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<!DOCTYPE html>
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<title>第 20 章 方差分量的推断方法 | 混乱数据分析：设计的实验</title>
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<meta name="twitter:title" content="第 20 章 方差分量的推断方法 | 混乱数据分析：设计的实验" />
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<meta name="author" content="Wang Zhen" />
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<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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<div id="chap20" class="section level1 hasAnchor" number="20">
<h1><span class="header-section-number">第 20 章</span> 方差分量的推断方法<a href="chap20.html#chap20" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<blockquote>
<p>“God not only plays dice. He also sometimes throws the dice where they cannot be seen.” - Stephen William Hawking</p>
</blockquote>
<p>当研究者设计一个涉及随机效应因素的实验时，她通常希望根据模型中指定的特定方差分量做出推断。特别是，如果 <span class="math inline">\(\sigma_u^2\)</span> 是对应于因素 A 水平分布的方差分量，实验者可能希望确定是否有足够的证据得出 <span class="math inline">\(\sigma_u^2>0\)</span> 的结论。可以通过以下方式做出适当的决定：1）检验假设 <span class="math inline">\(H_0{:}\sigma_u^2=0\mathrm{~vs~}{H_a}{:}\sigma_u^2>0\)</span>；2）构造关于 <span class="math inline">\(\sigma_u^2\)</span> 的置信区间；或 3）为 <span class="math inline">\(\sigma_u^2\)</span> 构造置信下限。本章讨论了随机效应模型的这些推断程序，其中假设检验方法在第 <a href="chap20.html#sec20-1">20.1</a> 节中描述，置信区间（下界）的构造在第 <a href="chap20.html#sec20-2">20.2</a> 节中描述。方差分量的置信区间构造一直是研究的一个活跃领域，许多作者已经为方差分量参数的特定函数开发了专门的置信区间。本章中描述的方法可在现行软件中使用，有些可用于特定问题。讨论并不全面，而是指出了已经解决的置信区间类型。在 Burdick and Graybill (1992) 以及当前统计期刊的论文中，有更为完整的讨论。</p>
<div id="sec20-1" class="section level2 hasAnchor" number="20.1">
<h2><span class="header-section-number">20.1</span> 假设检验<a href="chap20.html#sec20-1" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>有两种基本技术来检验关于方差分量的假设。第一种技术使用方差分析表中的平方和来构造 <span class="math inline">\(F\)</span> 统计量。对于大多数均衡模型，<span class="math inline">\(F\)</span> 统计量的分布与 <span class="math inline">\(F\)</span> 分布完全相同，而对于不均衡模型，分布由 <span class="math inline">\(F\)</span> 分布近似，随着设计变得越来越不均衡，近似值变得越来越差。第二种技术基于似然比检验 (likelihood rato test)，渐近分布于卡方分布。对于均衡设计，<span class="math inline">\(F\)</span> 统计量方法可能比似然比检验更好，而对于不均衡设计，没有明确的选择。读者可能希望在决定使用哪种方法来检验感兴趣的假设之前，使用与感兴趣的数据集类似的数据结构进行模拟实验，以研究检验统计量的分布。</p>
<div id="sec20-1-1" class="section level3 hasAnchor" number="20.1.1">
<h3><span class="header-section-number">20.1.1</span> 使用方差分析表<a href="chap20.html#sec20-1-1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>如果数据集是均衡的，那么通过常规方差分析获得的平方和独立分布于卡方随机变量的标量倍数<a href="#fn29" class="footnote-ref" id="fnref29"><sup>29</sup></a>。令 <span class="math inline">\(Q\)</span> 表示基于自由度 v 的平方和，其中其期望均方是四个方差分量的函数。也就是说，假设</p>
<p><span class="math display">\[E(Q/v)=\sigma_\varepsilon^2+k_1\sigma_1^2+k_2\sigma_2^2+k_3\sigma_3^2\]</span></p>
<p>然后，假设数据服从正态分布，</p>
<p><span class="math display">\[W=\frac Q{\sigma_\varepsilon^2+k_1\sigma_1^2+k_2\sigma_2^2+k_3\sigma_3^2}\]</span></p>
<p>通常分布为具有 n 个自由度的卡方随机变量。对于许多形如 <span class="math inline">\(H_0{:}\sigma_1^2=0\mathrm{~vs~}{H_a}{:}\sigma_1^2>0\)</span> 的假设，有两个独立的平方和，用 <span class="math inline">\(Q_1\)</span> 和 <span class="math inline">\(Q_2\)</span> 表示，分别基于 v<sub>1</sub> 和 v<sub>2</sub> 自由度，期望为</p>
<p><span class="math display">\[\begin{aligned}E(Q_1/n_1)&=\sigma_\varepsilon^2+k_1\sigma_1^2+k_2\sigma_2^2+k_3\sigma_3^2\\E(Q_2/n_2)&=\sigma_\varepsilon^2+k_2\sigma_2^2+k_3\sigma_3^2\end{aligned}\]</span></p>
<p>假设 <span class="math inline">\(H_0{:}\sigma_1^2=0\mathrm{~vs~}{H_a}{:}\sigma_1^2>0\)</span> 等价于</p>
<p><span class="math display">\[\begin{aligned}H_0\colon E\left(Q_1/v_1\right)&=E(Q_2/v_2)\text{ vs }H_a\colon E(Q_1/v_1)>E(Q_2/v_2)\end{aligned}\]</span></p>
<p>用于检验该假设的统计量为 <span class="math inline">\(F = (Q_1/v_1)/(Q_2/v_2)\)</span>，在 <span class="math inline">\(H_0\)</span> 条件下，该统计量通常分布为具有 v<sub>1</sub> 和 v<sub>2</sub> 自由度的中心 <span class="math inline">\(F\)</span> 分布。对于较大的 <span class="math inline">\(F\)</span> 值，该假设被拒绝。此过程涉及获取平方和，然后使用其期望均方来确定每个感兴趣假设的适当除数。以下两个示例演示了此程序。</p>
</div>
<div id="sec20-1-2" class="section level3 hasAnchor" number="20.1.2">
<h3><span class="header-section-number">20.1.2</span> 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 href="chap20.html#sec20-1-2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>在完全随机设计结构中，两个因素均为随机的双向处理结构的模型为</p>
<p><span class="math display">\[y_{ijk}=\mu+a_i+b_j+c_{ij}+\varepsilon_{ijk}\quad\mathrm{~for~}i=1,2,\ldots,a,j=1,2,\ldots,b,\mathrm{~and~}k=1,2,\ldots,n\]</span></p>
<p>其中 <span class="math inline">\(a_i\thicksim i.i.d.\,N(0,\sigma_a^2),b_j\thicksim i.i.d.\,N(0,\sigma_b^2),c_{ij}\thicksim i.i.d.\,N(0,\sigma_c^2),\varepsilon_{ijk}\thicksim i.i.d.\,N(0,\sigma_\varepsilon^2)\)</span>，且随机变量 <span class="math inline">\(a_i,b_j,c_{ij},\varepsilon_{ijk}\)</span> 独立分布。</p>
<p>表 <a href="chap20.html#tab:table20-1">20.1</a> 显示了模型的方差分析表以及模型的平方和和期望均方。通过检查期望均方来选择适当的分子和分母来构造检验统计量。用于检验假设 <span class="math inline">\(H_0{:}\sigma_a^2=0\mathrm{~vs~}{H_u}{:}\sigma_a^2>0\)</span> 的统计量是通过在 <span class="math inline">\(MSA\)</span> 的期望均方中设置 <span class="math inline">\(\sigma_a^2=0\)</span> 来构造的。接下来，当 <span class="math inline">\(H_0\)</span> 为真时，找到与 <span class="math inline">\(MSA\)</span> 具有相同期望均方的另一个均方，并使用该均方作为除数。要检验 <span class="math inline">\(H_0{:}\sigma_a^2=0\mathrm{~vs~}{H_u}{:}\sigma_a^2>0\)</span>，适当的除数是 <span class="math inline">\(MSAB\)</span>；要检验 <span class="math inline">\(H_0{:}\sigma_b^2=0\mathrm{~vs~}{H_u}{:}\sigma_b^2>0\)</span>，适当的除数是 <span class="math inline">\(MSAB\)</span>；要检验 <span class="math inline">\(H_0{:}\sigma_c^2=0\mathrm{~vs~}{H_u}{:}\sigma_c^2>0\)</span>，适当的除数是 <span class="math inline">\(MSResidual\)</span>. 决策规则是拒绝 <span class="math inline">\(H_0{:}\sigma_a^2=0\mathrm{~vs~}{H_u}{:}\sigma_a^2>0\)</span> 如果 <span class="math inline">\(F = MSA/MSAB>F_{\alpha,(a-1),(a-1)(b-1)}\)</span>，其中 <span class="math inline">\(\alpha\)</span> 为所选的 I 类错误率。对于 <span class="math inline">\(\sigma_b^2\)</span> 和 <span class="math inline">\(\sigma_c^2\)</span> 可以类似地确定检验统计量。表 <a href="chap20.html#tab:table20-1">20.1</a> 包含假设列表和相应的检验统计量。很可能，当 <span class="math inline">\(F\)</span> 统计量不超过指定的分位数时，结论并不是方差分量为零，而是与系统中其他变异源相比，方差分量的大小可以忽略不计。</p>
<table>
<caption>
<span id="tab:table20-1">表 20.1: </span>示例 <a href="chap20.html#sec20-1-2">20.1</a> 双向随机效应模型方差分析表
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<div id="sec20-1-3" class="section level3 hasAnchor" number="20.1.3">
<h3><span class="header-section-number">20.1.3</span> 示例 20.2：复杂三向随机效应检验统计量<a href="chap20.html#sec20-1-3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
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<span id="tab:table20-2">表 20.2: </span>示例 <a href="chap20.html#sec20-1-3">20.2</a> 数据
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<img src="table/table%2020.2.png">
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<p>表 <a href="chap20.html#tab:table20-2">20.2</a> 中的数据来自一个设计，其中处理结构中的三个因素的水平是随机效应，A 的水平与 B 的水平交叉，C 的水平嵌套在 B 的水平内，所有这些都在一个完全随机设计结构中。一个描述与表 <a href="chap20.html#tab:table20-2">20.2</a> 结构相似的大型数据集的一般模型是</p>
<p><span class="math display">\[\begin{aligned}&y_{ijkm}=\mu+a_i+b_j+(ab)_{ij}+c_{k(j)}+(ac)_{ik(j)}+\varepsilon_{ijkm}\\&\mathrm{for~}i=1,2,\ldots,a,j=1,2,\ldots,b,k=1,2,\ldots,c,\mathrm{~and~}m=1,2,\ldots,n\end{aligned}\]</span></p>
<p>参数 <span class="math inline">\(\mu\)</span> 表示总体均值，<span class="math inline">\(a_i\)</span> 表示因子 A 水平 i 的效应，<span class="math inline">\(b_j\)</span> 表示因子 B 水平 j 的效应，<span class="math inline">\((ab)_{ij}\)</span> 表示因子 A 和因子 B 水平之间的交互作用，<span class="math inline">\(c_{k(j)}\)</span> 表示嵌套在因子 B 第 j 个水平内的因子 C 水平 k 的效应，<span class="math inline">\((ac)_{ik(j)}\)</span> 表示因子 A 与嵌套在因子 B 内的因子 C 水平之间的交互作用，以及 <span class="math inline">\(\varepsilon_{ijkm}\)</span> 表示实验单元或抽样误差。在理想条件下，<span class="math inline">\(a_i\thicksim i.i.d.~N(0,\sigma_a^2),~b_j\thicksim i.i.d.~N(0,~\sigma_b^2),~(ab)_{ij}\thicksim i.i.d.~N(0,~\sigma_{ab}^2),~(ac)_{ik(j)}\thicksim i.i.d.~N(0,~\sigma_{ac(b)}^2)\)</span> 以及 <span class="math inline">\(\varepsilon_{ijkm}\thicksim i.i.d.~N(0,~\sigma_{\varepsilon}^2)\)</span>. 此外， <span class="math inline">\(a_i,b_j,(ab)_{ij},c_{k(j)},(ac)_{ik(j)}\)</span> 和 <span class="math inline">\(\varepsilon_{ijkm}\)</span> 独立分布。一般情况下的期望均方方差分析表如表 <a href="chap20.html#tab:table20-3">20.3</a> 所示。</p>
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<span id="tab:table20-3">表 20.3: </span>示例 <a href="chap20.html#sec20-1-3">20.2</a> 方差分析表
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<p>通过检查期望均方，可以构造 <span class="math inline">\(F\)</span> 统计量来检验每个方差分量的假设。用于检验以下假设的统计量是</p>
<ol style="list-style-type: decimal">
<li>要检验 <span class="math inline">\(H_0\colon{\sigma}_a^2=0\mathrm{~vs~}H_a\colon{\sigma}_a^2>0\)</span>，则 <span class="math inline">\(F_a=MSA/MSAB\)</span>.</li>
<li>要检验 <span class="math inline">\(H_0\colon{\sigma}_{ab}^2=0\mathrm{~vs~}H_a\colon{\sigma}_{ab}^2>0\)</span>，则 <span class="math inline">\(F_{ab}=MSAB/MSAC(B)\)</span>.</li>
<li>要检验 <span class="math inline">\(H_0\colon{\sigma}_{c(b)}^2=0\mathrm{~vs~}H_a\colon{\sigma}_{c(b)}^2>0\)</span>，则 <span class="math inline">\(F_{c(b)}=MSC(B)/MSAC(B)\)</span>.</li>
<li>要检验 <span class="math inline">\(H_0\colon{\sigma}_{ac(b)}^2=0\mathrm{~vs~}H_a\colon{\sigma}_{ac(b)}^2>0\)</span>，则 <span class="math inline">\(F_{ac(b)}=MSAC(B)/MSResidual\)</span>.</li>
</ol>
<p>然而，没有 <span class="math inline">\(F\)</span> 统计量来检验 <span class="math inline">\(H_0\colon{\sigma}_b^2=0\mathrm{~vs~}H_b\colon{\sigma}_a^2>0\)</span>，因为不涉及 <span class="math inline">\(\sigma^2_b\)</span> 的均方都不具有预期值 <span class="math inline">\(\sigma_\varepsilon^2+n\sigma_{ac(b)}^2+na\sigma_{c(b)}^2+nc\sigma_{ab}^2\)</span>，这是当 <span class="math inline">\(\sigma^2_b=0\)</span> 时 <span class="math inline">\(MSB\)</span> 的期望值。但存在一个均方的线性组合（不包括 <span class="math inline">\(MSB\)</span>）具有所需的期望值，即，<span class="math inline">\(E[MSC(B)+MSAB-MSAC(B)]=\sigma_\varepsilon^2+n\sigma_{ac(b)}^2+na\sigma_{c(b)}^2+nc\sigma_{ab}^2\)</span>. 令 <span class="math inline">\(Q=MSC(B)+MSAB-MSAC(B)\)</span>，那么用于检验 <span class="math inline">\(H_0\colon{\sigma}_b^2=0\mathrm{~vs~}H_b\colon{\sigma}_a^2>0\)</span> 的统计量为 <span class="math inline">\(F_b=MSB/Q\)</span>. <span class="math inline">\(F_b\)</span> 的抽样分布可以用自由度为 b-1 和 r 的 <span class="math inline">\(F\)</span> 分布来近似。分母自由度 r 的确定是通过使用第 <a href="chap2.html#chap2">2</a> 章中讨论的 Satterthwaite (1946) 近似，将 <span class="math inline">\(rQ/E(Q)\)</span> 的分布近似为卡方分布来完成的。Satterthwaite 近似用于近似 <span class="math inline">\(Q = q_1MS_1 + q_2MS_2 + \cdots + q_kMS_k\)</span> 的抽样分布，其中 <span class="math inline">\(MS_i\)</span> 表示基于自由度为 <span class="math inline">\(f_i\)</span> 的均方，均方独立分布，<span class="math inline">\(q_i\)</span> 是已知常数。那么 <span class="math inline">\(rQ/E(Q)\)</span> 近似分布为基于自由度为 r 的中心卡方随机变量，其中</p>
<p><span class="math display">\[r=\frac{(Q)^2}{\sum_{i=1}^k\frac{\left(q_iMS_i\right)^2}{f_i}}\]</span></p>
<p>假设 <span class="math inline">\(U\)</span> 是基于 f 个自由度的均方，独立分布于 <span class="math inline">\(MS_1,MS_2,\cdots,MS_k\)</span>，期望为 <span class="math inline">\(E(U) = E(Q) + k_0\sigma^2_0\)</span>. 检验 <span class="math inline">\(H_0\colon{\sigma}_0^2=0\mathrm{~vs~}H_a\colon{\sigma}_0^2>0\)</span> 的统计量为 <span class="math inline">\(F = U/Q\)</span>，其近似分布为具有 f 和 r 自由度的 <span class="math inline">\(F\)</span> 分布。</p>
<p>检验 <span class="math inline">\(H_0\colon{\sigma}_b^2=0\mathrm{~vs~}H_a\colon{\sigma}_b^2>0\)</span> 的统计量为 <span class="math inline">\(F_b = MSB/Q\)</span>，近似分布为具有 b-1 和 r 自由度的 <span class="math inline">\(F\)</span> 分布，其中</p>
<p><span class="math display">\[r=\frac{(Q)^2}{\frac{[MSC(B)]^2}{b(c-1)}+\frac{[MSAB]^2}{(a-1)(b-1)}+\frac{[MSAC(B)]^2}{b(a-1)(c-1)}}\]</span></p>
<table>
<caption>
<span id="tab:table20-4">表 20.4: </span>关示示例 <a href="chap20.html#sec20-1-3">20.2</a> 的数据，使用 I 型平方和的 Proc Mixed 代码的方差分析表
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<p>表 <a href="chap20.html#tab:table20-4">20.4</a> 包含表 <a href="chap20.html#tab:table20-3">20.3</a> 中数据的方差分析表，其中包括除数（用误差项表示）和用于检验各个假设的 <span class="math inline">\(F\)</span> 统计量。要检验 <span class="math inline">\(H_0\colon{\sigma}_b^2=0\mathrm{~vs~}H_a\colon{\sigma}_b^2>0\)</span>，令</p>
<p><span class="math display">\[\begin{aligned}
Q& =MSC(B)+MSAB-MSAC(B) \\
&=12.0625+10.5625-0.8125=21.8125
\end{aligned}\]</span></p>
<p><span class="math inline">\(Q\)</span> 对应的自由度为</p>
<p><span class="math display">\[\begin{aligned}
r& =\frac{(21.8125)^2}{(12.0625)^2/2+(10.5625)^2/1+(0.8125)^2/2} \\
&=\frac{475.7852}{184.9785}=2.57
\end{aligned}\]</span></p>
<p>检验统计量 <span class="math inline">\(F = 770.0625/21.8125 = 35.304\)</span> 基于 1 和 2.57 自由度。检验的显著性水平为 0.0144，表明有证据表明 <span class="math inline">\({\sigma}_b^2 > 0\)</span>，或者由于因子 B 水平总体引起的变异是系统总变异的重要组成部分。</p>
<p><strong>为了检验均衡设计中方差分量的假设，应尽可能使用根据两个均方之比构造的 <span class="math inline">\(F\)</span> 检验。当无法使用两个均方之比时，Satterthwaite 近似是可接受的替代方案</strong>。</p>
<p><strong>当设计不均衡时，几乎总是需要某种 Satterthwaite 近似来检验有关方差分量的假设</strong>。此外，方差分析表中的平方和可能不具有独立分布，尽管平方和集对于某些特殊情况可能是独立的。残差或误差平方和始终独立于方差分析表中的其他平方和。因此，对于任何期望为 <span class="math inline">\({\sigma}_{{\varepsilon}}^2+{k}_0{\sigma}_0^2\)</span> 的均方 <span class="math inline">\(U\)</span>，统计量 <span class="math inline">\(F_0 = U/MSResidual\)</span> 提供了对假设 <span class="math inline">\(H_0\colon{\sigma}_0^2=0\mathrm{~vs~}H_a\colon{\sigma}_0^2>0\)</span> 的检验。在 <span class="math inline">\(H_0\)</span> 的条件下，<span class="math inline">\(F\)</span> 分布为具有 u 和 v 自由度的中心 <span class="math inline">\(F\)</span> 分布，其中 u 是与 <span class="math inline">\(U\)</span> 相关的自由度，v 是与 <span class="math inline">\(MSResidual\)</span> 相关的自由度。</p>
<p>期望值涉及超过两个方差分量的均方通常不能用于获得具有精确 <span class="math inline">\(F\)</span> 抽样分布的单个方差分量的检验统计量。某些均衡设计会出现精确的 <span class="math inline">\(F\)</span> 分布，如前两个示例所示。比率不精确分布于 <span class="math inline">\(F\)</span> 的一个原因是各自的均方不是独立分布的。<strong>如果设计不太不均衡，那么使用 <span class="math inline">\(F\)</span> 分布作为近似应该是足够的</strong>。此外，当设计不均衡时，平方和（残差除外）不会分布为卡方分布的标量倍数。</p>
<p>一般来说，为了检验 <span class="math inline">\(H_0\colon{\sigma}_0^2=0\mathrm{~vs~}H_a\colon{\sigma}_0^2>0\)</span>，将有一个均方，记为 <span class="math inline">\(U_1\)</span>，期望为</p>
<p><span class="math display">\[E(U_1)=\sigma_\varepsilon^2+k_{1a}\sigma_a^2+k_{1b}\sigma_b^2+k_{1c}\sigma_c^2\]</span></p>
<p>但不会有其他均方具有期望 <span class="math inline">\(\sigma_\varepsilon^2+k_{1b}\sigma_b^2+k_{1c}\sigma_c^2\)</span>；也就是说，没有一个均方是合适的除数。方法是找到其他均方的线性组合，如 <span class="math inline">\(Q=\sum_{i=1}^kq_iMS_i\)</span> 其中 <span class="math inline">\(E(Q)=\sigma_\varepsilon^2+k_{1b}\sigma_b^2+k_{1c}\sigma_c^2\)</span>. Satterthwaite 近似可用于近似 <span class="math inline">\(Q\)</span> 的抽样分布，即，求 r，使得 <span class="math inline">\(rQ/E(Q)\)</span> 近似分布为具有 r 个自由度的卡方随机变量。该近似是双重的，因为 1) 自由度是近似的，2) 组成 <span class="math inline">\(Q\)</span> 的均方不一定按照近似所要求的那样独立分布为卡方随机变量。</p>
<p>SAS<sup>®</sup>-Mixed 代码和使用 III 型平方和得出的方差分析表，用于示例 <a href="chap19.html#sec19-1-2">19.2</a> 中的小麦虫害数据，如表 <a href="chap20.html#tab:table20-5">20.5</a> 所示。variaty 均方的期望值是 <span class="math inline">\(\sigma_\varepsilon^2+3.1795\sigma_\mathrm{var}^2\)</span>. 为了检验假设 <span class="math inline">\(H_0\colon\sigma_{\text{var}}^2=0\text{ vs }H_a\colon\sigma_{\text{var}}^2>0\)</span>，适当的除数是残差均方，它提供的 <span class="math inline">\(F\)</span> 统计量为 4.79. 将计算出的 <span class="math inline">\(F\)</span> 统计量与具有 3 和 9 个自由度的 <span class="math inline">\(F\)</span> 分布进行比较；其显著性水平为 0.0293. 由于这是一个单向实验，因此 I 型分析与 III 型分析相同。</p>
<table>
<caption>
<span id="tab:table20-5">表 20.5: </span>示例 <a href="chap19.html#sec19-1-2">19.2</a> 中数据含期望均方和 <span class="math inline">\(F\)</span> 统计量的方差分析表
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<table>
<caption>
<span id="tab:table20-6">表 20.6: </span>示例 <a href="chap19.html#sec19-1-3">19.3</a> 的双向随机效应数据的 I 型分析，其中预期均方和误差项用于计算 <span class="math inline">\(F\)</span> 统计量
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<p>表 <a href="chap20.html#tab:table20-6">20.6</a> 显示了根据示例 <a href="chap19.html#sec19-1-3">19.3</a> 的双向随机效应数据的 I 型平方和构造的 SAS-Mixed 代码和方差分析表。为了检验假设 <span class="math inline">\(H_0\colon{\sigma}_{\mathrm{row}\times\mathrm{col}}^2=0\text{ vs }H_a\colon{\sigma}_{\mathrm{row}\times\mathrm{col}}^2>0\)</span>，适当的除数是残差均方，它提供的 <span class="math inline">\(F\)</span> 统计量为 14.24. 将计算出的 <span class="math inline">\(F\)</span> 统计量与自由度为 2 和 8 的 <span class="math inline">\(F\)</span> 分布进行比较，提供的显著性水平为 0.0023. 没有精确的检验可用于检验 <span class="math inline">\(H_0\colon{\sigma^2}_{\text{row}}=0\text{ vs }H_a\colon{\sigma^2}_{\text{row}}>0\)</span> 和 <span class="math inline">\(H_0\colon{\sigma^2}_{\text{col}}=0\text{ vs }H_a\colon{\sigma^2}_{\text{col}}>0\)</span>，因此需要构造近似检验。用于检验 <span class="math inline">\(H_0\colon{\sigma^2}_{\text{row}}=0\text{ vs }H_a\colon{\sigma^2}_{\text{row}}>0\)</span> 的 <span class="math inline">\(MS_{\text{Row}}\)</span> 的适当除数计算如下</p>
<p><span class="math display">\[\begin{aligned}
Q_{row} =&\,\frac{0.1429}{4.5714}MSCol+\frac1{2.2588}{\left[2.4286-\frac{0.1429}{4.5714}\times2.3126\right]}MSRow\times Col \\
&+\left[1-\frac{0.1429}{4.5714}-\frac1{2.2588}{\left(2.4286-\frac{0.1429}{4.5714}\times2.3126\right)}\right]MSResidual \\
=&\,0.0313\times MSCol+1.0432 \,MSRow\times Col-0.0744\,MSResidual\\
=&\,55.7806
\end{aligned}\]</span></p>
<p>与 <span class="math inline">\(Q_{\mathrm{row}}\)</span> 相关的 Satterthwaite 近似自由度计算如下</p>
<p><span class="math display">\[\begin{aligned}df_{Q_{\mathrm{row}}}&=\frac{(Q_{\mathrm{row}})^2}{\frac{(0.0313\times MSCol)^2}2+\frac{(1.0432\times MSRow\times Col)^2}2+\frac{(0.0744\times MSResidual)^2}8}\\&=1.9961\end{aligned}\]</span></p>
<p>生成的 <span class="math inline">\(F\)</span> 统计量为 <span class="math inline">\(F_{\mathrm{row}}=MSRow/Q_{\mathrm{row}}=0.0113\)</span>，显著性水平为 0.9251.
用于检验 <span class="math inline">\(H_0\colon{\sigma^2}_{\text{col}}=0\text{ vs }H_a\colon{\sigma^2}_{\text{col}}>0\)</span> 的 <span class="math inline">\(MSCol\)</span> 的适当除数计算为</p>
<p><span class="math display">\[\begin{aligned}
Q_{\text{col}}& =\frac{2.3126}{2.2588}MSRow\times Col+\left[1-\frac{2.3126}{2.2588}\right]MSResidual \\
&=1.0238\,MSRow\times Col-0.0238\,MSResidual \\
&=56.8729
\end{aligned}\]</span></p>
<p>与 <span class="math inline">\(Q_{\mathrm{col}}\)</span> 相关的 Satterthwaite 近似自由度计算如下</p>
<p><span class="math display">\[\begin{aligned}df_{Q_{\mathrm{col}}}&=\frac{(Q_{\mathrm{col}})^2}{\frac{(1.0238\times MSRow\times Col)^2}2+\frac{(0.0238\times MSResidual)^2}8}\\&=1.9935\end{aligned}\]</span></p>
<p>生成的 <span class="math inline">\(F\)</span> 统计量为 <span class="math inline">\(F_{\mathrm{col}}=MSCol/Q_{\mathrm{col}}=0.1323\)</span>，显著性水平为 0.8832.</p>
</div>
<div id="sec20-1-4" class="section level3 hasAnchor" number="20.1.4">
<h3><span class="header-section-number">20.1.4</span> 似然比检验<a href="chap20.html#sec20-1-4" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>检验方差分量假设的第二种方法基于似然比程序，该程序涉及评估完全模型的似然函数值以及评估在 <span class="math inline">\(H_0\)</span> 条件下模型的似然函数值。</p>
<p>方程 <a href="chap18.html#eq:18-3">(18.3)</a> 的一般随机模型是</p>
<p><span class="math display">\[\boldsymbol y=\boldsymbol j_n\mu+\boldsymbol Z_1 \boldsymbol u_1+\boldsymbol Z_2\boldsymbol u_2+\cdots+\boldsymbol Z_k\boldsymbol u_k+\boldsymbol\varepsilon \]</span></p>
<p>其中 <span class="math inline">\(\boldsymbol{u}_1\thicksim N(0,{\sigma}_1^2\boldsymbol{I}_{t_1}),\boldsymbol{u}_2\thicksim N(0,{\sigma}_2^2\boldsymbol{I}_{t_2}),...,\boldsymbol{u}_r\thicksim N(0,{\sigma}_r^2\boldsymbol{I}_{t_r}),{\varepsilon}\thicksim N(0,{\sigma}_\varepsilon^2\boldsymbol{I}_N)\)</span> 且这些随机变量独立分布。分布假设意味着 <span class="math inline">\(\boldsymbol y\)</span> 的边际分布为 <span class="math inline">\(N(\boldsymbol j_n\mu,\boldsymbol \Sigma)\)</span> 其中 <span class="math inline">\(\boldsymbol \Sigma=\sigma_\varepsilon^2\boldsymbol I_n+\sigma_1^2\boldsymbol Z_1\boldsymbol Z_1^{\prime}+\sigma_2^2\boldsymbol Z_2\boldsymbol Z_2^{\prime}+\cdots+\sigma_k^2\boldsymbol Z_k\boldsymbol Z_k^{\prime}\)</span>. 似然方程是</p>
<p><span class="math display">\[L(\mu,\sigma_\varepsilon^2,\sigma_1^2,\sigma_2^2,\ldots,\sigma_k^2|\boldsymbol y)=(2\pi)^{-n/2}|\boldsymbol \Sigma|^{-1/2}\exp\left[-\frac12(\boldsymbol y-\boldsymbol j_n\mu)^{\prime}\boldsymbol\Sigma^{-1}(\boldsymbol y-\boldsymbol j_n\mu)\right]\]</span></p>
<p><span class="math inline">\(H_0{:{\sigma_1^2}}=0\)</span> 条件下的似然函数为</p>
<p><span class="math display">\[L_0(\mu,\sigma_\varepsilon^2,0,\sigma_2^2,\ldots,\sigma_k^2|\boldsymbol y)=(2\pi)^{-n/2}\left|\boldsymbol \Sigma_0\right|^{-1/2}\exp\left[-\frac12(\boldsymbol y-\boldsymbol j_n\mu)^{\prime}\boldsymbol \Sigma_0^{-1}(\boldsymbol y-\boldsymbol j_n\mu)\right]\]</span></p>
<p>其中 <span class="math inline">\(\boldsymbol \Sigma_0=\sigma_\varepsilon^2\boldsymbol I_n+\sigma_2^2\boldsymbol Z_2\boldsymbol Z_2^{\prime}+\sigma_3^2\boldsymbol Z_3\boldsymbol Z_3^{\prime}+\cdots+\sigma_k^2\boldsymbol Z_k\boldsymbol Z_k^{\prime}\)</span>.</p>
<p>该过程是获得两个似然函数的参数的最大似然估计，并根据其估计的值评估每个似然函数。似然比检验统计量为</p>
<p><span class="math display">\[LR(\sigma_1^2=0)=\frac{L_0(\hat{\mu}_0,\hat{\sigma}_{\varepsilon0}^2,0,\hat{\sigma}_{20}^2,\hat{\sigma}_{30}^2,\ldots,\hat{\sigma}_{k0}^2|\boldsymbol y)}{L(\hat{\mu},\hat{\sigma}_{\varepsilon}^2,\hat{\sigma}_1^2,\hat{\sigma}_2^2,\ldots,\hat{\sigma}_k^2|\boldsymbol y)}\]</span></p>
<p>其中 <span class="math inline">\(\hat{\sigma}_{i0}^2\)</span> 表示，在 <span class="math inline">\(H_0{:{\sigma_1^2}}=0\)</span> 条件下从似然函数中得到的 <span class="math inline">\({\sigma}_{i0}^2\)</span> 的最大似然估计。当 <span class="math inline">\(H_0{:{\sigma_1^2}}=0\)</span> 为真，</p>
<p><span class="math display">\[\begin{aligned}-2\log[LR(\sigma_1^2=0)]=&-2\log_{\mathrm{e}}[L_0(\hat{\mu}_0,\hat{\sigma}_{\varepsilon0}^2,0,\hat{\sigma}_{20}^2,\hat{\sigma}_{30}^2,\ldots,\hat{\sigma}_{k0}^2| \boldsymbol y)]\\&+2\log_{\mathrm{e}}[L(\hat{\mu},\hat{\sigma}_{\varepsilon}^2,\hat{\sigma}_1^2,\hat{\sigma}_2^2,\ldots,\hat{\sigma}_k^2|\boldsymbol y)]\end{aligned}\]</span></p>
<p>的渐近抽样分布为自由度为 1 的中心卡方分布。存在一个自由度的原因是 <span class="math inline">\(L_0(·)\)</span> 中的参数比 <span class="math inline">\(L_1(·)\)</span> 中少一个。<strong>决策规则是如果 <span class="math inline">\(-2\log[LR(\sigma_1^2=0)]>\chi_{\alpha,1}^2\)</span>，则拒绝 <span class="math inline">\(H_0\)</span></strong>. 似然比检验统计量可以使用 SAS-Mixed 计算，其中 METHOD = ML 用作方差分量估计程序。</p>
</div>
<div id="sec20-1-5" class="section level3 hasAnchor" number="20.1.5">
<h3><span class="header-section-number">20.1.5</span> 示例 20.3：小麦品种——单向随机效应模型<a href="chap20.html#sec20-1-5" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>表 <a href="chap20.html#tab:table20-7">20.7</a> 中给出了 SAS-Mixed 代码以及获得示例 <a href="chap19.html#sec19-1-2">19.2</a> 完全模型参数的最大似然估计结果。描述示例 <a href="chap19.html#sec19-1-2">19.2</a> 数据的模型的参数最大似然估计为 <span class="math inline">\(\hat{\mu}=3.9909,\hat{\sigma}_\varepsilon^2=0.05749\)</span> 和 <span class="math inline">\(\hat{\sigma}_\mathrm{var}^2=0.04855\)</span> 以及 <span class="math inline">\(-2\log_{\mathrm{e}}(\hat{\mu},\hat{\sigma}_{\varepsilon^{}}^2,\hat{\sigma}_{\mathrm{var}}^2|\boldsymbol y)\)</span> 的值为 4.96762. 缩减模型使用表 <a href="chap20.html#tab:table20-8">20.8</a> 中的 SAS-Mixed 代码拟合数据，其中从表 <a href="chap20.html#tab:table20-7">20.7</a> 的模型中删除了语句 “Random Variety;”. 在<span class="math inline">\(H_0{:{\sigma_1^2}}=0\)</span> 条件下，参数的最大似然估计为 <span class="math inline">\(\hat{\mu}=4.0269,\hat{\sigma}_\varepsilon^2=0.1014\)</span> 和 <span class="math inline">\(\hat{\sigma}_\mathrm{var}^2=0\)</span> 以及 <span class="math inline">\(-2\log_{\mathrm{e}}(\hat{\mu},\hat{\sigma}_{\varepsilon^{}}^2,\hat{\sigma}_{\mathrm{var}}^2|\boldsymbol y)\)</span> 的值为 7.13832. 检验 <span class="math inline">\(H_0\)</span> 的似然比检验的 -2log<sub>e</sub> 值为 <span class="math inline">\(7.13832 - 4.96762 = 2.1707\)</span>. 值 2.171 与具有一个自由度的中心卡方分布的分位数进行比较。此检验的显著性水平为 0.1407。使用表 <a href="chap20.html#tab:table20-5">20.5</a> 中的期望均方构造的 <span class="math inline">\(F\)</span> 检验提供了 0.0293 的显著性水平。<strong>似然比检验的抽样分布是渐近分布，对于大样本量来说是可以接受的</strong>，而在本例中，样本量很小。前面给出的 <span class="math inline">\(F\)</span> 检验的抽样分布是准确的。<strong>对于涉及超过两个方差分量的模型，其他方差分量的 <span class="math inline">\(F\)</span> 检验是小样本量的近似检验，通常相当充分，并且对于小样本量的情况，可能比基于渐近分布的检验更好</strong>。</p>
<table>
<caption>
<span id="tab:table20-7">表 20.7: </span>示例 <a href="chap19.html#sec19-1-2">19.2</a> 使用 Method = ML 评估似然函数的 Proc Mixed 代码和拟合完全模型的结果
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<caption>
<span id="tab:table20-8">表 20.8: </span>示例 <a href="chap19.html#sec19-1-2">19.2</a> 拟合缩减模型的 Proc Mixed 代码和结果，使用 Method = ML 评估当 <span class="math inline">\(\sigma^2_{\text{var}} = 0\)</span> 时的似然函数
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<div id="sec20-1-6" class="section level3 hasAnchor" number="20.1.6">
<h3><span class="header-section-number">20.1.6</span> 示例 20.4：不均衡双向<a href="chap20.html#sec20-1-6" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>为检验示例 <a href="chap19.html#sec19-1-3">19.3</a> 中的双向随机效应数据的假设 <span class="math inline">\(H_0\)</span>，似然比统计量是通过拟合一个在 random 语句中包含了所有项的模型，然后拟合一个在 random 语句中不包含 row × col 项的模型来获得的，如表 <a href="chap20.html#tab:table20-9">20.9</a> 所示。用于拟合这两个模型的 SAS-Mixed 代码包含在表 <a href="chap20.html#tab:table20-9">20.9</a> 中。不包含行效应和列效应的模型拟合结果也包含在内，但没有提供相应的 SAS-Mixed 代码。给出了方差分量的最大似然估计、截距估计和 -2log(likelihood) 值。用于假设 <span class="math inline">\(H_0:\sigma_{\mathrm{~row\times col}}^2=0\mathrm{~vs~}H_a{:}\sigma_{\mathrm{~row\times col}}^2>0\)</span> 的似然比统计量为 <span class="math inline">\(73.4-68.7=4.7\)</span>，其在 <span class="math inline">\(H_0\)</span> 条件下为具有单自由度的卡方抽样分布。显著性水平为 0.030，表明有足够的信息去相信 <span class="math inline">\(\sigma_{\mathrm{~row\times col}}^2>0\)</span>. 对于检验行和列方差分量是否等于零的假设，似然比统计量的值均为零。这是因为两个方差分量的最大似然估计均为零。因此，在检验行方差分量是否为零时，无论行是否包含在 random 语句中，-2log(likelihood) 的值都保持不变。</p>
<table>
<caption>
<span id="tab:table20-9">表 20.9: </span>各种模型的方差分量最大似然估计，以便计算似然比检验统计量来检验每个单独方差分量为零的假设
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<div id="sec20-2" class="section level2 hasAnchor" number="20.2">
<h2><span class="header-section-number">20.2</span> 构造置信区间<a href="chap20.html#sec20-2" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>有些程序可以为某些模型中的某些方差分量提供精确的置信区间，但大多数情况下，所获得的置信区间都是近似的，并依赖于某种类型的近似。</p>
<div id="sec20-2-1" class="section level3 hasAnchor" number="20.2.1">
<h3><span class="header-section-number">20.2.1</span> 残差方差 <span class="math inline">\(\sigma^2_\varepsilon\)</span><a href="chap20.html#sec20-2-1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>对于一般随机模型，<span class="math inline">\(\sigma^2_\varepsilon\)</span> 的 <span class="math inline">\((1 - \alpha)100\%\)</span> 置信区间为</p>