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  1. <!DOCTYPE html>
  2. <html class="TridactylThemeDark" lang="en"><head>
  3. <meta http-equiv="content-type" content="text/html; charset=UTF-8"><style class="darkreader darkreader--fallback" media="screen"></style><style class="darkreader darkreader--text" media="screen"></style><style class="darkreader darkreader--invert" media="screen">.jfk-bubble.gtx-bubble, .captcheck_answer_label > input + img, span#closed_text > img[src^="https://www.gstatic.com/images/branding/googlelogo"], span[data-href^="https://www.hcaptcha.com/"] > #icon, #bit-notification-bar-iframe, ::-webkit-calendar-picker-indicator {
  4.     filter: invert(100%) hue-rotate(180deg) contrast(90%) !important;
  5. }</style><style class="darkreader darkreader--inline" media="screen">[data-darkreader-inline-bgcolor] {
  6.   background-color: var(--darkreader-inline-bgcolor) !important;
  7. }
  8. [data-darkreader-inline-bgimage] {
  9.   background-image: var(--darkreader-inline-bgimage) !important;
  10. }
  11. [data-darkreader-inline-border] {
  12.   border-color: var(--darkreader-inline-border) !important;
  13. }
  14. [data-darkreader-inline-border-bottom] {
  15.   border-bottom-color: var(--darkreader-inline-border-bottom) !important;
  16. }
  17. [data-darkreader-inline-border-left] {
  18.   border-left-color: var(--darkreader-inline-border-left) !important;
  19. }
  20. [data-darkreader-inline-border-right] {
  21.   border-right-color: var(--darkreader-inline-border-right) !important;
  22. }
  23. [data-darkreader-inline-border-top] {
  24.   border-top-color: var(--darkreader-inline-border-top) !important;
  25. }
  26. [data-darkreader-inline-boxshadow] {
  27.   box-shadow: var(--darkreader-inline-boxshadow) !important;
  28. }
  29. [data-darkreader-inline-color] {
  30.   color: var(--darkreader-inline-color) !important;
  31. }
  32. [data-darkreader-inline-fill] {
  33.   fill: var(--darkreader-inline-fill) !important;
  34. }
  35. [data-darkreader-inline-stroke] {
  36.   stroke: var(--darkreader-inline-stroke) !important;
  37. }
  38. [data-darkreader-inline-outline] {
  39.   outline-color: var(--darkreader-inline-outline) !important;
  40. }
  41. [data-darkreader-inline-stopcolor] {
  42.   stop-color: var(--darkreader-inline-stopcolor) !important;
  43. }</style><style class="darkreader darkreader--variables" media="screen">:root {
  44.    --darkreader-neutral-background: #131516;
  45.    --darkreader-neutral-text: #d8d4cf;
  46.    --darkreader-selection-background: #004daa;
  47.    --darkreader-selection-text: #e8e6e3;
  48. }</style><style class="darkreader darkreader--root-vars" media="screen"></style><style class="darkreader darkreader--user-agent" media="screen">html {
  49.     background-color: #181a1b !important;
  50. }
  51. html {
  52.     color-scheme: dark !important;
  53. }
  54. html, body, input, textarea, select, button, dialog {
  55.     background-color: #181a1b;
  56. }
  57. html, body, input, textarea, select, button {
  58.     border-color: #736b5e;
  59.     color: #e8e6e3;
  60. }
  61. a {
  62.     color: #3391ff;
  63. }
  64. table {
  65.     border-color: #545b5e;
  66. }
  67. ::placeholder {
  68.     color: #b2aba1;
  69. }
  70. input:-webkit-autofill,
  71. textarea:-webkit-autofill,
  72. select:-webkit-autofill {
  73.     background-color: #404400 !important;
  74.     color: #e8e6e3 !important;
  75. }
  76. ::-webkit-scrollbar {
  77.     background-color: #202324;
  78.     color: #aba499;
  79. }
  80. ::-webkit-scrollbar-thumb {
  81.     background-color: #454a4d;
  82. }
  83. ::-webkit-scrollbar-thumb:hover {
  84.     background-color: #575e62;
  85. }
  86. ::-webkit-scrollbar-thumb:active {
  87.     background-color: #484e51;
  88. }
  89. ::-webkit-scrollbar-corner {
  90.     background-color: #181a1b;
  91. }
  92. * {
  93.     scrollbar-color: #454a4d #202324;
  94. }
  95. ::selection {
  96.     background-color: #004daa !important;
  97.     color: #e8e6e3 !important;
  98. }
  99. ::-moz-selection {
  100.     background-color: #004daa !important;
  101.     color: #e8e6e3 !important;
  102. }</style><script src="test.firefox_files/polyfill.min.js"></script><meta name="darkreader" content="07b83c52e9a44679bb53f234abcdf7aa"><style class="darkreader darkreader--override" media="screen">.vimvixen-hint {
  103.     background-color: #7b5300 !important;
  104.     border-color: #d8b013 !important;
  105.     color: #f3e8c8 !important;
  106. }
  107. #vimvixen-console-frame {
  108.     color-scheme: light !important
  109. }
  110. ::placeholder {
  111.     opacity: 0.5 !important;
  112. }
  113. #edge-translate-panel-body,
  114. .MuiTypography-body1,
  115. .nfe-quote-text {
  116.     color: var(--darkreader-neutral-text) !important;
  117. }
  118. gr-main-header {
  119.     background-color: #0f3a48 !important;
  120. }
  121. .tou-z65h9k,
  122. .tou-mignzq,
  123. .tou-1b6i2ox,
  124. .tou-lnqlqk {
  125.     background-color: var(--darkreader-neutral-background) !important;
  126. }
  127. .tou-75mvi {
  128.     background-color: #032029 !important;
  129. }
  130. .tou-ta9e87,
  131. .tou-1w3fhi0,
  132. .tou-1b8t2us,
  133. .tou-py7lfi,
  134. .tou-1lpmd9d,
  135. .tou-1frrtv8,
  136. .tou-17ezmgn {
  137.     background-color: #0a0a0a !important;
  138. }
  139. .tou-uknfeu {
  140.     background-color: #231603 !important;
  141. }
  142. .tou-6i3zyv {
  143.     background-color: #19576c !important;
  144. }
  145. div.mermaid-viewer-control-panel .btn {
  146.   fill: var(--darkreader-neutral-text);
  147.   background-color: var(--darkreader-neutral-background);
  148. }
  149. svg g rect.er {
  150.   fill: var(--darkreader-neutral-background) !important;
  151. }
  152. svg g rect.er.entityBox {
  153.   fill: var(--darkreader-neutral-background) !important;
  154. }
  155. svg g rect.er.attributeBoxOdd {
  156.   fill: var(--darkreader-neutral-background) !important;
  157. }
  158. svg g rect.er.attributeBoxEven {
  159.   fill-opacity: 0.8 !important;
  160.   fill: var(--darkreader-selection-background);
  161. }
  162. svg rect.er.relationshipLabelBox {
  163.   fill: var(--darkreader-neutral-background) !important;
  164. }
  165. svg g g.nodes rect, svg g g.nodes polygon {
  166.   fill: var(--darkreader-neutral-background) !important;
  167. }
  168. svg g rect.task {
  169.   fill: var(--darkreader-selection-background) !important;
  170. }
  171. svg line.messageLine0, svg line.messageLine1 {
  172.   stroke: var(--darkreader-neutral-text) !important;
  173. }
  174. div.mermaid .actor {
  175.   fill: var(--darkreader-neutral-background) !important;
  176. }
  177. embed[type="application/pdf"] { filter: invert(100%) contrast(90%); }</style>
  178. <script id="MathJax-script" async="" src="test.firefox_files/tex-mml-chtml.js"></script>
  179.  
  180. <meta charset="UTF-8">
  181. <meta http-equiv="X-UA-Compatible" content="IE=edge">
  182. <meta name="viewport" content="width=device-width, initial-scale=1.0">
  183. <meta name="generator" content="Asciidoctor 2.0.18">
  184. <title>Introduction</title>
  185. <link rel="stylesheet" href="test.firefox_files/css.css">
  186. <style>
  187. /*! Asciidoctor default stylesheet | MIT License | https://asciidoctor.org */
  188. /* Uncomment the following line when using as a custom stylesheet */
  189. /* @import "https://fonts.googleapis.com/css?family=Open+Sans:300,300italic,400,400italic,600,600italic%7CNoto+Serif:400,400italic,700,700italic%7CDroid+Sans+Mono:400,700"; */
  190. html{font-family:sans-serif;-webkit-text-size-adjust:100%}
  191. a{background:none}
  192. a:focus{outline:thin dotted}
  193. a:active,a:hover{outline:0}
  194. h1{font-size:2em;margin:.67em 0}
  195. b,strong{font-weight:bold}
  196. abbr{font-size:.9em}
  197. abbr[title]{cursor:help;border-bottom:1px dotted #dddddf;text-decoration:none}
  198. dfn{font-style:italic}
  199. hr{height:0}
  200. mark{background:#ff0;color:#000}
  201. code,kbd,pre,samp{font-family:monospace;font-size:1em}
  202. pre{white-space:pre-wrap}
  203. q{quotes:"\201C" "\201D" "\2018" "\2019"}
  204. small{font-size:80%}
  205. sub,sup{font-size:75%;line-height:0;position:relative;vertical-align:baseline}
  206. sup{top:-.5em}
  207. sub{bottom:-.25em}
  208. img{border:0}
  209. svg:not(:root){overflow:hidden}
  210. figure{margin:0}
  211. audio,video{display:inline-block}
  212. audio:not([controls]){display:none;height:0}
  213. fieldset{border:1px solid silver;margin:0 2px;padding:.35em .625em .75em}
  214. legend{border:0;padding:0}
  215. button,input,select,textarea{font-family:inherit;font-size:100%;margin:0}
  216. button,input{line-height:normal}
  217. button,select{text-transform:none}
  218. button,html input[type=button],input[type=reset],input[type=submit]{-webkit-appearance:button;cursor:pointer}
  219. button[disabled],html input[disabled]{cursor:default}
  220. input[type=checkbox],input[type=radio]{padding:0}
  221. button::-moz-focus-inner,input::-moz-focus-inner{border:0;padding:0}
  222. textarea{overflow:auto;vertical-align:top}
  223. table{border-collapse:collapse;border-spacing:0}
  224. *,::before,::after{box-sizing:border-box}
  225. html,body{font-size:100%}
  226. body{background:#fff;color:rgba(0,0,0,.8);padding:0;margin:0;font-family:"Noto Serif","DejaVu Serif",serif;line-height:1;position:relative;cursor:auto;-moz-tab-size:4;-o-tab-size:4;tab-size:4;word-wrap:anywhere;-moz-osx-font-smoothing:grayscale;-webkit-font-smoothing:antialiased}
  227. a:hover{cursor:pointer}
  228. img,object,embed{max-width:100%;height:auto}
  229. object,embed{height:100%}
  230. img{-ms-interpolation-mode:bicubic}
  231. .left{float:left!important}
  232. .right{float:right!important}
  233. .text-left{text-align:left!important}
  234. .text-right{text-align:right!important}
  235. .text-center{text-align:center!important}
  236. .text-justify{text-align:justify!important}
  237. .hide{display:none}
  238. img,object,svg{display:inline-block;vertical-align:middle}
  239. textarea{height:auto;min-height:50px}
  240. select{width:100%}
  241. .subheader,.admonitionblock td.content>.title,.audioblock>.title,.exampleblock>.title,.imageblock>.title,.listingblock>.title,.literalblock>.title,.stemblock>.title,.openblock>.title,.paragraph>.title,.quoteblock>.title,table.tableblock>.title,.verseblock>.title,.videoblock>.title,.dlist>.title,.olist>.title,.ulist>.title,.qlist>.title,.hdlist>.title{line-height:1.45;color:#7a2518;font-weight:400;margin-top:0;margin-bottom:.25em}
  242. div,dl,dt,dd,ul,ol,li,h1,h2,h3,#toctitle,.sidebarblock>.content>.title,h4,h5,h6,pre,form,p,blockquote,th,td{margin:0;padding:0}
  243. a{color:#2156a5;text-decoration:underline;line-height:inherit}
  244. a:hover,a:focus{color:#1d4b8f}
  245. a img{border:0}
  246. p{line-height:1.6;margin-bottom:1.25em;text-rendering:optimizeLegibility}
  247. p aside{font-size:.875em;line-height:1.35;font-style:italic}
  248. h1,h2,h3,#toctitle,.sidebarblock>.content>.title,h4,h5,h6{font-family:"Open Sans","DejaVu Sans",sans-serif;font-weight:300;font-style:normal;color:#ba3925;text-rendering:optimizeLegibility;margin-top:1em;margin-bottom:.5em;line-height:1.0125em}
  249. h1 small,h2 small,h3 small,#toctitle small,.sidebarblock>.content>.title small,h4 small,h5 small,h6 small{font-size:60%;color:#e99b8f;line-height:0}
  250. h1{font-size:2.125em}
  251. h2{font-size:1.6875em}
  252. h3,#toctitle,.sidebarblock>.content>.title{font-size:1.375em}
  253. h4,h5{font-size:1.125em}
  254. h6{font-size:1em}
  255. hr{border:solid #dddddf;border-width:1px 0 0;clear:both;margin:1.25em 0 1.1875em}
  256. em,i{font-style:italic;line-height:inherit}
  257. strong,b{font-weight:bold;line-height:inherit}
  258. small{font-size:60%;line-height:inherit}
  259. code{font-family:"Droid Sans Mono","DejaVu Sans Mono",monospace;font-weight:400;color:rgba(0,0,0,.9)}
  260. ul,ol,dl{line-height:1.6;margin-bottom:1.25em;list-style-position:outside;font-family:inherit}
  261. ul,ol{margin-left:1.5em}
  262. ul li ul,ul li ol{margin-left:1.25em;margin-bottom:0}
  263. ul.circle{list-style-type:circle}
  264. ul.disc{list-style-type:disc}
  265. ul.square{list-style-type:square}
  266. ul.circle ul:not([class]),ul.disc ul:not([class]),ul.square ul:not([class]){list-style:inherit}
  267. ol li ul,ol li ol{margin-left:1.25em;margin-bottom:0}
  268. dl dt{margin-bottom:.3125em;font-weight:bold}
  269. dl dd{margin-bottom:1.25em}
  270. blockquote{margin:0 0 1.25em;padding:.5625em 1.25em 0 1.1875em;border-left:1px solid #ddd}
  271. blockquote,blockquote p{line-height:1.6;color:rgba(0,0,0,.85)}
  272. @media screen and (min-width:768px){h1,h2,h3,#toctitle,.sidebarblock>.content>.title,h4,h5,h6{line-height:1.2}
  273. h1{font-size:2.75em}
  274. h2{font-size:2.3125em}
  275. h3,#toctitle,.sidebarblock>.content>.title{font-size:1.6875em}
  276. h4{font-size:1.4375em}}
  277. table{background:#fff;margin-bottom:1.25em;border:1px solid #dedede;word-wrap:normal}
  278. table thead,table tfoot{background:#f7f8f7}
  279. table thead tr th,table thead tr td,table tfoot tr th,table tfoot tr td{padding:.5em .625em .625em;font-size:inherit;color:rgba(0,0,0,.8);text-align:left}
  280. table tr th,table tr td{padding:.5625em .625em;font-size:inherit;color:rgba(0,0,0,.8)}
  281. table tr.even,table tr.alt{background:#f8f8f7}
  282. table thead tr th,table tfoot tr th,table tbody tr td,table tr td,table tfoot tr td{line-height:1.6}
  283. h1,h2,h3,#toctitle,.sidebarblock>.content>.title,h4,h5,h6{line-height:1.2;word-spacing:-.05em}
  284. h1 strong,h2 strong,h3 strong,#toctitle strong,.sidebarblock>.content>.title strong,h4 strong,h5 strong,h6 strong{font-weight:400}
  285. .center{margin-left:auto;margin-right:auto}
  286. .stretch{width:100%}
  287. .clearfix::before,.clearfix::after,.float-group::before,.float-group::after{content:" ";display:table}
  288. .clearfix::after,.float-group::after{clear:both}
  289. :not(pre).nobreak{word-wrap:normal}
  290. :not(pre).nowrap{white-space:nowrap}
  291. :not(pre).pre-wrap{white-space:pre-wrap}
  292. :not(pre):not([class^=L])>code{font-size:.9375em;font-style:normal!important;letter-spacing:0;padding:.1em .5ex;word-spacing:-.15em;background:#f7f7f8;border-radius:4px;line-height:1.45;text-rendering:optimizeSpeed}
  293. pre{color:rgba(0,0,0,.9);font-family:"Droid Sans Mono","DejaVu Sans Mono",monospace;line-height:1.45;text-rendering:optimizeSpeed}
  294. pre code,pre pre{color:inherit;font-size:inherit;line-height:inherit}
  295. pre>code{display:block}
  296. pre.nowrap,pre.nowrap pre{white-space:pre;word-wrap:normal}
  297. em em{font-style:normal}
  298. strong strong{font-weight:400}
  299. .keyseq{color:rgba(51,51,51,.8)}
  300. kbd{font-family:"Droid Sans Mono","DejaVu Sans Mono",monospace;display:inline-block;color:rgba(0,0,0,.8);font-size:.65em;line-height:1.45;background:#f7f7f7;border:1px solid #ccc;border-radius:3px;box-shadow:0 1px 0 rgba(0,0,0,.2),inset 0 0 0 .1em #fff;margin:0 .15em;padding:.2em .5em;vertical-align:middle;position:relative;top:-.1em;white-space:nowrap}
  301. .keyseq kbd:first-child{margin-left:0}
  302. .keyseq kbd:last-child{margin-right:0}
  303. .menuseq,.menuref{color:#000}
  304. .menuseq b:not(.caret),.menuref{font-weight:inherit}
  305. .menuseq{word-spacing:-.02em}
  306. .menuseq b.caret{font-size:1.25em;line-height:.8}
  307. .menuseq i.caret{font-weight:bold;text-align:center;width:.45em}
  308. b.button::before,b.button::after{position:relative;top:-1px;font-weight:400}
  309. b.button::before{content:"[";padding:0 3px 0 2px}
  310. b.button::after{content:"]";padding:0 2px 0 3px}
  311. p a>code:hover{color:rgba(0,0,0,.9)}
  312. #header,#content,#footnotes,#footer{width:100%;margin:0 auto;max-width:62.5em;*zoom:1;position:relative;padding-left:.9375em;padding-right:.9375em}
  313. #header::before,#header::after,#content::before,#content::after,#footnotes::before,#footnotes::after,#footer::before,#footer::after{content:" ";display:table}
  314. #header::after,#content::after,#footnotes::after,#footer::after{clear:both}
  315. #content{margin-top:1.25em}
  316. #content::before{content:none}
  317. #header>h1:first-child{color:rgba(0,0,0,.85);margin-top:2.25rem;margin-bottom:0}
  318. #header>h1:first-child+#toc{margin-top:8px;border-top:1px solid #dddddf}
  319. #header>h1:only-child,body.toc2 #header>h1:nth-last-child(2){border-bottom:1px solid #dddddf;padding-bottom:8px}
  320. #header .details{border-bottom:1px solid #dddddf;line-height:1.45;padding-top:.25em;padding-bottom:.25em;padding-left:.25em;color:rgba(0,0,0,.6);display:flex;flex-flow:row wrap}
  321. #header .details span:first-child{margin-left:-.125em}
  322. #header .details span.email a{color:rgba(0,0,0,.85)}
  323. #header .details br{display:none}
  324. #header .details br+span::before{content:"\00a0\2013\00a0"}
  325. #header .details br+span.author::before{content:"\00a0\22c5\00a0";color:rgba(0,0,0,.85)}
  326. #header .details br+span#revremark::before{content:"\00a0|\00a0"}
  327. #header #revnumber{text-transform:capitalize}
  328. #header #revnumber::after{content:"\00a0"}
  329. #content>h1:first-child:not([class]){color:rgba(0,0,0,.85);border-bottom:1px solid #dddddf;padding-bottom:8px;margin-top:0;padding-top:1rem;margin-bottom:1.25rem}
  330. #toc{border-bottom:1px solid #e7e7e9;padding-bottom:.5em}
  331. #toc>ul{margin-left:.125em}
  332. #toc ul.sectlevel0>li>a{font-style:italic}
  333. #toc ul.sectlevel0 ul.sectlevel1{margin:.5em 0}
  334. #toc ul{font-family:"Open Sans","DejaVu Sans",sans-serif;list-style-type:none}
  335. #toc li{line-height:1.3334;margin-top:.3334em}
  336. #toc a{text-decoration:none}
  337. #toc a:active{text-decoration:underline}
  338. #toctitle{color:#7a2518;font-size:1.2em}
  339. @media screen and (min-width:768px){#toctitle{font-size:1.375em}
  340. body.toc2{padding-left:15em;padding-right:0}
  341. #toc.toc2{margin-top:0!important;background:#f8f8f7;position:fixed;width:15em;left:0;top:0;border-right:1px solid #e7e7e9;border-top-width:0!important;border-bottom-width:0!important;z-index:1000;padding:1.25em 1em;height:100%;overflow:auto}
  342. #toc.toc2 #toctitle{margin-top:0;margin-bottom:.8rem;font-size:1.2em}
  343. #toc.toc2>ul{font-size:.9em;margin-bottom:0}
  344. #toc.toc2 ul ul{margin-left:0;padding-left:1em}
  345. #toc.toc2 ul.sectlevel0 ul.sectlevel1{padding-left:0;margin-top:.5em;margin-bottom:.5em}
  346. body.toc2.toc-right{padding-left:0;padding-right:15em}
  347. body.toc2.toc-right #toc.toc2{border-right-width:0;border-left:1px solid #e7e7e9;left:auto;right:0}}
  348. @media screen and (min-width:1280px){body.toc2{padding-left:20em;padding-right:0}
  349. #toc.toc2{width:20em}
  350. #toc.toc2 #toctitle{font-size:1.375em}
  351. #toc.toc2>ul{font-size:.95em}
  352. #toc.toc2 ul ul{padding-left:1.25em}
  353. body.toc2.toc-right{padding-left:0;padding-right:20em}}
  354. #content #toc{border:1px solid #e0e0dc;margin-bottom:1.25em;padding:1.25em;background:#f8f8f7;border-radius:4px}
  355. #content #toc>:first-child{margin-top:0}
  356. #content #toc>:last-child{margin-bottom:0}
  357. #footer{max-width:none;background:rgba(0,0,0,.8);padding:1.25em}
  358. #footer-text{color:hsla(0,0%,100%,.8);line-height:1.44}
  359. #content{margin-bottom:.625em}
  360. .sect1{padding-bottom:.625em}
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  1690. </style><style class="darkreader darkreader--sync" media="screen"></style></head>
  1691. <body class="article">
  1692. <div id="header">
  1693. </div>
  1694. <div id="content">
  1695. <div class="sect1">
  1696. <h2 id="_introduction">Introduction</h2>
  1697. <div class="sectionbody">
  1698. <div class="paragraph">
  1699. <p>In a oneway classification, the typical analysis of variance model for the value <strong><em>Y<sub>ij</sub></em></strong> of the <em>j<sup>th</sup></em> observation in the <em>i<sup>th</sup></em> class is given by,</p>
  1700. </div>
  1701. <div id="eq:lin-model" class="stemblock">
  1702. <div class="title">Basic linear model</div>
  1703. <div class="content">
  1704. <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.4%; position: relative;" display="true" tabindex="0" ctxtmenu_counter="0"><mjx-math display="true" style="margin-left: 0px; margin-right: 0px;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.182em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>Y</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>μ</mi><mi>i</mi></msub><mo>+</mo><msub><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container>
  1705. </div>
  1706. </div>
  1707. <div class="paragraph">
  1708. <p>where the <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.3%; position: relative;" tabindex="0" ctxtmenu_counter="1"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container> in the basic linear model at <a href="#eq:lin-model">Equation (1)</a>, represent the population means of the <strong><em>classes</em></strong> and the <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.3%; position: relative;" tabindex="0" ctxtmenu_counter="2"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container> are
  1709. the <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.3%; position: relative;" tabindex="0" ctxtmenu_counter="3"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: 0.363em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>j</mi><mrow data-mjx-texclass="ORD"><mi>t</mi><mi>h</mi></mrow></msup></math></mjx-assistive-mml></mjx-container> residuals for the <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.3%; position: relative;" tabindex="0" ctxtmenu_counter="4"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: 0.363em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>i</mi><mrow data-mjx-texclass="ORD"><mi>t</mi><mi>h</mi></mrow></msup></math></mjx-assistive-mml></mjx-container> class. But suppose that on each unit we have also measured another variable latexmah:[X_{ij}] that is linearly related to <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.3%; position: relative;" tabindex="0" ctxtmenu_counter="5"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.182em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container>. It is natural to set up the model,</p>
  1710. </div>
  1711. <div id="lin-model-X" class="stemblock">
  1712. <div class="title">Basic covariance model with linear regression of <strong>Y</strong> on covariate <strong>X</strong></div>
  1713. <div class="content">
  1714. <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.4%; position: relative;" display="true" tabindex="0" ctxtmenu_counter="6"><mjx-math display="true" style="margin-left: 0px; margin-right: 0px;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.182em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D6FD TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.024em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-texatom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom: 0.105em; padding-left: 0.515em; margin-bottom: -0.544em;"><mjx-mo class="mjx-n" style="width: 0px; margin-left: -0.25em;"><mjx-c class="mjx-cAF"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-texatom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D716 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>Y</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>μ</mi><mi>i</mi></msub><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><mi>X</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container>
  1715. </div>
  1716. </div>
  1717. <div id="lin-model-X-alpha" class="stemblock">
  1718. <div class="title">Basic covariance model in CRD with linear regression of <strong>Y</strong> on covariate <strong>X</strong></div>
  1719. <div class="content">
  1720. <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.4%; position: relative;" display="true" tabindex="0" ctxtmenu_counter="7"><mjx-math display="true" style="margin-left: 0px; margin-right: 0px;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44C TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.182em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D6FD TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.024em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-texatom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom: 0.105em; padding-left: 0.515em; margin-bottom: -0.544em;"><mjx-mo class="mjx-n" style="width: 0px; margin-left: -0.25em;"><mjx-c class="mjx-cAF"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-texatom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D716 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>Y</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>μ</mi><mo>+</mo><msub><mi>α</mi><mi>i</mi></msub><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><mi>X</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container>
  1721. </div>
  1722. </div>
  1723. <div class="paragraph">
  1724. <p>where, <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.3%; position: relative;" tabindex="0" ctxtmenu_counter="8"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mi>i</mi></msub><mo>=</mo><msub><mi>μ</mi><mi>i</mi></msub><mo>−</mo><mi>μ</mi></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.3%; position: relative;" tabindex="0" ctxtmenu_counter="9"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FD TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math></mjx-assistive-mml></mjx-container> is the regression coefficient of <strong><em>Y</em></strong> on <strong><em>X</em></strong>. The <a href="#lin-model-X">Equation (2)</a> is a typical <strong><em>model for the analysis of covariance after removing the general means <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.3%; position: relative;" tabindex="0" ctxtmenu_counter="10"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></mjx-assistive-mml></mjx-container> from group means <mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" style="font-size: 121.3%; position: relative;" tabindex="0" ctxtmenu_counter="11"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D707 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container></em></strong> and the <a href="#lin-model-X-alpha">Equation (3)</a> is a typical <strong><em>model for the analysis of covariance in CRD</em></strong>.</p>
  1725. </div>
  1726. </div>
  1727. </div>
  1728. </div>
  1729. <div id="footer">
  1730. <div id="footer-text">
  1731. Last updated 2023-08-07 14:32:39 +0530
  1732. </div>
  1733. </div>
  1734.  
  1735.  
  1736.  
  1737. <span class="cleanslate TridactylStatusIndicator TridactylModenormal">normal</span></body><iframe class="cleanslate hidden" src="test.firefox_files/commandline.html" id="cmdline_iframe" loading="lazy" style="height: 0px !important;"></iframe></html>
  1738.  
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