{"id":4892,"date":"2025-06-23T16:19:07","date_gmt":"2025-06-23T07:19:07","guid":{"rendered":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/?p=4892"},"modified":"2025-12-25T07:32:12","modified_gmt":"2025-12-24T22:32:12","slug":"%e7%ad%89%e5%88%86%e6%95%a3%e3%82%92%e4%bb%ae%e5%ae%9a%e3%81%97%e3%81%aa%e3%81%84%ef%bc%88welch%e7%89%88%ef%bc%89%e5%88%86%e6%95%a3%e5%88%86%e6%9e%90%e3%81%ae%e5%85%b7%e4%bd%93%e7%9a%84%e3%81%aa","status":"publish","type":"post","link":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/2025\/06\/23\/%e7%ad%89%e5%88%86%e6%95%a3%e3%82%92%e4%bb%ae%e5%ae%9a%e3%81%97%e3%81%aa%e3%81%84%ef%bc%88welch%e7%89%88%ef%bc%89%e5%88%86%e6%95%a3%e5%88%86%e6%9e%90%e3%81%ae%e5%85%b7%e4%bd%93%e7%9a%84%e3%81%aa\/","title":{"rendered":"\u7b49\u5206\u6563\u3092\u4eee\u5b9a\u3057\u306a\u3044\uff08Welch\u7248\uff09\u5206\u6563\u5206\u6790\u306e\u5177\u4f53\u7684\u306a\u8a08\u7b97"},"content":{"rendered":"\n\n\n\n\n\n<p class=\"wp-block-paragraph\">\u52c9\u5f37\u306e\u305f\u3081\u306e\u30e1\u30e2\u30022025\u5e746\u670825\u65e5\u66f4\u65b0\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u30c7\u30fc\u30bf<\/h2>\n\n\n\n<pre class=\"prettyprint\">#R\u5185\u3067\u306e\u4e0a\u306e\u30c7\u30fc\u30bf\u306e\u751f\u6210\nN=15\nset.seed(-6152847)\na &lt;- rnorm(N, mean=5.080351, sd=0.1375497)\nset.seed(3448258)\nb &lt;- rnorm(N, mean=5.185921, sd=0.05358547)\nset.seed(-9846648)\nc &lt;- rnorm(N, mean=5.186056, sd=0.05089308)\nset.seed(-1377657)\nd &lt;- rnorm(N, mean=5.186409, sd=0.06550713)\nset.seed(7612155)\ne &lt;- rnorm(N, mean=5.238295, sd=0.07842281)\nset.seed(NULL)\nvalue &lt;- c(a, b, c, d, e)\nvalue &lt;- round(value*10, digit=1)\ngroup &lt;- factor(c(rep(&#34;A&#34;, N), rep(&#34;B&#34;, N), rep(&#34;C&#34;, N), rep(&#34;D&#34;, N), rep(&#34;E&#34;, N)))<\/pre>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table class=\"has-fixed-layout\"><tbody><tr><td>\u7fa4<\/td><td>1<\/td><td>2<\/td><td>3<\/td><td>4<\/td><td>5<\/td><td>6<\/td><td>7<\/td><td>8<\/td><td>9<\/td><td>10<\/td><td>11<\/td><td>12<\/td><td>13<\/td><td>14<\/td><td>15<\/td><\/tr><tr><td>A<\/td><td>51.3<\/td><td>50.7<\/td><td>53.7<\/td><td>50.6<\/td><td>49.0<\/td><td>50.7<\/td><td>53.7<\/td><td>49.8<\/td><td>51.1<\/td><td>51.3<\/td><td>52.2<\/td><td>50.8<\/td><td>49.0<\/td><td>51.8<\/td><td>51.0<\/td><\/tr><tr><td>B<\/td><td>52.6<\/td><td>51.0<\/td><td>52.5<\/td><td>51.4<\/td><td>52.3<\/td><td>52.5<\/td><td>52.0<\/td><td>51.2<\/td><td>52.2<\/td><td>51.5<\/td><td>51.5<\/td><td>51.9<\/td><td>52.3<\/td><td>51.4<\/td><td>51.1<\/td><\/tr><tr><td>C<\/td><td>52.2<\/td><td>52.4<\/td><td>52.4<\/td><td>51.2<\/td><td>51.6<\/td><td>52.6<\/td><td>51.6<\/td><td>52.2<\/td><td>51.1<\/td><td>51.8&nbsp;<\/td><td>52.8<\/td><td>52.1<\/td><td>51.6&nbsp;<\/td><td>51.6<\/td><td>52.3<\/td><\/tr><tr><td>D<\/td><td>53.3<\/td><td>51.4<\/td><td>51.8<\/td><td>52.7<\/td><td>51.3<\/td><td>51.8<\/td><td>51.1<\/td><td>52.2<\/td><td>50.8<\/td><td>51.2<\/td><td>52.8<\/td><td>52.1<\/td><td>52.7<\/td><td>53.0<\/td><td>51.6<\/td><\/tr><tr><td>E<\/td><td>53.9<\/td><td>52.5<\/td><td>52.7<\/td><td>52.1<\/td><td>52.6<\/td><td>54.4<\/td><td>52.3<\/td><td>52.0<\/td><td>52.5<\/td><td>52.6<\/td><td>51.3<\/td><td>51.4<\/td><td>52.7<\/td><td>51.5<\/td><td>51.7<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table class=\"has-fixed-layout\"><tbody><tr><td>\u7fa4<\/td><td>\u30b5\u30a4\u30ba \\(n_i\\)<\/td><td>\u8a08 \\(T_i\\)<\/td><td>\u5e73\u5747 \\(\\bar{x}_i\\)<\/td><td>\u4e0d\u504f\u5206\u6563 \\(V_i\\)<\/td><\/tr><tr><td>A<\/td><td>15<\/td><td>766.7<\/td><td>51.11333<\/td><td>1.891238<\/td><\/tr><tr><td>B<\/td><td>15<\/td><td>777.4<\/td><td>51.82667<\/td><td>0.3078095<\/td><\/tr><tr><td>C<\/td><td>15<\/td><td>779.5<\/td><td>51.96667<\/td><td>0.2580952<\/td><\/tr><tr><td>D<\/td><td>15<\/td><td>779.8<\/td><td>51.98667<\/td><td>0.5955238<\/td><\/tr><tr><td>E<\/td><td>15<\/td><td>786.2<\/td><td>52.41333<\/td><td>0.7355238<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u5168\u30c7\u30fc\u30bf\u6570\\(N = 75\\)\u3002\u30c7\u30fc\u30bf\u306e\u7dcf\u548c\\(T = 3889.6\\)\u3002\u5168\u30c7\u30fc\u30bf\u306e\u5e73\u5747 \\(\\bar{\\bar{x}} = 51.86133\\)\u3002\u7fa4\u306e\u6570\\(k = 5\\)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u7b49\u5206\u6563\u3092\u4eee\u5b9a\u3059\u308b\u5206\u6563\u5206\u6790\u306e\u5b9f\u884c<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u5206\u6563\u5206\u6790\u306e\u6839\u5e79\u306f\u3001\u7dcf\u5e73\u65b9\u548c\u304c\u7fa4\u9593\u306e\u3070\u3089\u3064\u304d\uff08\u7fa4\u9593\u5e73\u65b9\u548c\uff09\u3068\u6a19\u672c\u8aa4\u5dee\u306e\u3070\u3089\u3064\u304d\uff08\u7fa4\u5185\u5e73\u65b9\u548c\uff09\u306b\u5206\u89e3\u3067\u304d\u308b\u3068\u3044\u3046\u4e8b\u5b9f\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u7528\u8a9e<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u7dcf\u5e73\u65b9\u548c<\/strong>\uff08\\(S_T\\)\uff09 \u2014 \u305d\u308c\u305e\u308c\u306e\u5024\u3068\u5168\u4f53\u306e\u5e73\u5747\u5024\u3068\u306e\u5dee\u3092\u4e8c\u4e57\u3057\u3066\u8db3\u3057\u5408\u308f\u305b\u305f\u3082\u306e\u3002<\/li>\n\n\n\n<li><strong>\u7fa4\u9593\u5e73\u65b9\u548c<\/strong>\uff08\\(S_A\\)\u3001A\u9593\u5e73\u65b9\u548c\uff09 \u2014 \u7fa4\u5185\u5e73\u5747\u3068\u5168\u4f53\u5e73\u5747\u306e\u5dee\u3092\u4e8c\u4e57\u3057\u3066\u8db3\u3057\u5408\u308f\u305b\u305f\u3082\u306e\u3002<\/li>\n\n\n\n<li><strong>\u7fa4\u5185\u5e73\u65b9\u548c<\/strong>\uff08\\(S_E\\)\u3001\u8aa4\u5dee\u5e73\u65b9\u548c\uff09 \u2014 \u305d\u308c\u305e\u308c\u306e\u5024\u3068\u5404\u7fa4\u5185\u306e\u5e73\u5747\u5024\u3068\u306e\u5dee\u3092\u4e8c\u4e57\u3057\u3066\u8db3\u3057\u5408\u308f\u305b\u305f\u3082\u306e\u3002<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">\u5404\u5e73\u65b9\u548c\u306e\u8a08\u7b97<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">$$ CT = \\frac{T^2}{N} = 201719.8$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$S_T = \\sum_{i=1}^k{} \\sum_{j=1}^{n_i}(x_{ij}-\\bar{\\bar{x}})^2 =&nbsp; \\sum_{i=1}^k{} \\sum_{j=1}^{n_i} x^2_{ij} - CT = 66.41787$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$S_A = \\sum_{i=1}^k{} \\sum_{j=1}^{n_i}(\\bar{x}_{i}-\\bar{\\bar{x}})^2 = \\sum_{i=1}^k{}\\frac{T_i^2}{n_i} - CT = 13.3832 $$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$S_E = \\sum_{i=1}^k{} \\sum_{j=1}^{n_i}(x_{ij}-\\bar{x})^2 = S_T - S_A = 53.03467$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u5206\u6563\u5206\u6790\u8868<\/h3>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table class=\"has-fixed-layout\"><tbody><tr><td>\u8981\u56e0<\/td><td>\u5e73\u65b9\u548c \\(S\\)<\/td><td>\u81ea\u7531\u5ea6 \\(\\phi\\)<\/td><td>\u5e73\u5747\u5e73\u65b9 \\(V\\)<\/td><td>\\(F_0\\)<\/td><\/tr><tr><td>\\(A\\)<\/td><td>13.3832<\/td><td>4<\/td><td>\\(S_A\/\\phi_A =\\) 3.3458<\/td><td>\\(V_A\/V_E\\) = 4.416093<\/td><\/tr><tr><td>\\(E\\)<\/td><td>53.03467<\/td><td>70<\/td><td>\\(S_E\/\\phi_E =\\) 0.7576381<\/td><td><\/td><\/tr><tr><td>\u8a08<\/td><td>66.41787<\/td><td>74<\/td><td><\/td><td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u30c7\u30fc\u30bf\u3092\u5e73\u57470\u3001\u5206\u65631\u306b\u6a19\u6e96\u5316 (standardization) \u3059\u308b\u306b\u306f\u3001\u5024\u3068\u5e73\u5747\u306e\u5dee\u3092\u53d6\u3063\u3066 (\\(X - \\bar{x}\\))\u3001\u6a19\u6e96\u504f\u5dee (\\(s\\)) \u3067\u5272\u308a\u307e\u3059\u3002\u3053\u306e\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u3055\u308c\u305f\u5024\\(\\frac{X-\\bar{x}}{s}\\) \u304c\u6b63\u898f\u5206\u5e03\uff08\u5e73\u57470\u3001\u5206\u65631\uff09\u304b\u3089\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3055\u308c\u3066\u3044\u308b\u3068\u3059\u308b\u3068\u3001\u305d\u306e\u5e73\u65b9\u548c\uff08\\(\\sum_{i=1}^k \\frac{(x_i - \\bar{x})^2}{s^2}\\)\uff09\u306f\u81ea\u7531\u5ea6 \\(k-1\\) \u306e\\(\\chi^2\\)\u5206\u5e03\uff08\u30ab\u30a4\u3058\u3058\u3087\u3046\u3076\u3093\u3077\uff09\u306b\u5f93\u3044\u307e\u3059\uff08\\(\\chi^2\\)\u5206\u5e03\u306e\u5b9a\u7fa9\uff09\u3002\u6b63\u898f\u5206\u5e03\u304b\u3089\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3055\u308c\u305f\u5024\u306e\u5e73\u65b9\u548c\u3092\u305d\u308c\u3089\u306e\u81ea\u7531\u5ea6\\(k_1, k_2\\)\u3067\u5272\u3063\u305f\u5024\u306e\u6bd4\u306f\u81ea\u7531\u5ea6 (\\(k_1, k_2\\)) \u306e<strong><em>F<\/em>\u5206\u5e03<\/strong>\u306b\u5f93\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001\u5e73\u65b9\u548c\\(S_A, S_E\\)\uff08\u3092\u4e0d\u504f\u5206\u6563\\(s^2\\)\u3067\u5272\u3063\u305f\u5024\uff09\u306f\u305d\u308c\u305e\u308c\u81ea\u7531\u5ea6\\(\\phi_A, \\phi_E\\) \u306e\\(\\chi^2\\)\u5206\u5e03\u306b\u5f93\u3044\u3001<em>F<\/em>\u5024\u306f\u5404\u7fa4\u306e\u5e73\u5747\u306e\u4e0d\u504f\u5206\u6563\uff08\\(S_A\/\\phi_A\\)\uff09\u3068\u8aa4\u5dee\u306e\u4e0d\u504f\u5206\u6563\uff08\\(S_E\/\\phi_E\\)\uff09\u306e\u6bd4$$F = \\frac{\\frac{S_A}{\\phi_A}}{\\frac{S_E}{\\phi_E}}$$\u3068\u3057\u3066\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\u4e0d\u504f\u5206\u6563\\(s^2\\)\u306f\u6bd4\u3092\u53d6\u3063\u305f\u6642\u306b\u6253\u3061\u6d88\u3055\u308c\u308b\u306e\u3067\u8a08\u7b97\u5f0f\u306b\u306f\u73fe\u308f\u308c\u3066\u3044\u307e\u305b\u3093\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"232\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2025\/06\/df_4_70-300x232.png\" alt=\"\" class=\"wp-image-4905\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2025\/06\/df_4_70-300x232.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2025\/06\/df_4_70.png 1496w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption class=\"wp-element-caption\">\u81ea\u7531\u5ea6(4,70) \u306e<em>F<\/em>\u5206\u5e03<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">R\u306epf\u95a2\u6570\u3067<em>p<\/em>\u5024\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p><pre class=\"prettyprint\">&gt; pf(4.416093, 4, 70, lower.tail=FALSE)\n&#091;1&#093; 0.003058812\n&gt; oneway.test(value ~ group, var.equal=TRUE) # \u30c7\u30d5\u30a9\u30eb\u30c8\u306fvar.equal=FALSE\u3067\u7b49\u5206\u6563\u3092\u4eee\u5b9a\u3057\u306a\u3044\n\n\tOne-way analysis of means\n\ndata:  value_ and group\nF = 4.4161, num df = 4, denom df = 70, p-value = 0.003059\n&gt; anova(aov(value ~ group)) # aov\u95a2\u6570\u306f\u7b49\u5206\u6563\u3092\u4eee\u5b9a\u3059\u308b\u5206\u6563\u5206\u6790\u3057\u304b\u5b9f\u884c\u3067\u304d\u306a\u3044\nAnalysis of Variance Table\n\nResponse: value\n          Df Sum Sq Mean Sq F value   Pr(&gt;F)   \ngroup      4 13.383  3.3458  4.4161 0.003059 **\nResiduals 70 53.035  0.7576                    \n---\nSignif. codes:  0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1<\/pre><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u7b49\u5206\u6563\u3092\u4eee\u5b9a\u3057\u306a\u3044\u5206\u6563\u5206\u6790\u306e\u5b9f\u884c<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/academic.oup.com\/biomet\/article\/38\/3-4\/330\/257875\">Welch, B.L. (1951) On the comparison of several mean values: an alternative approach. <em>Biometrika<\/em>, 38 (3-4), 330-336.<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$w_i = \\frac{n_i}{V_i}$$<br>$$\\hat{x} = \\frac{\\sum_i^k w_i \\bar{x}_i}{\\sum_i^k w_i}$$<br>\u3068\u3057\u307e\u3059\u3002\\(w_i = \\frac{n_i}{v_i}\\)\u306f\u7fa4\u5185\u306e\u4e0d\u504f\u6a19\u6e96\u8aa4\u5dee\\(\\frac{s_i}{\\sqrt{n_i}}\\)\u306e\u4e8c\u4e57\u306e\u9006\u6570\u3067\u3059\u3002\\(\\hat{x}\\)\u306f<strong>\u9006\u5206\u6563\u91cd\u307f\u4ed8\u304d\u5e73\u5747<\/strong>\uff08inverse-variance weighted mean\uff09\u307e\u305f\u306f<strong>\u5e73\u5747\u306e\u4e00\u822c\u5316\u6700\u5c0f\u4e8c\u4e57\u63a8\u5b9a\u91cf<\/strong>\uff08generalized least squares estimate of the mean\uff09\u3068\u547c\u3070\u308c\u308b\u91cf\u3067\u3059\u3002\\(n_i\\)\u304c\u5927\u304d\u304f\u3001\u5206\u6563\\(v_i\\)\u304c\u5c0f\u3055\u3044\u7fa4\u307b\u3069\u3001\u63a8\u5b9a\u91cf\uff08\u5168\u4f53\u5e73\u5747\uff09\u306b\u5927\u304d\u304f\u5bc4\u4e0e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u6bcd\u5e73\u5747\u304c\u5168\u3066\u7b49\u3057\u304f\u3001\u6a19\u672c\u30b5\u30a4\u30ba\uff08\\(n\\)\uff09\u304c\u5927\u304d\u3044\uff08\u3059\u306a\u308f\u3061\u7fa4\u5185\u81ea\u7531\u5ea6\u304c\u5927\u304d\u3044\uff09\u6642\u3001\u7d71\u8a08\u91cf$$\\sum_i^k w_i(\\bar{x}_i - \\hat{x})^2$$\u306f\u81ea\u7531\u5ea6\\(k-1\\)\u306e\\(\\chi^2\\)\u5206\u5e03\u306b\u5f93\u3044\u307e\u3059\u3002\u3053\u306e\u7d71\u8a08\u91cf\u306f<strong>\u7fa4\u5185\u5e73\u5747\\(x_i\\)\u306e\\(\\hat{x}\\)\u304b\u3089\u306e\u504f\u5dee\u306e\u91cd\u307f\u4ed8\u304d\u5e73\u65b9\u548c<\/strong>\u3067\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\\(k=2\\)\u306e\u6642<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\\(k=2\\)\u306e\u6642\u306f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$\\sum_{i=1}^k w_i(\\bar{x}_i - \\hat{x})^2 = w_1\\left(\\frac{w_1\\bar{x}_1 + w_2\\bar{x}_1 - w_1\\bar{x}_1 - w_2\\bar{x}_2}{w_1 + w_2}\\right)^2 + w_2\\left(\\frac{w_1\\bar{x}_2 + w_2\\bar{x}_2 - w_1\\bar{x}_1 - w_2\\bar{x}_2}{w_1 + w_2}\\right)^2$$<\/p>\n\n\n\n<p>$$= w_1 w_2^2 \\left(\\frac{\\bar{x}_1 - \\bar{x}_2}{w_1 + w_2}\\right)^2 + w_2 w_1^2 \\left(\\frac{ \\bar{x}_2 - \\bar{x}_1}{w_1 + w_2}\\right)^2&nbsp; = \\frac{w_1 w_2}{w_1+ w_2} (\\bar{x}_1 - \\bar{x}_2)^2 =\\frac{(\\bar{x}_1 - \\bar{x}_2)^2}{\\frac{s_1^2}{n_1} + \\frac{s_2^2}{n_2}} $$<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n<p>$$\\sum_{i=1}^k w_i(\\bar{x}_i - \\hat{x})^2&nbsp; = \\left( \\frac{\\bar{x}_1 - \\bar{x}_2}{\\sqrt{\\frac{V_1}{n_1}+\\frac{V_2}{n_2}}} \\right)^2 = v^2$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2\u3064\u306e\u7fa4\u306e\u6bcd\u5e73\u5747\u304c\u7b49\u3057\u3044\u6642\u306f\u3001\\(v\\)\u306e\u5206\u5b50\u306f\u6a19\u6e96\u504f\u5dee\\(\\sqrt{\\sigma_1^2\/n_1 + \\sigma_2^2\/n_2} \\)\u3092\u6301\u3064\u6b63\u898f\u504f\u5dee\u3002\\(v\\)\u306e\u5206\u6bcd\u306f\u6bcd\u5206\u6563\\(\\sigma^2\\)\u3092\u4e0d\u504f\u5206\u6563\\(s^2\\)\u3067\u7f6e\u304d\u5909\u3048\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u308b\u3053\u306e\u6a19\u6e96\u504f\u5dee\u306e\u63a8\u5b9a\u91cf\u3002\\(v\\)\u306fWelch\u306e<em>t<\/em>\u691c\u5b9a\u306b\u304a\u3051\u308b\u691c\u5b9a\u7d71\u8a08\u91cf<em>t<\/em>\u305d\u306e\u3082\u306e\u3067\u3059\u3002\\(\\chi^2\\)\u5206\u5e03\u306b\u5f93\u3046\u5909\u6570\u306e\u7dda\u5f62\u7d50\u5408\u3067\u3042\u308b\\(\\left(\\frac{s_1^2}{n_1} + \\frac{s_2^2}{n_2}\\right)\\) \u306e\u771f\u306e\u5206\u5e03\u3092\u89e3\u6790\u7684\u306b\u6c42\u3081\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u304c\u3001\u3053\u308c\u304c\u5358\u4e00\u306e\\(\\chi^2\\)\u5206\u5e03\u3067\u8fd1\u4f3c\u3067\u304d\u308b\u3068\u8003\u3048\u307e\u3059\u3002\u3053\u306e\\(\\chi^2\\)\u5206\u5e03\u306e\u81ea\u7531\u5ea6\u3068\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u56e0\u5b50\u306f\u3001\\(\\left(\\frac{s_1^2}{n_2} + \\frac{s_2^2}{n_2}\\right)\\) \u3068\u4e00\u6b21\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\uff08\u671f\u5f85\u5024\uff09\u3068\u4e8c\u6b21\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\uff08\u4e8c\u4e57\u306e\u671f\u5f85\u5024 = \u5206\u6563 + \u671f\u5f85\u5024\u306e\u4e8c\u4e57\uff09\u307e\u3067\u304c\u4e00\u81f4\u3059\u308b\u3088\u3046\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u7acb\u3066\u3066\u3001\u8fd1\u4f3c\u7684\u306b\u6c42\u3081\u307e\u3059\uff08Welch\u2013Satterthwaite\u306e\u5f0f\uff09\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\\(k&gt;2\\)\u306e\u6642<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">$$v^2 = \\frac{\\sum_{i=1}^k \\frac{w_i(\\bar{x}_i-\\hat{x})^2}{k-1}}{\\left[ 1 + \\frac{2(k-2)}{k^2-1}\\sum_{i=1}^k\\frac{1}{n_i-1}\\left(1-\\frac{w_i}{\\sum_{i=1}^k w_i}\\right)^2 \\right]}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u81ea\u7531\u5ea6\u306f<br>$$\\nu_1 = (k-1)$$<br>$$\\nu_2 = \\left[ \\frac{3}{(k^2-1)}\\sum_{i=1}^k\\frac{1}{n_i-1}\\left(1- \\frac{w_i}{\\sum_{i=1}^k w_i}\\right)^2 \\right]^{-1}$$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u8a08\u7b97<\/h4>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\">\n<table class=\"has-fixed-layout\" style=\"width: 100%; height: 130px;\">\n<tbody>\n<tr style=\"height: 20px;\">\n<td style=\"height: 20px; width: 6.54754%;\">\u7fa4<\/td>\n<td style=\"height: 20px; width: 15.4763%;\">\u30b5\u30a4\u30ba \\(n_i\\)<\/td>\n<td style=\"height: 20px; width: 19.5238%;\">\u5e73\u5747 \\(\\bar{x}_i\\)<\/td>\n<td style=\"height: 20px; width: 20.7143%;\">\u4e0d\u504f\u5206\u6563 \\(V_i\\)<\/td>\n<td style=\"height: 20px; width: 22.0238%;\">\u5e73\u5747\u5024\u306e\u63a8\u5b9a\u5206\u6563 \\(V_i\/n_i\\)<\/td>\n<td style=\"width: 15.7143%; height: 20px;\">\n<p>\\(w_i\\)<\/p>\n<p>(\\(n_i\/V_i)\\)<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"height: 20px; width: 6.54754%;\">A<\/td>\n<td style=\"height: 20px; width: 15.4763%;\">15<\/td>\n<td style=\"height: 20px; width: 19.5238%;\">51.1333<\/td>\n<td style=\"height: 20px; width: 20.7143%;\">1.891238<\/td>\n<td style=\"height: 20px; width: 22.0238%;\">0.12608254<\/td>\n<td style=\"width: 15.7143%; height: 20px;\">7.931312<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"height: 20px; width: 6.54754%;\">B<\/td>\n<td style=\"height: 20px; width: 15.4763%;\">15<\/td>\n<td style=\"height: 20px; width: 19.5238%;\">51.82667<\/td>\n<td style=\"height: 20px; width: 20.7143%;\">0.3078095<\/td>\n<td style=\"height: 20px; width: 22.0238%;\">0.02052063<\/td>\n<td style=\"width: 15.7143%; height: 20px;\">48.731436<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"height: 20px; width: 6.54754%;\">C<\/td>\n<td style=\"height: 20px; width: 15.4763%;\">15<\/td>\n<td style=\"height: 20px; width: 19.5238%;\">51.96667<\/td>\n<td style=\"height: 20px; width: 20.7143%;\">0.2580952<\/td>\n<td style=\"height: 20px; width: 22.0238%;\">0.01720635<\/td>\n<td style=\"width: 15.7143%; height: 20px;\">58.118081<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"height: 10px; width: 6.54754%;\">D<\/td>\n<td style=\"height: 10px; width: 15.4763%;\">15<\/td>\n<td style=\"height: 10px; width: 19.5238%;\">51.98667<\/td>\n<td style=\"height: 10px; width: 20.7143%;\">0.5955238<\/td>\n<td style=\"height: 10px; width: 22.0238%;\">0.0390159<\/td>\n<td style=\"width: 15.7143%; height: 10px;\">25.187910<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"height: 20px; width: 6.54754%;\">E<\/td>\n<td style=\"height: 20px; width: 15.4763%;\">15<\/td>\n<td style=\"height: 20px; width: 19.5238%;\">52.41333<\/td>\n<td style=\"height: 20px; width: 20.7143%;\">0.7355238<\/td>\n<td style=\"height: 20px; width: 22.0238%;\">0.04903492<\/td>\n<td style=\"width: 15.7143%; height: 20px;\">20.393629<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 6.54754%; height: 20px;\">\u8a08<\/td>\n<td style=\"width: 15.4763%; height: 20px;\">&nbsp;<\/td>\n<td style=\"width: 19.5238%; height: 20px;\">&nbsp;<\/td>\n<td style=\"width: 20.7143%; height: 20px;\">&nbsp;<\/td>\n<td style=\"width: 22.0238%; height: 20px;\">&nbsp;<\/td>\n<td style=\"width: 15.7143%; height: 20px;\">160.3624<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u3082\u3057\u5e73\u5747\u5024\u304c\u3059\u3079\u3066\u7b49\u3057\u3044\u306a\u3089\u3070\u3001\u5168\u4f53\u5e73\u5747\u306e\u6700\u826f\u306e\u63a8\u5b9a\u5024\u306f\\(\\hat{x} = (\\sum_{i=1}^k w_i\\bar{x}_i)\/(\\sum_{i=1}^k w_i) = 51.94186\\)\u3068\u306a\u308a\u307e\u3059\u3002\u7d71\u8a08\u91cf\u306e\u624b\u8a08\u7b97\u306b\u5fc5\u8981\u306a\u5024\u3092\u4e0b\u306e\u8868\u306b\u793a\u3057\u307e\u3059\u3002\u4efb\u610f\u306e\u539f\u70b9\u3068\u3057\u3066\\(x =52\\)\u304c\u4f7f\u7528\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\">\n<table class=\"has-fixed-layout\" style=\"width: 100%; height: 185px;\">\n<tbody>\n<tr style=\"height: 65px;\">\n<td style=\"width: 11.0714%; height: 65px;\">\u7fa4<\/td>\n<td style=\"width: 22.2619%; height: 65px;\">\\(w_i(\\bar{x}_i-52)\\)<\/td>\n<td style=\"width: 16.6667%; height: 65px;\">\\(w_i(\\bar{x}_i-52)^2\\)<\/td>\n<td style=\"width: 16.6667%; height: 65px;\">\\(w_i\/\\sum w_i\\)<\/td>\n<td style=\"width: 16.6667%; height: 65px;\">\\((1- w_i\/\\sum w_i)^2\\)<\/td>\n<td style=\"width: 16.6667%; height: 65px;\">\n<p>\\(\\frac{(1- w_i\/\\sum w_i)^2}{n_i-1}\\)<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 11.0714%; height: 20px;\">A<\/td>\n<td style=\"width: 22.2619%; height: 20px;\">-7.0324303<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">6.235421493<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.04945869<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.9035288<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.06453777<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 11.0714%; height: 20px;\">B<\/td>\n<td style=\"width: 22.2619%; height: 20px;\">-8.4467822<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">1.464108911<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.30388324<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.4845785<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.03461275<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 11.0714%; height: 20px;\">C<\/td>\n<td style=\"width: 22.2619%; height: 20px;\">-1.9372694<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.064575646<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.36241720<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.4065118<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.02903656<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 11.0714%; height: 20px;\">D<\/td>\n<td style=\"width: 22.2619%; height: 20px;\">-0.3358388<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.004477851<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.15706871<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.7105332<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.05075237<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 11.0714%; height: 20px;\">E<\/td>\n<td style=\"width: 22.2619%; height: 20px;\">8.4293668<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">3.484138288<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.12717216<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.7618284<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.05441632<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 11.0714%; height: 20px;\">\u8a08<\/td>\n<td style=\"width: 22.2619%; height: 20px;\">-9.322954<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">11.25272<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">&nbsp;<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">&nbsp;<\/td>\n<td style=\"width: 16.6667%; height: 20px;\">0.2333558<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">$$\\sum_{i=1}^k w_i(\\bar{x}_i - \\hat{x})^2 = 11.25272 - (9.322954)^2\/160.3624 = 10.71071$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(v^2\\)\u3092\u6c42\u3081\u308b\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$ v^2 = \\frac{1\/(k - 1) (10.71071) }{1 + \\frac{2(k-2)}{k^2-1}(0.2333558) } = \\frac{\\frac{1}{4}(10.71071)}{1+ \\frac{6}{24}(0.2333558)}=2.530076$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$\\nu_2 = \\left[\\frac{3}{24}(0.2333558)\\right]^{-1} = \\frac{8}{0.2333558} = 34.28241$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">R\u306epf\u95a2\u6570\u3067<em>p<\/em>\u5024\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p><pre class=\"prettyprint\">&gt; pf(2.530076, 4, 34.28241, lower.tail=FALSE)\n&#091;1&#093; 0.05823581\n&gt; oneway.test(value ~ group)\n\n\tOne-way analysis of means (not assuming equal variances)\n\ndata:  value_ and group\nF = 2.5301, num df = 4.000, denom df = 34.282, p-value = 0.05824<\/pre><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\uff08\u4e86\uff09<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u52c9\u5f37\u306e\u305f\u3081\u306e\u30e1\u30e2\u30022025\u5e746\u670825\u65e5\u66f4\u65b0\u3002 \u30c7\u30fc\u30bf #R\u5185 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