{"id":445,"date":"2019-05-02T16:00:00","date_gmt":"2019-05-02T07:00:00","guid":{"rendered":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/wilson-score-method-in-r\/"},"modified":"2024-12-29T18:09:17","modified_gmt":"2024-12-29T09:09:17","slug":"wilson-score-method-in-r","status":"publish","type":"post","link":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/wilson-score-method-in-r\/","title":{"rendered":"R \u3067\u30a6\u30a3\u30eb\u30bd\u30f3\u306e\u30b9\u30b3\u30a2\u6cd5\u3088\u308b\u4fe1\u983c\u533a\u9593\u3092\u8a08\u7b97\u3059\u308b\u65b9\u6cd5"},"content":{"rendered":"\n

\u70b9\u63a8\u5b9a\u5024\u304c100%\u306b\u8fd1\u3044\u3068\u304d\u3001\u3088\u308a\u9069\u5207\u306b\u5272\u5408\u306e\u4fe1\u983c\u533a\u9593\u3092\u8a08\u7b97\u3059\u308b\u65b9\u6cd5 Wilson’s score method\uff08\u30a6\u30a3\u30eb\u30bd\u30f3\u306e\u30b9\u30b3\u30a2\u6cd5\uff09\u306e\u89e3\u8aac\u3002<\/p>\n\n\n\n\n\n\n\n

\u5272\u5408\u306e\u4fe1\u983c\u533a\u9593\u306f\u8fd1\u4f3c\u3092\u4f7f\u3063\u3066\u3044\u308b<\/h2>\n\n\n\n

\u5272\u5408\u306e\u4fe1\u983c\u533a\u9593\u306f\u4ee5\u4e0b\u306e\u5f0f\u3067\u8a08\u7b97\u3067\u304d\u308b\u3002<\/p>\n\n\n\n

\u3053\u308c\u306f\u6b63\u898f\u5206\u5e03\u3092\u8fd1\u4f3c\u7684\u306b\u4f7f\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

$$ \\hat{p} \\pm z_{\\alpha\/2} \\sqrt{\\frac{\\hat{p}(1 – \\hat{p})}{n}} $$<\/p>\n\n\n\n\n\n\n\n

100\u4f8b\u306750%\u306e\u70b9\u63a8\u5b9a\u5024\u3060\u3063\u305f\u3068\u3059\u308b\u3068\u300195%\u4fe1\u983c\u533a\u9593\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n

$$ 0.5 \\pm 1.96 \\sqrt{\\frac{0.5(1 – 0.5)}{100}} = (0.402, 0.598) $$<\/p>\n\n\n\n\n\n\n\n

\u70b9\u63a8\u5b9a\u5024\u304c100%\u306b\u8fd1\u3044\u5834\u5408\u306f\u4fe1\u983c\u533a\u9593\u306f\u3069\u3046\u306a\u308b\u304b\uff1f<\/h2>\n\n\n\n

\u70b9\u63a8\u5b9a\u5024\u304c97%\u306e\u5834\u5408\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3002<\/p>\n\n\n\n

\u3053\u306e\u5834\u5408\u4fe1\u983c\u533a\u9593\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u3001\u9069\u5207\u306b\u8a08\u7b97\u3055\u308c\u306a\u3044\u3002<\/p>\n\n\n\n

\u4fe1\u983c\u533a\u9593\u306e\u4e0a\u9650\u304c1\u3092\u8d85\u3048\u3066\u3044\u308b\u3002<\/p>\n\n\n\n

$$ 0.97 \\pm 1.96 \\sqrt{\\frac{0.97(1 – 0.97)}{100}} = (0.9365649, 1.0034351) $$<\/p>\n\n\n\n\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>

\u70b9\u63a8\u5b9a\u5024\u304c100%\u306b\u8fd1\u3044\u5834\u5408\u306f\u30a6\u30a3\u30eb\u30bd\u30f3\u306e\u30b9\u30b3\u30a2\u6cd5\u3092\u4f7f\u3046<\/h2>\n\n\n\n

\u70b9\u63a8\u5b9a\u5024\u304c100%\u306b\u8fd1\u3044\u5834\u5408\u306f\u30a6\u30a3\u30eb\u30bd\u30f3\u306e\u30b9\u30b3\u30a2\u6cd5 Wilson’s score method \u3092\u4f7f\u3046\u3068\u9069\u5207\u306a\u4fe1\u983c\u533a\u9593\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

Wilson’s score method\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n

$$ \\frac{\\hat{p} + \\frac{z_{\\alpha\/2}^2}{2n}}{1 + \\frac{z_{\\alpha\/2}^2}{n}} \\pm \\frac{z_{\\alpha\/2}}{1 + \\frac{z_{\\alpha\/2}^2}{n}} \\sqrt{\\frac{\\hat{p}(1 – \\hat{p})}{n} + \\frac{z_{\\alpha\/2}^2}{4n^2}} $$<\/p>\n\n\n\n\n\n\n\n

R \u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3057\u3066\u307f\u308b\u3002<\/p>\n\n\n\n

p.hat <-<\/span> 0.97<\/span>\nalpha <-<\/span> 0.05<\/span>\nz <-<\/span> qnorm<\/span>(<\/span>alpha\/<\/span>2<\/span>,<\/span> lower.tail=<\/span>FALSE<\/span>)<\/span>\nn <-<\/span> 100<\/span>\n(<\/span>p.hat+<\/span>z^<\/span>2<\/span>\/<\/span>(<\/span>2<\/span>*<\/span>n))<\/span>\/<\/span>(<\/span>1<\/span>+<\/span>z^<\/span>2<\/span>\/<\/span>n)<\/span> +<\/span> c<\/span>(<\/span>-<\/span>1<\/span>,<\/span> 1<\/span>)<\/span>*<\/span> z\/<\/span>(<\/span>1<\/span>+<\/span>z^<\/span>2<\/span>\/<\/span>n)<\/span>*<\/span>sqrt<\/span>((<\/span>p.hat*<\/span>(<\/span>1<\/span>-<\/span>p.hat))<\/span>\/<\/span>n +<\/span> z^<\/span>2<\/span>\/<\/span>(<\/span>4<\/span>*<\/span>n^<\/span>2<\/span>))<\/span>\n<\/code><\/pre>\n\n\n\n
\u8a08\u7b97\u7d50\u679c\u306f\u3001\uff080.9154806, 0.9897455\uff09\u3067\u3001\u4fe1\u983c\u533a\u9593\u4e0a\u9650\u304c\uff11\u3092\u8d85\u3048\u3066\u304a\u3089\u305a\u3001\u6b63\u898f\u8fd1\u4f3c\u3088\u308a\u9069\u5207\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
><\/span> (<\/span>p.hat+<\/span>z^<\/span>2<\/span>\/<\/span>(<\/span>2<\/span>*<\/span>n))<\/span>\/<\/span>(<\/span>1<\/span>+<\/span>z^<\/span>2<\/span>\/<\/span>n)<\/span> +<\/span> c<\/span>(<\/span>-<\/span>1<\/span>,<\/span> 1<\/span>)<\/span>*<\/span> z\/<\/span>(<\/span>1<\/span>+<\/span>z^<\/span>2<\/span>\/<\/span>n)<\/span>*<\/span>sqrt<\/span>((<\/span>p.hat*<\/span>(<\/span>1<\/span>-<\/span>p.hat))<\/span>\/<\/span>n +<\/span> z^<\/span>2<\/span>\/<\/span>(<\/span>4<\/span>*<\/span>n^<\/span>2<\/span>))<\/span>\n[<\/span>1<\/span>]<\/span> 0.9154806<\/span> 0.9897455<\/span>\n<\/code><\/pre>\n\n\n\n
\u70b9\u63a8\u5b9a\u5024\u304c0\u306b\u8fd1\u3044\u3068\u304d\u306f\u3069\u3046\u306a\u308b\u304b\uff1f<\/h2>\n\n\n\n
\u70b9\u63a8\u5b9a\u5024\u304c0\u306b\u8fd1\u3044\u3068\u304d\u3001\u4f8b\u3048\u3070\u3001100\u4f8b\u4e2d3\u4f8b\u30013%\u306e95%\u306e\u4fe1\u983c\u533a\u9593\u306f\u3069\u3046\u306a\u308b\u304b\uff1f<\/p>\n\n\n\n
\u6b63\u898f\u8fd1\u4f3c\u3092\u4f7f\u3046\u3068\u3001\u4ee5\u4e0b\u306e\u901a\u308a\u4fe1\u983c\u533a\u9593\u306e\u4e0b\u9650\u304c0\u3092\u4e0b\u56de\u3063\u3066\u3057\u307e\u3046\u3002<\/p>\n\n\n\n
><\/span> 0.03<\/span> +<\/span> c<\/span>(<\/span>-<\/span>1<\/span>,<\/span>1<\/span>)<\/span>*<\/span>1.96<\/span>*<\/span>sqrt<\/span>(<\/span>0.03<\/span>*<\/span>(<\/span>1<\/span>-<\/span>0.03<\/span>)<\/span>\/<\/span>100<\/span>)<\/span>\n[<\/span>1<\/span>]<\/span> -<\/span>0.003435095<\/span>  0.063435095<\/span>\n<\/code><\/pre>\n\n\n\n
Wilson’s score method\u3092\u4f7f\u3046\u3068\u4fe1\u983c\u533a\u9593\u4e0b\u9650\u304c0\u3092\u4e0b\u56de\u3089\u306a\u3044\u7d50\u679c\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n\n\n\n
p.hat <-<\/span> 0.03<\/span>\nalpha <-<\/span> 0.05<\/span>\nz <-<\/span> qnorm<\/span>(<\/span>alpha\/<\/span>2<\/span>,<\/span> lower.tail=<\/span>FALSE<\/span>)<\/span>\nn <-<\/span> 100<\/span>\n(<\/span>p.hat+<\/span>z^<\/span>2<\/span>\/<\/span>(<\/span>2<\/span>*<\/span>n))<\/span>\/<\/span>(<\/span>1<\/span>+<\/span>z^<\/span>2<\/span>\/<\/span>n)<\/span> +<\/span> c<\/span>(<\/span>-<\/span>1<\/span>,<\/span> 1<\/span>)<\/span>*<\/span> z\/<\/span>(<\/span>1<\/span>+<\/span>z^<\/span>2<\/span>\/<\/span>n)<\/span>*<\/span>sqrt<\/span>((<\/span>p.hat*<\/span>(<\/span>1<\/span>-<\/span>p.hat))<\/span>\/<\/span>n +<\/span> z^<\/span>2<\/span>\/<\/span>(<\/span>4<\/span>*<\/span>n^<\/span>2<\/span>))<\/span>\n<\/code><\/pre>\n\n\n\n
\u7d50\u679c\u306f\u30010.01\u304b\u30890.08\u3068\u306a\u3063\u305f\u3002<\/p>\n\n\n\n
><\/span> (<\/span>p.hat+<\/span>z^<\/span>2<\/span>\/<\/span>(<\/span>2<\/span>*<\/span>n))<\/span>\/<\/span>(<\/span>1<\/span>+<\/span>z^<\/span>2<\/span>\/<\/span>n)<\/span> +<\/span> c<\/span>(<\/span>-<\/span>1<\/span>,<\/span> 1<\/span>)<\/span>*<\/span> z\/<\/span>(<\/span>1<\/span>+<\/span>z^<\/span>2<\/span>\/<\/span>n)<\/span>*<\/span>sqrt<\/span>((<\/span>p.hat*<\/span>(<\/span>1<\/span>-<\/span>p.hat))<\/span>\/<\/span>n +<\/span> z^<\/span>2<\/span>\/<\/span>(<\/span>4<\/span>*<\/span>n^<\/span>2<\/span>))<\/span>\n[<\/span>1<\/span>]<\/span> 0.01025452<\/span> 0.08451936<\/span>\n<\/code><\/pre>\n\n\n\n
\u5272\u5408\u306e\u4fe1\u983c\u533a\u9593\u30a8\u30af\u30bb\u30eb\u8a08\u7b97\u6a5f<\/h2>\n\n\n\n
\u30a8\u30af\u30bb\u30eb\u30d5\u30a1\u30a4\u30eb\u3092\u4f5c\u6210\u3057\u305f\u3002<\/p>\n\n\n\n
\u826f\u3051\u308c\u3070\u3069\u3046\u305e\u3002<\/p>\n\n\n\n
\u5272\u5408\u306e\u4fe1\u983c\u533a\u9593\u8a08\u7b97\u6a5f\u3010\u30a8\u30af\u30bb\u30eb\u8a08\u7b97\u6a5f\u3011 | TKER SHOP<\/a><\/p>\n\n\n\n
\u4f7f\u3044\u65b9\u52d5\u753b\u3092\u4f5c\u6210\u3057\u305f\u3002<\/p>\n\n\n\n
\u3053\u3061\u3089\u3082\u3088\u3051\u308c\u3070\u3002<\/p>\n\n\n\n