\n

\u504f\u56de\u5e30\u4fc2\u6570\u304c\u6bcd\u96c6\u56e3\u3067\u30bc\u30ed\u304b\u3069\u3046\u304b\u306e\u691c\u5b9a\u304c\u3001\u504f\u56de\u5e30\u4fc2\u6570\u306e\u691c\u5b9a\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u3082\u3057\u30bc\u30ed\u3067\u306a\u3044\u3053\u3068\u304c\u8a00\u3048\u306a\u3051\u308c\u3070\u3001\u305d\u306e\u504f\u56de\u5e30\u4fc2\u6570\u53ca\u3073\u305d\u308c\u304c\u304b\u304b\u3063\u3066\u3044\u308b\u5909\u6570\u306f\u3001\u3042\u3063\u3066\u3082\u306a\u304f\u3066\u3082\u610f\u5473\u304c\u306a\u3044\u3002<\/p>\n\n\n\n

\u6bcd\u96c6\u56e3\u306e\u504f\u56de\u5e30\u4fc2\u6570\u304c\u30bc\u30ed\u3067\u3042\u308b\u5e30\u7121\u4eee\u8aac\u304c\u68c4\u5374\u3055\u308c\u308c\u3070\u3001\u504f\u56de\u5e30\u4fc2\u6570\u306f\u7d71\u8a08\u5b66\u7684\u306b\u6709\u610f\uff08\u30bc\u30ed\u3067\u306f\u306a\u304f\u610f\u5473\u304c\u3042\u308b\uff09\u3068\u5224\u65ad\u3067\u304d\u308b\u3002<\/p>\n\n\n\n

\u504f\u56de\u5e30\u4fc2\u6570\u306e\u691c\u5b9a\u306e\u305f\u3081\u306b\u306f\u504f\u56de\u5e30\u4fc2\u6570\u306e\u6a19\u6e96\u8aa4\u5dee\u304c\u5fc5\u8981\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

$ i $ \u756a\u76ee\u306e\u504f\u56de\u5e30\u4fc2\u6570 $ \\beta_i $ \u306e\u6a19\u6e96\u8aa4\u5dee $ SE_{\\beta_i} $ \u306f\u3001\u4ee5\u4e0b\u306e\u5f0f\u3067\u8a08\u7b97\u3067\u304d\u308b\u3002<\/p>\n\n\n\n

$$ SE_{\\beta_i} = \\sqrt{s^{ii} MS_e} $$<\/p>\n\n\n\n\n\n\n\n

\u3053\u3053\u3067\u3001$ s^{ii} $ \u306f\u3001\u305a\u3063\u3068\u4e0a\u306e\u65b9\u3067\u8a08\u7b97\u3057\u305f\u5909\u52d5\u30fb\u5171\u5909\u52d5\u884c\u5217 $ \\boldsymbol S $ \u306e\u9006\u884c\u5217 $ \\boldsymbol S^{-1} $ \u306e \u5de6\u4e0a\u304b\u3089 $ i $ \u756a\u76ee\u306e\u5bfe\u89d2\u6210\u5206\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u4f8b\u3048\u3070\uff11\u756a\u76ee\u306e\u504f\u56de\u5e30\u4fc2\u6570\u3067\u3042\u308c\u3070\u3001\u9006\u884c\u5217 $ \\boldsymbol S^{-1} $ \u306e (1,1) \u306e\u5bfe\u89d2\u6210\u5206 $ s^{11} $ \u3068\u306a\u308b\u3002<\/p>\n\n\n\n

\u3064\u307e\u308a\u3001\u8aac\u660e\u5909\u6570\u304c\uff12\u3064\u306e\u5834\u5408\u306f\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n

$$
\\begin{array}{rcl}
SE_{\\beta_1} &=& \\sqrt{s^{11} MS_e} \\\\
SE_{\\beta_2} &=& \\sqrt{s^{22} MS_e}
\\end{array}
$$<\/p>\n\n\n\n\n\n\n\n

\u305d\u3057\u3066\u3001\u504f\u56de\u5e30\u4fc2\u6570\u3068\u6a19\u6e96\u8aa4\u5dee\u306e\u6bd4\u306e\u5024\u304c\u3001\u81ea\u7531\u5ea6 $ n-p-1 $\uff08\u6b8b\u5dee\u306e\u81ea\u7531\u5ea6\uff09\u306e\uff54\u5206\u5e03\u306b\u5f93\u3046\u6027\u8cea\u3092\u5229\u7528\u3057\u3066\u3001\u6709\u610f\u78ba\u7387\u3092\u6c42\u3081\u308b\u3002<\/p>\n\n\n\n

$$
\\begin{array}{rcl}
t_{\\beta_1} &=& \\frac{|\\beta_1|}{SE_{\\beta_1}} \\\\
t_{\\beta_2} &=& \\frac{|\\beta_2|}{SE_{\\beta_2}}
\\end{array}
$$<\/p>\n\n\n\n\n\n\n\n

\u56de\u5e30\u30e2\u30c7\u30eb\u306e\u524d\u63d0\u306f\u8aa4\u5dee\u304c\u6b63\u898f\u5206\u5e03\u3057\u3066\u3044\u308b\u3053\u3068<\/h2>\n\n\n\n

\u6700\u5f8c\u306b\u3001\u56de\u5e30\u30e2\u30c7\u30eb\u306e\u524d\u63d0\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u3066\u304a\u304f\u3002<\/p>\n\n\n\n

\u307b\u304b\u306e\u8a18\u4e8b\u3067\u3082\u3001\u307e\u3068\u3081\u305f\u3082\u306e\u304c\u3042\u308b\u306e\u3067\u3001\u53c2\u8003\u307e\u3067\u3002<\/p>\n\n\n

\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\u3042\u308f\u305b\u3066\u8aad\u307f\u305f\u3044<\/span>\n\t\t\t\t\t
\"\"<\/figure><\/div>\t\t\t\t\t
\n\t\t\t\t\t\t\u56de\u5e30\u5206\u6790\u306b\u304a\u3051\u308b 7 \u3064\u306e\u4eee\u5b9a<\/a>\n\t\t\t\t\t\t\u76ee\u7684\u5909\u6570\u304c\u6b63\u898f\u5206\u5e03\u3057\u3066\u3044\u308b\u5fc5\u8981\u306f\u306a\u3044\u3002 \u8aac\u660e\u5909\u6570\u3082\u6b63\u898f\u5206\u5e03\u3057\u3066\u3044\u308b\u5fc5\u8981\u306f\u306a\u3044\u3002 \u524d\u63d0\u77e5\u8b58 \u5b9f\u969b\u306b\u6e2c\u5b9a\u3055\u308c\u305f\u76ee\u7684\u5909\u6570\u3092\u5b9f\u6e2c\u5024\u3068\u8a00\u3046\u3002 \u56de\u5e30\u5f0f\u3067\u8a08\u7b97\u3055\u308c\u305f\u76ee\u7684\u5909\u6570\u3092…<\/span>\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\n\n\n\n\n\n

\u56de\u5e30\u30e2\u30c7\u30eb\u306e\u524d\u63d0\u3067\u4e00\u756a\u6c17\u306b\u306a\u308b\u70b9\u306f\u3001\u8aa4\u5dee\u9805 $ \\varepsilon $ \u304c\u5e73\u5747 0\u3001\u5206\u6563 $ \\sigma_e^2 $\uff08\u4f55\u3067\u3082\u3088\u3044\u304c\u4f55\u304b\u9069\u5207\u306a\u5024\uff09\u306e\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u3068\u3044\u3046\u3082\u306e\u3060\u3002<\/p>\n\n\n\n

\u8aac\u660e\u5909\u6570\u304c\uff12\u3064\u7248\u306e\u56de\u5e30\u5f0f\u3001<\/p>\n\n\n\n

$$ Y = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2 + \\varepsilon $$<\/p>\n\n\n\n\n\n\n\n

\u306e\u6700\u5f8c\u306e $ \\varepsilon $ \u304c \u5e73\u5747 0 \u3067\u3001\u5206\u6563\u306f\u4f55\u3089\u304b\u306e\u5927\u304d\u3055\u306e\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u3053\u3068\u304c\u6761\u4ef6\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

\u3053\u308c\u306f\u3001\u4e0a\u8a18\u306e\u56de\u5e30\u306e\u5206\u6563\u5206\u6790\u3084\u3001\u504f\u56de\u5e30\u4fc2\u6570\u306e\u691c\u5b9a\u306e\u969b\u3001\u691c\u5b9a\u7d71\u8a08\u91cf\u306e\u5206\u6bcd\u306b\u8aa4\u5dee\uff08\u6b8b\u5dee\uff09\u306e\u5e73\u5747\u5e73\u65b9\u304c\u6765\u3066\u3044\u308b\u3053\u3068\u304b\u3089\u3082\u7406\u306b\u304b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

\uff26\u5206\u5e03\u3084\uff54\u5206\u5e03\u3092\u4f7f\u3063\u305f\u691c\u5b9a\u306f\u3001\u6bcd\u96c6\u56e3\u304c\u6b63\u898f\u5206\u5e03\u3067\u3042\u308b\u3053\u3068\u3092\u4eee\u5b9a\u3057\u3066\u3044\u308b\u3068\u3044\u3046\u5927\u524d\u63d0\u304c\u3042\u308b\u3002<\/p>\n\n\n\n

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