{"id":452,"date":"2019-01-08T20:25:15","date_gmt":"2019-01-08T11:25:15","guid":{"rendered":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/analysis-of-variance-in-r\/"},"modified":"2024-10-12T12:39:26","modified_gmt":"2024-10-12T03:39:26","slug":"analysis-of-variance-in-r","status":"publish","type":"post","link":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/analysis-of-variance-in-r\/","title":{"rendered":"R \u3067\u4e00\u5143\u914d\u7f6e\u5206\u6563\u5206\u6790\u3092\u884c\u3046\u65b9\u6cd5"},"content":{"rendered":"\n

R\u3067\u3001\u4e00\u5143\u914d\u7f6e\u5206\u6563\u5206\u6790\u3092 step by step \u3067\u8a08\u7b97\u3057\u3066\u307f\u305f\u3002<\/p>\n\n\n\n

lm() \u3068 Anova() \u3092\u4f7f\u3048\u3070\u3042\u3063\u3068\u3044\u3046\u9593\u3060\u304c\u3001\u5177\u4f53\u7684\u306a\u4e00\u3064\u4e00\u3064\u306e\u8a08\u7b97\u3092\u81ea\u5206\u3067\u7d44\u307f\u7acb\u3066\u3066\u307f\u308b\u3068\u3069\u3046\u304b\uff1f<\/p>\n\n\n\n

\u6559\u79d1\u66f8\u306e\u4f8b\u984c\u306b\u6cbf\u3063\u3066\u78ba\u8a8d\u3057\u305f\u3002<\/p>\n\n\n\n\n\n\n\n

R \u3067\u4e00\u5143\u914d\u7f6e\u5206\u6563\u5206\u6790\u3092\u5b9f\u884c\u3059\u308b\u305f\u3081\u306e\u4f8b\u984c\u30c7\u30fc\u30bf<\/h2>\n\n\n\n

\u30b5\u30f3\u30d7\u30eb\u30c7\u30fc\u30bf\uff08\u8a18\u4e8b\u6700\u4e0b\u6bb5 \u53c2\u8003\u66f8\u7c4d \u4f8b\u984c8.1 \uff09\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n

A1 <-<\/span> c<\/span>(<\/span>64<\/span>,<\/span>72<\/span>,<\/span>68<\/span>,<\/span>77<\/span>)<\/span>\nA2 <-<\/span> c<\/span>(<\/span>82<\/span>,<\/span>78<\/span>,<\/span>77<\/span>,<\/span>85<\/span>)<\/span>\nA3 <-<\/span> c<\/span>(<\/span>55<\/span>,<\/span>64<\/span>,<\/span>66<\/span>,<\/span>49<\/span>)<\/span>\n# \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3059\u308b\u3002<\/span>\nX.all <-<\/span> c<\/span>(<\/span>A1,<\/span>A2,<\/span>A3)<\/span>\nA.all <-<\/span> c<\/span>(<\/span>rep<\/span>(<\/span>\"A1\"<\/span>,<\/span>length<\/span>(<\/span>A1)),<\/span>rep<\/span>(<\/span>\"A2\"<\/span>,<\/span>length<\/span>(<\/span>A2)),<\/span>rep<\/span>(<\/span>\"A3\"<\/span>,<\/span>length<\/span>(<\/span>A3)))<\/span>\n<\/code><\/pre>\n\n\n\n
\u5143\u306e\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u3002<\/p>\n\n\n\n
><\/span> data.frame<\/span>(<\/span>A1,<\/span>A2,<\/span>A3)<\/span>\nA1 A2 A3\n1<\/span> 64<\/span> 82<\/span> 55<\/span>\n2<\/span> 72<\/span> 78<\/span> 64<\/span>\n3<\/span> 68<\/span> 77<\/span> 66<\/span>\n4<\/span> 77<\/span> 85<\/span> 49<\/span>\n<\/code><\/pre>\n\n\n\n
\u4ee5\u4e0b\u304c\u89e3\u6790\u7528\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u3002<\/p>\n\n\n\n
><\/span> data.frame<\/span>(<\/span>cbind<\/span>(<\/span>X.all,<\/span> A.all))<\/span>\nX.all A.all\n1<\/span>     64<\/span>    A1\n2<\/span>     72<\/span>    A1\n3<\/span>     68<\/span>    A1\n4<\/span>     77<\/span>    A1\n5<\/span>     82<\/span>    A2\n6<\/span>     78<\/span>    A2\n7<\/span>     77<\/span>    A2\n8<\/span>     85<\/span>    A2\n9<\/span>     55<\/span>    A3\n10<\/span>    64<\/span>    A3\n11<\/span>    66<\/span>    A3\n12<\/span>    49<\/span>    A3\n<\/code><\/pre>\n\n\n\n
R \u3067\u4e00\u5143\u914d\u7f6e\u5206\u6563\u5206\u6790 step by step<\/h2>\n\n\n\n
\u4e00\u5143\u914d\u7f6e\u5206\u6563\u5206\u6790\u306f\u3001\u5206\u6563\u5206\u6790\u3068\u306f\u8a00\u3046\u304c\u3001\u8981\u56e0\u306e\u30b0\u30eb\u30fc\u30d7\uff08A1, A2, A3\uff09\u3054\u3068\u306e\u5e73\u5747\u5024\u304c\u3001\u5168\u90e8\u540c\u3058\u3068\u3044\u3046\u5e30\u7121\u4eee\u8aac\u3092\u691c\u5b9a\u3059\u308b\u3001\u4e09\u7fa4\u4ee5\u4e0a\u306e\u5e73\u5747\u5024\u306e\u5dee\u306e\u691c\u5b9a\u3060\u3002<\/p>\n\n\n\n
\u6e96\u5099\u3068\u3057\u3066\u3001\u5168\u4f53\u5e73\u5747\uff08Xbar\uff09\u3001\u30b0\u30eb\u30fc\u30d7\u306e\u5e73\u5747\uff08Xjbar\uff09\u3001\u30b0\u30eb\u30fc\u30d7\u306e\u30b5\u30f3\u30d7\u30eb\u30b5\u30a4\u30ba\uff08r\uff09\u3001\u30b0\u30eb\u30fc\u30d7\u306e\u6570\uff08a\uff09\u3001\u5168\u4f53\u306e\u30b5\u30f3\u30d7\u30eb\u30b5\u30a4\u30ba\uff08N\uff09\u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n
\u6b21\u306b\u3001\u4e09\u3064\u306e\u5e73\u65b9\u548c\u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n
\u5168\u4f53\u5e73\u65b9\u548c\uff08SS\uff09\u3001\u30b0\u30eb\u30fc\u30d7\u9593\u306e\u5e73\u65b9\u548c\uff08SSA\uff09\u3001\u30b0\u30eb\u30fc\u30d7\u5185\u306e\u5e73\u65b9\u548c\uff08SSE\uff09\u3002<\/p>\n\n\n\n
SS \u306f\u3001\u304a\u306e\u304a\u306e\u306e\u30c7\u30fc\u30bf\u3068\u5168\u4f53\u5e73\u5747\u306e\u5dee\u306e\u4e8c\u4e57\u548c\u3002<\/p>\n\n\n\n
SSA\u306f\u3001\u30b0\u30eb\u30fc\u30d7\u306e\u5e73\u5747\u3068\u5168\u4f53\u5e73\u5747\u306e\u5dee\u306e\u4e8c\u4e57\u548c\u3092\u30b0\u30eb\u30fc\u30d7\u306e\u30b5\u30f3\u30d7\u30eb\u30b5\u30a4\u30ba\u500d\u3057\u305f\u3082\u306e\u3002<\/p>\n\n\n\n
SSE\u306f\u3001SS \u304b\u3089 SSA \u3092\u5f15\u3044\u305f\u3082\u306e\u3002<\/p>\n\n\n\n
\u3055\u3089\u306b\u3001\u81ea\u7531\u5ea6\u306e\u8a08\u7b97\u3002<\/p>\n\n\n\n
\u5168\u4f53\uff08nu\uff09\u306f N-1\u3001\u30b0\u30eb\u30fc\u30d7\u9593\u5e73\u65b9\u548c\u306e\u81ea\u7531\u5ea6\uff08nuA\uff09\u306f a-1\u3001\u30b0\u30eb\u30fc\u30d7\u5185\u5e73\u65b9\u548c\u306e\u81ea\u7531\u5ea6\u306f N-a \u3068\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n
\u305d\u3057\u3066\u3001\u5e73\u65b9\u548c\u3092\u81ea\u7531\u5ea6\u3067\u5272\u3063\u3066\u3001\u5e73\u5747\u5e73\u65b9\u548c\u3092\u7b97\u51fa\u3059\u308b\u3002<\/p>\n\n\n\n
\u30b0\u30eb\u30fc\u30d7\u9593\uff08VA\uff09\u306f SSA\/nuA\u3001\u30b0\u30eb\u30fc\u30d7\u5185\uff08VE\uff09\u306f SSE\/nuE \u3067\u8a08\u7b97\u3067\u304d\u308b\u3002<\/p>\n\n\n\n
\u6700\u5f8c\u306b\u3001VA\u3068VE\u306e\u6bd4\u3092\u53d6\u308b\u3002<\/p>\n\n\n\n
\u3053\u308c\u304c\u691c\u5b9a\u7d71\u8a08\u91cf F\u5024\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\u3053\u306eF\u5024\u3068nuA, nuE\u3092\u4f7f\u3063\u3066\u3001\u6709\u610f\u78ba\u7387\u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n
one.way.anova <-<\/span> function<\/span>(<\/span>X,<\/span>A){<\/span>\nXbar  <-<\/span> mean<\/span>(<\/span>X)<\/span>\nXjbar <-<\/span> tapply<\/span>(<\/span>X,<\/span> A,<\/span> mean)<\/span>\nr     <-<\/span> matrix<\/span>(<\/span>table<\/span>(<\/span>A))[<\/span>1<\/span>]<\/span>\na     <-<\/span> length<\/span>(<\/span>table<\/span>(<\/span>A))<\/span>\nN     <-<\/span> length<\/span>(<\/span>X)<\/span>\nSS    <-<\/span> sum<\/span>((<\/span>X-<\/span>Xbar)<\/span>^<\/span>2<\/span>)<\/span>\nSSA   <-<\/span> r*<\/span>sum<\/span>((<\/span>Xjbar-<\/span>Xbar)<\/span>^<\/span>2<\/span>)<\/span>\nSSE   <-<\/span> SS-<\/span>SSA\nnu    <-<\/span> N-<\/span>1<\/span>\nnuA   <-<\/span> a-<\/span>1<\/span>\nnuE   <-<\/span> N-<\/span>a\nVA    <-<\/span> SSA\/<\/span>nuA\nVE    <-<\/span> SSE\/<\/span>nuE\nF<\/span>     <-<\/span> VA\/<\/span>VE\npF    <-<\/span> pf<\/span>(<\/span>F<\/span>,<\/span> nuA,<\/span> nuE,<\/span> lower.tail=<\/span>FALSE<\/span>)<\/span>\nlist<\/span>(<\/span>c<\/span>(<\/span>SS_A=<\/span>SSA,<\/span> nu_A=<\/span>nuA,<\/span> V_A=<\/span>VA,<\/span> SS_E=<\/span>SSE,<\/span> nu_E=<\/span>nuE,<\/span> V_E=<\/span>VE,<\/span>\nF<\/span>=<\/span>F<\/span>,<\/span> p_F=<\/span>pF,<\/span> SS=<\/span>SS,<\/span> nu=<\/span>nu))<\/span>\n}<\/span>\none.way.anova1 <-<\/span> unlist<\/span>(<\/span>one.way.anova<\/span>(<\/span>X=<\/span>X.all,<\/span>A=<\/span>A.all))<\/span>\none.way.anova1\n<\/code><\/pre>\n\n\n\n
\u7d50\u679c\u306f\u3053\u3061\u3089\u3002<\/p>\n\n\n\n
><\/span> one.way.anova1\nSS_A         nu_A          V_A         SS_E         nu_E\n9.695000e+02<\/span> 2.000000e+00<\/span> 4.847500e+02<\/span> 3.227500e+02<\/span> 9.000000e+00<\/span>\nV_E            F<\/span>          p_F           SS           nu\n3.586111e+01<\/span> 1.351743e+01<\/span> 1.944638e-03<\/span> 1.292250e+03<\/span> 1.100000e+01<\/span>\n<\/code><\/pre>\n\n\n\n
F\u5024\u306f13.52\u3001p\u5024\u306f0.001945\u3067\u3001\u7d71\u8a08\u5b66\u7684\u6709\u610f\u3002<\/p>\n\n\n\n
\u3064\u307e\u308a\u3001A1, A2, A3\u306e\u4e09\u30b0\u30eb\u30fc\u30d7\u306e\u5e73\u5747\u306f\u7b49\u3057\u304f\u306a\u3044\u3068\u8a00\u3048\u308b\u3002<\/p>\n\n\n\n
\u3061\u306a\u307f\u306b\u4e09\u30b0\u30eb\u30fc\u30d7\u306e\u5e73\u5747\u5024\u306f\u3001\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
A3\u3060\u3051\u4f4e\u304f\u898b\u3048\u308b\u3002<\/p>\n\n\n\n
><\/span> tapply<\/span>(<\/span>X.all,<\/span> A.all,<\/span> mean)<\/span>\nA1    A2    A3\n70.25<\/span> 80.50<\/span> 58.50<\/span>\n<\/code><\/pre>\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>
R \u3067\u4e00\u5143\u914d\u7f6e\u5206\u6563\u5206\u6790\u3092 lm() \u3092\u4f7f\u3063\u3066\u5b9f\u884c\u3059\u308b<\/h2>\n\n\n\n
Anova() \u3092\u4f7f\u3046\u305f\u3081\u3001car \u30d1\u30c3\u30b1\u30fc\u30b8\u3092\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3057\u3066\u3044\u306a\u3051\u308c\u3070\u3001\u5148\u306b\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3059\u308b\u3002<\/p>\n\n\n\n
install.packages<\/span>(<\/span>\"car\"<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u306f\u4e00\u56de\u306e\u307f\u3067 OK<\/p>\n\n\n\n
X.all \u3092\u5f93\u5c5e\u5909\u6570\u5074\u306b\u3001A.all\u3092\u72ec\u7acb\u5909\u6570\u5074\u306b\u66f8\u3044\u3066 \u30c1\u30eb\u30c0 \uff08~\uff09\u3067\u7d50\u3076\u3002<\/p>\n\n\n\n
\u5206\u6563\u5206\u6790\u8868\u3092 Anova() \u51fa\u529b\u3055\u305b\u308b\u3002<\/p>\n\n\n\n
library<\/span>(<\/span>car)<\/span> # \u4f7f\u3046\u3068\u304d\u306f\u3053\u306e library \u3067\u547c\u3073\u51fa\u3059<\/span>\nlm1 <-<\/span> lm<\/span>(<\/span>X.all ~<\/span> A.all)<\/span>\nAnova<\/span>(<\/span>lm1)<\/span>\n<\/code><\/pre>\n\n\n\n
\u7d50\u679c\u306f\u3053\u3061\u3089\u3002<\/p>\n\n\n\n
><\/span> Anova<\/span>(<\/span>lm1)<\/span>\nAnova Table <\/span>(<\/span>Type II tests)<\/span>\nResponse:<\/span> X.all\nSum Sq Df F<\/span> value   Pr<\/span>(<\/span>><\/span>F<\/span>)<\/span>\nA.all     969.50<\/span>  2<\/span>  13.517<\/span> 0.001945<\/span> **<\/span>\nResiduals 322.75<\/span>  9<\/span>\n---<\/span>\nSignif. codes:<\/span>  0<\/span> \u2018***<\/span>\u2019 0.001<\/span> \u2018**<\/span>\u2019 0.01<\/span> \u2018*<\/span>\u2019 0.05<\/span> \u2018.\u2019 0.1<\/span> \u2018 \u2019 1<\/span>\n<\/code><\/pre>\n\n\n\n
A.all \u306e Sum Sq\uff08Sum of Square\uff1d\u5e73\u65b9\u548c\uff09\u304c\u3001SSA\u306b\u4e00\u81f4\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
\u307e\u305f\u3001Residuals \u306e Sum Sq \u304c SSE \u306b\u4e00\u81f4\u3057\u3066\u3044\u308b\u306e\u304c\u308f\u304b\u308b\u3002<\/p>\n\n\n\n
F\u5024\u3082p\u5024\u3082\u540c\u4e00\u3060\u3002<\/p>\n\n\n\n
EZR \u3067\u4e00\u5143\u914d\u7f6e\u5206\u6563\u5206\u6790\u3092\u884c\u3046\u65b9\u6cd5<\/h2>\n\n\n\n
\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u306b\u66f8\u304d\u51fa\u3059<\/h3>\n\n\n\n
\u4e0a\u8a18\u306e\u4f8b\u984c\u30c7\u30fc\u30bf\u3092CSV\u30d5\u30a1\u30a4\u30eb\u306b\u66f8\u304d\u51fa\u3059\u3002<\/p>\n\n\n\n
A1 <-<\/span> c<\/span>(<\/span>64<\/span>,<\/span>72<\/span>,<\/span>68<\/span>,<\/span>77<\/span>)<\/span>\nA2 <-<\/span> c<\/span>(<\/span>82<\/span>,<\/span>78<\/span>,<\/span>77<\/span>,<\/span>85<\/span>)<\/span>\nA3 <-<\/span> c<\/span>(<\/span>55<\/span>,<\/span>64<\/span>,<\/span>66<\/span>,<\/span>49<\/span>)<\/span>\nX.all <-<\/span> c<\/span>(<\/span>A1,<\/span>A2,<\/span>A3)<\/span>\nA.all <-<\/span> c<\/span>(<\/span>rep<\/span>(<\/span>\"A1\"<\/span>,<\/span>length<\/span>(<\/span>A1)),<\/span>rep<\/span>(<\/span>\"A2\"<\/span>,<\/span>length<\/span>(<\/span>A2)),<\/span>rep<\/span>(<\/span>\"A3\"<\/span>,<\/span>length<\/span>(<\/span>A3)))<\/span>\ntab <-<\/span> data.frame<\/span>(<\/span>cbind<\/span>(<\/span>X.all,<\/span> A.all))<\/span>\nwrite.csv<\/span>(<\/span>tab,<\/span> \"data\/anova-data1.csv\"<\/span>,<\/span> row.names=<\/span>FALSE<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
\u3082\u3057\u304f\u306f\u3053\u308c\u3068\u540c\u3058\u5f62\u306bExcel\u306b\u5165\u529b\u3057\u3066CSV\u30d5\u30a1\u30a4\u30eb\u5f62\u5f0f\u3067\u4fdd\u5b58\u3059\u308b\u3002<\/p>\n\n\n\n
   X.all A.all\n1<\/span>     64<\/span>    A1\n2<\/span>     72<\/span>    A1\n3<\/span>     68<\/span>    A1\n4<\/span>     77<\/span>    A1\n5<\/span>     82<\/span>    A2\n6<\/span>     78<\/span>    A2\n7<\/span>     77<\/span>    A2\n8<\/span>     85<\/span>    A2\n9<\/span>     55<\/span>    A3\n10<\/span>    64<\/span>    A3\n11<\/span>    66<\/span>    A3\n12<\/span>    49<\/span>    A3\n<\/code><\/pre>\n\n\n\n
\u300c3\u7fa4\u4ee5\u4e0a\u306e\u9593\u306e\u5e73\u5747\u5024\u306e\u6bd4\u8f03\u300d\u3092\u7528\u3044\u3066\u89e3\u6790\u3059\u308b<\/h3>\n\n\n\n
anova-data1.csv\u3092EZR\u306b\u8aad\u307f\u8fbc\u3093\u3060\u5f8c\u3001\u300c\u7d71\u8a08\u89e3\u6790\u300d\u2192\u300c\u9023\u7d9a\u5909\u6570\u306e\u89e3\u6790\u300d\u2192\u300c3\u7fa4\u4ee5\u4e0a\u306e\u9593\u306e\u5e73\u5747\u5024\u306e\u6bd4\u8f03\uff08\u4e00\u5143\u914d\u7f6e\u5206\u6563\u5206\u6790 one-way ANOVA\uff09\u300d\u3092\u9078\u629e\u3059\u308b\u3002<\/p>\n\n\n\n
X.all\u3092\u76ee\u7684\u5909\u6570\u306b\u3057\u3066\u3001A.all\u3092\u6bd4\u8f03\u3059\u308b\u7fa4\u306b\u3057\u3066\u3001\u8a08\u7b97\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u7d50\u679c\u304c\u51fa\u529b\u3055\u308c\u308b\u3002<\/p>\n\n\n\n
lm() & Anova() \u306e\u7d50\u679c\u3068\u540c\u3058\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
><\/span> summary<\/span>(<\/span>AnovaModel.1)<\/span>\nDf Sum Sq Mean Sq F<\/span> value  Pr<\/span>(<\/span>><\/span>F<\/span>)<\/span>\nfactor<\/span>(<\/span>A.all)<\/span>  2<\/span>  969.5<\/span>   484.7<\/span>   13.52<\/span> 0.00194<\/span> **<\/span>\nResiduals      9<\/span>  322.7<\/span>    35.9<\/span>\n---<\/span>\nSignif. codes:<\/span>  0<\/span> '***'<\/span> 0.001<\/span> '**'<\/span> 0.01<\/span> '*'<\/span> 0.05<\/span> '.'<\/span> 0.1<\/span> ' '<\/span> 1<\/span>\n<\/code><\/pre>\n\n\n\n
EZR\u3067\u5206\u6563\u5206\u6790\u3010\u7121\u6599\u7d71\u8a08\u30bd\u30d5\u30c8EZR\u3067\u7c21\u5358\u7d71\u8a08\u3011\u3010\u52d5\u753b\u3011<\/h4>\n\n\n\n