\u5358\u56de\u5e30\u30e2\u30c7\u30eb\u306e\u304a\u3055\u3089\u3044<\/h2>\n\n\n\n
\u307e\u305a\u3001\u5358\u56de\u5e30\u30e2\u30c7\u30eb\u3092\u5b9a\u7fa9\u3057\u3088\u3046\u3002
\u76ee\u7684\u5909\u6570 $Y$ \u3068\u8aac\u660e\u5909\u6570 $X$ \u306e\u95a2\u4fc2\u3092\u3001\u4ee5\u4e0b\u306e\u7dda\u5f62\u30e2\u30c7\u30eb\u3067\u8868\u3059\u3002<\/p>\n\n\n\n
$$Y_i = \\beta_0 + \\beta_1 X_i + \\epsilon_i$$<\/p>\n\n\n\n
\u3053\u3053\u3067\u3001$i$ \u306f $i$ \u756a\u76ee\u306e\u89b3\u6e2c\u5024\u3092\u8868\u3057\u3001$\\beta_0$ \u306f\u5207\u7247\u3001$\\beta_1$ \u306f\u56de\u5e30\u4fc2\u6570\u3001$\\epsilon_i$ \u306f\u8aa4\u5dee\u9805\u3067\u3001\u5e73\u57470\u3001\u5206\u6563 $\\sigma^2$ \u306e\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u3068\u4eee\u5b9a\u3059\u308b\u3002<\/p>\n\n\n\n
\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306b\u3088\u308a\u63a8\u5b9a\u3055\u308c\u305f\u56de\u5e30\u4fc2\u6570 $\\hat{\\beta}_1$ \u306f\u3001\u4ee5\u4e0b\u306e\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002<\/p>\n\n\n\n
$$\\hat{\\beta}_1 = \\frac{\\sum_{i=1}^n (X_i – \\bar{X})(Y_i – \\bar{Y})}{\\sum_{i=1}^n (X_i – \\bar{X})^2}$$<\/p>\n\n\n\n
\u3053\u3053\u3067\u3001$\\bar{X}$ \u3068 $\\bar{Y}$ \u306f\u305d\u308c\u305e\u308c $X$ \u3068 $Y$ \u306e\u6a19\u672c\u5e73\u5747\u3067\u3042\u308b\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 \u21911\u4e07\u4eba\u4ee5\u4e0a\u306e\u533b\u7642\u5f93\u4e8b\u8005\u304c\u8cfc\u8aad\u4e2d<\/span><\/strong><\/span><\/p><\/div> \u56de\u5e30\u4fc2\u6570 $\\hat{\\beta}_1$ \u306e\u6709\u610f\u6027\u3092\u691c\u5b9a\u3059\u308b\u305f\u3081\u306e $t$ \u5024\u306f\u3001\u4ee5\u4e0b\u306e\u5f0f\u3067\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n $$t = \\frac{\\hat{\\beta}_1 – 0}{SE(\\hat{\\beta}_1)}$$<\/p>\n\n\n\n \u3053\u3053\u3067\u3001$SE(\\hat{\\beta}_1)$ \u306f $\\hat{\\beta}_1$ \u306e\u6a19\u6e96\u8aa4\u5dee\u3067\u3042\u308b\u3002\u901a\u5e38\u3001\u5e30\u7121\u4eee\u8aac\u306f $\\beta_1 = 0$ \u3067\u3042\u308b\u305f\u3081\u3001\u5206\u5b50\u306f $\\hat{\\beta}_1$ \u3068\u306a\u308b\u3002<\/p>\n\n\n\n $SE(\\hat{\\beta}_1)$ \u306f\u4ee5\u4e0b\u306e\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002<\/p>\n\n\n\n $$SE(\\hat{\\beta}_1) = \\frac{s_e}{\\sqrt{\\sum_{i=1}^n (X_i – \\bar{X})^2}}$$<\/p>\n\n\n\n \u3053\u3053\u3067\u3001$s_e$ \u306f\u3001\u6bcd\u96c6\u56e3\u306e\u8aa4\u5dee\u306e\u6a19\u6e96\u504f\u5dee\u306e\u63a8\u5b9a\u5024\u3068\u6349\u3048\u308c\u3070\u300c\u6b8b\u5dee\u306e\u6a19\u6e96\u8aa4\u5dee\u300d\uff08\u307e\u305f\u306f\u6b8b\u5dee\u306e\u3070\u3089\u3064\u304d\u306e\u6307\u6a19\u3068\u6349\u3048\u308c\u3070\u300c\u6b8b\u5dee\u306e\u6a19\u6e96\u504f\u5dee\u300d\uff09\u3067\u3042\u308a\u3001\u4ee5\u4e0b\u306e\u5f0f\u3067\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n $$s_e = \\sqrt{\\frac{SSE}{n-2}}$$<\/p>\n\n\n\n $SSE$ \u306f\u6b8b\u5dee\u5e73\u65b9\u548c (Sum of Squares Error) \u3067\u3042\u308a\u3001\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u3002<\/p>\n\n\n\n $$SSE = \\sum_{i=1}^n (Y_i – \\hat{Y}_i)^2$$<\/p>\n\n\n\n \u3053\u3053\u3067\u3001$\\hat{Y}_i = \\hat{\\beta}_0 + \\hat{\\beta}_1 X_i$ \u306f\u56de\u5e30\u30e2\u30c7\u30eb\u306b\u3088\u308b$Y_i$\u306e\u4e88\u6e2c\u5024\u3067\u3042\u308b\u3002<\/p>\n\n\n\n \u3053\u308c\u3089\u306e\u5f0f\u3092\u7d44\u307f\u5408\u308f\u305b\u308b\u3068\u3001$t$\u5024\u306e\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n\n\n\n $$t = \\frac{\\hat{\\beta}_1}{\\frac{\\sqrt{SSE\/(n-2)}}{\\sqrt{\\sum_{i=1}^n (X_i – \\bar{X})^2}}} = \\hat{\\beta}_1 \\frac{\\sqrt{\\sum_{i=1}^n (X_i – \\bar{X})^2}}{\\sqrt{SSE\/(n-2)}}$$<\/p>\n\n\n\n $t$ \u5024\u306e2\u4e57\u3092\u8003\u3048\u308b\u3068\u3001<\/p>\n\n\n\n $$t^2 = \\hat{\\beta}_1^2 \\frac{\\sum_{i=1}^n (X_i – \\bar{X})^2}{SSE\/(n-2)} \\quad (*)$$<\/p>\n\n\n\n \u6b21\u306b\u3001\u56de\u5e30\u30e2\u30c7\u30eb\u5168\u4f53\u306e\u6709\u610f\u6027\u3092\u691c\u5b9a\u3059\u308b\u305f\u3081\u306e $F$ \u5024\u3092\u898b\u3066\u307f\u3088\u3046\u3002\u5206\u6563\u5206\u6790\uff08ANOVA\uff09\u3067\u306f\u3001\u7dcf\u5e73\u65b9\u548c (Total Sum of Squares, $SST$) \u3092\u3001\u56de\u5e30\u5e73\u65b9\u548c (Regression Sum of Squares, $SSR$) \u3068\u6b8b\u5dee\u5e73\u65b9\u548c ($SSE$) \u306b\u5206\u89e3\u3059\u308b\u3002<\/p>\n\n\n\n $$SST = SSR + SSE$$<\/p>\n\n\n\n \u305d\u308c\u305e\u308c\u306e\u5b9a\u7fa9\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3042\u308b\u3002<\/p>\n\n\n\n $F$ \u5024\u306f\u3001\u56de\u5e30\u306e\u5e73\u5747\u5e73\u65b9 ($MSR$) \u3092\u6b8b\u5dee\u306e\u5e73\u5747\u5e73\u65b9 ($MSE$) \u3067\u5272\u308b\u3053\u3068\u3067\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n $$F = \\frac{MSR}{MSE}$$<\/p>\n\n\n\n \u3053\u3053\u3067\u3001<\/p>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u3001$F$ \u5024\u306e\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n\n\n\n $$F = \\frac{SSR\/1}{SSE\/(n-2)} = \\frac{SSR}{SSE\/(n-2)} \\quad (**)$$<\/p>\n\n\n\n \u3053\u3053\u3067\u3001$SSR$ \u3092 $\\hat{\\beta}_1$ \u3092\u7528\u3044\u3066\u8868\u73fe\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u3059\u3002\u5358\u56de\u5e30\u306e\u5834\u5408\u3001$SSR$ \u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n\n\n\n $$SSR = \\sum_{i=1}^n (\\hat{Y}_i – \\bar{Y})^2$$<\/p>\n\n\n\n $\\hat{Y}_i = \\hat{\\beta}_0 + \\hat{\\beta}_1 X_i$ \u3067\u3042\u308a\u3001$\\bar{Y} = \\hat{\\beta}_0 + \\hat{\\beta}_1 \\bar{X}$ (\u56de\u5e30\u76f4\u7dda\u306f\u6a19\u672c\u5e73\u5747 $(\\bar{X}, \\bar{Y})$ \u3092\u901a\u308b\u305f\u3081) \u3067\u3042\u308b\u3053\u3068\u3092\u5229\u7528\u3059\u308b\u3068\u3001<\/p>\n\n\n\n $$\\hat{Y}_i – \\bar{Y} = (\\hat{\\beta}_0 + \\hat{\\beta}_1 X_i) – (\\hat{\\beta}_0 + \\hat{\\beta}_1 \\bar{X}) = \\hat{\\beta}_1 (X_i – \\bar{X})$$<\/p>\n\n\n\n \u3053\u308c\u3092$SSR$\u306e\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001<\/p>\n\n\n\n $$SSR = \\sum_{i=1}^n [\\hat{\\beta}_1 (X_i – \\bar{X})]^2 = \\sum_{i=1}^n \\hat{\\beta}_1^2 (X_i – \\bar{X})^2 = \\hat{\\beta}_1^2 \\sum_{i=1}^n (X_i – \\bar{X})^2$$<\/p>\n\n\n\n \u3053\u308c\u3067\u5168\u3066\u306e\u30d4\u30fc\u30b9\u304c\u63c3\u3063\u305f\u3002\u5148\u307b\u3069\u6c42\u3081\u305f $SSR$ \u306e\u5f0f\u3092\u3001$F$ \u5024\u306e\u5f0f $(**)$ \u306b\u4ee3\u5165\u3057\u3066\u307f\u3088\u3046\u3002<\/p>\n\n\n\n $$F = \\frac{\\hat{\\beta}_1^2 \\sum_{i=1}^n (X_i – \\bar{X})^2}{SSE\/(n-2)}$$<\/p>\n\n\n\n \u3053\u306e$F$\u5024\u306e\u5f0f\u3068\u3001\u5148\u306b\u6c42\u3081\u305f $t^2$ \u306e\u5f0f $(*)$ \u3092\u6bd4\u8f03\u3057\u3066\u307f\u3066\u307b\u3057\u3044\u3002<\/p>\n\n\n\n $$t^2 = \\hat{\\beta}_1^2 \\frac{\\sum_{i=1}^n (X_i – \\bar{X})^2}{SSE\/(n-2)}$$<\/p>\n\n\n\n $$F = \\hat{\\beta}_1^2 \\frac{\\sum_{i=1}^n (X_i – \\bar{X})^2}{SSE\/(n-2)}$$<\/p>\n\n\n\n \u3054\u89a7\u306e\u901a\u308a\u3001\u4e21\u8005\u306f\u5168\u304f\u540c\u3058\u6570\u5f0f<\/strong>\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n $$t^2 = F$$<\/p>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u3001\u5358\u56de\u5e30\u5206\u6790\u306b\u304a\u3044\u3066\u3001\u56de\u5e30\u4fc2\u6570\u306e $t$ \u5024\u306e2\u4e57\u306f\u3001\u56de\u5e30\u30e2\u30c7\u30eb\u5168\u4f53\u306e\u6709\u610f\u6027\u691c\u5b9a\u306b\u304a\u3051\u308b $F$ \u5024\u3068\u6570\u5b66\u7684\u306b\u540c\u5024\u3067\u3042\u308b\u3053\u3068\u304c\u8a3c\u660e\u3055\u308c\u305f\u3002<\/p>\n\n\n\n \u5358\u56de\u5e30\u5206\u6790\u3067\u306f\u3001\u8aac\u660e\u5909\u6570\u304c1\u3064\u3057\u304b\u306a\u3044\u305f\u3081\u3001\u305d\u306e\u8aac\u660e\u5909\u6570\u306e\u52b9\u679c\u306e\u6709\u7121\u3092\u691c\u5b9a\u3059\u308b $t$ \u691c\u5b9a\u3068\u3001\u30e2\u30c7\u30eb\u5168\u4f53\u304c\u7d71\u8a08\u7684\u306b\u6709\u610f\u304b\u3069\u3046\u304b\u3092\u691c\u5b9a\u3059\u308b $F$ \u691c\u5b9a\u306f\u3001\u672c\u8cea\u7684\u306b\u540c\u3058\u3053\u3068\u3092\u8a55\u4fa1\u3057\u3066\u3044\u308b\u3002\u305d\u306e\u305f\u3081\u3001\u7d71\u8a08\u91cf\u3082\u6570\u5b66\u7684\u306b\u7d50\u3073\u3064\u3044\u3066\u3044\u308b\u306e\u3060\u3002\u3053\u306e\u95a2\u4fc2\u306f\u3001\u7d71\u8a08\u306e\u7f8e\u3057\u3055\u3068\u3001\u7570\u306a\u308b\u7d71\u8a08\u691c\u5b9a\u304c\u3069\u306e\u3088\u3046\u306b\u95a2\u9023\u3057\u5408\u3063\u3066\u3044\u308b\u304b\u3092\u793a\u3059\u826f\u3044\u4f8b\u3067\u3042\u308b\u3002<\/p>\n\n\n\n \u591a\u91cd\u56de\u5e30\u5206\u6790\u306b\u306a\u308b\u3068\u3001\u5404\u56de\u5e30\u4fc2\u6570\u306e $t$ \u5024\u306f\u305d\u306e\u4fc2\u6570\u5358\u72ec\u306e\u6709\u610f\u6027\u3092\u793a\u3057\u3001\u30e2\u30c7\u30eb\u5168\u4f53\u306e $F$ \u5024\u306f\u8907\u6570\u306e\u8aac\u660e\u5909\u6570\u304c\u307e\u3068\u3081\u3066\u30e2\u30c7\u30eb\u306b\u5bc4\u4e0e\u3057\u3066\u3044\u308b\u304b\u3092\u793a\u3059\u3002\u3053\u306e\u5834\u5408\u3001$t^2=F$\u306e\u95a2\u4fc2\u306f\u6210\u7acb\u3057\u306a\u3044\u304c\u3001\u305d\u308c\u306f\u307e\u305f\u5225\u306e\u8a71\u306b\u306a\u308b\u3002<\/p>\n\n\n\n \u4eca\u56de\u306e\u8a18\u4e8b\u3067\u3001\u56de\u5e30\u5206\u6790\u306e\u7406\u89e3\u304c\u6df1\u307e\u308c\u3070\u5e78\u3044\u3060\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u56de\u5e30\u5206\u6790\u306f\u3001\u5909\u6570\u9593\u306e\u95a2\u4fc2\u6027\u3092\u660e\u3089\u304b\u306b\u3059\u308b\u5f37\u529b\u306a\u7d71\u8a08\u30c4\u30fc\u30eb\u3060\u3002\u305d\u306e\u7d50\u679c\u3092\u89e3\u91c8\u3059\u308b\u969b\u3001$t$ \u5024\u3084 $F$ \u5024\u3068\u3044\u3063\u305f\u7d71\u8a08\u91cf\u3092\u76ee\u306b\u3059\u308b\u3053\u3068\u304c\u591a\u3044\u3060\u308d\u3046\u3002\u7279\u306b\u3001\u5358\u56de\u5e30\u5206\u6790\u306b\u304a\u3044\u3066\u306f\u3001\u56de\u5e30\u4fc2\u6570\u306e $t$ \u5024\u306e2\u4e57\u304c\u3001\u56de\u5e30\u30e2\u30c7\u30eb\u5168 […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"swell_btn_cv_data":"","_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[23],"tags":[],"class_list":["post-4270","post","type-post","status-publish","format-standard","hentry","category-23"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts\/4270","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/comments?post=4270"}],"version-history":[{"count":12,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts\/4270\/revisions"}],"predecessor-version":[{"id":4282,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/posts\/4270\/revisions\/4282"}],"wp:attachment":[{"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/media?parent=4270"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/categories?post=4270"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/best-biostatistics.com\/toukei-er\/wp-json\/wp\/v2\/tags?post=4270"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/best-biostatistics.com\/wp\/wp-content\/uploads\/2023\/11\/bn_r_03.png\")
<\/a>\r\n\u56de\u5e30\u4fc2\u6570 $\\hat{\\beta}_1$ \u306e $t$ \u5024<\/h2>\n\n\n\n
\u56de\u5e30\u30e2\u30c7\u30eb\u306e $F$ \u5024 (\u5206\u6563\u5206\u6790)<\/h2>\n\n\n\n
\n
$$SST = \\sum_{i=1}^n (Y_i – \\bar{Y})^2$$<\/li>\n\n\n\n
$$SSR = \\sum_{i=1}^n (\\hat{Y}_i – \\bar{Y})^2$$<\/li>\n\n\n\n
$$SSE = \\sum_{i=1}^n (Y_i – \\hat{Y}_i)^2$$<\/li>\n<\/ul>\n\n\n\n\n
$SSR$ \u3068 $\\hat{\\beta}_1$ \u306e\u95a2\u4fc2<\/h2>\n\n\n\n
\u8a3c\u660e\uff1a$t^2 = F$<\/h2>\n\n\n\n
\u307e\u3068\u3081<\/h2>\n\n\n\n