逻辑斯蒂回归

.title[
# 逻辑斯蒂回归
]
.subtitle[
## 社会学研究方法——第9讲
]
.author[
### 李代
]
.institute[
### 中国政法大学社会学院
]
.date[
### 2025-04-24
]

---

# 逻辑斯蒂回归

## 比数比

## 逻辑斯蒂回归

## 多分类逻辑斯蒂回归

---

# 比数比

## 线性回归

---

# 比数比

## 线性回归

---

# 比数比

## 线性回归

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  <thead>
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      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="label"><span class='gt_from_md'><strong>Characteristic</strong></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="estimate"><span class='gt_from_md'><strong>Beta</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;"><sup>1</sup></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="std.error"><span class='gt_from_md'><strong>SE</strong></span></th>
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    <tr><td headers="label" class="gt_row gt_left">(Intercept)</td>
<td headers="estimate" class="gt_row gt_center">0.26</td>
<td headers="std.error" class="gt_row gt_center">0.185</td></tr>
    <tr><td headers="label" class="gt_row gt_left">x</td>
<td headers="estimate" class="gt_row gt_center">0.32***</td>
<td headers="std.error" class="gt_row gt_center">0.018</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">R²</td>
<td headers="estimate" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">0.866</td>
<td headers="std.error" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">Adjusted R²</td>
<td headers="estimate" class="gt_row gt_center">0.863</td>
<td headers="std.error" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">df</td>
<td headers="estimate" class="gt_row gt_center">1</td>
<td headers="std.error" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">No. Obs.</td>
<td headers="estimate" class="gt_row gt_center">50</td>
<td headers="std.error" class="gt_row gt_center"><br /></td></tr>
  </tbody>
  <tfoot class="gt_sourcenotes">
    <tr>
      <td class="gt_sourcenote" colspan="3"><span class='gt_from_md'>Abbreviations: CI = Confidence Interval, SE = Standard Error</span></td>
    </tr>
  </tfoot>
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    <tr>
      <td class="gt_footnote" colspan="3"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;"><sup>1</sup></span> <span class='gt_from_md'>*p<0.05; **p<0.01; ***p<0.001</span></td>
    </tr>
  </tfoot>
</table>
</div>

---

# 比数比

## 线性回归

线性回归模型，是对变量之间相关关系的一种设想。它认为，变量之间存在“
线性”的关系。也就是说，一条直线可以描述自变量与因变量之间的关系。

该模型认为，x每增加1单位，y增加0.32单位。当x=0时，y=0.26。

---

# 比数比

## 最小二乘法

画一条直线，让每一个点和这条直线的垂直距离的平方和最小。

---

# 比数比

## 线性回归模型的使用条件

1. 因变量是连续变量（定距、定比）。

1. x和y之间的关系是线性的（能写成上面一次方程的形式）。

1. 独立：误差之间彼此不相关。（Durbin-Watson检验）

1. 同分布：误差分布相同，方差一样大，均值为0。

1. 正态分布：误差符合正态分布。（Q-Q图）

---

# 比数比

## 线性回归不适用怎么办？

如果因变量不是连续变量，应该怎么办？

例如：

二分类变量：性别、好恶、中外……

多分类变量：爱好、品牌、国籍……

定序变量：教育程度、满意程度……

---

# 比数比

## 比数比

发生比、比数：odds

What are the **odds**?

[国足仍有世界杯夺冠赔率 1赔3501倒数第二压泰国](https://www.163.com/sports/article/BJG7MT7700051C8M.html)

用1元钱赌中国队夺冠，中国队如果真的夺冠了，拿回来3501元钱。

odds: 1:3501

---

# 比数比

## 升学率

假设有n个学生考大学，考上大学的人数是x。

升学率，是一个百分比，proportion。 `$Proportion = \frac{x}{n}$`.

概率，probability。概率不可观测，但可以通过统计推断推算。 `$p = \frac{x}{n}$`.

比数，也就是事件发生的概率比事件不发生的概率： `$odds = \frac{p}{1-p}$`.

---

# 比数比

## 升学率

.pull-left[
<img src="method-logit_files/figure-html/unnamed-chunk-6-1.png" style="display: block; margin: auto;" />
]

.pull-left[
<img src="method-logit_files/figure-html/unnamed-chunk-7-1.png" style="display: block; margin: auto;" />
]

---

# 比数比

## 谁更占优势？

假设有m个男学生考大学，考上大学的男生人数是x。
假设有n个女学生考大学，考上大学的女生人数是y。

假如男生的升学率比女生大，是否说明男生更占优势？

$$ p_1 - p_2 = \frac{x}{m} - \frac{y}{n} $$

$$ \frac{p_1}{p_2} = \frac{\frac{x}{m}}{\frac{y}{n}} $$

---

# 比数比

## 谁更占优势？

---

# 比数比

## 比数比

odds ratio

男生升学的发生比是： `$\frac{p_m}{1 - p_m}$`

女生升学的发生比是： `$\frac{p_f}{1 - p_f}$`

比数比是： `$\frac{\frac{p_m}{1 - p_m}}{\frac{p_f}{1 - p_f}}$`

---

# 比数比

## 比数比

---

# 比数比

## 比数比的对数(log odds ratio)

---

# 逻辑斯蒂回归

---

# 逻辑斯蒂回归

## 泰坦尼克号

![](image/titanic-hundiendose.jpg)

---

# 逻辑斯蒂回归

## 泰坦尼克号

---

# 逻辑斯蒂回归

## 泰坦尼克号

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1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-12fad8c5{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-12fad8c6{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-1302500e'><thead><tr style="overflow-wrap:break-word;"><th class="cl-12fad8b0"><p class="cl-12fabd94"><span class="cl-12f77738">class</span></p></th><th class="cl-12fad8ba"><p class="cl-12fabd9e"><span class="cl-12f77738">No</span></p></th><th class="cl-12fad8ba"><p class="cl-12fabd9e"><span class="cl-12f77738">Yes</span></p></th><th class="cl-12fad8ba"><p class="cl-12fabd9e"><span class="cl-12f77738">p</span></p></th><th class="cl-12fad8ba"><p class="cl-12fabd9e"><span class="cl-12f77738">1-p</span></p></th><th class="cl-12fad8ba"><p class="cl-12fabd9e"><span class="cl-12f77738">odds</span></p></th><th class="cl-12fad8ba"><p class="cl-12fabd9e"><span class="cl-12f77738">or</span></p></th><th class="cl-12fad8ba"><p class="cl-12fabd9e"><span class="cl-12f77738">log or</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-12fad8bb"><p class="cl-12fabd94"><span class="cl-12f77738">1st</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">122</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">203</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">0.625</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">0.375</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">1.664</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">5.282</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">1.664</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-12fad8bb"><p class="cl-12fabd94"><span class="cl-12f77738">2nd</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">167</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">118</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">0.414</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">0.586</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">0.707</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">2.243</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">0.808</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-12fad8bb"><p class="cl-12fabd94"><span class="cl-12f77738">3rd</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">528</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">178</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">0.252</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">0.748</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">0.337</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">1.070</span></p></td><td class="cl-12fad8c4"><p class="cl-12fabd9e"><span class="cl-12f77738">0.068</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-12fad8c5"><p class="cl-12fabd94"><span class="cl-12f77738">Crew</span></p></td><td class="cl-12fad8c6"><p class="cl-12fabd9e"><span class="cl-12f77738">673</span></p></td><td class="cl-12fad8c6"><p class="cl-12fabd9e"><span class="cl-12f77738">212</span></p></td><td class="cl-12fad8c6"><p class="cl-12fabd9e"><span class="cl-12f77738">0.240</span></p></td><td class="cl-12fad8c6"><p class="cl-12fabd9e"><span class="cl-12f77738">0.760</span></p></td><td class="cl-12fad8c6"><p class="cl-12fabd9e"><span class="cl-12f77738">0.315</span></p></td><td class="cl-12fad8c6"><p class="cl-12fabd9e"><span class="cl-12f77738">1.000</span></p></td><td class="cl-12fad8c6"><p class="cl-12fabd9e"><span class="cl-12f77738">0.000</span></p></td></tr></tbody></table></div>

因为 `$OR > 1$` 或者 `$log or > 0$`，乘客跟船员比都更容易幸存，一等舱最容易幸存，其次是二等舱，再次是三等舱。

---

# 逻辑斯蒂回归

## 泰坦尼克号

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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="label"><span class='gt_from_md'><strong>Characteristic</strong></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="estimate"><span class='gt_from_md'><strong>log(OR)</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;"><sup>1</sup></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="std.error"><span class='gt_from_md'><strong>SE</strong></span></th>
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  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="label" class="gt_row gt_left">(Intercept)</td>
<td headers="estimate" class="gt_row gt_center">-1.2***</td>
<td headers="std.error" class="gt_row gt_center">0.079</td></tr>
    <tr><td headers="label" class="gt_row gt_left">Class</td>
<td headers="estimate" class="gt_row gt_center"><br /></td>
<td headers="std.error" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    Crew</td>
<td headers="estimate" class="gt_row gt_center">—</td>
<td headers="std.error" class="gt_row gt_center">—</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    1st</td>
<td headers="estimate" class="gt_row gt_center">1.7***</td>
<td headers="std.error" class="gt_row gt_center">0.139</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    2nd</td>
<td headers="estimate" class="gt_row gt_center">0.81***</td>
<td headers="std.error" class="gt_row gt_center">0.144</td></tr>
    <tr><td headers="label" class="gt_row gt_left">    3rd</td>
<td headers="estimate" class="gt_row gt_center">0.07</td>
<td headers="std.error" class="gt_row gt_center">0.117</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">AIC</td>
<td headers="estimate" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">2,597</td>
<td headers="std.error" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">No. Obs.</td>
<td headers="estimate" class="gt_row gt_center">2,201</td>
<td headers="std.error" class="gt_row gt_center"><br /></td></tr>
  </tbody>
  <tfoot class="gt_sourcenotes">
    <tr>
      <td class="gt_sourcenote" colspan="3"><span class='gt_from_md'>Abbreviations: CI = Confidence Interval, OR = Odds Ratio, SE = Standard Error</span></td>
    </tr>
  </tfoot>
  <tfoot class="gt_footnotes">
    <tr>
      <td class="gt_footnote" colspan="3"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;"><sup>1</sup></span> <span class='gt_from_md'>*p<0.05; **p<0.01; ***p<0.001</span></td>
    </tr>
  </tfoot>
</table>
</div>

---
# 星球大战

![](image/The-9-Star-Wars-Films-Ranked.jpg)

---

# 星球大战

角色的性别与身高

---

# 星球大战

线性关系可以很好的描述两个变量之间的关系吗？

---

# 逻辑斯蒂回归（logistic regression）

连接函数是logit函数的回归： `$logit(x) = log(\frac{x}{1-x})$`

---

# 逻辑斯蒂回归（logistic regression）

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      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="estimate"><span class='gt_from_md'><strong>log(OR)</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;"><sup>1</sup></span></th>
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    <tr><td headers="label" class="gt_row gt_left">(Intercept)</td>
<td headers="estimate" class="gt_row gt_center">-15**</td>
<td headers="std.error" class="gt_row gt_center">5.42</td></tr>
    <tr><td headers="label" class="gt_row gt_left">height</td>
<td headers="estimate" class="gt_row gt_center">0.09**</td>
<td headers="std.error" class="gt_row gt_center">0.031</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">AIC</td>
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<td headers="std.error" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;"><br /></td></tr>
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<td headers="estimate" class="gt_row gt_center">64</td>
<td headers="std.error" class="gt_row gt_center"><br /></td></tr>
  </tbody>
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      <td class="gt_sourcenote" colspan="3"><span class='gt_from_md'>Abbreviations: CI = Confidence Interval, OR = Odds Ratio, SE = Standard Error</span></td>
    </tr>
  </tfoot>
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      <td class="gt_footnote" colspan="3"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;"><sup>1</sup></span> <span class='gt_from_md'>*p<0.05; **p<0.01; ***p<0.001</span></td>
    </tr>
  </tfoot>
</table>
</div>

---

# 定类变量vs连续变量

泰坦尼克号的结果是把定类变量作为虚拟变量放入模型得到的。可以把它看成是logit函数的一种特殊情况。

假如自变量 `$x$`的取值是 `$0 = 船员, 1 = 一等舱$`。

当 `$x = 0$`时， `$log(\frac{p_0}{1-p_0}) = \alpha$`。此时对等式两边同时取指数，可得 `$or_0 = \frac{p_0}{1-p_0} = e^{\alpha}$`，这是船员幸存的比数。

当 `$x = 1$`时， `$log(\frac{p_1}{1-p_1}) = \alpha + \beta_1$`。此时对等式两边同时取指数，可得 `$or_1 = \frac{p_1}{1-p_1} = e^{\alpha + \beta_1}$`，这是一等舱乘客幸存的比数。

两个比数如果相除，会得到什么？

$$ \frac{or_1}{or_0} = \frac{e ^{\alpha + \beta_1}}{e^{\alpha }} = e^{\beta_1} $$

---

# 定类变量vs连续变量

所以说，通过logit函数作为连接函数得到模型系数，这个系数恰好是**比数比的对数**。在写作中，有时不报告模型系数而是**比数比**，亦可。

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<td headers="estimate" class="gt_row gt_center">0.00**</td>
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<td headers="estimate" class="gt_row gt_center">1.10**</td>
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</table>
</div>

---

# 多分类逻辑斯蒂回归

---

# 多分类逻辑斯蒂回归

multinomial logistic regression

逻辑斯蒂回归对应变量是二分变量，也就是因变量取值为0或1。其实我们已经看到，因变量取值在0-1之间的连续变量亦可用逻辑斯蒂回归。

如果变量是多分类的怎么办？

注意区分：

+ 多分类逻辑斯蒂回归的意思是因变量是取值超过2种情况的定类变量；
+ 多元逻辑斯蒂回归指的是可以放入多个自变量

---

# 多分类逻辑斯蒂回归

---

# 多分类逻辑斯蒂回归

---

# 有序逻辑斯蒂回归

因变量是定序变量时，采用有序逻辑斯蒂回归模型进行分析。
]
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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="label"><span class='gt_from_md'><strong>Characteristic</strong></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="estimate"><span class='gt_from_md'><strong>log(OR)</strong></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="conf.low"><span class='gt_from_md'><strong>95% CI</strong></span></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="label" class="gt_row gt_left">1|2</td>
<td headers="estimate" class="gt_row gt_center">-0.81</td>
<td headers="conf.low" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">2|3</td>
<td headers="estimate" class="gt_row gt_center">0.16</td>
<td headers="conf.low" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">3|4</td>
<td headers="estimate" class="gt_row gt_center">1.0</td>
<td headers="conf.low" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">4|5</td>
<td headers="estimate" class="gt_row gt_center">3.0</td>
<td headers="conf.low" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">lat</td>
<td headers="estimate" class="gt_row gt_center">-0.07</td>
<td headers="conf.low" class="gt_row gt_center">-0.08, -0.06</td></tr>
    <tr><td headers="label" class="gt_row gt_left">long</td>
<td headers="estimate" class="gt_row gt_center">-0.02</td>
<td headers="conf.low" class="gt_row gt_center">-0.02, -0.01</td></tr>
    <tr><td headers="label" class="gt_row gt_left" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">Log-likelihood</td>
<td headers="estimate" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">-5,794</td>
<td headers="conf.low" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">AIC</td>
<td headers="estimate" class="gt_row gt_center">11,599</td>
<td headers="conf.low" class="gt_row gt_center"><br /></td></tr>
    <tr><td headers="label" class="gt_row gt_left">No. Obs.</td>
<td headers="estimate" class="gt_row gt_center">4,803</td>
<td headers="conf.low" class="gt_row gt_center"><br /></td></tr>
  </tbody>
  <tfoot class="gt_sourcenotes">
    <tr>
      <td class="gt_sourcenote" colspan="3"><span class='gt_from_md'>Abbreviations: CI = Confidence Interval, OR = Odds Ratio</span></td>
    </tr>
  </tfoot>
  
</table>
</div>
]

---

# 代码

一般的逻辑斯蒂回归：使用自带的`stats`包的`glm`函数。最简单的形式是：
`glm(y~x, family = binomial, data = dt)`。其中，binomial表示因变量是二分类变量。

多分类的逻辑斯蒂回归：使用`nnet`包的`multinom`函数。最简单的形式是：
`multinom(y~x,  data = dt)`。此外还有`mlogit`包可用。`multinom`的结果不好用`modelsummary`、`flextable`、`gtsummary`之类的包处理，目前只能用`stargazer`来处理。

有序的逻辑斯蒂回归：使用`MASS`包的`polr`函数，最简单的形式是：
`polr(y~x, data = dt)`。模型自带结果不含 `$p$`值，不手动计算的话报告不带星星。

[本课代码](method-logit.Rmd)