chapter_8.html

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class: center, middle
&lt;span style="font-size: 50px;"&gt;**第八章**&lt;/span&gt; &lt;br&gt;
&lt;span style="font-size: 50px;"&gt;回归模型(一)&lt;/span&gt; &lt;br&gt;
&lt;span style="font-size: 30px;"&gt;胡传鹏&lt;/span&gt; &lt;br&gt;
&lt;span style="font-size: 20px;"&gt; &lt;/span&gt; &lt;br&gt;
&lt;span style="font-size: 30px;"&gt;2024-04-19&lt;/span&gt; &lt;br&gt;
&lt;span style="font-size: 20px;"&gt; Made with Rmarkdown&lt;/span&gt; &lt;br&gt;


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---
## 每个小组均有对应的助教同学负责


---
&lt;h1 lang="zh-CN" style="font-size: 60px;"&gt; &lt;/h1&gt;
&lt;br&gt;
&lt;br&gt;
## 纯粹的R代码学习 → 使用R语言来实现**统计知识** &lt;br&gt;
&lt;br&gt;
## (1) R: 更灵活的统计分析方法，与统计知识结合更加紧密&lt;br&gt;
&lt;br&gt;
## (2) 心理学/社会科学中常用的统计检验均是回归模型&lt;br&gt;

---
&lt;h1 lang="zh-CN" style="font-size: 60px;"&gt;研究问题&lt;/h1&gt;

&lt;br&gt;
## 在penguin数据中，恋爱状态(romantic)对被试核心体温(Temperature)的影响
&lt;br&gt;
&lt;br&gt;
## 在penguin数据中，距赤道距离(DEQ)和恋爱状态(romantic)对于被试核心体温(Temperature)的影响
&lt;br&gt;
&lt;br&gt;


---
&lt;h1 lang="en" style="font-size: 60px;"&gt;Contents&lt;/h1&gt;
&lt;br&gt;
&lt;span style="font-size: 45px;"&gt;8.1 *t*-test &amp; linear regression&lt;/span&gt;&lt;/center&gt; &lt;br&gt;
&lt;span style="font-size: 30px;"&gt;&amp;emsp;8.1.1 独立样本*t*检验&lt;/span&gt;&lt;/center&gt; &lt;br&gt;
&lt;span style="font-size: 30px;"&gt;&amp;emsp;8.1.2 线性回归&lt;/span&gt;&lt;/center&gt; &lt;br&gt;
&lt;span style="font-size: 30px;"&gt;&amp;emsp;8.1.3 单样本*t*检验&lt;/span&gt;&lt;/center&gt; &lt;br&gt;
&lt;span style="font-size: 30px;"&gt;&amp;emsp;8.1.4 配对样本*t*检验&lt;/span&gt;&lt;/center&gt; &lt;br&gt;
&lt;span style="font-size: 30px;"&gt;&amp;emsp;8.1.5 bruceR::TTEST&lt;/span&gt;&lt;/center&gt; &lt;br&gt;
&lt;br&gt;
&lt;span style="font-size: 45px;"&gt;8.2 ANOVA &amp; linear regression&lt;/span&gt;&lt;/center&gt; &lt;br&gt;
&lt;span style="font-size: 30px;"&gt;&amp;emsp;8.2.1 研究问题&lt;/span&gt;&lt;/center&gt; &lt;br&gt;
&lt;span style="font-size: 30px;"&gt;&amp;emsp;8.2.2 代码实操&lt;/span&gt;&lt;/center&gt; &lt;br&gt;
&lt;span style="font-size: 30px;"&gt;&amp;emsp;8.2.3 线性回归&lt;/span&gt;&lt;/center&gt; &lt;br&gt;
&lt;span style="font-size: 30px;"&gt;&amp;emsp;8.2.4 知识延申&lt;/span&gt;&lt;/center&gt; &lt;br&gt;


---
class: center, middle
&lt;span style="font-size: 60px;"&gt;8.1 *t*-test &amp; linear regression&lt;/span&gt; &lt;br&gt;

---
class: left, middle
&lt;span style="font-size: 60px;"&gt;8.1  研究问题&lt;/span&gt; &lt;br&gt;

![](picture/chp8/IJzerman2018fig.png)&lt;!-- --&gt;
(引自[IJzerman et al., 2018](https://doi.org/10.1525/collabra.165))

&lt;br&gt;
&lt;span style="font-size: 35px;"&gt;Q: 如何检查恋爱状态(romantic)对核心体温(Temperature)的影响？&lt;/span&gt; &lt;br&gt;
&lt;br&gt;
&lt;span style="font-size: 40px;"&gt;A:独立样本*t*检验&lt;/span&gt; &lt;br&gt;

---
# 8.1 *t*-test
## 8.1.1 独立样本*t*检验(independent *t*-test)

.panelset[
.panel[.panel-name[基础知识]

&lt;br&gt;
**比较两个独立样本群体的均值是否有显著差异。**
&lt;br&gt;
**前提条件**
* 正态性：两个样本数据都应该来自正态分布的总体。样本量足够大时，即使不严格服从正态分布，结果也是稳健的。
* 同方差性：两个样本的方差应该是相等的。
* 独立性：两个样本应该是独立的，即一个样本的观测值不应影响另一个样本的观测值。&lt;br&gt;
&lt;br&gt;

**假设**
* `\(H_0\)`: 两个独立样本群体的均值没有显著差异，即 `\(μ_1\)` = `\(μ_2\)`
* `\(H_1\)`: 两个独立样本群体的均值有显著差异，即 `\(μ_1\)` ≠ `\(μ_2\)`
&lt;br&gt;
`$$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$`

.panel[.panel-name[数据清理]


```r
df.penguin &lt;- bruceR::import(here::here('data', 'penguin', 'penguin_rawdata.csv')) %&gt;%
  dplyr::mutate(subjID = row_number()) %&gt;%
  dplyr::select(subjID,Temperature_t1, Temperature_t2, socialdiversity, 
                Site, DEQ, romantic, ALEX1:ALEX16) %&gt;%                                     # 选择变量
  dplyr::filter(!is.na(Temperature_t1) &amp; !is.na(Temperature_t2) &amp; !is.na(DEQ)) %&gt;%         # 处理缺失值
  dplyr::mutate(romantic = factor(romantic, levels = c(1,2), labels = c("恋爱", "单身")),  # 转化为因子
                Temperature = rowMeans(select(., starts_with("Temperature"))),             # 计算两次核心温度的均值
                ALEX4  = case_when(TRUE ~ 6 - ALEX4),
                ALEX12 = case_when(TRUE ~ 6 - ALEX12),
                ALEX14 = case_when(TRUE ~ 6 - ALEX14),
                ALEX16 = case_when(TRUE ~ 6 - ALEX16),
                ALEX   = rowSums(select(., starts_with("ALEX")))) # 反向计分后计算总分
```

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.panel[.panel-name[代码实操]


```r
stats::t.test(data = df.penguin,      # 数据框
              Temperature ~ romantic, # 因变量~自变量
              var.equal = TRUE) %&gt;%
  capture.output()                    # 将输出变整齐
```

```
##  [1] ""                                                                                                    
##  [2] "\tTwo Sample t-test"                                                                                 
##  [3] ""                                                                                                    
##  [4] "data:  Temperature by romantic"                                                                      
##  [5] "t = -0.34664, df = 1425, p-value = 0.7289"                                                           
##  [6] "alternative hypothesis: true difference in means between group 恋爱 and group 单身 is not equal to 0"
##  [7] "95 percent confidence interval:"                                                                     
##  [8] " -0.0555949  0.0388971"                                                                              
##  [9] "sample estimates:"                                                                                   
## [10] "mean in group 恋爱 mean in group 单身 "                                                              
## [11] "          36.38498           36.39333 "                                                              
## [12] ""
```


]]]]

---
# 8.1 *t*-test作为回归模型的特例
## 8.1.2 线性回归(linear regression)
&lt;br&gt;
**回归分析**
* 回归分析用于研究一个或多个自变量（预测变量）与一个因变量（响应变量）之间的关系。&lt;br&gt;
&lt;br&gt;
&lt;br&gt;
--

**线性回归**
* 线性回归的基本思想是通过数据拟合一条直线，使得这条直线尽可能地接近所有的数据点，从而实现对新数据点的预测。&lt;br&gt;
* 线性回归模型可以表示为：
`$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_p x_p + \epsilon$$`

* 其中，$y$是因变量，x是自变量， `\(\beta_i\)`  是模型参数，表示截距和斜率，$\epsilon$是误差项，表示模型未能解释的随机误差。

---
# 8.1 *t*-test &amp; linear regression
## 8.1.2 线性回归(linear regression)
&lt;br&gt;
* 独立样本*t*检验是线性模型的特殊形式(自变量为二分变量）
--
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/indet-lm.png" width="55%"&gt;
&lt;/p&gt;


---
# 8.1 *t*-test &amp; linear regression

.pull-left[

```r
# t检验
stats::t.test(
  data = df.penguin,
  Temperature ~ romantic,
  var.equal = TRUE) %&gt;%
  capture.output() # 将输出变整齐
```

&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/compare1.1.png" width="100%"&gt;
&lt;/p&gt;
]

.pull-right[

```r
# 线性回归
model.inde &lt;- stats::lm(
  data = df.penguin,
  formula = Temperature ~ 1 + romantic
  )
summary(model.inde)
```

&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/compare1.2.png" width="100%"&gt;
&lt;/p&gt;

]


---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.3 单样本*t*检验(one sample *t*-test)
* 例如：在penguin数据中，全体被试的核心体温(Temperature)是否等于36.6？&lt;br&gt;
&lt;br&gt;
--
&lt;br&gt;
**比较单个样本的平均值(m)与已知的总体平均值(μ)之间是否存在显著差异**&lt;br&gt;
&lt;br&gt;

**前提条件**
* 正态性：样本数据应来自正态分布的总体。样本量足够大时，即使不严格服从正态分布，结果也是稳健的。
* 独立性：样本中的观测值必须是独立的，即一个观测值不应影响另一个观测值。&lt;br&gt;
&lt;br&gt;

**假设**
* `\(H_0\)`: 样本的均值(m)与给定的总体均值或假设的总体均值(μ)之间没有显著差异。
* `\(H_1\)`: 样本的均值(m)与给定的总体均值或假设的总体均值(μ)之间有显著差异。
&lt;br&gt;

`$$t = \frac{\bar{X} - \mu}{s / \sqrt{n}}$$`

---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.3 单样本*t*检验(one sample *t*-test)
.pull-left[
&lt;br&gt;
&lt;br&gt;

`$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_p x_p + \epsilon$$`

单样本*t*检验中，仅截距不为0。此时公式为：&lt;br&gt;
--
`$$y = \beta_0$$`
`$$H_0: \beta_0 = 0$$`
]
.pull-right[
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/singlet-lm.png" width="75%"&gt;
&lt;/p&gt;
]
---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.3 单样本*t*检验(one sample *t*-test)

.pull-left[

```r
stats::t.test(
  x = df.penguin$Temperature, # 核心体温均值
  mu = 36.6) 
```

&lt;br&gt;
&lt;br&gt;
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/compare2.1.png" width="100%"&gt;
&lt;/p&gt;

]
.pull-right[

```r
model.single &lt;- lm(
  data = df.penguin,
  formula = Temperature - 36.6 ~ 1
  ) 
summary(model.single) 
```

&lt;br&gt;
&lt;br&gt;
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/compare2.2.png" width="100%"&gt;
&lt;/p&gt;

]

---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.4 配对样本*t*检验(paired *t*-test)
* 例如，在penguin数据中，被试报告的两次核心温度(Temperature_t1, Temperature_t2)是否有显著差异？&lt;br&gt;
&lt;br&gt;
--
**比较两个相关的样本组（例如，同一组受试者在不同条件下的测量）的平均值是否存在显著差异。**&lt;br&gt;
&lt;br&gt;
**前提条件**
* 正态性：样本数据应来自正态分布的总体。样本量足够大时，即使不严格服从正态分布，结果也是稳健的。
* 独立性：配对样本中的观测值必须是独立的，即每一对观测值不应影响其他对的观测值。
* 配对设计：数据必须是以配对形式收集的。&lt;br&gt;
&lt;br&gt;
**假设**
* `\(H_0\)`: 配对样本的总体平均差与零没有显著差异（两个配对样本的均值没有显著差异）。
* `\(H_1\)`: 配对样本的总体平均差与零有显著差异（两个配对样本的均值存在显著差异）。
&lt;br&gt;

`$$t = \frac{\bar{X} - \mu}{s / \sqrt{n}}$$`
---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.4 配对样本*t*检验(paired *t*-test)

$$y_1 - y_2 = \beta_0 $$
$$H_0: \beta_0 = 0 $$
--
可以将配对样本*t*检验理解为对差值进行的单样本*t*检验，即单独用一个数字来预测对应的差值（见图的右半部分）。&lt;br&gt;
也可以认为这些组间之差是斜率（见图的左半部分）。&lt;br&gt;

&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/pairt-lm.png" width="70%"&gt;
&lt;/p&gt;

---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.4 配对样本*t*检验(paired *t*-test)
.pull-left[


```r
stats::t.test(
  x = df.penguin$Temperature_t1, 
  y = df.penguin$Temperature_t2, 
  paired = TRUE
  )
```

&lt;br&gt;
&lt;br&gt;
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/compare3.1.png" width="100%"&gt;
&lt;/p&gt;

]
.pull-right[


```r
model.paired &lt;- lm(
  Temperature_t1 - Temperature_t2 ~ 1,
  data = df.penguin
  )
summary(model.paired)
```

&lt;br&gt;
&lt;br&gt;
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/compare3.2.png" width="100%"&gt;
&lt;/p&gt;

]

---
# 8.1 *t*-test
## 8.1.5 bruceR::TTEST
* 如果偏好按传统方式使用*t*检验，推荐`bruceR::TTEST`函数

&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/TTEST.png" width="70%"&gt;
&lt;/p&gt;

中文帮助文档：https://zhuanlan.zhihu.com/p/281150493
---
# 8.1 *t*-test
## 8.1.5 bruceR::TTEST
.panelset[
.panel[.panel-name[独立样本t检验]
.pull-left[


```r
stats::t.test(
  data = df.penguin, 
  Temperature ~ romantic,
  var.equal = TRUE
  ) 
```


&lt;br&gt;
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/compare1.1.png" width="100%"&gt;
&lt;/p&gt;
]
.pull-right[


```r
bruceR::TTEST(
  data = df.penguin, # 数据
  y = "Temperature", # 因变量
  x = "romantic"     # 自变量
) 
```

&lt;br&gt;
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/leven.png" width="100%"&gt;
&lt;/p&gt;


]
.panel[.panel-name[单样本t检验]
.pull-left[


```r
stats::t.test(
  x = df.penguin$Temperature, 
  mu = 36.6
  )
```


&lt;br&gt;
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/compare2.1.png" width="100%"&gt;
&lt;/p&gt;
]
.pull-right[


```r
bruceR::TTEST(
  data = df.penguin, # 数据
  y = "Temperature", # 确定变量
  test.value = 36.6, # 固定值
  test.sided = "=")  # 假设的方向
```

&lt;br&gt;
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/sample.png" width="100%"&gt;
&lt;/p&gt;

]
.panel[.panel-name[配对样本t检验]

.pull-left[


```r
stats::t.test(
  x = df.penguin$Temperature_t1, #第1次
  y = df.penguin$Temperature_t2, #第2次
  paired = TRUE) 
```


&lt;br&gt;
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/compare3.1.png" width="100%"&gt;
&lt;/p&gt;
]
.pull-right[


```r
bruceR::TTEST(
  data = df.penguin, # 数据
  y = c("Temperature_t1",  
        "Temperature_t2"), # 变量为两次核心体温
  paired = T)  # 配对数据，默认是FALSE
```

&lt;br&gt;
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/pair.png" width="100%"&gt;
&lt;/p&gt;
]
]]]]

---
# 8.1 *t*-test系列均为回归模型的特例
&lt;br&gt;
&lt;br&gt;

|       | R自带函数 | 线性模型 | 解释 |
|-------|-------|-------|-------|
| 单样本*t* | t.test(y, mu = 0) | lm(y ~ 1)| 仅有截距的回归模型 |
| 独立样本*t* | t.test($y_1$, `\(y_2\)`) | lm(y ~ 1 + `\(G_2\)`)| 自变量为二分变量的回归模型 |
| 配对样本*t* | t.test($y_1$, `\(y_2\)`, paired=T)  | lm($y_1$ - `\(y_2\)` ~ 1)| 仅有截距的回归模型)|


---
class: center, middle
&lt;span style="font-size: 60px;"&gt;8.2  ANOVA &amp; linear regression&lt;/span&gt; &lt;br&gt;


---
class: left, middle
&lt;span style="font-size: 60px;"&gt;8.2.1  研究问题&lt;/span&gt; &lt;br&gt;

&lt;br&gt;
&lt;span style="font-size: 35px;"&gt;Q: 如何同时检验距赤道距离(DEQ)和恋爱状态(romantic)对于被试体温的影响&lt;/span&gt; &lt;br&gt;
&lt;br&gt;
&lt;span style="font-size: 40px;"&gt;A:双因素被试间方差分析&lt;/span&gt; &lt;br&gt;


---
# 8.2  ANOVA &amp; linear regression
## 8.2.2 代码实操 &amp; 知识回顾
.pull-left[
&lt;br&gt;
**方法简介：**

当研究者想要比较两个或多个组之间的均值差异时，可使用方差分析(Analysis of Variance，简称ANOVA)。&lt;br&gt;
&lt;br&gt;
它由英国统计学家R.A.Fisher提出，基本思想是将测量数据的总变异（即总方差）按照变异来源分为处理（组间）效应和误差（组内）效应，并作出其数量估计，从而确定实验处理对研究结果影响力的大小。

**假设：**
* `\(H_0\)`: 各因素各个水平下，因变量的均值完全相同
* `\(H_1\)`: 各因素各个水平下，因变量的均值不完全相同
]

.pull-right[
&lt;br&gt;
&lt;br&gt;
&lt;br&gt;
&lt;br&gt;
**前提条件：**

* 可加性：各效应可加，即观测值是由各主效应，交互作用以及误差通过相加得到的&lt;br&gt;
* 随机性：各样本（观测值）是随机样本&lt;br&gt;
* 正态性：各样本来自于正态分布的总体&lt;br&gt;
* 独立性：各样本观测值互相独立&lt;br&gt;
* 方差齐性：各样本来自的总体方差相同&lt;br&gt;
* 因变量应为连续变量&lt;br&gt;

]

---
# 8.2  ANOVA 
## 8.2.2 代码实操|数据预处理
.panelset[
.panel[.panel-name[vars]


```r
summary(df.penguin$DEQ)
```

```
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.293  34.433  39.912  39.842  51.317  60.391
```


```r
# 设定分割点
# [0-23.5 热带， 23.5-35 亚热带]， [35-40 暖温带， 40-50 中温带]， [50-66.5 寒温带]
breaks &lt;- c(0, 35, 50, 66.5)

# 设定相应的标签
labels &lt;- c('热带', '温带', '寒温带')

# 创建新的变量
df.penguin$climate &lt;- cut(df.penguin$DEQ, 
                          breaks = breaks, 
                          labels = labels)
summary(df.penguin$climate)
```

```
##   热带   温带 寒温带 
##    396    592    439
```

.panel[.panel-name[tidy data]


```r
df &lt;- df.penguin %&gt;% 
  select(subjID, climate, romantic, Temperature) 
```

<div class="datatables html-widget html-fill-item" id="htmlwidget-bec36c95e7b22d5fe1bc" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-bec36c95e7b22d5fe1bc">{"x":{"filter":"none","vertical":false,"fillContainer":true,"data":[["1","2","3","4","5","6"],[1,2,3,4,5,6],["热带","寒温带","寒温带","寒温带","温带","寒温带"],["单身","恋爱","恋爱","恋爱","恋爱","恋爱"],[36.75,35.40000000000001,35.1,35.95,36.08333333,35.59999999999999]],"container":"<table class=\"display fill-container\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>subjID<\/th>\n      <th>climate<\/th>\n      <th>romantic<\/th>\n      <th>Temperature<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"columnDefs":[{"className":"dt-right","targets":[1,4]},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"subjID","targets":1},{"name":"climate","targets":2},{"name":"romantic","targets":3},{"name":"Temperature","targets":4}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script>

]]]


---
# 8.2  ANOVA &amp; linear regression
## 8.2.2 代码实操|正态性检验
.panelset[
.panel[.panel-name[KS检验]


```r
# 正态性检验-Kolmogorov-Smirnov检验
# 若p &gt;.05，不能拒绝数据符合正态分布的零假设
ks.test(df$Temperature, 'pnorm')
```

```
## 
## 	Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  df$Temperature
## D = 1, p-value &lt; 0.00000000000000022
## alternative hypothesis: two-sided
```


```r
# 进行数据转换，转换后仍非正态分布
df$Temperature_log &lt;- log(df$Temperature)
ks.test(df$Temperature_log, 'pnorm')
```

```
## 
## 	Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  df$Temperature_log
## D = 0.99981, p-value &lt; 0.00000000000000022
## alternative hypothesis: two-sided
```

.panel[.panel-name[qq图]

```r
# 正态性检验-qq图
qqnorm(df$Temperature)
qqline(df$Temperature, col = "red") # 添加理论正态分布线
```

![](chapter_8_files/figure-html/unnamed-chunk-20-1.png)&lt;!-- --&gt;

.panel[.panel-name[直方图]

```r
ggplot(df, aes(Temperature)) +
    geom_histogram(aes(y =..density..), color='black', fill='white', bins=30) +
    geom_density(alpha=.5, fill='red')
```

![](chapter_8_files/figure-html/unnamed-chunk-21-1.png)&lt;!-- --&gt;

]]]]


---
# 8.2  ANOVA &amp; linear regression
## 8.2.2 代码实操|双因素被试间方差分析
.panelset[
.panel[.panel-name[stats::aov()]


```r
aov1 &lt;- stats::aov(Temperature ~ climate * romantic, data = df)
summary(aov1)
```

```
##                    Df Sum Sq Mean Sq F value               Pr(&gt;F)    
## climate             2  18.82   9.408  49.392 &lt; 0.0000000000000002 ***
## romantic            1   0.24   0.244   1.280              0.25807    
## climate:romantic    2   1.91   0.955   5.014              0.00676 ** 
## Residuals        1421 270.65   0.190                                 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

.panel[.panel-name[SPSS]
![](picture/chp8/SPSS.png)&lt;!-- --&gt;

]]]


---
class: center, middle
&lt;span style="font-size: 60px;"&gt;结果为什么不相同？&lt;/span&gt; &lt;br&gt;


---
# 8.2  ANOVA &amp; linear regression
## 8.2.2 代码实操|平方和(SS)的计算
&lt;br&gt;
在平衡设计中，三种类型的平方和的结果会很清晰，并且方差分析的结果独立于平方和的类型；&lt;br&gt;
而在非平衡设计中，尤其是当各组样本量差距较大时，三种类型的平方和计算结果可能会不同，此时需要根据具体研究设计和问题来选择使用哪一种类型的平方和。&lt;br&gt;
&lt;br&gt;
对于 Y ~ A + B + A * B&lt;br&gt;
&lt;br&gt;
.panelset[
.panel[.panel-name[Type I SS]
&lt;br&gt;
解释变量的顺序会影响到类型I平方和的计算结果，通常用于顺序重要的模型。&lt;br&gt;
效应根据表达式中先出现的效应做调整。A不做调整，B根据A调整，A:B交互项根据A和B调整。&lt;br&gt;
&lt;br&gt;
**stats::aov()函数默认采用的就是Type I SS，它逐步将每一个因子引入模型进行计算。**&lt;br&gt;

.panel[.panel-name[Type II SS]
&lt;br&gt;
忽略了因子之间可能存在的交互作用，适用于所有主效应不涉及交互效应的情况。&lt;br&gt;
假定所有的因子都是同时进入模型的，并且它们都是等价的。&lt;br&gt;
效应根据同水平或低水平的效应做调整。A根据B调整，B依据A调整，A:B交互项同时根据A和B调整。&lt;br&gt;
&lt;br&gt;
**car::Anova()函数默认计算Type II SS，可以通过type = 3调整为Type III SS。**&lt;br&gt;

.panel[.panel-name[Type III SS]
&lt;br&gt;
更全面，假定所有因子（以及它们的交互项）都是重要的，并考虑所有因素。&lt;br&gt;
每个效应根据模型其他各效应做相应调整。A根据B和A:B做调整，A:B交互项根据A和B调整。&lt;br&gt;
&lt;br&gt;
**SPSS默认采用Type III SS。**&lt;br&gt;
**bruceR::MANOVA*()函数也默认采用Type III SS，可以通过ss.type = 2调整为Type II SS。**&lt;br&gt;

]]]]


---
# 8.2  ANOVA &amp; linear regression
## 8.2.2 代码实操|双因素被试间方差分析
.panelset[
.panel[.panel-name[car::Anova()]

```r
# 结果不一致，原因PPT显示不全，请回到rmd文档查看
aov1 &lt;- car::Anova(stats::aov(Temperature ~ climate * romantic, data = df))
aov1
```

```
## Anova Table (Type II tests)
## 
## Response: Temperature
##                   Sum Sq   Df F value                Pr(&gt;F)    
## climate           19.034    2 49.9680 &lt; 0.00000000000000022 ***
## romantic           0.244    1  1.2801              0.258072    
## climate:romantic   1.910    2  5.0136              0.006765 ** 
## Residuals        270.654 1421                                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```


```r
# 原因debug
# 查看R的默认对比设置
options("contrasts")
```

```
## $contrasts
##         unordered           ordered 
## "contr.treatment"      "contr.poly"
```

```r
# 从输出结果可知，无序默认为contr.treatment()，有序默认为contr.poly()
# factor()函数来创建无序因子，ordered()函数创建有序因子

is.factor(df$climate)
```

```
## [1] TRUE
```

```r
is.ordered(df$climate)
```

```
## [1] FALSE
```

```r
# climate是无序因子

# 创建一个3水平的因子的基准对比
c1 &lt;- contr.treatment(3)

# 创建一个新的对比，这个编码假设分类水平之间的差异被等分，每一个水平与总均值的差异等于1/3
my.coding &lt;- matrix(rep(1/3, 6), ncol=2)
# 将对比调整为每个水平与第一个水平的振幅减去1/3
# 可能的原因：除了关心每个水平对应的效果，同时也关心水平与水平之间的效果
my.simple &lt;- c1-my.coding
my.simple
```

```
##            2          3
## 1 -0.3333333 -0.3333333
## 2  0.6666667 -0.3333333
## 3 -0.3333333  0.6666667
```

```r
# 更改climate的对比
contrasts(df$climate) &lt;- my.simple

# 将数据集df的romantic列的对比设为等距对比，它假设分类水平之间的差异为等距离
contrasts(df$romantic) &lt;- contr.sum(2)/2

# 方差分析
aov1 &lt;- car::Anova(lm(Temperature ~ climate * romantic, data = df),
                   type = 3)
aov1
```

```
## Anova Table (Type III tests)
## 
## Response: Temperature
##                   Sum Sq   Df      F value                Pr(&gt;F)    
## (Intercept)      1768309    1 9284069.6498 &lt; 0.00000000000000022 ***
## climate               19    2      50.8832 &lt; 0.00000000000000022 ***
## romantic               0    1       1.0006              0.317336    
## climate:romantic       2    2       5.0136              0.006765 ** 
## Residuals            271 1421                                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

.panel[.panel-name[afex::aov_ez()]

```r
afex::aov_ez(id = "subjID", 
             dv = "Temperature", 
             data = df, 
             between = c("climate", "romantic"), 
             type = 3)
```

```
## Anova Table (Type 3 tests)
## 
## Response: Temperature
##             Effect      df  MSE         F   ges p.value
## 1          climate 2, 1421 0.19 50.88 ***  .067   &lt;.001
## 2         romantic 1, 1421 0.19      1.00 &lt;.001    .317
## 3 climate:romantic 2, 1421 0.19   5.01 **  .007    .007
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1
```


```r
# afex中的其他函数可以得到同样的结果
afex::aov_car(Temperature ~ climate * romantic + Error(subjID), data = df, type = 3)
afex::aov_4(Temperature ~ climate * romantic + (1|subjID), data = df)
```


]]]

---
# 8.2  ANOVA &amp; linear regression
## 8.2.3 线性回归

.pull-left[

&lt;br&gt;
&lt;br&gt;
&lt;br&gt;
这里的ANOVA仍然是线性回归模型的特例，即自变量是离散变量的情况。
&lt;br&gt;
如果使用哑变量（dummy coding）对自变量进行编码后进入回归方程，线性模型中的斜率即是对组间差异的估计。&lt;br&gt;
&lt;br&gt;
**双因素方差分析可以看作是一种特殊的线性回归模型，自变量为两个分类变量。**&lt;br&gt;

]

.pull-right[
&lt;br&gt;
&lt;br&gt;
![](picture/chp8/aovLM.png)&lt;!-- --&gt;
]

---
# 8.2  ANOVA &amp; linear regression
## 8.2.3 线性回归
.pull-left[


```r
aov1 &lt;- car::Anova(
  aov(Temperature ~ climate * romantic, 
      data = df), 
  type = 3
  )
aov1
```
&lt;br&gt;
![](./picture/chp8/compare8231.png)
]


.pull-right[


```r
lm1 &lt;- car::Anova(
  lm(Temperature ~ climate * romantic, 
     data = df), 
  type = 3
  )
lm1
```
&lt;br&gt;
![](./picture/chp8/compare8232.png)

]


---
# 8.2  ANOVA &amp; linear regression
## 8.2.2 代码实操: `bruceR`
* 使用`bruceR`可以更简单地实现心理学中习惯的ANOVA，但要注意数据格式
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/ANOVA data.png" width="65%"&gt;
&lt;/p&gt;


---
# 8.2  ANOVA &amp; linear regression
## 8.2.2 代码实操: `bruceR::MANOVA`
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/MANOVA.png" width="70%"&gt;
&lt;/p&gt;


---
# 8.2  ANOVA &amp; linear regression
## 8.2.2 代码实操: `bruceR::MANOVA`

```r
res1 &lt;- bruceR::MANOVA(data = df, 
                       dv = "Temperature", 
                       between = c("climate", "romantic"))
```

```
## [1] "Anova Table (Type III tests)"                                  
## [2] ""                                                              
## [3] "Response: Temperature"                                         
## [4] "            Effect      df  MSE         F   ges p.value"       
## [5] "1          climate 2, 1421 0.19 50.88 ***  .067   &lt;.001"       
## [6] "2         romantic 1, 1421 0.19      1.00 &lt;.001    .317"       
## [7] "3 climate:romantic 2, 1421 0.19   5.01 **  .007    .007"       
## [8] "---"                                                           
## [9] "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1"
```


---
# 8.2  ANOVA &amp; linear regression
## 8.2.2 代码实操: `bruceR::EMMEANS`
更进一步进行简单效应分析与多重比较
&lt;p align="center"&gt;
  &lt;img src="./picture/chp8/EMMEANS.png" width="70%"&gt;
&lt;/p&gt;


---
# 8.2  ANOVA &amp; linear regression
## 8.2.2 代码实操: `bruceR::EMMEANS`

```r
sim_eff &lt;- res1 %&gt;% 
  bruceR::EMMEANS("climate", by = "romantic") %&gt;%
  bruceR::EMMEANS("romantic", by = "climate")
```

------ EMMEANS (effect = "climate") ------

Joint Tests of "climate":
─────────────────────────────────────────────────────────────────
  Effect "romantic" df1  df2      F     p     η²p [90% CI of η²p]
─────────────────────────────────────────────────────────────────
 climate       恋爱   2 1421 13.165 &lt;.001 ***   .018 [.008, .031]
 climate       单身   2 1421 41.817 &lt;.001 ***   .056 [.037, .075]
─────────────────────────────────────────────────────────────────
Note. Simple effects of repeated measures with 3 or more levels
are different from the results obtained with SPSS MANOVA syntax.

Estimated Marginal Means of "climate":
─────────────────────────────────────────────────────
 "climate" "romantic"   Mean [95% CI of Mean]    S.E.
─────────────────────────────────────────────────────
      热带       恋爱 36.510 [36.446, 36.575] (0.033)
      温带       恋爱 36.393 [36.346, 36.440] (0.024)
    寒温带       恋爱 36.296 [36.245, 36.347] (0.026)
      热带       单身 36.590 [36.532, 36.648] (0.029)
      温带       单身 36.356 [36.303, 36.409] (0.027)
    寒温带       单身 36.182 [36.114, 36.250] (0.035)
─────────────────────────────────────────────────────

Pairwise Comparisons of "climate":
────────────────────────────────────────────────────────────────────────────────────────
      Contrast "romantic" Estimate    S.E.   df      t     p     Cohen’s d [95% CI of d]
────────────────────────────────────────────────────────────────────────────────────────
   温带 - 热带       恋爱   -0.117 (0.041) 1421 -2.879  .012 *   -0.268 [-0.491, -0.045]
 寒温带 - 热带       恋爱   -0.214 (0.042) 1421 -5.111 &lt;.001 *** -0.490 [-0.720, -0.260]
 寒温带 - 温带       恋爱   -0.097 (0.035) 1421 -2.741  .019 *   -0.222 [-0.417, -0.028]
   温带 - 热带       单身   -0.234 (0.040) 1421 -5.853 &lt;.001 *** -0.536 [-0.756, -0.317]
 寒温带 - 热带       单身   -0.409 (0.046) 1421 -8.969 &lt;.001 *** -0.936 [-1.186, -0.686]
 寒温带 - 温带       单身   -0.174 (0.044) 1421 -3.966 &lt;.001 *** -0.400 [-0.641, -0.158]
────────────────────────────────────────────────────────────────────────────────────────
Pooled SD for computing Cohen’s d: 0.436
P-value adjustment: Bonferroni method for 3 tests.

Disclaimer:
By default, pooled SD is Root Mean Square Error (RMSE).
There is much disagreement on how to compute Cohen’s d.
You are completely responsible for setting `sd.pooled`.
You might also use `effectsize::t_to_d()` to compute d.

------ EMMEANS (effect = "romantic") ------

Joint Tests of "romantic":
────────────────────────────────────────────────────────────────
   Effect "climate" df1  df2     F     p     η²p [90% CI of η²p]
────────────────────────────────────────────────────────────────
 romantic      热带   1 1421 3.287  .070 .     .002 [.000, .008]
 romantic      温带   1 1421 1.055  .305       .001 [.000, .005]
 romantic    寒温带   1 1421 6.966  .008 **    .005 [.001, .013]
────────────────────────────────────────────────────────────────
Note. Simple effects of repeated measures with 3 or more levels
are different from the results obtained with SPSS MANOVA syntax.

Estimated Marginal Means of "romantic":
─────────────────────────────────────────────────────
 "romantic" "climate"   Mean [95% CI of Mean]    S.E.
─────────────────────────────────────────────────────
       恋爱      热带 36.510 [36.446, 36.575] (0.033)
       单身      热带 36.590 [36.532, 36.648] (0.029)
       恋爱      温带 36.393 [36.346, 36.440] (0.024)
       单身      温带 36.356 [36.303, 36.409] (0.027)
       恋爱    寒温带 36.296 [36.245, 36.347] (0.026)
       单身    寒温带 36.182 [36.114, 36.250] (0.035)
─────────────────────────────────────────────────────

Pairwise Comparisons of "romantic":
─────────────────────────────────────────────────────────────────────────────────────
    Contrast "climate" Estimate    S.E.   df      t     p     Cohen’s d [95% CI of d]
─────────────────────────────────────────────────────────────────────────────────────
 单身 - 恋爱      热带    0.080 (0.044) 1421  1.813  .070 .    0.183 [-0.015,  0.382]
 单身 - 恋爱      温带   -0.037 (0.036) 1421 -1.027  .305     -0.085 [-0.247,  0.077]
 单身 - 恋爱    寒温带   -0.115 (0.043) 1421 -2.639  .008 **  -0.262 [-0.458, -0.067]
─────────────────────────────────────────────────────────────────────────────────────
Pooled SD for computing Cohen’s d: 0.436
No need to adjust p values.

Disclaimer:
By default, pooled SD is Root Mean Square Error (RMSE).
There is much disagreement on how to compute Cohen’s d.
You are completely responsible for setting `sd.pooled`.
You might also use `effectsize::t_to_d()` to compute d.

---
# 8.2  ANOVA &amp; linear regression
## 8.2.4 知识延申|单因素方差分析示例

```r
# DEQ对Temperature的影响
res2 &lt;- bruceR::MANOVA(
  data = df,
  dv = "Temperature",
  between = "climate") 
```

```
## 
## ====== ANOVA (Between-Subjects Design) ======
## 
## Descriptives:
## ─────────────────────────────
##  "climate"   Mean    S.D.   n
## ─────────────────────────────
##       热带 36.555 (0.411) 396
##       温带 36.377 (0.401) 592
##     寒温带 36.255 (0.504) 439
## ─────────────────────────────
## Total sample size: N = 1427
## 
## ANOVA Table:
## Dependent variable(s):      Temperature
## Between-subjects factor(s): climate
## Within-subjects factor(s):  –
## Covariate(s):               –
## ───────────────────────────────────────────────────────────────────────
##             MS   MSE df1  df2      F     p     η²p [90% CI of η²p]  η²G
## ───────────────────────────────────────────────────────────────────────
## climate  9.408 0.192   2 1424 49.106 &lt;.001 ***   .065 [.045, .085] .065
## ───────────────────────────────────────────────────────────────────────
## MSE = mean square error (the residual variance of the linear model)
## η²p = partial eta-squared = SS / (SS + SSE) = F * df1 / (F * df1 + df2)
## ω²p = partial omega-squared = (F - 1) * df1 / (F * df1 + df2 + 1)
## η²G = generalized eta-squared (see Olejnik &amp; Algina, 2003)
## Cohen’s f² = η²p / (1 - η²p)
## 
## Levene’s Test for Homogeneity of Variance:
## ──────────────────────────────────────────────
##                  Levene’s F df1  df2     p    
## ──────────────────────────────────────────────
## DV: Temperature      18.207   2 1424 &lt;.001 ***
## ──────────────────────────────────────────────
```


---
# 8.2  ANOVA &amp; linear regression
## 8.2.4 知识延申|总结
&lt;br&gt;
&lt;br&gt;

|       | R自带函数 | 线性模型 | 解释 |
|-------|-------|-------|-------|
| 单因素ANOVA | aov(y ~ G) | lm(y ~ 1 + `\(G_1\)` + `\(G_2\)` + ...)| 一个离散自变量的回归模型 |
| 多因素ANOVA | aovt(y ~ G * S) | lm(y ~ `\(G_1\)` + `\(G_2\)` + ... + `\(S_1\)` + `\(S_2\)` + ...)| 多个离散自变量的回归模型 |


---
&lt;h1 lang="zh-CN" style="font-size: 60px;"&gt;课堂总结&lt;/h1&gt;

![](./picture/chp8/route.png)


---
&lt;h1 lang="zh-CN" style="font-size: 60px;"&gt;课堂总结&lt;/h1&gt;

|       | R自带函数 | 线性模型 | 解释 |
|-------|-------|-------|-------|
| 单样本*t* | t.test(y, mu = 0) | lm(y ~ 1)| 仅有截距的回归模型 |
| 独立样本*t* | t.test($y_1$, `\(y_2\)`) | lm(y ~ 1 + `\(G_2\)`)| 自变量为二分变量的回归模型 |
| 配对样本*t* | t.test($y_1$, `\(y_2\)`, paired=T)  | lm($y_1$ - `\(y_2\)` ~ 1)| 仅有截距的回归模型)|
| 单因素ANOVA | aov(y ~ G) | lm(y ~ 1 + `\(G_1\)` + `\(G_2\)` + ...)| 一个离散自变量的回归模型 |
| 多因素ANOVA | aovt(y ~ G * S) | lm(y ~ `\(G_1\)` + `\(G_2\)` + ... + `\(S_1\)` + `\(S_2\)` + ...)| 多个离散自变量的回归模型 |

参考：https://lindeloev.github.io/tests-as-linear/#61_one-way_anova_and_kruskal-wallis


---
class: center, middle
&lt;span style="font-size: 60px;"&gt;思考&lt;/span&gt; &lt;br&gt;
&lt;span style="font-size: 55px;"&gt;如何从回归模型的角度理解重复测量方差分析呢？&lt;/span&gt; &lt;br&gt;


---
&lt;h1 lang="zh-CN" style="font-size: 60px;"&gt;课堂练习&lt;/h1&gt;
&lt;br&gt;
&lt;br&gt;
&lt;br&gt;
&lt;h2 lang="zh-CN"&gt;1. 在Penguin数据中，处在不同恋爱状态下(romantic)的被试在社交复杂度(socialdiversity)上是否有显著差异？&lt;/h2&gt;
&lt;br&gt;
&lt;br&gt;
&lt;h2 lang="zh-CN"&gt;2. 在Penguin数据中，不同气候带(DEQ)与恋爱状态(romantic)对被试社交复杂度(socialdiversity)的影响.&lt;/h2&gt;


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