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<!DOCTYPE html>
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<title>第 27 章 不满足理想条件时重复测量实验的分析 | 混乱数据分析：设计的实验</title>
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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="chap27" class="section level1 hasAnchor" number="27">
<h1><span class="header-section-number">第 27 章</span> 不满足理想条件时重复测量实验的分析<a href="chap27.html#chap27" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<blockquote>
<p>“He uses statistics as a drunken man uses lamp-posts–for support rather than illumination.” - Andrew Lang</p>
</blockquote>
<p>重复测量设计涉及一个或多个步骤，其中研究人员不能将一个或更多因素的水平随机分配给实验单元。使用时间作为因素是无法使用随机化的最常见实验情况。例如，当在同一实验单元的多个时间点收集数据时，无法随机化时间点的顺序。时间 1 必须是第一个，时间 2 必须是第二个，依此类推。重复测量因子的这种非随机分配会影响实验单元之间的方差和协方差，第 <a href="chap26.html#chap26">26</a> 章中描述的理想条件可能无效。本章介绍了当第 <a href="chap26.html#chap26">26</a> 章中给出的理想条件不成立时，分析重复测量实验数据的策略。此外，还描述了允许检查第 <a href="chap26.html#chap26">26</a> 章中描述的理想条件是否满足的程序。</p>
<div id="sec27-1" class="section level2 hasAnchor" number="27.1">
<h2><span class="header-section-number">27.1</span> 介绍<a href="chap27.html#sec27-1" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>考虑类似于表 <a href="chap26.html#tab:table26-1">26.1</a> 中描述的实验情况。设 <span class="math inline">\(y_{ijk}\)</span> 代表处理组 i 中个体<a href="#fn50" class="footnote-ref" id="fnref50"><sup>50</sup></a> k 在时间 j 观测到的响应，并设</p>
<p><span class="math display">\[\boldsymbol{y}_{ik}=\begin{bmatrix}{y}_{i1k}\\{y}_{i2k}\\\vdots\\{y}_{ipk}\end{bmatrix}\]</span></p>
<p>表示处理组 i 中个体 k 的响应向量。</p>
<p>可以用来描述这些数据的模型是</p>
<p><span class="math display">\[y_{ijk}=\mu+\alpha_i+\tau_j+\gamma_{ij}+\varepsilon_{ijk}^*,i=1,2,\ldots,t;~j=1,2,\ldots,p;~k=1,2,\ldots,n_i\]</span></p>
<p>令</p>
<p><span class="math display">\[\boldsymbol{\varepsilon}_{ik}^*=\begin{bmatrix}{\varepsilon}_{i1k}^*\\{\varepsilon}_{i2k}^*\\\vdots\\{\varepsilon}_{ipk}^*\end{bmatrix}\]</span></p>
<p>表示处理组 i 中个体 k 的误差向量。假设 <span class="math inline">\(\boldsymbol{\varepsilon}_{ik}^*\)</span> 独立同分布于均值为 <span class="math inline">\(\boldsymbol 0\)</span> 和协方差矩阵为 <span class="math inline">\(\boldsymbol \Sigma\)</span> 的 p 元正态分布。即 <span class="math inline">\(\boldsymbol \varepsilon_{ik}^*\sim i.i.d.N(\boldsymbol{0},\boldsymbol{\Sigma}),i=1,2,\ldots,t;k=1,2,\ldots,n_i\)</span></p>
<p>令</p>
<p><span class="math display">\[\boldsymbol{\Sigma}=\begin{bmatrix}{\sigma}_{11}&{\sigma}_{12}&\cdots&{\sigma}_{1p}\\{\sigma}_{21}&{\sigma}_{22}&\cdots&{\sigma}_{2p}\\\vdots&\vdots&\ddots&\vdots\\{\sigma}_{p1}&{\sigma}_{p2}&\cdots&{\sigma}_{pp}\end{bmatrix}\]</span></p>
<p>表示重复测量向量的协方差阵。</p>
<div class="definition">
<p><span id="def:27-1" class="definition"><strong>定义 27.1 </strong></span>♦</p>
<p>若 <span class="math inline">\(\boldsymbol{\Sigma}={\lambda}\boldsymbol{I}_t+\boldsymbol{\eta}\boldsymbol{j}'+\boldsymbol{j}\boldsymbol{\eta}'\)</span>，其中 <span class="math inline">\(\boldsymbol j\)</span> 是全 1 的 p × 1 向量，<span class="math inline">\(\boldsymbol \eta\)</span> 是由常数组成的 p × 1 向量，则重复测量被认为满足 Huynh–Feldt (H–F) 条件（参见 Huynh and Feldt, 1970）。</p>
</div>
<p>当定义 27.1 成立时，<span class="math inline">\(\boldsymbol{\Sigma}\)</span> 具有如下形式</p>
<p><span class="math display">\[\boldsymbol{\Sigma}=\begin{bmatrix}\lambda+2{\eta}_1&{\eta}_1+{\eta}_2&\cdots&{\eta}_1+{\eta}_p\\{\eta}_2+{\eta}_1&\lambda+2{\eta}_2&\cdots&{\eta}_2+{\eta}_p\\\vdots&\vdots&\ddots&\vdots\\{\eta}_p+{\eta}_1&{\eta}_p+{\eta}_2&\cdots&\lambda+2{\eta}_p\end{bmatrix}\]</span></p>
<p>H-F 条件的一个特例是当重复测量具有<strong>复合对称协方差结构</strong> (compound
symmetry covariance structure, <strong>CS</strong>)，则对于 <span class="math inline">\(\sigma^2,\rho\)</span> 的某种取值，</p>
<p><span class="math display">\[\boldsymbol{\Sigma}={\sigma}^2{\begin{bmatrix}1&{\rho}&\cdots&{\rho}\\{\rho}&1&\cdots&{\rho}\\\vdots&\vdots&\ddots&\vdots\\{\rho}&{\rho}&\cdots&1\end{bmatrix}}\]</span></p>
<p>请注意，第 <a href="chap26.html#sec26-1">26.1</a> 节中讨论的<u>时间的裂区分析</u> (split-plot-in-time analysis) 描述的理想条件是具有复合对称性的重复测量的协方差阵的特例，其中 <span class="math inline">\(\sigma^2=\sigma_\delta^2+\sigma_\varepsilon^2\)</span> 以及 <span class="math inline">\(\rho=\sigma_\delta^2/(\sigma_\delta^2+\sigma_\varepsilon^2)\)</span>. <strong>复合对称结构比时间的裂区结构更通用，因为复合对称结构中的 <span class="math inline">\(\rho\)</span> 可以为负</strong>。</p>
<div class="rmdnote">
<p>如果满足 H-F 条件，那么许多涉及时间比较的重要问题可以通过分析重复测量实验来回答，就像分析满足第 <a href="chap26.html#sec26-1">26.1</a> 节给出的理想条件的重复测量实验一样，并且如果重复测量满足复合对称性，则可以使用第 <a href="chap26.html#chap26">26</a> 章中给出的<u>时间的裂区分析</u>方法。特别是，<strong>当且仅当重复测量满足定义 27.1 中给出的 H-F 条件时，时间主效应和时间 × 处理交互效应的<u>时间的裂区检验</u>才可以在统计上证明有效</strong>。此外，使用<u>时间的裂区分析</u>时，比较时间主效应的对比以及比较处理变量特定值内时间效应的对比都具有统计有效性。如果关注双向均值或边际均值，<u>时间的裂区分析</u>可以提供均值的正确估计，但标准误的估计是不正确的。</p>
</div>
<div class="rmdnote">
<p>如果重复测量具有复合对称性且 <span class="math inline">\(\rho>0\)</span>，则<u>时间的裂区分析</u>给出的所有结果都是正确的。在这种情况下，可以说 <span class="math inline">\(\varepsilon_{ik}^*=\delta_{ik}+\varepsilon_{ijk}\)</span>，其中 <span class="math inline">\(\delta_{ik}\sim i.i.d.N(0,\sigma_\delta^2),\varepsilon_{ijk}\sim i.i.d.N(0,\sigma_\varepsilon^2)\)</span>，以及 <span class="math inline">\(\delta_{ik}\)</span> 和 <span class="math inline">\(\varepsilon_{ijk}\)</span> 是独立的。</p>
</div>
<p><strong>提问：如果 H-F 条件不满足呢？</strong></p>
<p>在不满足 H–F 条件的情况下，可以考虑几种分析方法。一种总是合适的方法是将重复测量的向量视为多元响应向量，并使用<strong>多元方差分析</strong> (multivariate analysis of variance, MANOVA) 方法。第二种是使用<u>时间的裂区分析</u>，但通过调整与相关效应均方相对应的自由度来调整 <span class="math inline">\(p\)</span> 值<a href="#fn51" class="footnote-ref" id="fnref51"><sup>51</sup></a>。第三种方法是使用 SAS<sup>®</sup>-Mixed 和 SAS<sup>®</sup>-Gilimix 程序中可用的混合模型方法，并对协方差结构进行建模。</p>
<p>多元方差分析方法在第 <a href="chap27.html#sec27-2">27.2</a> 节中描述，调整自由度方法在第 <a href="chap27.html#sec27-3">27.3</a> 节中讨论，混合模型方法将在第 <a href="chap27.html#sec27-4">27.4</a> 节中讨论。</p>
</div>
<div id="sec27-2" class="section level2 hasAnchor" number="27.2">
<h2><span class="header-section-number">27.2</span> MANOVA 法<a href="chap27.html#sec27-2" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>本节考虑使用多元方差分析法来分析重复测量实验。这些方法<strong>总是</strong>适用于以下情况</p>
<p><span class="math display">\[\boldsymbol\varepsilon_{ik}^*\sim i.i.d.\mathrm{~}N(\boldsymbol0,\boldsymbol{\Sigma}),\quad i=1,2,\ldots,t;\mathrm{~}k=1,2,\ldots,n_i\]</span></p>
<p><strong>MANOVA 方法只能应用于给定实验单元上的所有重复测量都存在或全部缺失的实验</strong>，因为所使用的统计软件将自动从分析中删除一个或多个重复测量值缺失的任何实验单元。然而，对于其他两种方法，这并不存在问题。（对于它们，）分配给每个处理的实验单元数量无需均衡。要充分利用这一章节的内容，读者应需理解第 @chap(6) 章讨论的模型的矩阵形式。</p>
<p>具有类似于表 <a href="chap26.html#tab:table26-1">26.1</a> 中给出的结构的重复测量实验的一般多元模型是式 <a href="chap26.html#eq:26-1">(26.1)</a> 中定义的模型的矩阵形式的推广。多元模型可以表示为</p>
<p><span class="math display" id="eq:27-1">\[\begin{equation}
\boldsymbol Y=\boldsymbol {XB}+\boldsymbol E
\tag{27.1}
\end{equation}\]</span></p>
<p>其中 <span class="math inline">\(\boldsymbol Y\)</span> 表示在实验中测量的所有数据。数据矩阵 <span class="math inline">\(\boldsymbol Y\)</span> 的每一行对应于特定的实验单元，每一列对应于一个重复测量中。因此 <span class="math inline">\(\boldsymbol Y\)</span> 是 N × p 矩阵，其中 <span class="math inline">\(N=\sum_{i=1}^t n_i\)</span>. 矩阵 <span class="math inline">\(\boldsymbol X\)</span> 是假设秩为 t 的 N × r 设计矩阵。<span class="math inline">\(\boldsymbol B\)</span> 的每一列都是未知参数的 r × 1 向量，每一列对应于特定的重复测量。矩阵 <span class="math inline">\(\boldsymbol E\)</span> 是不可观测随机误差的 N × p 矩阵。假设 <span class="math inline">\(\boldsymbol E\)</span> 的行独立分布于 <span class="math inline">\(N(\boldsymbol{0},\boldsymbol{\Sigma})\)</span>. 因此，当 <span class="math inline">\(\boldsymbol E\)</span> 中的行是独立的时，一行中的元素可以彼此相关 (correlated) 且可以具有不同的方差。</p>
<p>对于多元模型 <a href="chap27.html#eq:27-1">(27.1)</a>，可以检验以下形式的一般假设</p>
<p><span class="math display" id="eq:27-2">\[\begin{equation}
H_0{:\boldsymbol{CBM}}=0\mathrm{~vs~}H_a{:\boldsymbol{CBM}}\neq0
\tag{27.2}
\end{equation}\]</span></p>
<p>其中 <span class="math inline">\(\boldsymbol C\)</span> 是秩为 g 的 g × r 矩阵，<span class="math inline">\(\boldsymbol M\)</span> 是秩为 q 的 p × q 矩阵。</p>
<p>为了检验式 <a href="chap27.html#eq:27-2">(27.2)</a> 中的假设，首先需要求得矩阵 <span class="math inline">\(\boldsymbol B\)</span> 中参数的最小二乘估计以及观测到的残差平方和与叉乘矩阵 (cross-products matrix)<a href="#fn52" class="footnote-ref" id="fnref52"><sup>52</sup></a>。分别记作 <span class="math inline">\(\hat{\boldsymbol B}\)</span> 和 <span class="math inline">\(\hat{\boldsymbol E}\)</span> ，并由以下式给出：</p>
<p><span class="math display" id="eq:27-3">\[\begin{equation}
\hat{\boldsymbol B}=(\boldsymbol X^{\prime}\boldsymbol X)^-\boldsymbol X^{\prime}\boldsymbol Y\quad\mathrm{and}\quad\hat{\boldsymbol E}=\boldsymbol Y^{\prime}[\boldsymbol I-\boldsymbol X(\boldsymbol X^{\prime}\boldsymbol X)^-\boldsymbol X^{\prime}]\boldsymbol Y
\tag{27.3}
\end{equation}\]</span></p>
<p>用于检验等式 <a href="chap27.html#eq:27-2">(27.2)</a> 中的假设，似然比检验统计量 (likelihood ratio test statistic) 由下式给出</p>
<p><span class="math display" id="eq:27-4">\[\begin{equation}
\Lambda=\frac{|\boldsymbol R|}{\left|\boldsymbol H+\boldsymbol R\right|}
\tag{27.4}
\end{equation}\]</span></p>
<p>其中</p>
<p><span class="math display">\[\boldsymbol R=\boldsymbol M'\hat{\boldsymbol E}\boldsymbol M,\quad \boldsymbol H=\boldsymbol M^{\prime}\hat{\boldsymbol B}\boldsymbol C^{\prime}\left[\boldsymbol C(\boldsymbol X^{\prime}\boldsymbol X)^{-}\boldsymbol C^{\prime}\right]^{-1}\boldsymbol C \hat{\boldsymbol B} \boldsymbol M\]</span></p>
<p>以及 <span class="math inline">\(|\boldsymbol W|\)</span> 表示矩阵 <span class="math inline">\(\boldsymbol W\)</span> 的行列式。</p>
<p>这种统计量被称为 Wilks’ 似然比准则 (likelihood ratio criterion) (Morrison, 1976). <span class="math inline">\(\Lambda\)</span> 的抽样分布相当复杂，但对于大多数实际目的，可以获得一个近似 <span class="math inline">\(\alpha\)</span> 水平的检验，拒绝 <span class="math inline">\(H_0\)</span> 当</p>
<p><span class="math display">\[-\left(N-t-\frac{|q-g|+1}2\right)\mathrm{log}_{\mathrm{e}}(\Lambda)>\chi_{\alpha,qg}^2\]</span></p>
<p>只有当 <span class="math inline">\(q\)</span> 和 <span class="math inline">\(g\)</span> 都大于 <span class="math inline">\(2\)</span> 才可使用的一个更好的近似是，拒绝 <span class="math inline">\(H_0\)</span> 当</p>
<p><span class="math display">\[F>F_{\alpha,qg,ab-c}\]</span></p>
<p>其中</p>
<p><span class="math display">\[F=\frac{(1-\Lambda^{1/b})(ab-c)}{qg\Lambda^{1/b}}\]</span></p>
<p>以及</p>
<p><span class="math display" id="eq:27-5">\[\begin{align}
&a=N-t-\frac{|q-s|+1}2 \\
&b=\left(\frac{q^{2}s^{2}-4}{q^{2}+s^{2}-5}\right)^{1/2} \\
& c=\frac{qs-2}2 \\
&s=\min(q,g)
\tag{27.5}
\end{align}\]</span></p>
<p>而只要 <span class="math inline">\(q = 1, 2\)</span> 或 <span class="math inline">\(g = 1, 2\)</span>，式 <a href="chap27.html#eq:27-2">(27.2)</a> 就存在精确的 <span class="math inline">\(F\)</span> 检验。这些检验如下</p>
<ol style="list-style-type: decimal">
<li><p>对于 <span class="math inline">\(g = 1\)</span> 以及任何 <span class="math inline">\(q\)</span>，拒绝 <span class="math inline">\(H_0\)</span> 如果
<span class="math display" id="eq:27-6">\[\begin{equation}
F=\left(\frac{1-\Lambda}\Lambda\right)\left(\frac{N-t-q+1}q\right)>F_{\alpha,q,N-t-q+1}
\tag{27.6}
\end{equation}\]</span></p></li>
<li><p>对于 <span class="math inline">\(q = 1\)</span> 以及任何 <span class="math inline">\(g\)</span>，拒绝 <span class="math inline">\(H_0\)</span> 如果
<span class="math display" id="eq:27-7">\[\begin{equation}
F=\left(\frac{1-\Lambda}\Lambda\right)\left(\frac{N-t}g\right)>F_{\alpha,g,N-t}
\tag{27.7}
\end{equation}\]</span></p></li>
<li><p>对于 <span class="math inline">\(g = 2\)</span> 以及任何 <span class="math inline">\(q > 1\)</span>，拒绝 <span class="math inline">\(H_0\)</span> 如果
<span class="math display" id="eq:27-8">\[\begin{equation}
F=\left(\frac{1-\sqrt{\Lambda}}{\sqrt{\Lambda}}\right){\left(\frac{N-t-q+1}q\right)}{>F_{\alpha,2q,2(N-t-q+1)}}
\tag{27.8}
\end{equation}\]</span></p></li>
<li><p>对于 <span class="math inline">\(q = 2\)</span> 以及任何 <span class="math inline">\(g > 1\)</span>，拒绝 <span class="math inline">\(H_0\)</span> 如果
<span class="math display" id="eq:27-9">\[\begin{equation}
F=\left(\frac{1-\sqrt{\Lambda}}{\sqrt{\Lambda}}\right)\left(\frac{N-t-1}g\right)>F_{\alpha,2g,2(N-t-1)}
\tag{27.9}
\end{equation}\]</span></p></li>
</ol>
<p>多元方法的一个缺点是必须满足 <span class="math inline">\(p < N – t\)</span>. 当 <span class="math inline">\(p \ge N – t\)</span> 时，通常可以将相邻的重复测量组合成 <span class="math inline">\(p^*\)</span> 个新变量，或者仅分析重复测量尺寸为 <span class="math inline">\(p^*\)</span> 的子集，其中 <span class="math inline">\(p^* < N – t\)</span>.</p>
<p>为了说明本节中描述的分析，考虑进行一项实验，研究四种高粱 (sorghum) 品种和五种肥料 (fertilizer) 水平在叶面积指数<a href="#fn53" class="footnote-ref" id="fnref53"><sup>53</sup></a> (leaf area index) 上的差异，其中四种高粱品种用 V1, V2, V3 和 V4 表示，五个肥料水平用 1, 2, 3, 4 和 5 表示。还假设这 20 个品种 × 肥料的组合被随机分配到田间 20 个区 (plots). 对于此示例，假设肥料水平和品种之间不存在交互作用，并且可以使用基本的双向加性模型<a href="#fn54" class="footnote-ref" id="fnref54"><sup>54</sup></a>来分析数据。最后，假设从植物出苗后两周开始，在五周内的每周对每个品种 × 肥料的区中进行叶面积指数的测量。获得的数据列于表 <a href="chap27.html#tab:table27-1">27.1</a> 中。</p>
<table>
<caption>
<span id="tab:table27-1">表 27.1: </span>四个高粱品种的叶面积指数
</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%2027.1.png">
</td>
</tr>
</tbody>
</table>
<p>对于表 <a href="chap27.html#tab:table27-1">27.1</a> 中的数据，数据矩阵为</p>
<p><span class="math display">\[\boldsymbol Y=\begin{bmatrix}5.00&4.84&4.02&3.75&3.13\\4.42&4.30&3.67&3.29&2.83\\4.42&4.10&3.46&3.09&2.82\\4.01&3.89&3.21&2.89&2.56\\3.36&3.10&2.67&2.47&2.16\\5.82&5.60&5.05&4.72&4.46\\5.73&5.59&5.00&4.65&4.42\\5.31&5.19&4.86&4.44&4.22\\4.92&4.66&4.56&4.16&3.99\\3.96&3.86&3.50&3.13&2.95\\5.65&5.97&5.27&5.07&4.52\\5.39&5.49&5.08&4.87&4.32\\5.15&5.28&4.93&4.67&4.15\\4.50&4.89&4.74&4.49&4.10\\3.75&3.74&3.55&3.28&3.00\\5.86&5.60&5.37&5.00&4.37\\5.82&5.55&5.29&4.95&4.07\\5.26&5.06&4.76&4.48&3.94\\4.87&4.75&4.55&4.33&3.83\\3.96&3.76&3.56&3.18&2.96\end{bmatrix}\]</span></p>
<p>参数矩阵为</p>
<p><span class="math display">\[\boldsymbol{B}=\begin{bmatrix}
\mu^{(1)}&\mu^{(2)}&\mu^{(3)}&\mu^{(4)}&\mu^{(5)}\\
\tau_1^{(1)}&\tau_1^{(2)}&\tau_1^{(3)}&\tau_1^{(4)}&\tau_1^{(5)}\\
\tau_2^{(1)}&\tau_2^{(2)}&\tau_2^{(3)}&\tau_2^{(4)}&\tau_2^{(5)}\\
\tau_3^{(1)}&\tau_3^{(2)}&\tau_3^{(3)}&\tau_3^{(4)}&\tau_3^{(5)}\\
\tau_4^{(1)}&\tau_4^{(2)}&\tau_4^{(3)}&\tau_4^{(4)}&\tau_4^{(5)}\\
\beta_1^{(1)}&\beta_1^{(2)}&\beta_1^{(3)}&\beta_1^{(4)}&\beta_1^{(5)}\\
\beta_2^{(1)}&\beta_2^{(2)}&\beta_2^{(3)}&\beta_2^{(4)}&\beta_2^{(5)}\\
\beta_3^{(1)}&\beta_3^{(2)}&\beta_3^{(3)}&\beta_3^{(4)}&\beta_3^{(5)}\\
\beta_4^{(1)}&\beta_4^{(2)}&\beta_4^{(3)}&\beta_4^{(4)}&\beta_4^{(5)}\\
\beta_5^{(1)}&\beta_5^{(2)}&\beta_5^{(3)}&\beta_5^{(4)}&\beta_5^{(5)}\\
\end{bmatrix}\]</span></p>
<p>其中 <span class="math inline">\(\tau\)</span> 对应不同品种，<span class="math inline">\(\beta\)</span> 对应不同肥料水平，设计矩阵为</p>
<p><span class="math display">\[\boldsymbol X=\begin{bmatrix}
1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1
\end{bmatrix} \]</span></p>
<p>注意，<span class="math inline">\(\boldsymbol B\)</span> 的每一列代表数据矩阵 <span class="math inline">\(\boldsymbol Y\)</span> 的第 <span class="math inline">\(j(j=1,2,3,4,5)\)</span> 个响应列的双向加性模型所需的参数，设计矩阵 <span class="math inline">\(\boldsymbol X\)</span> 是一个 20 × 10 矩阵且秩等于 8；因此 <span class="math inline">\(N = 20,p = 10\)</span> 且 <span class="math inline">\(t = 8\)</span>。式 <a href="chap27.html#eq:27-3">(27.3)</a> 给出的 <span class="math inline">\(\hat{\boldsymbol B}\)</span> 的值为</p>
<p><span class="math display">\[\hat{\boldsymbol{B}}=\begin{bmatrix}3.350&3.283&3.003&2.788&2.150\\0.222&0.106&-0.198&-0.260&-0.312\\1.128&1.040&0.990&0.874&0.996\\0.868&1.134&1.110&1.130&1.006\\1.134&1.004&1.102&1.042&0.822\\1.395&1.398&1.173&1.115&0.982\\1.152&1.128&1.006&0.940&0.772\\0.847&0.803&0.748&0.685&0.645\\0.387&0.443&0.511&0.483&0.482\\-0.430&-0.489&-0.434&-0.470&-0.370\end{bmatrix}\]</span></p>
<p>以及式 <a href="chap27.html#eq:27-3">(27.3)</a> 中的 <span class="math inline">\(\boldsymbol E\)</span> 的值为</p>
<p><span class="math display">\[\hat{\boldsymbol{E}}=\begin{bmatrix}0.237&0.171&0.162&0.228&0.129\\0.171&0.247&0.163&0.231&0.135\\0.162&0.163&0.268&0.303&0.184\\0.228&0.231&0.303&0.392&0.241\\0.129&0.135&0.184&0.241&0.247\end{bmatrix}\]</span></p>
<p>等品种主效应均值检验由式 <a href="chap27.html#eq:27-4">(27.4)</a> 得出，通过取</p>
<p><span class="math display">\[\boldsymbol{C}=\begin{bmatrix}0&1&-1&0&0&0&0&0&0&0\\0&1&0&-1&0&0&0&0&0&0\\0&1&0&0&-1&0&0&0&0&0\end{bmatrix}\quad\mathrm{and}\quad\boldsymbol{M}=\begin{bmatrix}1\\1\\1\\1\\1\\1\end{bmatrix}\]</span></p>
<p>那么 <span class="math inline">\(\Lambda= 0.04345,g = 3,q = 1\)</span>. 由于 <span class="math inline">\(q = 1\)</span>，可以使用式 <a href="chap27.html#eq:27-7">(27.7)</a>，得到</p>
<p><span class="math display">\[F=\frac{1-0.04345}{0.04345}\times\frac{12}3=88.06\]</span></p>
<p>具有 3 个和 12 个自由度。观察到的显著性水平 <span class="math inline">\(\hat\alpha\)</span> 小于 0.0001.</p>
<p>等时间主效应均值的检验由式 <a href="chap27.html#eq:27-4">(27.4)</a> 得出，通过取</p>
<p><span class="math display">\[\boldsymbol{C}=\begin{bmatrix}1&\frac14&\frac14&\frac14&\frac14&\frac15&\frac15&\frac15&\frac15&\frac15\\\end{bmatrix}\quad\mathrm{and}\quad\boldsymbol{M}=\begin{bmatrix}1&1&1&1\\-1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}\]</span></p>
<p>可得到 <span class="math inline">\(\Lambda= 0.04345,g=1,q=4\)</span>. 由于 <span class="math inline">\(g = 1\)</span>，因此可以使用式 <a href="chap27.html#eq:27-6">(27.6)</a>，并得到</p>
<p><span class="math display">\[\begin{aligned}F=&\frac{1-0.00502}{0.00502}\times\frac94=445.96\end{aligned}\]</span></p>
<p>具有 4 个和 9 个自由度。观察到的显著性水平 <span class="math inline">\(\hat\alpha\)</span> 小于 0.0001.</p>
<p>品种 × 时间交互作用的检验由式 <a href="chap27.html#eq:27-4">(27.4)</a> 得出，通过取</p>
<p><span class="math display">\[\boldsymbol{C}=\begin{bmatrix}0&1&-1&0&0&0&0&0&0&0\\0&1&0&-1&0&0&0&0&0&0\\0&1&0&0&-1&0&0&0&0&0\end{bmatrix}\quad\mathrm{and}\quad\boldsymbol{M}=\begin{bmatrix}1&1&1&1\\-1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}\]</span></p>
<p>可得到 <span class="math inline">\(\Lambda= 0.01426,g=1,q=4\)</span>. 由于 g 和 q 都大于 1，可以使用式 <a href="chap27.html#eq:27-5">(27.5)</a>，其中 <span class="math inline">\(s = 3,a = 11,b = 2.646\)</span> 以及 <span class="math inline">\(c = 5\)</span>. 那么</p>
<p><span class="math display">\[F=\frac{(1-0.01426^{1/2.646})(11\times2.646-5)}{4\times3\times0.01426^{1/2.646}}=\frac{(1-0.2006)(24.1060)}{12(0.2006)}=8.00\]</span></p>
<p>具有 12 个和 24.1 个自由度。观察到的显著性水平 <span class="math inline">\(\hat\alpha\)</span> 小于 0.0001.</p>
<p>请注意，等品种均值的检验与通过<u>时间的裂区分析</u>获得的检验相同。这是三个检验中唯一与通过<u>时间的裂区分析</u>获得的相应检验相同的检验。以类似的方式可以得到肥料主效应（<span class="math inline">\(F=94.36\)</span>）和肥料 × 时间交互效应（<span class="math inline">\(F=1.91\)</span>）的检验统计量。</p>
<p><span class="math inline">\(\boldsymbol{B}\)</span> 中参数的许多特殊函数是可估的，并且可以对这些可估函数进行推断。<span class="math inline">\(\boldsymbol{B}\)</span> 中参数的大多数有趣的线性函数都可以写成 <span class="math inline">\(\boldsymbol{c'Bm}\)</span> 的形式，其中 <span class="math inline">\(\boldsymbol{c}\)</span> 是 r × 1 向量，<span class="math inline">\(\boldsymbol{m}\)</span> 是 p × 1 向量。正如第 <a href="chap6.html#chap6">6</a> 章中那样，当且仅当存在一个向量 <span class="math inline">\(\boldsymbol{u}\)</span> 使得 <span class="math inline">\(\boldsymbol{X^{\prime}}\boldsymbol{X}\boldsymbol{u}=\boldsymbol{c}\)</span> 时，<span class="math inline">\(\boldsymbol{c'Bm}\)</span> 才是可估的。对于 <span class="math inline">\(\boldsymbol m\)</span> 没有限制。</p>
<p><span class="math inline">\(\boldsymbol{c'Bm}\)</span> 的最佳估计是 <span class="math inline">\(\boldsymbol{c'\hat Bm}\)</span>，其中 <span class="math inline">\(\boldsymbol{\hat B}\)</span> 由式 <a href="chap27.html#eq:27-3">(27.3)</a> 中给出。 <span class="math inline">\(\boldsymbol{c'\hat Bm}\)</span> 的标准误估计由下式给出</p>
<p><span class="math display">\[\widehat{s.e.}(\boldsymbol c^{\prime}\hat{\boldsymbol B}\boldsymbol m)=\sqrt{\boldsymbol c^{\prime}(\boldsymbol X^{\prime}\boldsymbol X)^-\boldsymbol c\cdot\frac{{\boldsymbol m^{\prime}\hat{\boldsymbol E}\boldsymbol m}}{N-t}}\]</span></p>
<p>相应的自由度为 N-t. 因此，<span class="math inline">\(\boldsymbol{c'Bm}\)</span> 的 <span class="math inline">\((1 - \alpha)100\%\)</span> 置信区间由下式给出</p>
<p><span class="math display" id="eq:27-10">\[\begin{equation}
\boldsymbol{c^{\prime}\hat{B}m}\pm t_{\alpha,N-t}\left[\widehat{s.e.}(\boldsymbol{c^{\prime}\hat{B}m})\right]
\tag{27.10}
\end{equation}\]</span></p>
<p>用于检验 <span class="math inline">\(H_0:\boldsymbol{c'Bm}=a_0\)</span> 的 <span class="math inline">\(t\)</span> 统计量由下式给出</p>
<p><span class="math display" id="eq:27-11">\[\begin{equation}
t=\frac{\boldsymbol{c'\hat Bm}-a_0}{\widehat{s.e.}(\boldsymbol{c'\hat Bm})}
\tag{27.11}
\end{equation}\]</span></p>
<p>如果 <span class="math inline">\(t>t_{\alpha/2,N-t}\)</span> 则拒绝 <span class="math inline">\(H_0\)</span>.</p>
<p>例如，考虑估计 V1 的边际均值。对于此边际均值，</p>
<p><span class="math display">\[\boldsymbol c'=\begin{bmatrix}1&1&0&0&0&0.2&0.2&0.2&0.2&0.2\end{bmatrix}\quad\mathrm{and}\quad\boldsymbol{m}=\begin{bmatrix}0.2\\0.2\\0.2\\0.2\\0.2\end{bmatrix}\]</span></p>
<p><span class="math inline">\(\boldsymbol{c'\hat Bm}\)</span> 的值为 3.496，其标准误估计为 0.0594. 利用式 <a href="chap27.html#eq:27-10">(27.10)</a> 可以得到 V1 边际均值的 95% 置信区间为 3.496 ± (2.179) (0.0594).</p>
<p>作为第二个例子，考虑估计表 <a href="chap27.html#tab:table27-1">27.1</a> 中数据的时间 1 的 边际均值。对于此边际均值，</p>
<p><span class="math display">\[\boldsymbol c^{\prime}=[1\quad0.25\quad0.25\quad0.25\quad0.2\quad0.2\quad0.2\quad0.2\quad0.2]\quad\mathrm{and}\quad\boldsymbol{m}=\begin{bmatrix}1\\0\\0\\0\\0\end{bmatrix}\]</span></p>
<p><span class="math inline">\(\boldsymbol{c'\hat Bm}\)</span> 的值为 4.858，其标准误估计为 0.0315.</p>
<p>作为第三个示例，考虑估计 V1 和 V2 边际均值之差。此时</p>
<p><span class="math display">\[\boldsymbol c^{\prime}=\begin{bmatrix}0&1&-1&0&0&0&0&0&0&0\end{bmatrix}\quad\mathrm{and}\quad\boldsymbol{m}=\begin{bmatrix}0.2\\0.2\\0.2\\0.2\\0.2\end{bmatrix}\]</span></p>
<p><span class="math inline">\(\boldsymbol{c'\hat Bm}\)</span> 的值为 -1.094，其标准误估计为 0.0840. 用于比较这两个边际平均值的 <span class="math inline">\(t\)</span> 统计量为 <span class="math inline">\(t = -1.094/0.0840 = 13.02\)</span>，其观察到的显著性水平为 <span class="math inline">\(\hat\alpha < 0.0001\)</span>.</p>
<p>当使用 <code>MANOVA</code> 选项及其 <code>M =</code> 选项时，可以从 SAS<sup>®</sup>-GLM 程序获得上面给出的许多推断结果。为了说明这一点，使用表 <a href="chap27.html#tab:table27-2">27.2</a> 中给出的 SAS 命令重新分析表 <a href="chap27.html#tab:table27-1">27.1</a> 中的数据。第一个 <code>MANOVA</code> 选项用于获取 E^，第二个 <code>MANOVA</code> 选项用于获取比较品种主效应均值和肥料主效应均值的检验【注意，M = (0.2 0.2 0.2 0.2 0.2) 告诉 <code>MANOVA</code> 选项计算五次重复的时间测量平均值】。肥料 × 时间交互作用和品种 × 时间交互作用的检验是从第三个 <code>MANOVA</code> 选项获得的，其中 <span class="math inline">\(\boldsymbol M\)</span> 是时间对比的 4 × 5 矩阵。</p>
<table>
<caption>
<span id="tab:table27-2">表 27.2: </span>使用 MANOVA 分析表 <a href="chap27.html#tab:table27-1">27.1</a> 中的数据的 SAS-GLM 代码
</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%2027.2.png">
</td>
</tr>
</tbody>
</table>
<p>表 <a href="chap27.html#tab:table27-3">27.3</a> 给出了从表 <a href="chap27.html#tab:table27-2">27.2</a> 的多元方差分析中获得的 <span class="math inline">\(\hat{\boldsymbol E}\)</span> 值。表 <a href="chap27.html#tab:table27-4">27.4</a> 给出了肥料和品种主效应的 MANOVA 检验，表 <a href="chap27.html#tab:table27-5">27.5</a> 给出了 MANOVA 交互检验。</p>
<table>
<caption>
<span id="tab:table27-3">表 27.3: </span>误差平方和和叉乘矩阵
</caption>
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<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%2027.3.png">
</td>
</tr>
</tbody>
</table>
<table>
<caption>
<span id="tab:table27-4">表 27.4: </span>肥料和品种主效应检验
</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%2027.4.png">
</td>
</tr>
</tbody>
</table>
<table>
<caption>
<span id="tab:table27-5">表 27.5: </span>MANOVA 交互检验
</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%2027.5.png">
</td>
</tr>
</tbody>