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  1. {
  2. "cells": [
  3. {
  4. "cell_type": "markdown",
  5. "metadata": {},
  6. "source": [
  7. "##The analysis of variance (ANOVA)\n",
  8. "***\n",
  9. "- Grand mean\n",
  10. " - $a$ = number of groups\n",
  11. "\n",
  12. " - $n$ = number of observations within each group\n",
  13. "\n",
  14. " - $Y$ = single observation\n",
  15. " \n",
  16. "$\\overline{\\overline{Y}}=\\Large\\frac{1}{an}\\sum \\limits_{a}\\sum \\limits_{n}Y$\n",
  17. "\n",
  18. "- *Sum of squares among groups*\n",
  19. " - (estimate of the variation among groups)\n",
  20. " \n",
  21. "$SS_{among}=n\\sum \\limits_{a}(\\overline{Y}-\\overline{\\overline{Y}})^{2}$\n",
  22. "\n",
  23. "- *Sum of squares within groups*\n",
  24. " - (estimate of the variation among observations within groups)\n",
  25. " \n",
  26. "$SS_{within}=\\sum \\limits_{a}\\sum \\limits_{n}(Y-\\overline{Y})^{2}$\n"
  27. ]
  28. },
  29. {
  30. "cell_type": "code",
  31. "execution_count": 1,
  32. "metadata": {
  33. "collapsed": false
  34. },
  35. "outputs": [
  36. {
  37. "data": {
  38. "text/html": [
  39. "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
  40. "<table border=\"1\" class=\"dataframe\">\n",
  41. " <thead>\n",
  42. " <tr style=\"text-align: right;\">\n",
  43. " <th></th>\n",
  44. " <th>Group</th>\n",
  45. " <th>Seed 1</th>\n",
  46. " <th>Seed 2</th>\n",
  47. " <th>Ancestor</th>\n",
  48. " <th>Dominant</th>\n",
  49. " </tr>\n",
  50. " </thead>\n",
  51. " <tbody>\n",
  52. " <tr>\n",
  53. " <th>0 </th>\n",
  54. " <td> 20k</td>\n",
  55. " <td> 1000</td>\n",
  56. " <td> 1100</td>\n",
  57. " <td> 1.615690</td>\n",
  58. " <td> 406.0690</td>\n",
  59. " </tr>\n",
  60. " <tr>\n",
  61. " <th>1 </th>\n",
  62. " <td> 20k</td>\n",
  63. " <td> 1000</td>\n",
  64. " <td> 1101</td>\n",
  65. " <td> 1.615690</td>\n",
  66. " <td> 455.7070</td>\n",
  67. " </tr>\n",
  68. " <tr>\n",
  69. " <th>2 </th>\n",
  70. " <td> 20k</td>\n",
  71. " <td> 1000</td>\n",
  72. " <td> 1102</td>\n",
  73. " <td> 1.615690</td>\n",
  74. " <td> 438.2780</td>\n",
  75. " </tr>\n",
  76. " <tr>\n",
  77. " <th>3 </th>\n",
  78. " <td> 20k</td>\n",
  79. " <td> 1000</td>\n",
  80. " <td> 1103</td>\n",
  81. " <td> 1.615690</td>\n",
  82. " <td> 453.4130</td>\n",
  83. " </tr>\n",
  84. " <tr>\n",
  85. " <th>4 </th>\n",
  86. " <td> 20k</td>\n",
  87. " <td> 1000</td>\n",
  88. " <td> 1104</td>\n",
  89. " <td> 1.615690</td>\n",
  90. " <td> 437.8480</td>\n",
  91. " </tr>\n",
  92. " <tr>\n",
  93. " <th>5 </th>\n",
  94. " <td> 20k</td>\n",
  95. " <td> 1001</td>\n",
  96. " <td> 1200</td>\n",
  97. " <td> 0.417722</td>\n",
  98. " <td> 455.8450</td>\n",
  99. " </tr>\n",
  100. " <tr>\n",
  101. " <th>6 </th>\n",
  102. " <td> 20k</td>\n",
  103. " <td> 1001</td>\n",
  104. " <td> 1201</td>\n",
  105. " <td> 0.417722</td>\n",
  106. " <td> 414.4760</td>\n",
  107. " </tr>\n",
  108. " <tr>\n",
  109. " <th>7 </th>\n",
  110. " <td> 20k</td>\n",
  111. " <td> 1001</td>\n",
  112. " <td> 1202</td>\n",
  113. " <td> 0.417722</td>\n",
  114. " <td> 484.0730</td>\n",
  115. " </tr>\n",
  116. " <tr>\n",
  117. " <th>8 </th>\n",
  118. " <td> 20k</td>\n",
  119. " <td> 1001</td>\n",
  120. " <td> 1203</td>\n",
  121. " <td> 0.417722</td>\n",
  122. " <td> 477.8950</td>\n",
  123. " </tr>\n",
  124. " <tr>\n",
  125. " <th>9 </th>\n",
  126. " <td> 20k</td>\n",
  127. " <td> 1001</td>\n",
  128. " <td> 1204</td>\n",
  129. " <td> 0.417722</td>\n",
  130. " <td> 459.4160</td>\n",
  131. " </tr>\n",
  132. " <tr>\n",
  133. " <th>10</th>\n",
  134. " <td> 20k</td>\n",
  135. " <td> 1002</td>\n",
  136. " <td> 1300</td>\n",
  137. " <td> 8.099260</td>\n",
  138. " <td> 18.6589</td>\n",
  139. " </tr>\n",
  140. " <tr>\n",
  141. " <th>11</th>\n",
  142. " <td> 20k</td>\n",
  143. " <td> 1002</td>\n",
  144. " <td> 1301</td>\n",
  145. " <td> 8.099260</td>\n",
  146. " <td> 18.8127</td>\n",
  147. " </tr>\n",
  148. " <tr>\n",
  149. " <th>12</th>\n",
  150. " <td> 20k</td>\n",
  151. " <td> 1002</td>\n",
  152. " <td> 1302</td>\n",
  153. " <td> 8.099260</td>\n",
  154. " <td> 18.7135</td>\n",
  155. " </tr>\n",
  156. " <tr>\n",
  157. " <th>13</th>\n",
  158. " <td> 20k</td>\n",
  159. " <td> 1002</td>\n",
  160. " <td> 1303</td>\n",
  161. " <td> 8.099260</td>\n",
  162. " <td> 19.0716</td>\n",
  163. " </tr>\n",
  164. " <tr>\n",
  165. " <th>14</th>\n",
  166. " <td> 20k</td>\n",
  167. " <td> 1002</td>\n",
  168. " <td> 1304</td>\n",
  169. " <td> 8.099260</td>\n",
  170. " <td> 18.6282</td>\n",
  171. " </tr>\n",
  172. " <tr>\n",
  173. " <th>15</th>\n",
  174. " <td> 20k</td>\n",
  175. " <td> 1003</td>\n",
  176. " <td> 1400</td>\n",
  177. " <td> 0.382609</td>\n",
  178. " <td> 462.7690</td>\n",
  179. " </tr>\n",
  180. " <tr>\n",
  181. " <th>16</th>\n",
  182. " <td> 20k</td>\n",
  183. " <td> 1003</td>\n",
  184. " <td> 1401</td>\n",
  185. " <td> 0.382609</td>\n",
  186. " <td> 461.3630</td>\n",
  187. " </tr>\n",
  188. " <tr>\n",
  189. " <th>17</th>\n",
  190. " <td> 20k</td>\n",
  191. " <td> 1003</td>\n",
  192. " <td> 1402</td>\n",
  193. " <td> 0.382609</td>\n",
  194. " <td> 368.4690</td>\n",
  195. " </tr>\n",
  196. " <tr>\n",
  197. " <th>18</th>\n",
  198. " <td> 20k</td>\n",
  199. " <td> 1003</td>\n",
  200. " <td> 1403</td>\n",
  201. " <td> 0.382609</td>\n",
  202. " <td> 29.5652</td>\n",
  203. " </tr>\n",
  204. " <tr>\n",
  205. " <th>19</th>\n",
  206. " <td> 20k</td>\n",
  207. " <td> 1003</td>\n",
  208. " <td> 1404</td>\n",
  209. " <td> 0.382609</td>\n",
  210. " <td> 30.4262</td>\n",
  211. " </tr>\n",
  212. " <tr>\n",
  213. " <th>20</th>\n",
  214. " <td> 20k</td>\n",
  215. " <td> 1004</td>\n",
  216. " <td> 1500</td>\n",
  217. " <td> 0.217002</td>\n",
  218. " <td> 24.2383</td>\n",
  219. " </tr>\n",
  220. " <tr>\n",
  221. " <th>21</th>\n",
  222. " <td> 20k</td>\n",
  223. " <td> 1004</td>\n",
  224. " <td> 1501</td>\n",
  225. " <td> 0.217002</td>\n",
  226. " <td> 246.3490</td>\n",
  227. " </tr>\n",
  228. " <tr>\n",
  229. " <th>22</th>\n",
  230. " <td> 20k</td>\n",
  231. " <td> 1004</td>\n",
  232. " <td> 1502</td>\n",
  233. " <td> 0.217002</td>\n",
  234. " <td> 27.7736</td>\n",
  235. " </tr>\n",
  236. " <tr>\n",
  237. " <th>23</th>\n",
  238. " <td> 20k</td>\n",
  239. " <td> 1004</td>\n",
  240. " <td> 1503</td>\n",
  241. " <td> 0.217002</td>\n",
  242. " <td> 28.6697</td>\n",
  243. " </tr>\n",
  244. " <tr>\n",
  245. " <th>24</th>\n",
  246. " <td> 20k</td>\n",
  247. " <td> 1004</td>\n",
  248. " <td> 1504</td>\n",
  249. " <td> 0.217002</td>\n",
  250. " <td> 23.5402</td>\n",
  251. " </tr>\n",
  252. " </tbody>\n",
  253. "</table>\n",
  254. "<p>25 rows × 5 columns</p>\n",
  255. "</div>"
  256. ],
  257. "text/plain": [
  258. " Group Seed 1 Seed 2 Ancestor Dominant\n",
  259. "0 20k 1000 1100 1.615690 406.0690\n",
  260. "1 20k 1000 1101 1.615690 455.7070\n",
  261. "2 20k 1000 1102 1.615690 438.2780\n",
  262. "3 20k 1000 1103 1.615690 453.4130\n",
  263. "4 20k 1000 1104 1.615690 437.8480\n",
  264. "5 20k 1001 1200 0.417722 455.8450\n",
  265. "6 20k 1001 1201 0.417722 414.4760\n",
  266. "7 20k 1001 1202 0.417722 484.0730\n",
  267. "8 20k 1001 1203 0.417722 477.8950\n",
  268. "9 20k 1001 1204 0.417722 459.4160\n",
  269. "10 20k 1002 1300 8.099260 18.6589\n",
  270. "11 20k 1002 1301 8.099260 18.8127\n",
  271. "12 20k 1002 1302 8.099260 18.7135\n",
  272. "13 20k 1002 1303 8.099260 19.0716\n",
  273. "14 20k 1002 1304 8.099260 18.6282\n",
  274. "15 20k 1003 1400 0.382609 462.7690\n",
  275. "16 20k 1003 1401 0.382609 461.3630\n",
  276. "17 20k 1003 1402 0.382609 368.4690\n",
  277. "18 20k 1003 1403 0.382609 29.5652\n",
  278. "19 20k 1003 1404 0.382609 30.4262\n",
  279. "20 20k 1004 1500 0.217002 24.2383\n",
  280. "21 20k 1004 1501 0.217002 246.3490\n",
  281. "22 20k 1004 1502 0.217002 27.7736\n",
  282. "23 20k 1004 1503 0.217002 28.6697\n",
  283. "24 20k 1004 1504 0.217002 23.5402\n",
  284. "\n",
  285. "[25 rows x 5 columns]"
  286. ]
  287. },
  288. "execution_count": 1,
  289. "metadata": {},
  290. "output_type": "execute_result"
  291. }
  292. ],
  293. "source": [
  294. "import numpy as np\n",
  295. "import pandas as pd\n",
  296. "import matplotlib.pyplot as plt\n",
  297. "\n",
  298. "from pandas import *\n",
  299. "\n",
  300. "df = DataFrame.from_csv('data.csv',index_col=False)\n",
  301. "df[:25]"
  302. ]
  303. },
  304. {
  305. "cell_type": "code",
  306. "execution_count": 2,
  307. "metadata": {
  308. "collapsed": false
  309. },
  310. "outputs": [
  311. {
  312. "name": "stdout",
  313. "output_type": "stream",
  314. "text": [
  315. "Grand mean = [251.2027240000001, 203.75499199999999, 243.52518800000007]\n"
  316. ]
  317. }
  318. ],
  319. "source": [
  320. "a = 5.0\n",
  321. "n = 5.0\n",
  322. "grand_mean = []\n",
  323. "for i in range(3):\n",
  324. " grand_mean.append((1.0/(a*n))*(df[(i*25):25+(i*25)].sum(numeric_only=True)[3]))\n",
  325. "print 'Grand mean =',grand_mean"
  326. ]
  327. },
  328. {
  329. "cell_type": "code",
  330. "execution_count": 3,
  331. "metadata": {
  332. "collapsed": false
  333. },
  334. "outputs": [
  335. {
  336. "name": "stdout",
  337. "output_type": "stream",
  338. "text": [
  339. "SS among = [825428.52578417759, 881710.70758277329, 826902.13975996012]\n"
  340. ]
  341. }
  342. ],
  343. "source": [
  344. "ss_among = []\n",
  345. "for i in range(3):\n",
  346. " group = 0\n",
  347. " for j in range(5):\n",
  348. " group += (df[(j*5):5+(j*5)].mean()[3] - grand_mean[i])**2\n",
  349. " ss_among.append(n*(group))\n",
  350. "print 'SS among =',ss_among"
  351. ]
  352. },
  353. {
  354. "cell_type": "code",
  355. "execution_count": 4,
  356. "metadata": {
  357. "collapsed": false
  358. },
  359. "outputs": [
  360. {
  361. "name": "stdout",
  362. "output_type": "stream",
  363. "text": [
  364. "SS within = [1210341.92795164, 3114449.8073218209, 4240090.8150265003]\n"
  365. ]
  366. }
  367. ],
  368. "source": [
  369. "ss_within = []\n",
  370. "for i in range(3):\n",
  371. " group = 0\n",
  372. " for j in range(5):\n",
  373. " for k in range(5):\n",
  374. " group += (df.iloc[k+(j*5)+(i*25),4] - df[(j*5)+(i*25):5+(j*5)+(i*25)].mean()[3])**2\n",
  375. " ss_within.append(n*(group))\n",
  376. "print 'SS within =',ss_within"
  377. ]
  378. },
  379. {
  380. "cell_type": "code",
  381. "execution_count": 5,
  382. "metadata": {
  383. "collapsed": false
  384. },
  385. "outputs": [
  386. {
  387. "name": "stdout",
  388. "output_type": "stream",
  389. "text": [
  390. "SS total = [2035770.4537358177, 3996160.514904594, 5066992.9547864608]\n"
  391. ]
  392. }
  393. ],
  394. "source": [
  395. "ss_total = []\n",
  396. "for i in range(3):\n",
  397. " ss_total.append(ss_among[i] + ss_within[i])\n",
  398. "print 'SS total =', ss_total"
  399. ]
  400. },
  401. {
  402. "cell_type": "code",
  403. "execution_count": null,
  404. "metadata": {
  405. "collapsed": true
  406. },
  407. "outputs": [],
  408. "source": []
  409. }
  410. ],
  411. "metadata": {
  412. "kernelspec": {
  413. "display_name": "Python 2",
  414. "language": "python",
  415. "name": "python2"
  416. },
  417. "language_info": {
  418. "codemirror_mode": {
  419. "name": "ipython",
  420. "version": 2
  421. },
  422. "file_extension": ".py",
  423. "mimetype": "text/x-python",
  424. "name": "python",
  425. "nbconvert_exporter": "python",
  426. "pygments_lexer": "ipython2",
  427. "version": "2.7.6"
  428. }
  429. },
  430. "nbformat": 4,
  431. "nbformat_minor": 0
  432. }
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