示例：交互作用

使用 effects 函数了解交互作用在实验中的角色。

1. 使用 fullfact 函数来创建两个因子的设计矩阵。

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2. 在矩阵 Vals 中指定实验的实数值。Gender (性别) 因子分为 male (男性) 和 female (女性) 两个水平。Age (年龄) 因子分为 young (青年)、middle (中年) 和 old (老年) 三个水平。由于这两个因子具有的水平数量不同，因此将在 Vals 的第一行插入 NaN 以填充空元素。

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3. 使用 doelabel 函数根据 gender (性别) 和 age (年龄) 对实验结果进行排序。

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矩阵 X 和 D 相同，但 X 显示因子的代码值而 D 显示其实际值。

无交互作用 - 仅有一个显著因子

1. 记录各组在学习任务中花费的平均时间。对于 Run 1，青年男性的平均学习时间为 9 分钟。

2. 调用 effects 函数以查看 gender (性别)、age (年龄) 及其交互作用的效果。

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Gender 子矩阵证明 gender (性别) 不会影响学习时间。因此，不会报告 age (年龄) 和 gender (性别) 之间的交互作用效果。

3. 创建效果图以查看 age (年龄) 如何影响学习时间。最年轻的一组在学习任务中花费的平均时间为 9 分钟。

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无交互作用 - 有两个显著因子

1. 记录各组在学习第二个任务中花费的平均时间。

2. 调用 effects 函数以查看 gender (性别)、age (年龄) 及其交互作用的效果。

Age (年龄) 对学习过程的影响与对第一个任务的影响相同。但此次 male (男性) 在学习任务中花费的时间要比 female (女性) 多。

3. 要计算 AB 在 A 和 B 不同水平处的交互作用效果，可将 i 和 j 定义为 A 和 B 的水平，将 mABi, j 定义为 AB 在 i 和 j 处的平均响应。

4. 使用 mean 函数来计算此实验的整体平均时间以及各个因子的平均时间。

5. 计算每个因子的水平效果。

6. 针对 A 和 B 的因子水平，使用 augment 函数计算累加因子效果，这些效果为总体实验均值，即针对 A 和 B 的各个水平的 A 水平效果与 B 水平效果之和。

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7. 计算交互作用效果，或 AB 的平均响应与 A 和 B 每个水平的累加因子效果之间的差值。

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对于本任务，age (年龄) 与 gender (性别) 之间不存在交互作用效果。

8. 绘制每种 gender (性别) 平均学习时间的图像。青年女性在学习第二个任务中花费的平均时间为 7 分钟。因为 gender (性别) 和 age (年龄) 之间不存在交互作用效果，因此两条曲线平行。

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具有重要的交互作用

1. 记录各组在学习第三个任务中花费的平均时间。调用 effects 函数以查看 gender (性别)、age (年龄) 及其交互作用的效果。

age (年龄) 对第一个和第二个任务的影响相同，但 gender (性别) 对第二个任务的影响较小。对于第三个任务，age (年龄) 和 gender (性别) 之间存在交互作用效果。

2. 计算交互作用效果。

3. 使用 mean 函数来计算此实验的整体平均时间以及各个因子的平均时间。

4. 针对 A 和 B 的因子水平，使用 augment 函数计算累加因子效果，这些效果为总体实验均值，即针对 A 和 B 的各个水平的 A 水平效果与 B 水平效果之和。

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5. 计算交互作用效果，或 AB 的平均响应与 A 和 B 每个水平的累加因子效果之间的差值。

交互作用 AB 的平均响应与累加因子效果之间存在差值。

6. 绘制每种 gender (性别) 平均学习时间的图像。虽然青年男性和青年女性之间的完成时间一样，但老年男性学习第三个任务花费的时间要比老年女性长。两条曲线不平行，因为 age (年龄) 和 gender (性别) 之间存在重要的交互作用。

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具有不重要的交互作用

1. 记录各组在学习第四个任务中花费的平均时间。

2. 调用 effects 函数以查看 gender (性别)、age (年龄) 及其交互作用的效果。

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Gender (性别) 和 age (年龄) 的影响对第三个任务的影响非常相似。

3. 绘制每种 gender (性别) 平均学习时间的图像。两条曲线几乎平行，表示尽管 age (年龄) 和 gender (性别) 之间存在交互作用，但此交互作用并不重要。

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参考文献

Neter, J., Kutner, M.H., Nachtsheim, C.J., Wasserman, W., Applied Linear Statistical Models, 4th ed., McGraw-Hill/Irwin, Boston, 1996, pp. 803.