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<!DOCTYPE html>
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<title>第 6 章 模型的矩阵形式 | 混乱数据分析：设计的实验</title>
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<meta property="og:description" content="Analysis of Messy Data Volume 1: Designed Experiments的翻译" />
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<meta name="twitter:title" content="第 6 章 模型的矩阵形式 | 混乱数据分析：设计的实验" />
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<meta name="author" content="Wang Zhen" />
<meta name="date" content="2024-03-15" />
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<ul class="summary">
<li><a href="./">混乱数据分析：设计的实验</a></li>
<li class="divider"></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>介绍</a></li>
<li class="part"><span><b>I 热身</b></span></li>
<li class="chapter" data-level="1" data-path="chap1.html"><a href="chap1.html"><i class="fa fa-check"></i><b>1</b> 最简单的情况：具有同质误差的完全随机设计结构中的单向处理结构</a>
<ul>
<li class="chapter" data-level="1.1" data-path="chap1.html"><a href="chap1.html#sec1-1"><i class="fa fa-check"></i><b>1.1</b> 模型定义和假设</a></li>
<li class="chapter" data-level="1.2" data-path="chap1.html"><a href="chap1.html#sec1-2"><i class="fa fa-check"></i><b>1.2</b> 参数估计</a></li>
<li class="chapter" data-level="1.3" data-path="chap1.html"><a href="chap1.html#sec1-3"><i class="fa fa-check"></i><b>1.3</b> 线性组合的推断：检验与置信区间</a></li>
<li class="chapter" data-level="1.4" data-path="chap1.html"><a href="chap1.html#sec1-4"><i class="fa fa-check"></i><b>1.4</b> 示例：任务和脉搏率</a></li>
<li class="chapter" data-level="1.5" data-path="chap1.html"><a href="chap1.html#sec1-5"><i class="fa fa-check"></i><b>1.5</b> 几个线性组合的同时检验</a></li>
<li class="chapter" data-level="1.6" data-path="chap1.html"><a href="chap1.html#sec1-6"><i class="fa fa-check"></i><b>1.6</b> 示例：任务和脉搏率（续）</a></li>
<li class="chapter" data-level="1.7" data-path="chap1.html"><a href="chap1.html#sec1-7"><i class="fa fa-check"></i><b>1.7</b> 检验所有均值相等</a></li>
<li class="chapter" data-level="1.8" data-path="chap1.html"><a href="chap1.html#sec1-8"><i class="fa fa-check"></i><b>1.8</b> 示例：任务和脉搏率（续）</a></li>
<li class="chapter" data-level="1.9" data-path="chap1.html"><a href="chap1.html#sec1-9"><i class="fa fa-check"></i><b>1.9</b> 比较两种模型的一般方法：条件误差原理</a></li>
<li class="chapter" data-level="1.10" data-path="chap1.html"><a href="chap1.html#sec1-10"><i class="fa fa-check"></i><b>1.10</b> 示例：任务和脉搏率（续）</a></li>
<li class="chapter" data-level="1.11" data-path="chap1.html"><a href="chap1.html#sec1-11"><i class="fa fa-check"></i><b>1.11</b> 计算机分析</a></li>
<li class="chapter" data-level="1.12" data-path="chap1.html"><a href="chap1.html#sec1-12"><i class="fa fa-check"></i><b>1.12</b> 结束语</a></li>
<li class="chapter" data-level="1.13" data-path="chap1.html"><a href="chap1.html#sec1-13"><i class="fa fa-check"></i><b>1.13</b> 练习</a></li>
<li class="chapter" data-level="1.14" data-path="chap1.html"><a href="chap1.html#sec1-14"><i class="fa fa-check"></i><b>1.14</b> R 代码</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="chap2.html"><a href="chap2.html"><i class="fa fa-check"></i><b>2</b> 具有异质误差的完全随机设计结构中的单向处理结构</a>
<ul>
<li class="chapter" data-level="2.1" data-path="chap2.html"><a href="chap2.html#sec2-1"><i class="fa fa-check"></i><b>2.1</b> 模型定义和假设</a></li>
<li class="chapter" data-level="2.2" data-path="chap2.html"><a href="chap2.html#sec2-2"><i class="fa fa-check"></i><b>2.2</b> 参数估计</a></li>
<li class="chapter" data-level="2.3" data-path="chap2.html"><a href="chap2.html#sec2-3"><i class="fa fa-check"></i><b>2.3</b> 方差齐性检验</a>
<ul>
<li class="chapter" data-level="2.3.1" data-path="chap2.html"><a href="chap2.html#sec2-3-1"><i class="fa fa-check"></i><b>2.3.1</b> Hartley’s <em>F</em>-Max Test</a></li>
<li class="chapter" data-level="2.3.2" data-path="chap2.html"><a href="chap2.html#sec2-3-2"><i class="fa fa-check"></i><b>2.3.2</b> Bartlett’s Test</a></li>
<li class="chapter" data-level="2.3.3" data-path="chap2.html"><a href="chap2.html#sec2-3-3"><i class="fa fa-check"></i><b>2.3.3</b> Levene’s Test</a></li>
<li class="chapter" data-level="2.3.4" data-path="chap2.html"><a href="chap2.html#sec2-4-4"><i class="fa fa-check"></i><b>2.3.4</b> Brown and Forsythe’s Test</a></li>
<li class="chapter" data-level="2.3.5" data-path="chap2.html"><a href="chap2.html#sec2-3-5"><i class="fa fa-check"></i><b>2.3.5</b> O’Brien’s Test</a></li>
<li class="chapter" data-level="2.3.6" data-path="chap2.html"><a href="chap2.html#sec2-3-6"><i class="fa fa-check"></i><b>2.3.6</b> 一些建议</a></li>
</ul></li>
<li class="chapter" data-level="2.4" data-path="chap2.html"><a href="chap2.html#sec2-4"><i class="fa fa-check"></i><b>2.4</b> 示例：药物和错误</a></li>
<li class="chapter" data-level="2.5" data-path="chap2.html"><a href="chap2.html#sec2-5"><i class="fa fa-check"></i><b>2.5</b> 关于线性组合的推断</a></li>
<li class="chapter" data-level="2.6" data-path="chap2.html"><a href="chap2.html#sec2-6"><i class="fa fa-check"></i><b>2.6</b> 示例：药物和错误（续）</a></li>
<li class="chapter" data-level="2.7" data-path="chap2.html"><a href="chap2.html#sec2-7"><i class="fa fa-check"></i><b>2.7</b> 自由度的一般 Satterthwaite 近似</a></li>
<li class="chapter" data-level="2.8" data-path="chap2.html"><a href="chap2.html#sec2-8"><i class="fa fa-check"></i><b>2.8</b> 比较所有均值</a></li>
<li class="chapter" data-level="2.9" data-path="chap2.html"><a href="chap2.html#sec2-9"><i class="fa fa-check"></i><b>2.9</b> 结束语</a></li>
<li class="chapter" data-level="2.10" data-path="chap2.html"><a href="chap2.html#sec2-10"><i class="fa fa-check"></i><b>2.10</b> 练习</a></li>
<li class="chapter" data-level="2.11" data-path="chap2.html"><a href="chap2.html#sec2-11"><i class="fa fa-check"></i><b>2.11</b> R 代码</a></li>
</ul></li>
<li class="part"><span><b>II 磨刀</b></span></li>
<li class="chapter" data-level="3" data-path="chap3.html"><a href="chap3.html"><i class="fa fa-check"></i><b>3</b> 同时推断程序和多重比较</a>
<ul>
<li class="chapter" data-level="3.1" data-path="chap3.html"><a href="chap3.html#sec3-1"><i class="fa fa-check"></i><b>3.1</b> 错误率</a></li>
<li class="chapter" data-level="3.2" data-path="chap3.html"><a href="chap3.html#sec3-2"><i class="fa fa-check"></i><b>3.2</b> 建议</a></li>
<li class="chapter" data-level="3.3" data-path="chap3.html"><a href="chap3.html#sec3-3"><i class="fa fa-check"></i><b>3.3</b> 最小显著差异</a></li>
<li class="chapter" data-level="3.4" data-path="chap3.html"><a href="chap3.html#sec3-4"><i class="fa fa-check"></i><b>3.4</b> Fisher’s LSD Procedure</a></li>
<li class="chapter" data-level="3.5" data-path="chap3.html"><a href="chap3.html#sec3-5"><i class="fa fa-check"></i><b>3.5</b> Bonferroni’s Method</a></li>
<li class="chapter" data-level="3.6" data-path="chap3.html"><a href="chap3.html#sec3-6"><i class="fa fa-check"></i><b>3.6</b> Scheffé’s Procedure</a></li>
<li class="chapter" data-level="3.7" data-path="chap3.html"><a href="chap3.html#sec3-7"><i class="fa fa-check"></i><b>3.7</b> Tukey–Kramer Method</a></li>
<li class="chapter" data-level="3.8" data-path="chap3.html"><a href="chap3.html#sec3-8"><i class="fa fa-check"></i><b>3.8</b> 模拟方法</a></li>
<li class="chapter" data-level="3.9" data-path="chap3.html"><a href="chap3.html#sec3-9"><i class="fa fa-check"></i><b>3.9</b> Šidák Procedure</a></li>
<li class="chapter" data-level="3.10" data-path="chap3.html"><a href="chap3.html#sec3-10"><i class="fa fa-check"></i><b>3.10</b> 示例：成对比较</a></li>
<li class="chapter" data-level="3.11" data-path="chap3.html"><a href="chap3.html#sec3-11"><i class="fa fa-check"></i><b>3.11</b> Dunnett’s Procedure</a></li>
<li class="chapter" data-level="3.12" data-path="chap3.html"><a href="chap3.html#sec3-12"><i class="fa fa-check"></i><b>3.12</b> 示例：与对照比较</a></li>
<li class="chapter" data-level="3.13" data-path="chap3.html"><a href="chap3.html#sec3-13"><i class="fa fa-check"></i><b>3.13</b> 多元 <span class="math inline">\(t\)</span></a></li>
<li class="chapter" data-level="3.14" data-path="chap3.html"><a href="chap3.html#sec3-14"><i class="fa fa-check"></i><b>3.14</b> 示例：线性独立比较</a></li>
<li class="chapter" data-level="3.15" data-path="chap3.html"><a href="chap3.html#sec3-15"><i class="fa fa-check"></i><b>3.15</b> 序贯拒绝方法</a>
<ul>
<li class="chapter" data-level="3.15.1" data-path="chap3.html"><a href="chap3.html#sec3-15-1"><i class="fa fa-check"></i><b>3.15.1</b> Bonferroni–Holm Method</a></li>
<li class="chapter" data-level="3.15.2" data-path="chap3.html"><a href="chap3.html#sec3-15-2"><i class="fa fa-check"></i><b>3.15.2</b> Šidák–Holm Method</a></li>
<li class="chapter" data-level="3.15.3" data-path="chap3.html"><a href="chap3.html#sec3-15-3"><i class="fa fa-check"></i><b>3.15.3</b> 控制 FDR 的 Benjamini 和 Hochberg Method</a></li>
</ul></li>
<li class="chapter" data-level="3.16" data-path="chap3.html"><a href="chap3.html#sec3-16"><i class="fa fa-check"></i><b>3.16</b> 示例：线性相关比较</a></li>
<li class="chapter" data-level="3.17" data-path="chap3.html"><a href="chap3.html#sec3-17"><i class="fa fa-check"></i><b>3.17</b> 多重极差检验</a>
<ul>
<li class="chapter" data-level="3.17.1" data-path="chap3.html"><a href="chap3.html#sec3-17-1"><i class="fa fa-check"></i><b>3.17.1</b> Student–Newman–Keul’s Method</a></li>
<li class="chapter" data-level="3.17.2" data-path="chap3.html"><a href="chap3.html#sec3-17-2"><i class="fa fa-check"></i><b>3.17.2</b> Duncan’s New Multiple Range Method</a></li>
</ul></li>
<li class="chapter" data-level="3.18" data-path="chap3.html"><a href="chap3.html#sec3-18"><i class="fa fa-check"></i><b>3.18</b> Waller–Duncan Procedure</a></li>
<li class="chapter" data-level="3.19" data-path="chap3.html"><a href="chap3.html#sec3-19"><i class="fa fa-check"></i><b>3.19</b> 示例：成对比较的多重极差</a></li>
<li class="chapter" data-level="3.20" data-path="chap3.html"><a href="chap3.html#sec3-20"><i class="fa fa-check"></i><b>3.20</b> 警示</a></li>
<li class="chapter" data-level="3.21" data-path="chap3.html"><a href="chap3.html#sec3-21"><i class="fa fa-check"></i><b>3.21</b> 结束语</a></li>
<li class="chapter" data-level="3.22" data-path="chap3.html"><a href="chap3.html#sec3-22"><i class="fa fa-check"></i><b>3.22</b> 练习</a></li>
<li class="chapter" data-level="3.23" data-path="chap3.html"><a href="chap3.html#sec3-23"><i class="fa fa-check"></i><b>3.23</b> R 代码</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="chap4.html"><a href="chap4.html"><i class="fa fa-check"></i><b>4</b> 实验设计基础</a>
<ul>
<li class="chapter" data-level="4.1" data-path="chap4.html"><a href="chap4.html#sec4-1"><i class="fa fa-check"></i><b>4.1</b> 介绍基本概念</a></li>
<li class="chapter" data-level="4.2" data-path="chap4.html"><a href="chap4.html#sec4-2"><i class="fa fa-check"></i><b>4.2</b> 设计实验的结构</a>
<ul>
<li class="chapter" data-level="4.2.1" data-path="chap4.html"><a href="chap4.html#sec4-2-1"><i class="fa fa-check"></i><b>4.2.1</b> 设计结构类型</a></li>
<li class="chapter" data-level="4.2.2" data-path="chap4.html"><a href="chap4.html#sec4-2-2"><i class="fa fa-check"></i><b>4.2.2</b> 处理结构类型</a></li>
</ul></li>
<li class="chapter" data-level="4.3" data-path="chap4.html"><a href="chap4.html#sec4-3"><i class="fa fa-check"></i><b>4.3</b> 不同设计实验的示例</a>
<ul>
<li class="chapter" data-level="4.3.1" data-path="chap4.html"><a href="chap4.html#sec4-3-1"><i class="fa fa-check"></i><b>4.3.1</b> 示例 4.1: 饮食</a></li>
<li class="chapter" data-level="4.3.2" data-path="chap4.html"><a href="chap4.html#sec4-3-2"><i class="fa fa-check"></i><b>4.3.2</b> 示例 4.2: 房屋油漆</a></li>
<li class="chapter" data-level="4.3.3" data-path="chap4.html"><a href="chap4.html#sec4-3-3"><i class="fa fa-check"></i><b>4.3.3</b> 示例 4.3: 钢板</a></li>
<li class="chapter" data-level="4.3.4" data-path="chap4.html"><a href="chap4.html#sec4-3-4"><i class="fa fa-check"></i><b>4.3.4</b> 示例 4.4: 氮和钾的水平</a></li>
<li class="chapter" data-level="4.3.5" data-path="chap4.html"><a href="chap4.html#sec4-3-5"><i class="fa fa-check"></i><b>4.3.5</b> 示例 4.5: 区组和重复</a></li>
<li class="chapter" data-level="4.3.6" data-path="chap4.html"><a href="chap4.html#sec4-3-6"><i class="fa fa-check"></i><b>4.3.6</b> 示例 4.6：行区组和列区组</a></li>
</ul></li>
<li class="chapter" data-level="4.4" data-path="chap4.html"><a href="chap4.html#sec4-4"><i class="fa fa-check"></i><b>4.4</b> 结束语</a></li>
<li class="chapter" data-level="4.5" data-path="chap4.html"><a href="chap4.html#sec4-5"><i class="fa fa-check"></i><b>4.5</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="5" data-path="chap5.html"><a href="chap5.html"><i class="fa fa-check"></i><b>5</b> 多水平设计：裂区、条区、重复测量及其组合</a>
<ul>
<li class="chapter" data-level="5.1" data-path="chap5.html"><a href="chap5.html#sec5-1"><i class="fa fa-check"></i><b>5.1</b> 识别实验单元的尺寸——四种基本设计结构</a></li>
<li class="chapter" data-level="5.2" data-path="chap5.html"><a href="chap5.html#sec5-2"><i class="fa fa-check"></i><b>5.2</b> 分层设计：一种多水平的设计结构</a></li>
<li class="chapter" data-level="5.3" data-path="chap5.html"><a href="chap5.html#sec5-3"><i class="fa fa-check"></i><b>5.3</b> 裂区设计结构：两水平设计结构</a>
<ul>
<li class="chapter" data-level="5.3.1" data-path="chap5.html"><a href="chap5.html#sec5-3-1"><i class="fa fa-check"></i><b>5.3.1</b> 示例 5.1：烹饪大豆——最简单的裂区或两水平设计结构</a></li>
<li class="chapter" data-level="5.3.2" data-path="chap5.html"><a href="chap5.html#sec5-3-2"><i class="fa fa-check"></i><b>5.3.2</b> 示例 5.2：磨小麦——通常的裂区或两水平设计结构</a></li>
<li class="chapter" data-level="5.3.3" data-path="chap5.html"><a href="chap5.html#sec5-3-3"><i class="fa fa-check"></i><b>5.3.3</b> 示例 5.3：烘焙面包——具有不完全块设计结构的裂区</a></li>
<li class="chapter" data-level="5.3.4" data-path="chap5.html"><a href="chap5.html#sec5-3-4"><i class="fa fa-check"></i><b>5.3.4</b> 示例 5.4：展示柜中的肉——复杂裂区或四水平设计</a></li>
</ul></li>
<li class="chapter" data-level="5.4" data-path="chap5.html"><a href="chap5.html#sec5-4"><i class="fa fa-check"></i><b>5.4</b> 条区设计结构：一种无层次的多水平设计</a>
<ul>
<li class="chapter" data-level="5.4.1" data-path="chap5.html"><a href="chap5.html#sec5-4-1"><i class="fa fa-check"></i><b>5.4.1</b> 示例 5.5：制作奶酪</a></li>
</ul></li>
<li class="chapter" data-level="5.5" data-path="chap5.html"><a href="chap5.html#sec5-5"><i class="fa fa-check"></i><b>5.5</b> 重复测量设计</a>
<ul>
<li class="chapter" data-level="5.5.1" data-path="chap5.html"><a href="chap5.html#sec5-5-1"><i class="fa fa-check"></i><b>5.5.1</b> 示例 5.6：马足——基本重复测量设计</a></li>
<li class="chapter" data-level="5.5.2" data-path="chap5.html"><a href="chap5.html#sec5-5-2"><i class="fa fa-check"></i><b>5.5.2</b> 示例 5.7：舒适度研究——重复测量设计</a></li>
<li class="chapter" data-level="5.5.3" data-path="chap5.html"><a href="chap5.html#示例-5.8交叉或转换设计"><i class="fa fa-check"></i><b>5.5.3</b> 示例 5.8：交叉或转换设计</a></li>
</ul></li>
<li class="chapter" data-level="5.6" data-path="chap5.html"><a href="chap5.html#sec5-6"><i class="fa fa-check"></i><b>5.6</b> 涉及嵌套因素的设计</a>
<ul>
<li class="chapter" data-level="5.6.1" data-path="chap5.html"><a href="chap5.html#sec5-6-1"><i class="fa fa-check"></i><b>5.6.1</b> 示例 5.9：动物遗传学</a></li>
<li class="chapter" data-level="5.6.2" data-path="chap5.html"><a href="chap5.html#sec5-6-2"><i class="fa fa-check"></i><b>5.6.2</b> 示例 5.10：大豆的生育期组</a></li>
<li class="chapter" data-level="5.6.3" data-path="chap5.html"><a href="chap5.html#sec5-6-3"><i class="fa fa-check"></i><b>5.6.3</b> 示例 5.11：飞机引擎</a></li>
<li class="chapter" data-level="5.6.4" data-path="chap5.html"><a href="chap5.html#sec5-6-4"><i class="fa fa-check"></i><b>5.6.4</b> 示例 5.12：简单的舒适度实验</a></li>
<li class="chapter" data-level="5.6.5" data-path="chap5.html"><a href="chap5.html#sec5-6-5"><i class="fa fa-check"></i><b>5.6.5</b> 示例 5.13：重复测量的多地点研究</a></li>
</ul></li>
<li class="chapter" data-level="5.7" data-path="chap5.html"><a href="chap5.html#sec5-7"><i class="fa fa-check"></i><b>5.7</b> 结束语</a></li>
<li class="chapter" data-level="5.8" data-path="chap5.html"><a href="chap5.html#sec5-8"><i class="fa fa-check"></i><b>5.8</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="6" data-path="chap6.html"><a href="chap6.html"><i class="fa fa-check"></i><b>6</b> 模型的矩阵形式</a>
<ul>
<li class="chapter" data-level="6.1" data-path="chap6.html"><a href="chap6.html#sec6-1"><i class="fa fa-check"></i><b>6.1</b> 基本符号</a>
<ul>
<li class="chapter" data-level="6.1.1" data-path="chap6.html"><a href="chap6.html#sec6-1-1"><i class="fa fa-check"></i><b>6.1.1</b> 简单线性回归模型</a></li>
<li class="chapter" data-level="6.1.2" data-path="chap6.html"><a href="chap6.html#sec6-1-2"><i class="fa fa-check"></i><b>6.1.2</b> 单向处理结构模型</a></li>
<li class="chapter" data-level="6.1.3" data-path="chap6.html"><a href="chap6.html#sec6-1-3"><i class="fa fa-check"></i><b>6.1.3</b> 双向处理结构模型</a></li>
<li class="chapter" data-level="6.1.4" data-path="chap6.html"><a href="chap6.html#sec6-1-4"><i class="fa fa-check"></i><b>6.1.4</b> 示例 6.1：双向处理结构的均值模型</a></li>
</ul></li>
<li class="chapter" data-level="6.2" data-path="chap6.html"><a href="chap6.html#sec6-2"><i class="fa fa-check"></i><b>6.2</b> 最小二乘估计</a>
<ul>
<li class="chapter" data-level="6.2.1" data-path="chap6.html"><a href="chap6.html#sec6-2-1"><i class="fa fa-check"></i><b>6.2.1</b> 最小二乘方程组</a></li>
<li class="chapter" data-level="6.2.2" data-path="chap6.html"><a href="chap6.html#sec6-2-2"><i class="fa fa-check"></i><b>6.2.2</b> 零和限制</a></li>
<li class="chapter" data-level="6.2.3" data-path="chap6.html"><a href="chap6.html#sec6-2-3"><i class="fa fa-check"></i><b>6.2.3</b> 置零限制</a></li>
<li class="chapter" data-level="6.2.4" data-path="chap6.html"><a href="chap6.html#sec6-2-4"><i class="fa fa-check"></i><b>6.2.4</b> 示例 6.2：单向处理结构</a></li>
</ul></li>
<li class="chapter" data-level="6.3" data-path="chap6.html"><a href="chap6.html#sec6-3"><i class="fa fa-check"></i><b>6.3</b> 可估性和连通的设计</a>
<ul>
<li class="chapter" data-level="6.3.1" data-path="chap6.html"><a href="chap6.html#sec6-3-1"><i class="fa fa-check"></i><b>6.3.1</b> 可估函数</a></li>
<li class="chapter" data-level="6.3.2" data-path="chap6.html"><a href="chap6.html#sec6-3-2"><i class="fa fa-check"></i><b>6.3.2</b> 连通性</a></li>
</ul></li>
<li class="chapter" data-level="6.4" data-path="chap6.html"><a href="chap6.html#sec6-4"><i class="fa fa-check"></i><b>6.4</b> 关于线性模型参数的检验假设</a></li>
<li class="chapter" data-level="6.5" data-path="chap6.html"><a href="chap6.html#sec6-5"><i class="fa fa-check"></i><b>6.5</b> 总体边际均值</a></li>
<li class="chapter" data-level="6.6" data-path="chap6.html"><a href="chap6.html#sec6-6"><i class="fa fa-check"></i><b>6.6</b> 结束语</a></li>
<li class="chapter" data-level="6.7" data-path="chap6.html"><a href="chap6.html#sec6-7"><i class="fa fa-check"></i><b>6.7</b> 练习</a></li>
<li class="chapter" data-level="6.8" data-path="chap6.html"><a href="chap6.html#sec6-8"><i class="fa fa-check"></i><b>6.8</b> R 代码</a></li>
</ul></li>
<li class="part"><span><b>III 砍柴</b></span></li>
<li class="chapter" data-level="7" data-path="chap7.html"><a href="chap7.html"><i class="fa fa-check"></i><b>7</b> 均衡双向处理结构</a>
<ul>
<li class="chapter" data-level="7.1" data-path="chap7.html"><a href="chap7.html#sec7-1"><i class="fa fa-check"></i><b>7.1</b> 模型定义和假设</a>
<ul>
<li class="chapter" data-level="7.1.1" data-path="chap7.html"><a href="chap7.html#sec7-1-1"><i class="fa fa-check"></i><b>7.1.1</b> 均值模型</a></li>
<li class="chapter" data-level="7.1.2" data-path="chap7.html"><a href="chap7.html#sec7-1-2"><i class="fa fa-check"></i><b>7.1.2</b> 效应模型</a></li>
</ul></li>
<li class="chapter" data-level="7.2" data-path="chap7.html"><a href="chap7.html#sec7-2"><i class="fa fa-check"></i><b>7.2</b> 参数估计</a></li>
<li class="chapter" data-level="7.3" data-path="chap7.html"><a href="chap7.html#sec7-3"><i class="fa fa-check"></i><b>7.3</b> 交互作用及它们的重要性</a></li>
<li class="chapter" data-level="7.4" data-path="chap7.html"><a href="chap7.html#sec7-4"><i class="fa fa-check"></i><b>7.4</b> 主效应</a></li>
<li class="chapter" data-level="7.5" data-path="chap7.html"><a href="chap7.html#sec7-5"><i class="fa fa-check"></i><b>7.5</b> 计算机分析</a></li>
<li class="chapter" data-level="7.6" data-path="chap7.html"><a href="chap7.html#sec7-6"><i class="fa fa-check"></i><b>7.6</b> 结束语</a></li>
<li class="chapter" data-level="7.7" data-path="chap7.html"><a href="chap7.html#sec7-7"><i class="fa fa-check"></i><b>7.7</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="chap8.html"><a href="chap8.html"><i class="fa fa-check"></i><b>8</b> 案例研究：均衡双向实验的完整分析</a>
<ul>
<li class="chapter" data-level="8.1" data-path="chap8.html"><a href="chap8.html#sec8-1"><i class="fa fa-check"></i><b>8.1</b> 主效应均值对比</a></li>
<li class="chapter" data-level="8.2" data-path="chap8.html"><a href="chap8.html#sec8-2"><i class="fa fa-check"></i><b>8.2</b> 交互对比</a></li>
<li class="chapter" data-level="8.3" data-path="chap8.html"><a href="chap8.html#sec8-3"><i class="fa fa-check"></i><b>8.3</b> 油漆铺路示例</a></li>
<li class="chapter" data-level="8.4" data-path="chap8.html"><a href="chap8.html#sec8-4"><i class="fa fa-check"></i><b>8.4</b> 分析定量处理因素</a></li>
<li class="chapter" data-level="8.5" data-path="chap8.html"><a href="chap8.html#sec8-5"><i class="fa fa-check"></i><b>8.5</b> 多重检验</a></li>
<li class="chapter" data-level="8.6" data-path="chap8.html"><a href="chap8.html#sec8-6"><i class="fa fa-check"></i><b>8.6</b> 结束语</a></li>
<li class="chapter" data-level="8.7" data-path="chap8.html"><a href="chap8.html#sec8-7"><i class="fa fa-check"></i><b>8.7</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="9" data-path="chap9.html"><a href="chap9.html"><i class="fa fa-check"></i><b>9</b> 使用均值模型分析子类数不等的均衡双向处理结构</a>
<ul>
<li class="chapter" data-level="9.1" data-path="chap9.html"><a href="chap9.html#sec9-1"><i class="fa fa-check"></i><b>9.1</b> 模型定义和假设</a></li>
<li class="chapter" data-level="9.2" data-path="chap9.html"><a href="chap9.html#sec9-2"><i class="fa fa-check"></i><b>9.2</b> 参数估计</a></li>
<li class="chapter" data-level="9.3" data-path="chap9.html"><a href="chap9.html#sec9-3"><i class="fa fa-check"></i><b>9.3</b> 检验所有均值是否相等</a></li>
<li class="chapter" data-level="9.4" data-path="chap9.html"><a href="chap9.html#sec9-4"><i class="fa fa-check"></i><b>9.4</b> 交互作用和主效应假设</a></li>
<li class="chapter" data-level="9.5" data-path="chap9.html"><a href="chap9.html#sec9-5"><i class="fa fa-check"></i><b>9.5</b> 总体边际均值</a></li>
<li class="chapter" data-level="9.6" data-path="chap9.html"><a href="chap9.html#sec9-6"><i class="fa fa-check"></i><b>9.6</b> 同时推断与多重比较</a></li>
<li class="chapter" data-level="9.7" data-path="chap9.html"><a href="chap9.html#sec9-7"><i class="fa fa-check"></i><b>9.7</b> 结束语</a></li>
<li class="chapter" data-level="9.8" data-path="chap9.html"><a href="chap9.html#sec9-8"><i class="fa fa-check"></i><b>9.8</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="chap10.html"><a href="chap10.html"><i class="fa fa-check"></i><b>10</b> 使用效应模型分析子类数不等的均衡双向处理结构</a>
<ul>
<li class="chapter" data-level="10.1" data-path="chap10.html"><a href="chap10.html#sec10-1"><i class="fa fa-check"></i><b>10.1</b> 模型定义</a></li>
<li class="chapter" data-level="10.2" data-path="chap10.html"><a href="chap10.html#sec10-2"><i class="fa fa-check"></i><b>10.2</b> 参数估计和 I 型分析</a></li>
<li class="chapter" data-level="10.3" data-path="chap10.html"><a href="chap10.html#sec10-3"><i class="fa fa-check"></i><b>10.3</b> 在 SAS 中使用可估函数</a></li>
<li class="chapter" data-level="10.4" data-path="chap10.html"><a href="chap10.html#sec10-4"><i class="fa fa-check"></i><b>10.4</b> I–IV 型假设</a></li>
<li class="chapter" data-level="10.5" data-path="chap10.html"><a href="chap10.html#sec10-5"><i class="fa fa-check"></i><b>10.5</b> 在 SAS-GLM 中使用 I–IV 型可估函数</a></li>
<li class="chapter" data-level="10.6" data-path="chap10.html"><a href="chap10.html#sec10-6"><i class="fa fa-check"></i><b>10.6</b> 总体边际均值与最小二乘均值</a></li>
<li class="chapter" data-level="10.7" data-path="chap10.html"><a href="chap10.html#sec10-7"><i class="fa fa-check"></i><b>10.7</b> 计算机分析</a></li>
<li class="chapter" data-level="10.8" data-path="chap10.html"><a href="chap10.html#sec10-8"><i class="fa fa-check"></i><b>10.8</b> 结束语</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="chap11.html"><a href="chap11.html"><i class="fa fa-check"></i><b>11</b> 分析子类数不等的大型均衡双向实验</a>
<ul>
<li class="chapter" data-level="11.1" data-path="chap11.html"><a href="chap11.html#sec11-1"><i class="fa fa-check"></i><b>11.1</b> 可行性问题</a></li>
<li class="chapter" data-level="11.2" data-path="chap11.html"><a href="chap11.html#sec11-2"><i class="fa fa-check"></i><b>11.2</b> 未加权均值法</a></li>
<li class="chapter" data-level="11.3" data-path="chap11.html"><a href="chap11.html#sec11-3"><i class="fa fa-check"></i><b>11.3</b> 同时推断与多重比较</a></li>
<li class="chapter" data-level="11.4" data-path="chap11.html"><a href="chap11.html#sec11-4"><i class="fa fa-check"></i><b>11.4</b> 未加权均值的示例</a></li>
<li class="chapter" data-level="11.5" data-path="chap11.html"><a href="chap11.html#sec11-5"><i class="fa fa-check"></i><b>11.5</b> 计算机分析</a></li>
<li class="chapter" data-level="11.6" data-path="chap11.html"><a href="chap11.html#sec11-6"><i class="fa fa-check"></i><b>11.6</b> 结束语</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="chap12.html"><a href="chap12.html"><i class="fa fa-check"></i><b>12</b> 案例研究：子类数不等的均衡双向处理结构</a>
<ul>
<li class="chapter" data-level="12.1" data-path="chap12.html"><a href="chap12.html#sec12-1"><i class="fa fa-check"></i><b>12.1</b> 脂肪-表面活性剂示例</a></li>
<li class="chapter" data-level="12.2" data-path="chap12.html"><a href="chap12.html#sec12-2"><i class="fa fa-check"></i><b>12.2</b> 结束语</a></li>
</ul></li>
<li class="chapter" data-level="13" data-path="chap13.html"><a href="chap13.html"><i class="fa fa-check"></i><b>13</b> 使用均值模型分析缺失处理组合的双向处理结构</a>
<ul>
<li class="chapter" data-level="13.1" data-path="chap13.html"><a href="chap13.html#sec13-1"><i class="fa fa-check"></i><b>13.1</b> 参数估计</a></li>
<li class="chapter" data-level="13.2" data-path="chap13.html"><a href="chap13.html#sec13-2"><i class="fa fa-check"></i><b>13.2</b> 假设检验和置信区间</a>
<ul>
<li class="chapter" data-level="13.2.1" data-path="chap13.html"><a href="chap13.html#sec13-2-1"><i class="fa fa-check"></i><b>13.2.1</b> 示例 13.1</a></li>
</ul></li>
<li class="chapter" data-level="13.3" data-path="chap13.html"><a href="chap13.html#sec13-3"><i class="fa fa-check"></i><b>13.3</b> 计算机分析</a></li>
<li class="chapter" data-level="13.4" data-path="chap13.html"><a href="chap13.html#sec13-4"><i class="fa fa-check"></i><b>13.4</b> 结束语</a></li>
<li class="chapter" data-level="13.5" data-path="chap13.html"><a href="chap13.html#sec13-5"><i class="fa fa-check"></i><b>13.5</b> 练习</a></li>
<li class="chapter" data-level="13.6" data-path="chap13.html"><a href="chap13.html#sec13-6"><i class="fa fa-check"></i><b>13.6</b> R 代码</a></li>
</ul></li>
<li class="chapter" data-level="14" data-path="chap14.html"><a href="chap14.html"><i class="fa fa-check"></i><b>14</b> 使用效应模型分析缺失处理组合的双向处理结构</a>
<ul>
<li class="chapter" data-level="14.1" data-path="chap14.html"><a href="chap14.html#i-型和-ii-型假设"><i class="fa fa-check"></i><b>14.1</b> I 型和 II 型假设</a></li>
<li class="chapter" data-level="14.2" data-path="chap14.html"><a href="chap14.html#iii-型假设"><i class="fa fa-check"></i><b>14.2</b> III 型假设</a></li>
<li class="chapter" data-level="14.3" data-path="chap14.html"><a href="chap14.html#sec14-3"><i class="fa fa-check"></i><b>14.3</b> IV 型假设</a></li>
<li class="chapter" data-level="14.4" data-path="chap14.html"><a href="chap14.html#sec14-4"><i class="fa fa-check"></i><b>14.4</b> 总体边际均值和最小二乘均值</a></li>
<li class="chapter" data-level="14.5" data-path="chap14.html"><a href="chap14.html#sec14-5"><i class="fa fa-check"></i><b>14.5</b> 计算机分析</a></li>
<li class="chapter" data-level="14.6" data-path="chap14.html"><a href="chap14.html#sec14-6"><i class="fa fa-check"></i><b>14.6</b> 结束语</a></li>
<li class="chapter" data-level="14.7" data-path="chap14.html"><a href="chap14.html#sec14-7"><i class="fa fa-check"></i><b>14.7</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="15" data-path="chap15.html"><a href="chap15.html"><i class="fa fa-check"></i><b>15</b> 案例研究：缺失处理组合的双向处理结构</a>
<ul>
<li class="chapter" data-level="15.1" data-path="chap15.html"><a href="chap15.html#sec15-1"><i class="fa fa-check"></i><b>15.1</b> 案例研究</a></li>
<li class="chapter" data-level="15.2" data-path="chap15.html"><a href="chap15.html#sec15-2"><i class="fa fa-check"></i><b>15.2</b> 结束语</a></li>
</ul></li>
<li class="chapter" data-level="16" data-path="chap16.html"><a href="chap16.html"><i class="fa fa-check"></i><b>16</b> 分析三向和高阶处理结构</a>
<ul>
<li class="chapter" data-level="16.1" data-path="chap16.html"><a href="chap16.html#sec16-1"><i class="fa fa-check"></i><b>16.1</b> 一般策略</a></li>
<li class="chapter" data-level="16.2" data-path="chap16.html"><a href="chap16.html#sec16-2"><i class="fa fa-check"></i><b>16.2</b> 均衡和不均衡实验</a></li>
<li class="chapter" data-level="16.3" data-path="chap16.html"><a href="chap16.html#sec16-3"><i class="fa fa-check"></i><b>16.3</b> I 型和 II 型分析</a></li>
<li class="chapter" data-level="16.4" data-path="chap16.html"><a href="chap16.html#sec16-4"><i class="fa fa-check"></i><b>16.4</b> 结束语</a></li>
<li class="chapter" data-level="16.5" data-path="chap16.html"><a href="chap16.html#sec16-5"><i class="fa fa-check"></i><b>16.5</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="chap17.html"><a href="chap17.html"><i class="fa fa-check"></i><b>17</b> 案例研究：具有许多缺失处理组合的三向处理结构</a>
<ul>
<li class="chapter" data-level="17.1" data-path="chap17.html"><a href="chap17.html#sec17-1"><i class="fa fa-check"></i><b>17.1</b> 营养评分示例</a></li>
<li class="chapter" data-level="17.2" data-path="chap17.html"><a href="chap17.html#sec17-2"><i class="fa fa-check"></i><b>17.2</b> SAS-GLM 分析</a></li>
<li class="chapter" data-level="17.3" data-path="chap17.html"><a href="chap17.html#sec17-3"><i class="fa fa-check"></i><b>17.3</b> 一个完整的分析</a></li>
<li class="chapter" data-level="17.4" data-path="chap17.html"><a href="chap17.html#sec17-4"><i class="fa fa-check"></i><b>17.4</b> 结束语</a></li>
<li class="chapter" data-level="17.5" data-path="chap17.html"><a href="chap17.html#sec17-5"><i class="fa fa-check"></i><b>17.5</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="18" data-path="chap18.html"><a href="chap18.html"><i class="fa fa-check"></i><b>18</b> 随机效应模型和方差分量</a>
<ul>
<li class="chapter" data-level="18.1" data-path="chap18.html"><a href="chap18.html#sec18-1"><i class="fa fa-check"></i><b>18.1</b> 介绍</a>
<ul>
<li class="chapter" data-level="18.1.1" data-path="chap18.html"><a href="chap18.html#sec18-1-1"><i class="fa fa-check"></i><b>18.1.1</b> 示例 18.1：随机效应嵌套处理结构</a></li>
</ul></li>
<li class="chapter" data-level="18.2" data-path="chap18.html"><a href="chap18.html#sec18-2"><i class="fa fa-check"></i><b>18.2</b> 矩阵表示法中的一般随机效应模型</a>
<ul>
<li class="chapter" data-level="18.2.1" data-path="chap18.html"><a href="chap18.html#sec18-2-1"><i class="fa fa-check"></i><b>18.2.1</b> 示例 18.2：单向随机效应模型</a></li>
</ul></li>
<li class="chapter" data-level="18.3" data-path="chap18.html"><a href="chap18.html#sec18-3"><i class="fa fa-check"></i><b>18.3</b> 计算期望均方</a>
<ul>
<li class="chapter" data-level="18.3.1" data-path="chap18.html"><a href="chap18.html#sec18-3-1"><i class="fa fa-check"></i><b>18.3.1</b> 代数方法</a></li>
<li class="chapter" data-level="18.3.2" data-path="chap18.html"><a href="chap18.html#sec18-3-2"><i class="fa fa-check"></i><b>18.3.2</b> Hartley 综合法的计算</a></li>
</ul></li>
<li class="chapter" data-level="18.4" data-path="chap18.html"><a href="chap18.html#sec18-4"><i class="fa fa-check"></i><b>18.4</b> 结束语</a></li>
<li class="chapter" data-level="18.5" data-path="chap18.html"><a href="chap18.html#sec18-5"><i class="fa fa-check"></i><b>18.5</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="19" data-path="chap19.html"><a href="chap19.html"><i class="fa fa-check"></i><b>19</b> 方差分量的估计方法</a>
<ul>
<li class="chapter" data-level="19.1" data-path="chap19.html"><a href="chap19.html#sec19-1"><i class="fa fa-check"></i><b>19.1</b> 矩法</a>
<ul>
<li class="chapter" data-level="19.1.1" data-path="chap19.html"><a href="chap19.html#sec19-1-1"><i class="fa fa-check"></i><b>19.1.1</b> 应用。示例 19.1：不均衡单向模型</a></li>
<li class="chapter" data-level="19.1.2" data-path="chap19.html"><a href="chap19.html#sec19-1-2"><i class="fa fa-check"></i><b>19.1.2</b> 示例 19.2：单向随机效应模型中的小麦品种</a></li>
<li class="chapter" data-level="19.1.3" data-path="chap19.html"><a href="chap19.html#sec19-1-3"><i class="fa fa-check"></i><b>19.1.3</b> 示例 19.3：表 18.2 中的双向设计数据</a></li>
</ul></li>
<li class="chapter" data-level="19.2" data-path="chap19.html"><a href="chap19.html#sec19-2"><i class="fa fa-check"></i><b>19.2</b> 最大似然</a>
<ul>
<li class="chapter" data-level="19.2.1" data-path="chap19.html"><a href="chap19.html#sec19-2-1"><i class="fa fa-check"></i><b>19.2.1</b> 示例 19.4：均衡单向模型的最大似然解</a></li>
</ul></li>
<li class="chapter" data-level="19.3" data-path="chap19.html"><a href="chap19.html#sec19-3"><i class="fa fa-check"></i><b>19.3</b> 受限或残差最大似然估计</a>
<ul>
<li class="chapter" data-level="19.3.1" data-path="chap19.html"><a href="chap19.html#sec19-3-1"><i class="fa fa-check"></i><b>19.3.1</b> 示例 19.5：均衡单向模型的 REML 解</a></li>
</ul></li>
<li class="chapter" data-level="19.4" data-path="chap19.html"><a href="chap19.html#sec19-4"><i class="fa fa-check"></i><b>19.4</b> MIVQUE 法</a>
<ul>
<li class="chapter" data-level="19.4.1" data-path="chap19.html"><a href="chap19.html#sec19-4-1"><i class="fa fa-check"></i><b>19.4.1</b> 方法说明</a></li>
<li class="chapter" data-level="19.4.2" data-path="chap19.html"><a href="chap19.html#sec19-4-2"><i class="fa fa-check"></i><b>19.4.2</b> 应用。示例 19.6：MIVQUE 用于不均衡单向设计</a></li>
</ul></li>
<li class="chapter" data-level="19.5" data-path="chap19.html"><a href="chap19.html#sec19-5"><i class="fa fa-check"></i><b>19.5</b> 使用 JMP 估计方差分量</a></li>
<li class="chapter" data-level="19.6" data-path="chap19.html"><a href="chap19.html#sec19-6"><i class="fa fa-check"></i><b>19.6</b> 结束语</a></li>
<li class="chapter" data-level="19.7" data-path="chap19.html"><a href="chap19.html#sec19-7"><i class="fa fa-check"></i><b>19.7</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="20" data-path="chap20.html"><a href="chap20.html"><i class="fa fa-check"></i><b>20</b> 方差分量的推断方法</a>
<ul>
<li class="chapter" data-level="20.1" data-path="chap20.html"><a href="chap20.html#sec20-1"><i class="fa fa-check"></i><b>20.1</b> 假设检验</a>
<ul>
<li class="chapter" data-level="20.1.1" data-path="chap20.html"><a href="chap20.html#sec20-1-1"><i class="fa fa-check"></i><b>20.1.1</b> 使用方差分析表</a></li>
<li class="chapter" data-level="20.1.2" data-path="chap20.html"><a href="chap20.html#sec20-1-2"><i class="fa fa-check"></i><b>20.1.2</b> 示例 20.1：完全随机设计结构中的双向随机效应检验统计量</a></li>
<li class="chapter" data-level="20.1.3" data-path="chap20.html"><a href="chap20.html#sec20-1-3"><i class="fa fa-check"></i><b>20.1.3</b> 示例 20.2：复杂三向随机效应检验统计量</a></li>
<li class="chapter" data-level="20.1.4" data-path="chap20.html"><a href="chap20.html#sec20-1-4"><i class="fa fa-check"></i><b>20.1.4</b> 似然比检验</a></li>
<li class="chapter" data-level="20.1.5" data-path="chap20.html"><a href="chap20.html#sec20-1-5"><i class="fa fa-check"></i><b>20.1.5</b> 示例 20.3：小麦品种——单向随机效应模型</a></li>
<li class="chapter" data-level="20.1.6" data-path="chap20.html"><a href="chap20.html#sec20-1-6"><i class="fa fa-check"></i><b>20.1.6</b> 示例 20.4：不均衡双向</a></li>
</ul></li>
<li class="chapter" data-level="20.2" data-path="chap20.html"><a href="chap20.html#sec20-2"><i class="fa fa-check"></i><b>20.2</b> 构造置信区间</a>
<ul>
<li class="chapter" data-level="20.2.1" data-path="chap20.html"><a href="chap20.html#sec20-2-1"><i class="fa fa-check"></i><b>20.2.1</b> 残差方差 <span class="math inline">\(\sigma^2_\varepsilon\)</span></a></li>
<li class="chapter" data-level="20.2.2" data-path="chap20.html"><a href="chap20.html#sec20-2-2"><i class="fa fa-check"></i><b>20.2.2</b> 一般 Satterthwaite 近似</a></li>
<li class="chapter" data-level="20.2.3" data-path="chap20.html"><a href="chap20.html#sec20-2-3"><i class="fa fa-check"></i><b>20.2.3</b> 方差分量函数的近似置信区间</a></li>
<li class="chapter" data-level="20.2.4" data-path="chap20.html"><a href="chap20.html#sec20-2-4"><i class="fa fa-check"></i><b>20.2.4</b> 方差分量的 Wald 型置信区间</a></li>
<li class="chapter" data-level="20.2.5" data-path="chap20.html"><a href="chap20.html#sec20-2-5"><i class="fa fa-check"></i><b>20.2.5</b> 一些精确的置信区间</a></li>
<li class="chapter" data-level="20.2.6" data-path="chap20.html"><a href="chap20.html#sec20-2-6"><i class="fa fa-check"></i><b>20.2.6</b> 示例 20.5：均衡单向随机效应处理结构</a></li>
<li class="chapter" data-level="20.2.7" data-path="chap20.html"><a href="chap20.html#sec20-2-7"><i class="fa fa-check"></i><b>20.2.7</b> 示例 20.6</a></li>
<li class="chapter" data-level="20.2.8" data-path="chap20.html"><a href="chap20.html#sec20-2-8"><i class="fa fa-check"></i><b>20.2.8</b> 示例 20.6 （续）</a></li>
</ul></li>
<li class="chapter" data-level="20.3" data-path="chap20.html"><a href="chap20.html#sec20-3"><i class="fa fa-check"></i><b>20.3</b> 模拟研究</a></li>
<li class="chapter" data-level="20.4" data-path="chap20.html"><a href="chap20.html#sec20-4"><i class="fa fa-check"></i><b>20.4</b> 结束语</a></li>
<li class="chapter" data-level="20.5" data-path="chap20.html"><a href="chap20.html#sec20-5"><i class="fa fa-check"></i><b>20.5</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="21" data-path="chap21.html"><a href="chap21.html"><i class="fa fa-check"></i><b>21</b> 案例研究：随机效应模型分析</a>
<ul>
<li class="chapter" data-level="21.1" data-path="chap21.html"><a href="chap21.html#sec21-1"><i class="fa fa-check"></i><b>21.1</b> 数据集</a></li>
<li class="chapter" data-level="21.2" data-path="chap21.html"><a href="chap21.html#sec21-2"><i class="fa fa-check"></i><b>21.2</b> 估计</a></li>
<li class="chapter" data-level="21.3" data-path="chap21.html"><a href="chap21.html#sec21-3"><i class="fa fa-check"></i><b>21.3</b> 模型构建</a></li>
<li class="chapter" data-level="21.4" data-path="chap21.html"><a href="chap21.html#sec21-4"><i class="fa fa-check"></i><b>21.4</b> 缩减模型</a></li>
<li class="chapter" data-level="21.5" data-path="chap21.html"><a href="chap21.html#sec21-5"><i class="fa fa-check"></i><b>21.5</b> 置信区间</a></li>
<li class="chapter" data-level="21.6" data-path="chap21.html"><a href="chap21.html#sec21-6"><i class="fa fa-check"></i><b>21.6</b> 使用 JMP 进行计算</a></li>
<li class="chapter" data-level="21.7" data-path="chap21.html"><a href="chap21.html#sec21-7"><i class="fa fa-check"></i><b>21.7</b> 结束语</a></li>
<li class="chapter" data-level="21.8" data-path="chap21.html"><a href="chap21.html#sec21-8"><i class="fa fa-check"></i><b>21.8</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="22" data-path="chap22.html"><a href="chap22.html"><i class="fa fa-check"></i><b>22</b> 混合模型的分析</a>
<ul>
<li class="chapter" data-level="22.1" data-path="chap22.html"><a href="chap22.html#sec22-1"><i class="fa fa-check"></i><b>22.1</b> 混合模型简介</a></li>
<li class="chapter" data-level="22.2" data-path="chap22.html"><a href="chap22.html#sec22-2"><i class="fa fa-check"></i><b>22.2</b> 混合模型随机效应部分的分析</a>
<ul>
<li class="chapter" data-level="22.2.1" data-path="chap22.html"><a href="chap22.html#sec22-2-1"><i class="fa fa-check"></i><b>22.2.1</b> 矩法</a></li>
<li class="chapter" data-level="22.2.2" data-path="chap22.html"><a href="chap22.html#sec22-2-2"><i class="fa fa-check"></i><b>22.2.2</b> 最大似然方法</a></li>
<li class="chapter" data-level="22.2.3" data-path="chap22.html"><a href="chap22.html#sec22-2-3"><i class="fa fa-check"></i><b>22.2.3</b> 残差最大似然法</a></li>
<li class="chapter" data-level="22.2.4" data-path="chap22.html"><a href="chap22.html#sec22-2-4"><i class="fa fa-check"></i><b>22.2.4</b> MINQUE 法</a></li>
</ul></li>
<li class="chapter" data-level="22.3" data-path="chap22.html"><a href="chap22.html#sec22-3"><i class="fa fa-check"></i><b>22.3</b> 混合模型固定效应部分的分析</a>
<ul>
<li class="chapter" data-level="22.3.1" data-path="chap22.html"><a href="chap22.html#sec22-3-1"><i class="fa fa-check"></i><b>22.3.1</b> 估计</a></li>
<li class="chapter" data-level="22.3.2" data-path="chap22.html"><a href="chap22.html#sec22-3-2"><i class="fa fa-check"></i><b>22.3.2</b> 置信区间的构建</a></li>
<li class="chapter" data-level="22.3.3" data-path="chap22.html"><a href="chap22.html#sec22-3-3"><i class="fa fa-check"></i><b>22.3.3</b> 假设检验</a></li>
</ul></li>
<li class="chapter" data-level="22.4" data-path="chap22.html"><a href="chap22.html#sec22-4"><i class="fa fa-check"></i><b>22.4</b> 最佳线性无偏预测</a></li>
<li class="chapter" data-level="22.5" data-path="chap22.html"><a href="chap22.html#sec22-5"><i class="fa fa-check"></i><b>22.5</b> 混合模型方程组</a></li>
<li class="chapter" data-level="22.6" data-path="chap22.html"><a href="chap22.html#sec22-6"><i class="fa fa-check"></i><b>22.6</b> 结束语</a></li>
<li class="chapter" data-level="22.7" data-path="chap22.html"><a href="chap22.html#sec22-7"><i class="fa fa-check"></i><b>22.7</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="23" data-path="chap23.html"><a href="chap23.html"><i class="fa fa-check"></i><b>23</b> 案例研究：混合模型</a>
<ul>
<li class="chapter" data-level="23.1" data-path="chap23.html"><a href="chap23.html#sec23-1"><i class="fa fa-check"></i><b>23.1</b> 双向混合模型</a></li>
<li class="chapter" data-level="23.2" data-path="chap23.html"><a href="chap23.html#sed23-2"><i class="fa fa-check"></i><b>23.2</b> 不均衡双向混合模型</a></li>
<li class="chapter" data-level="23.3" data-path="chap23.html"><a href="chap23.html#sec23-3"><i class="fa fa-check"></i><b>23.3</b> 不均衡双向数据集的 JMP 分析</a></li>
<li class="chapter" data-level="23.4" data-path="chap23.html"><a href="chap23.html#sec23-4"><i class="fa fa-check"></i><b>23.4</b> 结束语</a></li>
<li class="chapter" data-level="23.5" data-path="chap23.html"><a href="chap23.html#sec23-5"><i class="fa fa-check"></i><b>23.5</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="24" data-path="chap24.html"><a href="chap24.html"><i class="fa fa-check"></i><b>24</b> 分析裂区型设计的方法</a>
<ul>
<li class="chapter" data-level="24.1" data-path="chap24.html"><a href="chap24.html#sec24-1"><i class="fa fa-check"></i><b>24.1</b> 介绍</a>
<ul>
<li class="chapter" data-level="24.1.1" data-path="chap24.html"><a href="chap24.html#sec24-1-1"><i class="fa fa-check"></i><b>24.1.1</b> 示例 24.1：面包配方和烘焙温度</a></li>
<li class="chapter" data-level="24.1.2" data-path="chap24.html"><a href="chap24.html#sec24-1-2"><i class="fa fa-check"></i><b>24.1.2</b> 示例 24.2：在不同肥力条件下生长的小麦品种</a></li>
</ul></li>
<li class="chapter" data-level="24.2" data-path="chap24.html"><a href="chap24.html#sec24-2"><i class="fa fa-check"></i><b>24.2</b> 模型定义和参数估计</a></li>
<li class="chapter" data-level="24.3" data-path="chap24.html"><a href="chap24.html#sec24-3"><i class="fa fa-check"></i><b>24.3</b> 均值间比较的标准误</a></li>
<li class="chapter" data-level="24.4" data-path="chap24.html"><a href="chap24.html#sec24-4"><i class="fa fa-check"></i><b>24.4</b> 计算均值差标准误的一般方法</a>
<ul>
<li class="chapter" data-level="24.4.1" data-path="chap24.html"><a href="chap24.html#sec24-5"><i class="fa fa-check"></i><b>24.4.1</b> 通过一般对比进行比较</a></li>
</ul></li>
<li class="chapter" data-level="24.5" data-path="chap24.html"><a href="chap24.html#sec24-6"><i class="fa fa-check"></i><b>24.5</b> 其他示例</a>
<ul>
<li class="chapter" data-level="24.5.1" data-path="chap24.html"><a href="chap24.html#sec24-6-1"><i class="fa fa-check"></i><b>24.5.1</b> 示例 24.3：水分和肥料</a></li>
<li class="chapter" data-level="24.5.2" data-path="chap24.html"><a href="chap24.html#sec24-6-2"><i class="fa fa-check"></i><b>24.5.2</b> 示例 24.4：具有裂区误差的回归</a></li>
<li class="chapter" data-level="24.5.3" data-path="chap24.html"><a href="chap24.html#sec24-6-3"><i class="fa fa-check"></i><b>24.5.3</b> 示例 24.5：混乱的裂区设计</a></li>
<li class="chapter" data-level="24.5.4" data-path="chap24.html"><a href="chap24.html#sec24-6-4"><i class="fa fa-check"></i><b>24.5.4</b> 示例 24.6：裂-裂区设计</a></li>
</ul></li>
<li class="chapter" data-level="24.6" data-path="chap24.html"><a href="chap24.html#sec24-7"><i class="fa fa-check"></i><b>24.6</b> 样本量和功效考虑</a></li>
<li class="chapter" data-level="24.7" data-path="chap24.html"><a href="chap24.html#sec24-8"><i class="fa fa-check"></i><b>24.7</b> 使用 JMP 进行计算：示例 24.7</a></li>
<li class="chapter" data-level="24.8" data-path="chap24.html"><a href="chap24.html#sec24-9"><i class="fa fa-check"></i><b>24.8</b> 结束语</a></li>
<li class="chapter" data-level="24.9" data-path="chap24.html"><a href="chap24.html#sec24-10"><i class="fa fa-check"></i><b>24.9</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="25" data-path="chap25.html"><a href="chap25.html"><i class="fa fa-check"></i><b>25</b> 分析条区型设计的方法</a>
<ul>
<li class="chapter" data-level="25.1" data-path="chap25.html"><a href="chap25.html#sec25-1"><i class="fa fa-check"></i><b>25.1</b> 条区设计和模型的描述</a></li>
<li class="chapter" data-level="25.2" data-path="chap25.html"><a href="chap25.html#sec25-2"><i class="fa fa-check"></i><b>25.2</b> 推断技术</a></li>
<li class="chapter" data-level="25.3" data-path="chap25.html"><a href="chap25.html#sec25-3"><i class="fa fa-check"></i><b>25.3</b> 示例：氮与灌溉</a></li>
<li class="chapter" data-level="25.4" data-path="chap25.html"><a href="chap25.html#sec25-4"><i class="fa fa-check"></i><b>25.4</b> 示例：含裂区的条区 1</a></li>
<li class="chapter" data-level="25.5" data-path="chap25.html"><a href="chap25.html#sec25-5"><i class="fa fa-check"></i><b>25.5</b> 示例：含裂区的条区 2</a></li>
<li class="chapter" data-level="25.6" data-path="chap25.html"><a href="chap25.html#sec25-6"><i class="fa fa-check"></i><b>25.6</b> 示例：含裂区的条区 3</a></li>
<li class="chapter" data-level="25.7" data-path="chap25.html"><a href="chap25.html#sec25-7"><i class="fa fa-check"></i><b>25.7</b> 示例：含裂区的条区 4</a></li>
<li class="chapter" data-level="25.8" data-path="chap25.html"><a href="chap25.html#sec25-8"><i class="fa fa-check"></i><b>25.8</b> 条-条区的设计与分析：基于 JMP7</a></li>
<li class="chapter" data-level="25.9" data-path="chap25.html"><a href="chap25.html#sec25-9"><i class="fa fa-check"></i><b>25.9</b> 结束语</a></li>
<li class="chapter" data-level="25.10" data-path="chap25.html"><a href="chap25.html#sec25-10"><i class="fa fa-check"></i><b>25.10</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="26" data-path="chap26.html"><a href="chap26.html"><i class="fa fa-check"></i><b>26</b> 分析重复测量实验的方法</a>
<ul>
<li class="chapter" data-level="26.1" data-path="chap26.html"><a href="chap26.html#sec26-1"><i class="fa fa-check"></i><b>26.1</b> 模型指定和理想条件</a></li>
<li class="chapter" data-level="26.2" data-path="chap26.html"><a href="chap26.html#sec26-2"><i class="fa fa-check"></i><b>26.2</b> 时间的裂区分析</a>
<ul>
<li class="chapter" data-level="26.2.1" data-path="chap26.html"><a href="chap26.html#sec26-2-1"><i class="fa fa-check"></i><b>26.2.1</b> 示例 26.1：药物对心率的影响</a></li>
<li class="chapter" data-level="26.2.2" data-path="chap26.html"><a href="chap26.html#sec26-2-2"><i class="fa fa-check"></i><b>26.2.2</b> 示例 26.2：一个复杂的舒适度实验</a></li>
<li class="chapter" data-level="26.2.3" data-path="chap26.html"><a href="chap26.html#sec26-2-3"><i class="fa fa-check"></i><b>26.2.3</b> 示例 26.3：家庭态度</a></li>
</ul></li>
<li class="chapter" data-level="26.3" data-path="chap26.html"><a href="chap26.html#sec26-3"><i class="fa fa-check"></i><b>26.3</b> 使用 SAS-Mixed 程序的数据分析</a>
<ul>
<li class="chapter" data-level="26.3.1" data-path="chap26.html"><a href="chap26.html#sec26-3-1"><i class="fa fa-check"></i><b>26.3.1</b> 示例 26.1</a></li>
<li class="chapter" data-level="26.3.2" data-path="chap26.html"><a href="chap26.html#sec26-3-2"><i class="fa fa-check"></i><b>26.3.2</b> 示例 26.2</a></li>
<li class="chapter" data-level="26.3.3" data-path="chap26.html"><a href="chap26.html#sec26-3-3"><i class="fa fa-check"></i><b>26.3.3</b> 示例 26.3</a></li>
</ul></li>
<li class="chapter" data-level="26.4" data-path="chap26.html"><a href="chap26.html#sec26-4"><i class="fa fa-check"></i><b>26.4</b> 结束语</a></li>
<li class="chapter" data-level="26.5" data-path="chap26.html"><a href="chap26.html#sec26-5"><i class="fa fa-check"></i><b>26.5</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="27" data-path="chap27.html"><a href="chap27.html"><i class="fa fa-check"></i><b>27</b> 不满足理想条件时重复测量实验的分析</a>
<ul>
<li class="chapter" data-level="27.1" data-path="chap27.html"><a href="chap27.html#sec27-1"><i class="fa fa-check"></i><b>27.1</b> 介绍</a></li>
<li class="chapter" data-level="27.2" data-path="chap27.html"><a href="chap27.html#sec27-2"><i class="fa fa-check"></i><b>27.2</b> MANOVA 法</a></li>
<li class="chapter" data-level="27.3" data-path="chap27.html"><a href="chap27.html#sec27-3"><i class="fa fa-check"></i><b>27.3</b> <span class="math inline">\(p\)</span> 值调整法</a></li>
<li class="chapter" data-level="27.4" data-path="chap27.html"><a href="chap27.html#sec27-4"><i class="fa fa-check"></i><b>27.4</b> 混合模型法</a>
<ul>
<li class="chapter" data-level="27.4.1" data-path="chap27.html"><a href="chap27.html#sec27-4-1"><i class="fa fa-check"></i><b>27.4.1</b> 最大似然法</a></li>
<li class="chapter" data-level="27.4.2" data-path="chap27.html"><a href="chap27.html#sec27-4-2"><i class="fa fa-check"></i><b>27.4.2</b> 受限最大似然法</a></li>
</ul></li>
<li class="chapter" data-level="27.5" data-path="chap27.html"><a href="chap27.html#sec27-5"><i class="fa fa-check"></i><b>27.5</b> 总结</a></li>
<li class="chapter" data-level="27.6" data-path="chap27.html"><a href="chap27.html#sec27-6"><i class="fa fa-check"></i><b>27.6</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="28" data-path="chap28.html"><a href="chap28.html"><i class="fa fa-check"></i><b>28</b> 案例研究：重复测量的复杂例子</a>
<ul>
<li class="chapter" data-level="28.1" data-path="chap28.html"><a href="chap28.html#sec28-1"><i class="fa fa-check"></i><b>28.1</b> 复杂舒适度实验</a></li>
<li class="chapter" data-level="28.2" data-path="chap28.html"><a href="chap28.html#sec28-2"><i class="fa fa-check"></i><b>28.2</b> 家庭态度实验</a></li>
<li class="chapter" data-level="28.3" data-path="chap28.html"><a href="chap28.html#sec28-3"><i class="fa fa-check"></i><b>28.3</b> 多地点研究</a></li>
<li class="chapter" data-level="28.4" data-path="chap28.html"><a href="chap28.html#sec28-4"><i class="fa fa-check"></i><b>28.4</b> 练习</a></li>
</ul></li>
<li class="chapter" data-level="29" data-path="chap29.html"><a href="chap29.html"><i class="fa fa-check"></i><b>29</b> 交叉设计的分析</a>
<ul>
<li class="chapter" data-level="29.1" data-path="chap29.html"><a href="chap29.html#sec29-1"><i class="fa fa-check"></i><b>29.1</b> 定义，假设和模型</a></li>
<li class="chapter" data-level="29.2" data-path="chap29.html"><a href="chap29.html#sec29-2"><i class="fa fa-check"></i><b>29.2</b> 两时期/两处理交叉设计</a></li>
<li class="chapter" data-level="29.3" data-path="chap29.html"><a href="chap29.html#sec29-3"><i class="fa fa-check"></i><b>29.3</b> 具有两个以上时期的交叉设计</a></li>
<li class="chapter" data-level="29.4" data-path="chap29.html"><a href="chap29.html#sec29-4"><i class="fa fa-check"></i><b>29.4</b> 具有两种以上处理的交叉设计</a></li>
<li class="chapter" data-level="29.5" data-path="chap29.html"><a href="chap29.html#sec29-5"><i class="fa fa-check"></i><b>29.5</b> 小结</a></li>
</ul></li>
<li class="chapter" data-level="30" data-path="chap30.html"><a href="chap30.html"><i class="fa fa-check"></i><b>30</b> 嵌套设计的分析</a>
<ul>
<li class="chapter" data-level="30.1" data-path="chap30.html"><a href="chap30.html#sec30-1"><i class="fa fa-check"></i><b>30.1</b> 定义，假设和模型</a>
<ul>
<li class="chapter" data-level="30.1.1" data-path="chap30.html"><a href="chap30.html#sec30-1-1"><i class="fa fa-check"></i><b>30.1.1</b> 示例 30.1：公司和杀虫剂</a></li>
<li class="chapter" data-level="30.1.2" data-path="chap30.html"><a href="chap30.html#sec30-1-2"><i class="fa fa-check"></i><b>30.1.2</b> 示例 30.2：舒适度实验回顾</a></li>
<li class="chapter" data-level="30.1.3" data-path="chap30.html"><a href="chap30.html#sec30-1-3"><i class="fa fa-check"></i><b>30.1.3</b> 示例 30.3：咖啡价格示例回顾</a></li>
</ul></li>
<li class="chapter" data-level="30.2" data-path="chap30.html"><a href="chap30.html#sec30-2"><i class="fa fa-check"></i><b>30.2</b> 参数估计</a>
<ul>
<li class="chapter" data-level="30.2.1" data-path="chap30.html"><a href="chap30.html#sec30-2-1"><i class="fa fa-check"></i><b>30.2.1</b> 示例 30.1：继续</a></li>
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<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">混乱数据分析：设计的实验</a>
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<h1><span class="header-section-number">第 6 章</span> 模型的矩阵形式<a href="chap6.html#chap6" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<blockquote>
<p>“The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work” - John Von Neumann</p>
</blockquote>
<p>当使用不均衡 (unbalanced) 的固定效应模型、随机效应模型或混合效应模型时，求和符号变得非常费力，有时甚至几乎无法使用。这个问题可以通过使用模型的矩阵形式表达来解决。本章讨论了模型矩阵形式的构建，并描述了如何使用矩阵获得最小二乘估计、检验假设、计算最小二乘均值或总体边际均值以及构建置信区间。在 <a href="chap6.html#sec6-3">6.3</a> 节中讨论了可估性 (estimability) 的概念。</p>
<div id="sec6-1" class="section level2 hasAnchor" number="6.1">
<h2><span class="header-section-number">6.1</span> 基本符号<a href="chap6.html#sec6-1" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>模型的矩阵形式可以表示为</p>
<p><span class="math display" id="eq:6-1">\[\begin{equation}
\underset{n\times1}{\boldsymbol y}=\underset{n × p}{\vphantom{\boldsymbol y}\boldsymbol X} \,\,\underset{p × 1}{\vphantom{\boldsymbol y}\boldsymbol \beta}+\underset{n × 1}{\vphantom{\boldsymbol y}\boldsymbol\varepsilon}
\tag{6.1}
\end{equation}\]</span></p>
<p>其中 <span class="math inline">\(\boldsymbol y\)</span> 表示 <span class="math inline">\(n \times 1\)</span> 观测向量，<span class="math inline">\(\boldsymbol X\)</span> 表示 <span class="math inline">\(n \times p\)</span> 已知常数矩阵，称为<strong>设计矩阵</strong> (designed matrix)，<span class="math inline">\(\boldsymbol \beta\)</span> 表示 <span class="math inline">\(p \times 1\)</span> 未知参数向量，<span class="math inline">\(\boldsymbol \varepsilon\)</span> 表示 <span class="math inline">\(n \times 1\)</span> 未观测的误差向量。第 <span class="math inline">\(i\)</span> 个观测值（<span class="math inline">\(\boldsymbol y\)</span> 的第 <span class="math inline">\(i\)</span> 个元素）的模型具有以下形式</p>
<p><span class="math display" id="eq:6-2">\[\begin{equation}
y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\cdots+\beta_{p-1}x_{ip-1}+\varepsilon_i,\quad i=1,2,\ldots,n
\tag{6.2}
\end{equation}\]</span></p>
<p>用于将模型 <a href="chap6.html#eq:6-2">(6.2)</a> 表示为矩阵模型 <a href="chap6.html#eq:6-1">(6.1)</a> 的向量和矩阵是</p>
<p><span class="math display" id="eq:6-3">\[\begin{equation}
\boldsymbol{y}=\begin{bmatrix}y_1\\y_2\\\vdots\\y_n\end{bmatrix},\quad\boldsymbol{X}=\begin{bmatrix}1&x_{11}&x_{12}&\cdots&x_{1p-1}\\1&x_{21}&x_{22}&\cdots&x_{2p-1}\\\vdots&\vdots&\vdots&\ddots&\vdots\\1&x_{n1}&x_{n2}&\cdots&x_{np-1}\end{bmatrix},\quad\boldsymbol{\beta}=\begin{bmatrix}\beta_0\\\beta_1\\\beta_2\\\vdots\\\beta_{p-1}\end{bmatrix},\quad\mathrm{and}\quad\boldsymbol{\varepsilon}=\begin{bmatrix}\varepsilon_1\\\varepsilon_2\\\vdots\\\varepsilon_n\end{bmatrix}
\tag{6.3}
\end{equation}\]</span></p>
<p>式 <a href="chap6.html#eq:6-3">(6.3)</a> 中的矩阵可用于表示多种模型类型，包括单向模型、双向模型、析因模型和部分析因模型等设计模型以及回归模型、协方差分析模型、随机模型等。通过指定 <span class="math inline">\(\boldsymbol X\)</span> 的适当元素以及 <span class="math inline">\(\boldsymbol \beta\)</span> 和 <span class="math inline">\(\boldsymbol \varepsilon\)</span> 的适当假设，可以构建效应模型、混合效应模型、裂区模型、重复测量模型和随机系数回归模型。以下各节介绍了各种实验情况下的一些矩阵模型。</p>
<div id="sec6-1-1" class="section level3 hasAnchor" number="6.1.1">
<h3><span class="header-section-number">6.1.1</span> 简单线性回归模型<a href="chap6.html#sec6-1-1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>简单线性回归模型可以表示为 <span class="math inline">\(y_i=\beta_0+\beta_1x_i+\varepsilon_i,i=1,2,\ldots,n\)</span> 并且可以用矩阵形式表示为</p>
<p><span class="math display">\[\begin{bmatrix}y_1\\y_2\\\vdots\\y_n\end{bmatrix}=\begin{bmatrix}1&x_1\\1&x_2\\\vdots&\vdots\\1&x_n\end{bmatrix}\begin{bmatrix}{\beta}_0\\{\beta}_1\end{bmatrix}+\begin{bmatrix}{\varepsilon}_1\\{\varepsilon}_2\\\vdots\\{\varepsilon}_n\end{bmatrix}\]</span></p>
<p><span class="math inline">\(\boldsymbol X\)</span> 中的全 1 列对应于回归模型的截距 <span class="math inline">\(\boldsymbol \beta_0\)</span>，<span class="math inline">\(x_i\)</span> 列对应于回归模型的斜率。</p>
</div>
<div id="sec6-1-2" class="section level3 hasAnchor" number="6.1.2">
<h3><span class="header-section-number">6.1.2</span> 单向处理结构模型<a href="chap6.html#sec6-1-2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>为了在完全随机设计结构中表示具有 <span class="math inline">\(t\)</span> 个处理的单向处理结构的模型，其中第 <span class="math inline">\(i\)</span> 个处理有 <span class="math inline">\(n_i\)</span> 个观测结果，设自变量 <span class="math inline">\(x_{ij}\)</span> 定义为</p>
<p><span class="math display">\[x_{kij}=\begin{cases}0&\text{ if the }ij\text{th observation is not from the }k\text{th treatment}\\1&\text{ if the }ij\text{th observation is from the }k\text{th treatment}&\end{cases}\]</span></p>
<p>其中 <span class="math inline">\(i=1,2,\cdots,t\)</span>，<span class="math inline">\(j=1,2,\cdots,n_i\)</span>. 变量 <span class="math inline">\(x_{kij}\)</span> 被称为<strong>指示变量</strong> (indicator variable)，因为当它的值为 1 时，它表明观测结果来自处理 k. 当它的值为 0 时，它表明观测结果不来自处理 k.</p>
<p><strong>均值模型</strong> (means model) 可表示为 <span class="math inline">\(y_{ij}={\mu}_1x_{1ij}+{\mu}_2x_{2ij}+\cdots+{\mu}_tx_{tij}+{\varepsilon}_{ij}\)</span>，其中 <span class="math inline">\(i=1,2,\cdots,t\)</span>，<span class="math inline">\(j=1,2,\cdots,n_i\)</span>. 或者以矩阵表示法</p>
<p><span class="math display">\[\begin{bmatrix}{y}_{11}\\{y}_{12}\\\vdots\\{y}_{1n_1}\\{y}_{22}\\\vdots\\{y}_{2{n}_2}\\\vdots\\{y}_{t1}\\\vdots\\{y}_{t{n}_t}\end{bmatrix}=\begin{bmatrix}1&0&\cdots&0\\1&0&\cdots&0\\\vdots&\vdots&\vdots&\vdots\\1&0&\cdots&0\\0&1&\cdots&0\\0&1&\cdots&0\\\vdots&\vdots&\vdots&\vdots\\0&1&\cdots&0\\\vdots&\vdots&\vdots&\vdots\\0&0&\cdots&1\\\vdots&\vdots&\ddots&\vdots\\0&0&\cdots&1\end{bmatrix}\begin{bmatrix}\mu_1\\\mu_2\\\vdots\\\mu_t\end{bmatrix}+\boldsymbol \varepsilon\]</span></p>
<p>均值模型通常表示为 <span class="math inline">\(y_{ij}=\mu_i+\varepsilon_{ij}\)</span>，其中 <span class="math inline">\(i=1,2,\cdots,t\)</span>，<span class="math inline">\(j=1,2,\cdots,n_i\)</span>.</p>
<p><strong>效应模型</strong> (effect model) 可表示为 <span class="math inline">\(y_{ij}=\mu+\tau_1x_{1ij}+\tau_2x_{2ij}+\cdots+\tau_tx_{tij}+\varepsilon_{ij}\)</span>，其中 <span class="math inline">\(i=1,2,\cdots,t\)</span>，<span class="math inline">\(j=1,2,\cdots,n_i\)</span>. 或者以矩阵表示法</p>
<p><span class="math display">\[\begin{bmatrix}{y}_{11}\\{y}_{12}\\\vdots\\{y}_{1{n}_1}\\{y}_{21}\\{y}_{22}\\\vdots\\{y}_{2{n}_2}\\\vdots\\{y}_{t1}\\\vdots\\{y}_{t{n}_t}\end{bmatrix}=\begin{bmatrix}1&1&0&\cdots&0\\1&1&0&\cdots&0\\\vdots&\vdots&\vdots&\vdots&\vdots\\1&1&0&\cdots&0\\1&0&1&\cdots&0\\1&0&1&\cdots&0\\\vdots&\vdots&\vdots&\vdots&\vdots\\1&0&1&\cdots&0\\\vdots&\vdots&\vdots&\vdots&\vdots\\1&0&0&\cdots&1\\\vdots&\vdots&\vdots&\ddots&\vdots\\1&0&0&\cdots&1\end{bmatrix}\begin{bmatrix}\mu\\\tau_1\\\tau_2\\\vdots\\\tau_t\end{bmatrix}+\boldsymbol \varepsilon \]</span></p>
<p>效应模型通常表示为 <span class="math inline">\(y_{ij}=\mu+\tau_i+\varepsilon_{ij}\)</span>，其中 <span class="math inline">\(i=1,2,\cdots,t\)</span>，<span class="math inline">\(j=1,2,\cdots,n_i\)</span>.</p>
<p>均值模型和效应模型之间的区别在于，效应模型的设计矩阵包含模型截距 <span class="math inline">\(\mu\)</span> 的全 1 列，而均值模型不包含全 1 列。</p>
</div>
<div id="sec6-1-3" class="section level3 hasAnchor" number="6.1.3">
<h3><span class="header-section-number">6.1.3</span> 双向处理结构模型<a href="chap6.html#sec6-1-3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>在完全随机设计结构中，对于具有 <span class="math inline">\(t\)</span> 行处理和 <span class="math inline">\(b\)</span> 列处理的双向处理结构，模型的一种形式为</p>
<p><span class="math display" id="eq:6-4">\[\begin{equation}
y_{ijk}=\mu_{ij}+\varepsilon_{ijk}\quad i=1,2,\ldots,t,j=1,2,\ldots,b,\mathrm{~and~}k=1,2,\ldots,n_{ij}
\tag{6.4}
\end{equation}\]</span></p>
<p>式子 <a href="chap6.html#eq:6-4">(6.4)</a> 中使用的模型称为均值模型，可以用矩阵形式表示为</p>
<p><span class="math display">\[\begin{bmatrix}y_{111}\\y_{112}\\\vdots\\y_{11n_1}\\y_{121}\\\vdots\\y_{12n_2}\\\vdots\\y_{1b1}\\\vdots\\y_{1bn_{1b}}
\\y_{211}\\\vdots\\y_{21n_{21}}\\\vdots\\y_{tb1}\\\vdots\\y_{tbn_{tb}}
\end{bmatrix}=\begin{bmatrix}
1&0&\cdots&0&0&\cdots&0\\
1&0&\cdots&0&0&\cdots&0\\
\vdots&\vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\
1&0&\cdots&0&0&\cdots&0\\
0&1&\cdots&0&0&\cdots&0\\
\vdots&\vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\
0&1&\cdots&0&0&\cdots&0\\
\vdots&\vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\
0&0&\cdots&1&0&\cdots&0\\
\vdots&\vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\
0&0&\cdots&1&0&\cdots&0\\
0&0&\cdots&0&1&\cdots&0\\
\vdots&\vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\
0&0&\cdots&0&1&\cdots&0\\
\vdots&\vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\
0&0&\cdots&0&0&\cdots&1\\
\vdots&\vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\
0&0&\cdots&0&0&\cdots&1\end{bmatrix}
\begin{bmatrix}\mu_{11}\\\mu_{12}\\\vdots\\\mu_{1b}\\\mu_{21}\\\vdots\\\mu_{tb}\end{bmatrix} +
\boldsymbol \varepsilon \]</span></p>
<p>双向效应模型可表示为</p>
<p><span class="math display" id="eq:6-5">\[\begin{equation}
y_{ijk}=\mu+\tau_i+\beta_j+\gamma_{ij}+\varepsilon_{ijk}\quad i=1,2,\ldots,t,j=1,2,\ldots,b,k=1,2,\ldots,n_{ij}
\tag{6.5}
\end{equation}\]</span></p>
<p>双向效应模型的矩阵形式为</p>
<p><span class="math display">\[
\begin{bmatrix}y_{111}\\y_{112}\\\vdots\\y_{11n_1}\\y_{121}\\\vdots\\y_{12n_2}\\\vdots\\y_{1b1}\\\vdots\\y_{1bn_{1b}}
\\y_{211}\\\vdots\\y_{21n_{21}}\\\vdots\\y_{tb1}\\\vdots\\y_{tbn_{tb}}
\end{bmatrix}=\begin{bmatrix}
1 & 0 & \cdots & 0 & 0 & \cdots & 0 \\
1 & 0 & \cdots & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \cdots & \vdots & \vdots & \cdots & \vdots \\
1 & 0 & \cdots & 0 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \cdots & \vdots & \vdots & \cdots & \vdots \\
0 & 1 & \cdots & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \cdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & \cdots & 1 & 0 & \cdots & 0 \\
\vdots & \vdots & \cdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & \cdots & 1 & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \cdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & \cdots & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \cdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & \cdots & 0 & 0 & \cdots & 1 \\
\vdots & \vdots & \cdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & \cdots & 0 & 0 & \cdots & 1 \\
\end{bmatrix} \begin{bmatrix}\mu_{11}\\\mu_{12}\\\vdots\\\mu_{1b}\\\mu_{21}\\\vdots\\\mu_{tb}\end{bmatrix}+
\boldsymbol \varepsilon
\]</span></p>
<p>本书强调与实验设计情境相对应的模型，而不是纯粹的回归情境。接下来的两个例子将展示如何从数据结构构建这样的模型。</p>
</div>
<div id="sec6-1-4" class="section level3 hasAnchor" number="6.1.4">
<h3><span class="header-section-number">6.1.4</span> 示例 6.1：双向处理结构的均值模型<a href="chap6.html#sec6-1-4" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>表 <a href="chap6.html#tab:table6-1">6.1</a> 中的信息代表了一个完全随机设计结构中的双向处理结构的数据，其中有三行处理和三列处理，每个单元格有一个或两个观测值。表 <a href="chap6.html#tab:table6-1">6.1</a> 中数据的均值模型的矩阵形式为：</p>
<p><span class="math display">\[\begin{bmatrix}3\\6\\9\\10\\2\\5\\3\\8\\4\\2\\6\end{bmatrix}=\begin{bmatrix}
1&0&0&0&0&0&0&0&0\\
1&0&0&0&0&0&0&0&0\\
0&1&0&0&0&0&0&0&0\\
0&0&1&0&0&0&0&0&0\\
0&0&0&1&0&0&0&0&0\\
0&0&0&0&1&0&0&0&0\\
0&0&0&0&1&0&0&0&0\\
0&0&0&0&0&1&0&0&0\\
0&0&0&0&0&0&1&0&0\\
0&0&0&0&0&0&0&1&0\\
0&0&0&0&0&0&0&0&1\end{bmatrix}\begin{bmatrix}\mu_{11}\\\mu_{12}\\\mu_{13}\\\mu_{21}\\\mu_{22}\\\mu_{23}\\\mu_{31}\\\mu_{32}\\\mu_{33}\end{bmatrix}+\boldsymbol \varepsilon\]</span></p>
<p>表 <a href="chap6.html#tab:table6-1">6.1</a> 中数据的效应模型的矩阵形式为</p>
<p><span class="math display">\[
\begin{bmatrix}3\\6\\9\\10\\2\\5\\3\\8\\4\\2\\6\end{bmatrix}=\begin{bmatrix}
1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
\mu_1 \\
\tau_1 \\
\tau_2 \\
\tau_3 \\
\beta_1 \\
\beta_2 \\
\beta_3 \\
\gamma_{11} \\
\gamma_{12} \\
\gamma_{13} \\
\gamma_{21} \\
\gamma_{22} \\
\gamma_{23} \\
\gamma_{31} \\
\gamma_{32} \\
\gamma_{33}
\end{bmatrix}
+\boldsymbol \varepsilon
\]</span></p>
<p>或</p>
<p><span class="math display">\[\boldsymbol y=\boldsymbol j\boldsymbol\mu+\boldsymbol X_1\boldsymbol \tau+\boldsymbol X_2\boldsymbol\beta+\boldsymbol X_3\boldsymbol \gamma+\boldsymbol\varepsilon \]</span></p>
<p>其中，<span class="math inline">\(\boldsymbol j\)</span> 是与上述设计矩阵的第一列对应的 11×1 向量，<span class="math inline">\(\boldsymbol X_1\)</span> 是与第 2-4 列对应的 11×3 矩阵，<span class="math inline">\(\boldsymbol X_2\)</span> 是与第 5-7 列对应的 11×3 矩阵，<span class="math inline">\(\boldsymbol X_3\)</span> 是与以上设计矩阵的最后九列对应的 11×9 矩阵。</p>
<table class="table table-condensed" style="margin-left: auto; margin-right: auto;">
<caption>
<span id="tab:table6-1">表 6.1: </span>CRD 结构中双向处理结构的数据
</caption>
<thead>
<tr>
<th style="text-align:center;color: white !important;background-color: white !important;font-size: 0px;">
x
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center;">
<img src="table/table%206.1.png">
</td>
</tr>
</tbody>
</table>
<p>其他处理和设计结构的设计矩阵也是以类似的方式构建的。幸运的是，大多数用于拟合非均衡数据结构的软件都会使用上述类型的表示形式，并在模型中指定分类效应 (categorical effects) 时自动生成设计矩阵的必要列。</p>
</div>
</div>
<div id="sec6-2" class="section level2 hasAnchor" number="6.2">
<h2><span class="header-section-number">6.2</span> 最小二乘估计<a href="chap6.html#sec6-2" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>一旦模型以矩阵形式指定，分析的下一步是获得参数向量 <span class="math inline">\(\boldsymbol \beta\)</span> 的<strong>最小二乘估计</strong> (least squares estimator)。最小二乘法可用于估计模型的参数。要使用此方法，假设模型可以表示为</p>
<p><span class="math display" id="eq:6-6">\[\begin{equation}
y_i=f(x_i;\boldsymbol{\beta})+\varepsilon_i\quad\mathrm{~for~}i=1,2,\ldots,n
\tag{6.6}
\end{equation}\]</span></p>
<p>其中 <span class="math inline">\(y_i=f(x_i;\boldsymbol{\beta})\)</span> 是由 <span class="math inline">\(x_i\)</span> 表示的设计变量向量的函数，且依赖于参数向量 <span class="math inline">\(\boldsymbol \beta\)</span>. <span class="math inline">\(\boldsymbol \beta\)</span> 的最小二乘估计通常用 <span class="math inline">\(\hat {\boldsymbol \beta}\)</span> 表示，它能最小化平方和</p>
<p><span class="math display" id="eq:6-7">\[\begin{equation}
SS(\boldsymbol{\beta})=\sum_{i=1}^n[y_i-f(x_i;\boldsymbol{\beta})]^2
\tag{6.7}
\end{equation}\]</span></p>
<p>如果除了假设模型的形式为 <a href="chap6.html#eq:6-6">(6.6)</a> 之外，还额外假设 <span class="math inline">\(\varepsilon_i \sim i.i.d\,N(0,\sigma^2),i＝1,2,\cdots,n\)</span>，则 <span class="math inline">\(\boldsymbol \beta\)</span> 的最小二乘估计也是<strong>最大似然估计</strong> (maximum likelihood estimator).</p>
<p>例如，完全随机设计中单向处理结构的均值模型的模型函数为</p>
<p><span class="math display">\[f(x_{ij};\boldsymbol{\beta})=\mu_i,\quad i=1,2,\ldots,t;\quad j=1,2,\ldots,n_i\]</span></p>
<p><span class="math inline">\(\mu_i\)</span> 的最小二乘估计记作 <span class="math inline">\(\hat \mu_1,\cdots,\hat \mu_t\)</span>，它们是这样的值，即能够最小化</p>
<p><span class="math display">\[\begin{aligned}SS({\mu})=\sum_{i=1}^t\sum_{j=1}^{n_j}(y_{ij}-\mu_i)^2\end{aligned}\]</span></p>
<p>完全随机设计中双向处理结构的均值模型的模型函数为</p>
<p><span class="math display">\[f(x_{ijk};\boldsymbol{\beta})=\mu_{ij}\quad i=1,2,\ldots,t;j=1,2,\ldots,b;k=1,2,\ldots,n_{ij}\]</span></p>
<p><span class="math inline">\(\mu_{ij}\)</span> 的最小二乘估计记作 <span class="math inline">\(\hat \mu_{11},\cdots,\hat \mu_{tb}\)</span>，它们是这样的值，即能够最小化</p>
<p><span class="math display">\[\begin{aligned}SS({\mu})=\sum_{i=1}^{t}\sum_{j=1}^{b}\sum_{k=1}^{n_{ij}}(y_{ijk}-\mu_{ij})^2\end{aligned}\]</span></p>
<p>完全随机设计中双向处理结构的效应模型的模型函数为</p>
<p><span class="math display">\[f(x_{ijk};\boldsymbol{\beta})=\mu+\tau_i+{\beta}_j+\gamma_{ij},\quad i=1,2,\ldots,t;j=1,2,\ldots,b;k=1,2,\ldots,n_{ij}\]</span></p>
<p><span class="math inline">\(\mu,\tau_i,\beta_j,\gamma_{ij}\)</span> 的最小二乘估计是通过最小化下式来获得的</p>
<p><span class="math display">\[SS(\mu,\tau_i,\beta_j,\gamma_{ij})=\sum_{i=1}^t\sum_{j=1}^b\sum_{k=1}^{n_{ij}}(y_{ijk}-\mu-\tau_i-\beta_j-\gamma_{ij})^2\]</span></p>
<p>通常，模型可以写成矩阵形式，如式 <a href="chap6.html#eq:6-1">(6.1)</a> 中的矩阵形式，并且 <span class="math inline">\(\boldsymbol \beta\)</span> 的最小二乘估计是这样的 <span class="math inline">\(\hat {\boldsymbol \beta}\)</span> ，即能最小化平方和</p>
<p><span class="math display" id="eq:6-8">\[\begin{equation}
SS(\boldsymbol\beta)=(\boldsymbol y-\boldsymbol X \boldsymbol\beta)^{\prime}(\boldsymbol y-\boldsymbol X \boldsymbol\beta)
\tag{6.8}
\end{equation}\]</span></p>
<div id="sec6-2-1" class="section level3 hasAnchor" number="6.2.1">
<h3><span class="header-section-number">6.2.1</span> 最小二乘方程组<a href="chap6.html#sec6-2-1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>矩阵表示和计算可用于确定能使残差平方和最小化的 <span class="math inline">\(\boldsymbol\beta\)</span> 值。当进行最小化时，会得到一组 <span class="math inline">\(\hat {\boldsymbol \beta}\)</span> 必须满足的方程，这些方程被称为模型的<strong>最小二乘方程组</strong> (least squares
equations) 或<strong>正规方程组</strong> (normal equations). 对于模型 <a href="chap6.html#eq:6-1">(6.1)</a>，正规方程组由下式给出：</p>
<p><span class="math display" id="eq:6-9">\[\begin{equation}
\boldsymbol X^{\prime} \boldsymbol X\hat{\boldsymbol{\beta}}=\boldsymbol X^{\prime}\boldsymbol{y}
\tag{6.9}
\end{equation}\]</span></p>
<p>任何满足正规方程组的向量 <span class="math inline">\(\hat {\boldsymbol \beta}\)</span> 都是 <span class="math inline">\(\boldsymbol\beta\)</span> 的最小二乘估计。对于一些模型，最小二乘估计不必是唯一的。为了帮助读者更加熟悉正规方程组，下面提供了第 <a href="chap6.html#sec6-1">6.1</a> 节中讨论的模型的正规方程组。</p>
<p>单向均值模型的正规方程组为</p>
<p><span class="math display">\[\begin{bmatrix}n_1&0&\cdots&0\\0&n_2&\cdots&0\\\vdots&\vdots&\ddots&\vdots\\0&0&\cdots&n_t\end{bmatrix}\begin{bmatrix}\hat{\mu}_1\\\hat{\mu}_2\\\vdots\\\hat{\mu}_t\end{bmatrix}=\begin{bmatrix}y_{1\cdot}\\y_{2\cdot}\\\vdots\\y_{t\cdot}\end{bmatrix}\mathrm{~where~}y_{i\cdot}=\sum_{i=1}^{n_i}y_{ij}\]</span></p>
<p>单向效应模型的正规方程组为</p>
<p><span class="math display">\[\begin{bmatrix}n_{\cdot}&n_1&n_2&\cdots&n_t\\n_1&n_1&0&\cdots&0\\n_2&0&n_2&\cdots&0\\\vdots&\vdots&\vdots&\ddots&\vdots\\n_t&0&0&\cdots&n_t\end{bmatrix}\begin{bmatrix}\hat{\mu}\\\hat{\tau}_1\\\hat \tau_2\\\vdots\\\hat{\tau}_t\end{bmatrix}=\begin{bmatrix}y_{\cdot\cdot}\\y_{1\cdot}\\y_{2\cdot}\\\vdots\\y_{t\cdot}\end{bmatrix}\quad\mathrm{where~}y_{\cdot\cdot}=\sum_{i=1}^t\sum_{j=1}^{n_i}y_{ij}\quad\mathrm{and}\quad n_{\cdot}=\sum_{i=1}^tn_t\]</span></p>
<p>使用表 <a href="chap6.html#tab:table6-1">6.1</a> 中数据的双向均值模型的正规方程组为</p>
<p><span class="math display">\[\begin{bmatrix}2&0&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0&0\\0&0&1&0&0&0&0&0&0\\0&0&0&1&0&0&0&0&0\\0&0&0&0&2&0&0&0&0\\0&0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&0&1\end{bmatrix}\begin{bmatrix}\hat{\mu}_{11}\\\hat{\mu}_{12}\\\hat{\mu}_{13}\\\hat{\mu}_{21}\\\hat{\mu}_{22}\\\hat{\mu}_{23}\\\hat{\mu}_{31}\\\hat{\mu}_{32}\\\hat{\mu}_{33}\end{bmatrix}=\begin{bmatrix}y_{11\cdot}\\y_{12\cdot}\\y_{13\cdot}\\y_{21\cdot}\\y_{22\cdot}\\y_{23\cdot}\\y_{31\cdot}\\y_{32\cdot}\\y_{33\cdot}\end{bmatrix}\]</span></p>
<p>其中</p>
<p><span class="math display">\[y_{ij\cdot}=\sum_{k=1}^{n_{ij}}y_{ijk}\]</span></p>
<p>与表 <a href="chap6.html#tab:table6-1">6.1</a> 中的数据相对应的双向效应模型的正规方程组为</p>
<p>$$$$</p>
<p><span class="math display">\[\begin{bmatrix}
11&4&4&3&4&4&3&2&1&1&1&2&1&1&1&1\\
4&4&0&0&2&1&1&2&1&1&0&0&0&0&0&0\\
4&0&4&0&1&2&1&0&0&0&1&2&1&0&0&0\\
3&0&0&3&1&1&1&0&0&0&0&0&0&1&1&1\\
4&2&1&1&4&0&0&2&0&0&1&0&0&1&0&0\\
4&1&2&1&0&4&0&0&1&0&0&2&0&0&1&0\\
3&1&1&1&0&0&3&0&0&1&0&0&1&0&0&1\\
2&2&0&0&2&0&0&2&0&1&0&0&0&0&0&0\\
1&1&0&0&0&1&0&0&1&0&0&0&0&0&0&0\\
1&1&0&0&0&0&1&0&0&1&0&0&0&0&0&0\\
1&0&1&0&1&0&0&0&0&0&1&0&0&0&0&0\\
2&0&2&0&0&2&0&0&0&0&0&2&0&0&0&0\\
1&0&1&0&0&0&1&0&0&0&0&0&1&0&0&0\\
1&0&0&1&1&0&0&0&0&0&0&0&0&1&0&0\\
1&0&0&1&0&1&0&0&0&0&0&0&0&0&1&0\\
1&0&0&1&0&0&1&0&0&0&0&0&0&0&0&1\end{bmatrix}
\begin{bmatrix}\hat\mu\\\hat\tau_{1}\\\hat{\tau}_2\\\hat\tau_{3}\\\hat\beta_{1}\\\hat\beta_{2}\\\hat\beta_{3}\\\hat\gamma_{11}\\\hat\gamma_{12}\\\hat\gamma_{13}\\\hat\gamma_{21}\\\hat\gamma_{22}\\\hat\gamma_{23}\\\hat\gamma_{31}\\\hat\gamma_{32}\\\hat\gamma_{33}\end{bmatrix}=\begin{bmatrix}y_{\cdot\cdot\cdot}\\y_{1\cdot\cdot}\\y_{2\cdot\cdot}\\y_{3\cdot\cdot}\\y_{\cdot1\cdot}\\y_{\cdot2\cdot}\\y_{\cdot3\cdot}\\y_{11\cdot}\\y_{12\cdot}\\y_{13\cdot}\\y_{21\cdot}\\y_{22\cdot}\\y_{23\cdot}\\y_{31\cdot}\\y_{32\cdot}\\y_{33\cdot}\end{bmatrix}\]</span></p>
<p>其中</p>
<p><span class="math display">\[y_{\cdot\cdot\cdot}=\sum_{i=1}^t\sum_{j=1}^b\sum_{k=1}^{n_{ij}}y_{ijk},\quad y_{i\cdot\cdot}=\sum_{j=1}^b\sum_{k=1}^{n_{ij}}y_{ijk},\quad y_{\cdot j\cdot}=\sum_{i=1}^t\sum_{k=1}^{n_{ij}}y_{ijk},\quad\mathrm{and}\quad y_{ij\cdot}=\sum_{k=1}^{n_{ij}}y_{ijk}\]</span></p>
<p>当 <span class="math inline">\(\boldsymbol X^\prime \boldsymbol X\)</span> 满秩时 (Graybill, 1976)，即 <span class="math inline">\(\boldsymbol X^\prime \boldsymbol X\)</span> 是非奇异的，则 <span class="math inline">\(\boldsymbol X^\prime \boldsymbol X\)</span> 的逆存在，并且 <span class="math inline">\({\boldsymbol \beta}\)</span> 的最小二乘估计（方程 <a href="chap6.html#eq:6-9">(6.9)</a> 中 <span class="math inline">\(\hat {\boldsymbol \beta}\)</span> 的解）为</p>
<p><span class="math display" id="eq:6-10">\[\begin{equation}
\hat {\boldsymbol \beta}=(\boldsymbol X^\prime \boldsymbol X)^{-1}\boldsymbol{X^{\prime}y}
\tag{6.10}
\end{equation}\]</span></p>
<p>当 <span class="math inline">\(\boldsymbol X^\prime \boldsymbol X\)</span> 是满秩时，最小二乘估计是唯一的。计算 <span class="math inline">\(\boldsymbol X^\prime \boldsymbol X\)</span> 的逆通常不是一项容易的任务。计算软件开发的最重要方面之一是，现在统计学家可以求超大型矩阵的逆，这在计算机出现之前是不可能的。然而，当 <span class="math inline">\(\boldsymbol X^\prime \boldsymbol X\)</span> 中存在某些模式时，可以利用这些模式更容易地求逆。对于单向均值模型和双向均值模型的正规方程组，<span class="math inline">\(\boldsymbol X^\prime \boldsymbol X\)</span> 是对角矩阵（所有对角元非零，非对角元为零），并且 <span class="math inline">\(\boldsymbol X^\prime \boldsymbol X\)</span> 的逆是通过简单地将每个对角元替换为其倒数来获得的。因此，单向均值模型的 <span class="math inline">\(\mu_i\)</span> 的最小二乘估计为</p>
<p><span class="math display">\[\begin{bmatrix}\hat{{\mu}}_1\\\hat{{\mu}}_2\\\vdots\\\hat{{\mu}}_t\end{bmatrix}=\begin{bmatrix}\frac1{n_1}&0&\cdots&0\\0&\frac1{n_2}&\cdots&0\\\vdots&\vdots&\ddots&\vdots\\0&0&\cdots&\frac1{n_t}\end{bmatrix}\begin{bmatrix}y_{1\cdot}\\y_{2\cdot}\\\vdots\\y_{t\cdot}\end{bmatrix}\]</span></p>
<p>或等价地</p>
<p><span class="math display">\[\hat{\mu}_i=\frac{y_{i\cdot}}{n_i}=\bar{y}_{i\cdot}\quad i=1,2,\ldots,t\]</span></p>
<p>类似地，双向均值模型 <span class="math inline">\(\mu_{ij}\)</span> 的最小二乘估计为</p>
<p><span class="math display">\[\hat{\mu}_{ij}=\frac{y_{ij\cdot}}{n_{ij}}=\bar{y}_{ij\cdot}\quad i=1,2,\ldots,t,\quad j=1,2,\ldots,b\]</span></p>
<p>与均值模型的正规方程组不同，效应模型的 <span class="math inline">\(\boldsymbol X^\prime \boldsymbol X\)</span> 是奇异的，并且 <span class="math inline">\(\boldsymbol X^\prime \boldsymbol X\)</span> 的逆不存在。在这种情况下，正规方程组有许多解（实际上是无穷多个最小二乘解）。效应模型被称为<strong>过度指定</strong> (overspecified models) 或<strong>奇异模型</strong> (sigular models)，因为与从收集的数据中可以唯一估计的参数相比，这些模型具有更多的参数。通常使用过度指定的模型，并且有几种方法可以求解其相应的正规方程组。以下讨论涉及完全随机设计结构中的双向处理结构，但类似的技术也可用于其他因素效应模型。</p>
<p>理论上，广义逆可用于求解关于 <span class="math inline">\({\boldsymbol \beta}\)</span> 的正规方程组 (Graybill, 1976)，但求解过度指定模型的正规方程组的常用方法是对模型中的参数进行限制（这实际上产生了 g-inverse 解）。对模型参数施加限制可以通过多种方式实现，这里考虑其中两种方式。</p>
</div>
<div id="sec6-2-2" class="section level3 hasAnchor" number="6.2.2">
<h3><span class="header-section-number">6.2.2</span> 零和限制<a href="chap6.html#sec6-2-2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>一种常见的技术是要求某些参数的和等于零。这种方法自方差分析之初就被用来求解正规方程组。对于使用表 <a href="chap6.html#tab:table6-1">6.1</a> 中数据的双向效应模型，<strong>零和限制</strong> (Sum-to-Zero Restrictions) 为：</p>
<p><span class="math display">\[\begin{aligned}
&\begin{aligned}\sum_{i=1}^3\tau_i=0,\sum_{j=1}^3\beta_j=0,\sum_{i=1}^3\gamma_{i1}=0,\sum_{i=1}^3\gamma_{i2}=0,\sum_{i=1}^3\gamma_{i3}=0\end{aligned} \\
&\sum_{j=1}^3\gamma_{1j}=0,\sum_{j=1}^3\gamma_{2j}=0,\,\text{and }\sum_{j=1}^3\gamma_{3j}=0
\end{aligned}\]</span></p>
<p>接下来，通过在考虑限制的情况下根据其他参数求解一些参数，然后将表达式替换回模型，将这些限制合并到模型中。例如，可以替换的参数有</p>
<p><span class="math display">\[\begin{aligned}\tau_3&=-\tau_1-\tau_2,\quad\beta_3=-\beta_1-\beta_2,\quad\gamma_{13}=-\gamma_{11}-\gamma_{12}\\\gamma_{23}&=-\gamma_{21}-\gamma_{22},\quad\gamma_{31}=-\gamma_{11}-\gamma_{21},\quad\gamma_{32}=-\gamma_{12}-\gamma_{22}\\\gamma_{33}&=-\gamma_{13}-\gamma_{23}=-\gamma_{31}-\gamma_{32}=\gamma_{11}+\gamma_{12}+\gamma_{21}+\gamma_{22}\end{aligned}\]</span></p>
<p>因此，替换模型中的 <span class="math inline">\(\tau_3,~\beta_3,~\gamma_{13},~\gamma_{23},~\gamma_{33},~\gamma_{31}\)</span> 和 <span class="math inline">\(\gamma_{32}\)</span>，以获得重新参数化的模型</p>
<p><span class="math display">\[
\begin{bmatrix}y_{111}\\y_{112}\\y_{121}\\y_{131}\\y_{211}\\y_{221}\\y_{222}\\y_{231}\\y_{311}\\y_{321}\\y_{331}\end{bmatrix}=
\begin{bmatrix}1&1&0&1&0&1&0&0&0\\1&1&0&1&0&1&0&0&0\\1&1&0&0&1&0&1&0&0\\1&1&0&0&1&0&1&0&0\\1&1&0&-1&-1&-1&-1&0&0\\1&0&1&1&0&0&0&1&0\\1&0&1&0&1&0&0&0&1\\1&0&1&0&1&0&0&0&1\\1&0&1&-1&-1&0&0&-1&-1\\1&-1&-1&0&0&-1&0&-1&0\\1&-1&-1&0&1&0&-1&0&-1\\1&-1&-1&-1&-1&1&1&1&1\end{bmatrix}\begin{bmatrix}\mu^*\\\tau_1^*\\\tau_2^*\\\beta_1^*\\\beta_2^*\\\gamma_{11}^*\\\gamma_{12}^*\\\gamma_{21}^*\\\gamma_{22}^*\end{bmatrix}+\boldsymbol{\varepsilon}
\]</span></p>
<p>其表示为</p>
<p><span class="math display">\[\begin{aligned}\boldsymbol{y}=\boldsymbol{X}^*\boldsymbol{\beta}^*+\boldsymbol{\varepsilon}\end{aligned}\]</span></p>
<p>表 <a href="chap6.html#tab:table6-1">6.1</a> 中数据对应的零和限制正态方程的解为 <span class="math inline">\(\hat{\boldsymbol{\beta^*}}=(\boldsymbol X^{*\prime} \boldsymbol X^*)^{-1}\boldsymbol X^{*\prime}\boldsymbol{y}\)</span>. 可得到</p>
<p><span class="math display">\[\begin{aligned}
\hat{\boldsymbol \beta}^{*\prime}&=[\hat{\mu}^*,\hat{\tau}_1^*,\hat{\tau}_2^*,\hat{\beta}_1^*,\hat{\beta}_2^*,\hat{\gamma}_{11}^*,\hat{\gamma}_{12}^*,\hat{\gamma}_{21}^*,\hat{\gamma}_{22}^*] \\
&=[5.500,2.333,-0.833,-2.000,-0.5000,-1.333,1.667,-0.667,-0.167]