05-01-relationship.knit

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# 统计学原理(Statistic)

### 胡华平

### 西北农林科技大学

### 经济管理学院数量经济教研室

### huhuaping01@hotmail.com

### 2023-05-08

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---
class: center, middle, duke-orange,hide_logo
name:chapter
exclude: FALSE

# 第五章 相关和回归分析

### [.white[5.1 变量间关系的度量]](#corl)

### [5.2 回归分析的基本思想](#concept)

### [5.3 OLS方法与参数估计](#ols)

### [5.4 假设检验](#hypothesis)

### [5.5 拟合优度与残差分析](#goodness)

### [5.6 回归预测分析](#forecast)

### [5.7 回归报告解读](#report)

---
layout: false
class: center, middle, duke-softblue,hide_logo
name: corl

# 5.1 变量间关系的度量

### [变量间的关系](#corl-vars)

### [相关关系的描述与测度](#corl-measure)

### [相关系数的显著性检验](#corl-test)

---
layout: true

<div class="my-footer"><span>huhuaping@  &emsp;&emsp; <a href="#chapter"> 第05章 相关和回归分析 </a>
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#corl"> 5.1 变量间关系的度量 </a> </span></div>

---

### （示例）变量间的关系：经济学专业解读

> “我们数据不少，做了很严格的回归，但异常值略多略多，符合理论的数值反而难找……”

---

### （示例）变量间的关系：金融学专业解读

> “我们的数据多如牛毛，无孔不入。即使做完回归，也会发现异常值和符合理论的数值多得不忍直视。”

---

### （示例）变量间的关系：土木工程专业解读

> “我们得要设计余量，所以理论设计得远高于实际承受……”

---

### （示例）变量间的关系：物理学专业解读

> “我们的理论和数据严丝合缝，bingo！”

---

### （示例）变量间的关系：环境科学专业解读

> “我们的理论和数据大致吻合，就是……应用范围有点蛋疼。”

---

### （示例）变量间的关系：历史学专业解读

> “数据虽然很多，可我们能用理论把他们统统连起来！”

---

### （示例）变量间的关系：政治学专业解读

> “世界大势一日三变，尽管我们数据不少，可……我们的理论跟数据趋势是反着来的……”

---

### （示例）变量间的关系：社会学专业解读

> “学海无涯苦作舟。那么多数据，那么多理论，慢慢学，恩……”

---

### （示例）变量间的关系：数学专业解读

> “数据很少，但能建立理论～”

---

### （示例）变量间的关系：新闻学专业解读

> （示例）“只有一个数据，也能建立理论……”

---

### （示例）变量间的关系：哲学专业解读

> “没有数据，依然建立理论……”

---

### （示例）变量间的关系：文学批评专业解读

> “如图所示，你懂的……”

---
name: corl-vars

## 变量间的关系：函数关系

两个变量若存在是一一对应的确定关系，则称之为二者具有**函数关系**。

<div class="notes">
<p>设有两个变量 <span class="math inline">$X$</span>和 <span
class="math inline">$Y$</span>，变量 <span
class="math inline">$Y$</span>随变量 <span
class="math inline">$X$</span>一起变化，并完全依赖于 <span
class="math inline">$X$</span>，当变量 <span
class="math inline">$X$</span>取某个数值时， <span
class="math inline">$Y$</span>依确定的关系取相应的值，则称 <span
class="math inline">$Y$</span>是 <span
class="math inline">$X$</span>的函数，记为 <span
class="math inline">$Y = f(X)$</span>，其中 <span
class="math inline">$X$</span>称为自变量， <span
class="math inline">$Y$</span>称为因变量。</p>
</div>

> 从**几何学**角度来看，数据集各观测点会落在一条曲线上。

---

### （示例）函数关系

某种商品的销售额
`$Y$`与销售量
`$X$`之间的关系可表示为(
`$P$`为单价)：

`$$Y_i = P_i\cdot X_i$$`

圆的面积
`$S$`与半径
`$R$`之间的关系可表示为：

`$$S = \pi R^2$$`

企业的原材料消耗额
`$Y$`与产量
`$X1$` 、单位产量消耗
`$X2$` 、原材料价格
`$X3$`之间的关系可表示为：

`$$Y = X_1 \cdot X_2 \cdot X_3$$`

---

## 变量间的关系：相关关系(correlation)

<div class="figure" style="text-align: center">
<img src="../pic/chpt05-measure-rel.png" alt="相关关系的类型" width="90%" />
<p class="caption">相关关系的类型</p>
</div>

---

### （示例）相关关系

<div class="case">
<ul>
<li><p>父亲身高 <span class="math inline">$Y$</span>与子女身高 <span
class="math inline">$X$</span>之间的关系</p></li>
<li><p>收入水平 <span class="math inline">$Y$</span>与受教育程度 <span
class="math inline">$X$</span>之间的关系</p></li>
<li><p>粮食单位面积产量 <span class="math inline">$Y$</span>与施肥量
<span class="math inline">$X1$</span> 、降雨量 <span
class="math inline">$X2$</span>、温度 <span
class="math inline">$X3$</span>之间的关系</p></li>
<li><p>商品的消费量 <span class="math inline">$Y$</span>与居民收入
<span class="math inline">$X$</span>之间的关系</p></li>
<li><p>商品销售额 <span class="math inline">$Y$</span>与广告费支出
<span class="math inline">$X$</span>之间的关系</p></li>
</ul>
</div>

---

## 相关关系的描述与测度：问题与假定

相关系数的显著性检验步骤：

1）提出假设：
`$H_0: \rho =0; H_1: \rho \neq 0$`

2）计算样本统计量

`$$T^{\ast} = |r|\sqrt{\frac{n-2}{1-r^2}} \quad \sim t(n-2)$$`

3）给定显著性水平
`$\alpha$`，确定t理论分布值
`$t_{1-\alpha/2}(n-2)$`。

4）得到假设检验结论：

- 若
`$T^{\ast}> t_{1-\alpha/2}(n-2)$`，则拒绝
`$H_0$`，认为显著存在相关关系；

- 若
`$T^{\ast} < t_{1-\alpha/2}(n-2)$`，则无法拒绝
`$H_0$`，认为相关关系不显著。

---

### 附录：假设检验的分布及统计量证明1/3

$$
`\begin{align}
\sum_y y h(y \mid x)& =\sum_y y \frac{f(x, y)}{f_X(x)}= \beta_1 + \beta_2 x && \text{(1)}
\end{align}`
$$

$$
`\begin{align}
\sum_y y f(x, y)&=(\beta_1+ \beta_2 x) f_X(x) && \text{(2)}\\
\sum_{x } \sum_y y f(x, y) &=\sum_{x }(\beta_1+ \beta_2 x) f_X(x) && \text{(3)} \\
\sum_{x } \sum_y x y f(x, y) &=\sum_{x }\left(\beta_1 x+ \beta_2 x^2\right) f_X(x) && \text{(4)}  \\
E(X Y)&=\beta_1 E(X)+ \beta_2 E\left(X^2\right) && \text{(5)}\\
\end{align}`
$$

$$
`\begin{align}
\mu_Y&=\beta_1 + \beta_2 \mu_X && \text{(6 <--2)} \\
\mu_X \mu_Y+\rho \sigma_X \sigma_Y &=\beta_1 \mu_X+\beta_2\left(\mu_X^2+\sigma_X^2\right) && \text{(7 <--5)}
\end{align}`
$$

.footnote[

当然，故事其实比上面还要更复杂。大家可以深入思考和讨论。参看：[1] Hogg R V, Tanis E A, Zimmerman D L.  Probability and statistical inference[M]. 第10版.   NJ,Hoboken:Pearson, 2020. pg 148 (section 4.3)

]

---

### 附录：假设检验的分布及统计量证明2/3

利用上述二元一次方程组，可以解出参数：

$$
`\begin{align}
\beta_1 &=\mu_Y-\rho \frac{\sigma_Y}{\sigma_X} \mu_X && \text{(8)} \\
\beta_2 &=\rho \frac{\sigma_Y}{\sigma_X}  && \text{(9)}
\end{align}`
$$

$$
`\begin{align}
E(Y \mid X_i)= \beta_1 +\beta_2X_i = \beta_1 + \rho \frac{\sigma_Y}{\sigma_X}  X_i && \text{(10)}
\end{align}`
$$

相关系数
`$\rho$`的显著性检验等价于一元线性回归分析中斜率参数
`$\beta_2$`的t检验过程，也即：
`$H_0: \rho = 0; \quad H_1: \rho \neq 0$`；等价于
`$H_0: \beta_2 = 0; \quad H_1: \beta_2 \neq 0$`

---

### 附录：假设检验的分布及统计量证明3/3

一元线性回归
`$Y_i = \beta_1 +\beta_2X_i + u_i$`；斜率系数t检验
`$H_0: \beta_2 = 0; H_1: \beta_2 \neq 0$`

$$
`\begin{align}
t=\frac{\hat{\beta_2} -\beta_2}{\hat{\sigma}_{\hat{\beta}_2}}
=\frac{\hat{\beta_2}}{\sqrt{\frac{\hat{\sigma}^2}{\sum\left(X_i-\bar{X}\right)^2}}}
=\frac{\hat{\beta_2}-0}{\sqrt{\frac{\mathrm{MSE}}{\sum\left(X_i-\bar{X}\right)^2}}}
=\frac{r \cdot \left(S_Y / S_X\right)}{\sqrt{\frac{(n-1) S_Y^2 \left(1-r^2\right)}{(n-2) } \cdot \frac{1}{(n-1)S_X^2}}}
=\frac{r \sqrt{n-2}}{\sqrt{1-r^2}}
\end{align}`
$$

$$
`\begin{align}
r &= \frac{S_{XY}}{S_X S_Y} \\
\hat{\beta}_2 &=\frac{\frac{1}{n-1} \sum_{i=1}^n\left(X_i-\bar{X}\right)\left(Y_i-\bar{Y}\right)}{\frac{1}{n-1} \sum_{i=1}^n\left(X_i-\bar{X}\right)^2}=\frac{S_{XY}}{S_X^2}=r \cdot \frac{S_Y}{S_X}\\
\hat{\beta}_1 & = \overline{Y} - \hat{\beta}_2 \overline{X} \\
M S E \equiv \sigma^2 &= \frac{\sum_{i=1}^n\left(Y_i-\hat{Y}_i\right)^2}{n-2}=\frac{\sum_{i=1}^n\left[Y_i-\left(\bar{Y}+\frac{S_{XY}}{S_X^2}\left(X_i-\bar{X}\right)\right)\right]^2}{n-2}=\frac{(n-1) S_Y^2\left(1-r^2\right)}{n-2}
\end{align}`
$$

???

参看队长问答[Why Test Statistic for the Pearson Correlation Coefficient](https://stats.stackexchange.com/questions/270612/why-test-statistic-for-the-pearson-correlation-coefficient-is-frac-r-sqrtn-2/321382#321382)

终极回答见[Three Tests for Rho](https://web.archive.org/web/20130625153233/https://onlinecourses.science.psu.edu/stat414/node/254)

---

exclude: true

## （案例）银行贷款

---

### （案例）银行贷款：案例数据

**案例说明**：某银行共有25家分行，分行及所在地区的相关变量数据如下表所示。

<div id="htmlwidget-aa37b2d0577873f29911" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-aa37b2d0577873f29911">{"x":{"filter":"none","vertical":false,"data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25"],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[0.9,1.1,4.8,3.2,7.8,2.7,1.6,12.5,1,2.6,0.3,4,0.8,3.5,10.2,3,0.2,0.4,1,6.8,11.6,1.6,1.2,7.2,3.2],[67.3,111.3,173,80.8,199.7,16.2,107.4,185.4,96.1,72.8,64.2,132.2,58.6,174.6,263.5,79.3,14.8,73.5,24.7,139.4,368.2,95.7,109.6,196.2,102.2],[6.8,19.8,7.7,7.2,16.5,2.2,10.7,27.1,1.7,9.1,2.1,11.2,6,12.7,15.6,8.9,0.6,5.9,5,7.2,16.8,3.8,10.3,15.8,12],[5,16,17,10,19,1,17,18,10,14,11,23,14,26,34,15,2,11,4,28,32,10,14,16,10],[51.9,90.9,73.7,14.5,63.2,2.2,20.2,43.8,55.9,64.3,42.7,76.7,22.8,117.1,146.7,29.9,42.1,25.3,13.4,64.3,163.9,44.5,67.9,39.7,97.1]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>ID.bank<\/th>\n      <th>loan.bad<\/th>\n      <th>loan.surplus<\/th>\n      <th>loan.receivable<\/th>\n      <th>loan.numbers<\/th>\n      <th>investment.fixed<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0}],"pageLength":7,"scrollX":true,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[7,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

.footnote[**说明**：上述变量的含义分别是ID.bank（分行编号）、loan.bad（不良贷款）、loan.surplus（各项贷款余额
）、loan.receivable（本年累计应收贷款）、loan.numbers（贷款项目个数）、investment.fixed（本年固定资产投资额）。]

---

### （案例）银行贷款：不良贷款VS贷款余额的散点图

<div class="figure" style="text-align: center">
<img src="05-01-relationship_files/figure-html/unnamed-chunk-30-1.png" alt="不良贷款VS贷款余额散点图"  />
<p class="caption">不良贷款VS贷款余额散点图</p>
</div>

---

### （案例）银行贷款：不良贷款VS贷款余额的相关系数（大FF）

<div id="htmlwidget-ea46ab7a2497f65566a7" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-ea46ab7a2497f65566a7">{"x":{"filter":"none","vertical":false,"caption":"<caption>大FF计算表<\/caption>","data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26"],["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","Total"],[0.9,1.1,4.8,3.2,7.8,2.7,1.6,12.5,1,2.6,0.3,4,0.8,3.5,10.2,3,0.2,0.4,1,6.8,11.6,1.6,1.2,7.2,3.2,93.2],[67.3,111.3,173,80.8,199.7,16.2,107.4,185.4,96.1,72.8,64.2,132.2,58.6,174.6,263.5,79.3,14.8,73.5,24.7,139.4,368.2,95.7,109.6,196.2,102.2,3006.7],[60.57,122.43,830.4,258.56,1557.66,43.74,171.84,2317.5,96.1,189.28,19.26,528.8,46.88,611.1,2687.7,237.9,2.96,29.4,24.7,947.92,4271.12,153.12,131.52,1412.64,327.04,17080.14],[4529.29,12387.69,29929,6528.64,39880.09,262.44,11534.76,34373.16,9235.21,5299.84,4121.64,17476.84,3433.96,30485.16,69432.25,6288.49,219.04,5402.25,610.09,19432.36,135571.24,9158.49,12012.16,38494.44,10444.84,516543.37],[0.81,1.21,23.04,10.24,60.84,7.29,2.56,156.25,1,6.76,0.09,16,0.64,12.25,104.04,9,0.04,0.16,1,46.24,134.56,2.56,1.44,51.84,10.24,660.1]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>ID.bank<\/th>\n      <th>Y<\/th>\n      <th>X<\/th>\n      <th>XY<\/th>\n      <th>X_sqr<\/th>\n      <th>Y_sqr<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"targets":4,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 2, 3, \",\", \".\");\n  }"},{"targets":5,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 2, 3, \",\", \".\");\n  }"},{"targets":6,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 2, 3, \",\", \".\");\n  }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0}],"pageLength":7,"scrollX":true,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[7,10,25,50,100]}},"evals":["options.columnDefs.0.render","options.columnDefs.1.render","options.columnDefs.2.render"],"jsHooks":[]}</script>

---

### （案例）银行贷款：不良贷款VS贷款余额的相关系数（大FF）

相关系数
`$r$`的大FF计算公式（`eq01`）：

`$$\begin{align}
r & = \frac{n \sum X_i Y_i -\sum X_i \sum Y_i}{\sqrt{n \sum X_i^{2}-\left(\sum X_i\right)^{2}} \cdot \sqrt{n \sum Y_i^{2}-\left(\sum Y_i\right)^{2}}} \\
& = \frac{25 \times 17080.14 - 3006.7 \times 93.2}{\sqrt{25 \times 516543.37-\left(3006.7\right)^2} \cdot \sqrt{25 \times 660.1-\left( 93.2\right)^{2}}}  \\
& = 0.8436
\end{align}$$`

---

### （案例）银行贷款：不良贷款VS贷款余额的相关系数（小ff）

<div id="htmlwidget-d24dc8648028f66c5193" style="width:100%;height:auto;" class="datatables html-widget"></div>
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---

### （案例）银行贷款：不良贷款VS贷款余额的相关系数

相关系数
`$r$`的小FF计算公式（`eq02`）：

`$$\begin{align}
r & = \frac{ \sum{\left( (X_i - \overline{X})(Y_i - \overline{Y})\right ) } }{\sqrt{\sum{(X_i - \overline{X})^2 (Y_i - \overline{Y})^2}}} \\
& = \frac{\sum{x_i y_i}}{\sqrt{\sum{x_i^2}\sum{y_i^2}}} \\
& = \frac{5871.16}{\sqrt{ 154933.57 \times 312.65}} \\
& = 0.8436
\end{align}$$`

---

### （案例）银行贷款：相关系数矩阵表(Pearson)

```r
corl_pearson<- round(cor(df_loan[,-1], method = "pearson"),4) 
corl_pearson[upper.tri(corl_pearson)]<- NA
```

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Pearson相关系数矩阵

</caption><thead><tr style="overflow-wrap:break-word;"><td class="cl-e36083b5"><p class="cl-e35f992e"><span class="cl-e35f4b18"> </span></p></td><td class="cl-e360aaaa"><p class="cl-e35f992e"><span class="cl-e35f4b18">loan.bad</span></p></td><td class="cl-e360aaa8"><p class="cl-e35f992e"><span class="cl-e35f4b18">loan.surplus</span></p></td><td class="cl-e360aaa9"><p class="cl-e35f992e"><span class="cl-e35f4b18">loan.receivable</span></p></td><td class="cl-e36083b6"><p class="cl-e35f992e"><span class="cl-e35f4b18">loan.numbers</span></p></td><td class="cl-e36083b5"><p class="cl-e35f992e"><span class="cl-e35f4b18">investment.fixed</span></p></td></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-e3605c9c"><p class="cl-e35f992e"><span class="cl-e35f4b18">loan.bad</span></p></td><td class="cl-e3605c9f"><p class="cl-e35f992e"><span class="cl-e35f4b18">1.0000</span></p></td><td class="cl-e3605c9e"><p class="cl-e35f992e"><span class="cl-e35f4b1b"></span></p></td><td class="cl-e3605ca0"><p class="cl-e35f992e"><span class="cl-e35f4b1b"></span></p></td><td class="cl-e3605c9d"><p class="cl-e35f992e"><span class="cl-e35f4b1b"></span></p></td><td class="cl-e3605c9c"><p class="cl-e35f992e"><span class="cl-e35f4b1b"></span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-e3605ca3"><p class="cl-e35f992e"><span class="cl-e35f4b18">loan.surplus</span></p></td><td class="cl-e3605ca2"><p class="cl-e35f992e"><span class="cl-e35f4b19">0.8436</span></p></td><td class="cl-e3605ca4"><p class="cl-e35f992e"><span class="cl-e35f4b18">1.0000</span></p></td><td class="cl-e3605ca5"><p class="cl-e35f992e"><span class="cl-e35f4b1b"></span></p></td><td class="cl-e3605ca1"><p class="cl-e35f992e"><span class="cl-e35f4b1b"></span></p></td><td class="cl-e3605ca3"><p class="cl-e35f992e"><span class="cl-e35f4b1b"></span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-e3605ca6"><p class="cl-e35f992e"><span class="cl-e35f4b18">loan.receivable</span></p></td><td class="cl-e36083ae"><p class="cl-e35f992e"><span class="cl-e35f4b1a">0.7315</span></p></td><td class="cl-e36083ad"><p class="cl-e35f992e"><span class="cl-e35f4b18">0.6788</span></p></td><td class="cl-e36083ac"><p class="cl-e35f992e"><span class="cl-e35f4b18">1.0000</span></p></td><td class="cl-e36083af"><p class="cl-e35f992e"><span class="cl-e35f4b1b"></span></p></td><td class="cl-e3605ca6"><p class="cl-e35f992e"><span class="cl-e35f4b1b"></span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-e3605ca6"><p class="cl-e35f992e"><span class="cl-e35f4b18">loan.numbers</span></p></td><td class="cl-e36083ae"><p class="cl-e35f992e"><span class="cl-e35f4b1a">0.7003</span></p></td><td class="cl-e36083ad"><p class="cl-e35f992e"><span class="cl-e35f4b19">0.8484</span></p></td><td class="cl-e36083ac"><p class="cl-e35f992e"><span class="cl-e35f4b18">0.5858</span></p></td><td class="cl-e36083af"><p class="cl-e35f992e"><span class="cl-e35f4b18">1.0000</span></p></td><td class="cl-e3605ca6"><p class="cl-e35f992e"><span class="cl-e35f4b1b"></span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-e36083b1"><p class="cl-e35f992e"><span class="cl-e35f4b18">investment.fixed</span></p></td><td class="cl-e36083b3"><p class="cl-e35f992e"><span class="cl-e35f4b18">0.5185</span></p></td><td class="cl-e36083b0"><p class="cl-e35f992e"><span class="cl-e35f4b1a">0.7797</span></p></td><td class="cl-e36083b4"><p class="cl-e35f992e"><span class="cl-e35f4b18">0.4724</span></p></td><td class="cl-e36083b2"><p class="cl-e35f992e"><span class="cl-e35f4b1a">0.7466</span></p></td><td class="cl-e36083b1"><p class="cl-e35f992e"><span class="cl-e35f4b18">1.0000</span></p></td></tr></tbody></table></div></template>
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---

### （案例）银行贷款：相关系数矩阵(Spearman)

```r
corl_spearman<- round(cor(df_loan[,-1], method = "spearman"),4) 
corl_spearman[upper.tri(corl_spearman)] <- NA
```

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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-e3e67c78{width:135.7pt;background-color:transparent;vertical-align: middle;border-bottom: 2pt solid rgba(102, 102, 102, 1.00);border-top: 2pt solid rgba(102, 102, 102, 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-e3e6a37e{width:123pt;background-color:transparent;vertical-align: middle;border-bottom: 2pt solid rgba(102, 102, 102, 1.00);border-top: 2pt solid rgba(102, 102, 102, 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-e3e6a37f{width:148.3pt;background-color:transparent;vertical-align: middle;border-bottom: 2pt solid rgba(102, 102, 102, 1.00);border-top: 2pt solid rgba(102, 102, 102, 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-e3e6a380{width:93.4pt;background-color:transparent;vertical-align: middle;border-bottom: 2pt solid rgba(102, 102, 102, 1.00);border-top: 2pt solid rgba(102, 102, 102, 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 class='cl-e4035348'><caption class="Table Caption">

Spearman相关系数矩阵

</caption><thead><tr style="overflow-wrap:break-word;"><td class="cl-e3e67c77"><p class="cl-e3e5e060"><span class="cl-e3e5b932"> </span></p></td><td class="cl-e3e6a380"><p class="cl-e3e5e060"><span class="cl-e3e5b932">loan.bad</span></p></td><td class="cl-e3e6a37e"><p class="cl-e3e5e060"><span class="cl-e3e5b932">loan.surplus</span></p></td><td class="cl-e3e6a37f"><p class="cl-e3e5e060"><span class="cl-e3e5b932">loan.receivable</span></p></td><td class="cl-e3e67c78"><p class="cl-e3e5e060"><span class="cl-e3e5b932">loan.numbers</span></p></td><td class="cl-e3e67c77"><p class="cl-e3e5e060"><span class="cl-e3e5b932">investment.fixed</span></p></td></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-e3e65568"><p class="cl-e3e5e060"><span class="cl-e3e5b932">loan.bad</span></p></td><td class="cl-e3e6556b"><p class="cl-e3e5e060"><span class="cl-e3e5b932">1.0000</span></p></td><td class="cl-e3e6556a"><p class="cl-e3e5e060"><span class="cl-e3e5b935"></span></p></td><td class="cl-e3e6556c"><p class="cl-e3e5e060"><span class="cl-e3e5b935"></span></p></td><td class="cl-e3e65569"><p class="cl-e3e5e060"><span class="cl-e3e5b935"></span></p></td><td class="cl-e3e65568"><p class="cl-e3e5e060"><span class="cl-e3e5b935"></span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-e3e6556f"><p class="cl-e3e5e060"><span class="cl-e3e5b932">loan.surplus</span></p></td><td class="cl-e3e6556e"><p class="cl-e3e5e060"><span class="cl-e3e5b933">0.8339</span></p></td><td class="cl-e3e65570"><p class="cl-e3e5e060"><span class="cl-e3e5b932">1.0000</span></p></td><td class="cl-e3e65571"><p class="cl-e3e5e060"><span class="cl-e3e5b935"></span></p></td><td class="cl-e3e6556d"><p class="cl-e3e5e060"><span class="cl-e3e5b935"></span></p></td><td class="cl-e3e6556f"><p class="cl-e3e5e060"><span class="cl-e3e5b935"></span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-e3e65572"><p class="cl-e3e5e060"><span class="cl-e3e5b932">loan.receivable</span></p></td><td class="cl-e3e67c70"><p class="cl-e3e5e060"><span class="cl-e3e5b934">0.7331</span></p></td><td class="cl-e3e67c6f"><p class="cl-e3e5e060"><span class="cl-e3e5b932">0.8148</span></p></td><td class="cl-e3e67c6e"><p class="cl-e3e5e060"><span class="cl-e3e5b932">1.0000</span></p></td><td class="cl-e3e67c71"><p class="cl-e3e5e060"><span class="cl-e3e5b935"></span></p></td><td class="cl-e3e65572"><p class="cl-e3e5e060"><span class="cl-e3e5b935"></span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-e3e65572"><p class="cl-e3e5e060"><span class="cl-e3e5b932">loan.numbers</span></p></td><td class="cl-e3e67c70"><p class="cl-e3e5e060"><span class="cl-e3e5b934">0.7172</span></p></td><td class="cl-e3e67c6f"><p class="cl-e3e5e060"><span class="cl-e3e5b933">0.8559</span></p></td><td class="cl-e3e67c6e"><p class="cl-e3e5e060"><span class="cl-e3e5b932">0.7393</span></p></td><td class="cl-e3e67c71"><p class="cl-e3e5e060"><span class="cl-e3e5b932">1.0000</span></p></td><td class="cl-e3e65572"><p class="cl-e3e5e060"><span class="cl-e3e5b935"></span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-e3e67c73"><p class="cl-e3e5e060"><span class="cl-e3e5b932">investment.fixed</span></p></td><td class="cl-e3e67c75"><p class="cl-e3e5e060"><span class="cl-e3e5b932">0.4407</span></p></td><td class="cl-e3e67c72"><p class="cl-e3e5e060"><span class="cl-e3e5b934">0.6582</span></p></td><td class="cl-e3e67c76"><p class="cl-e3e5e060"><span class="cl-e3e5b932">0.5469</span></p></td><td class="cl-e3e67c74"><p class="cl-e3e5e060"><span class="cl-e3e5b934">0.5975</span></p></td><td class="cl-e3e67c73"><p class="cl-e3e5e060"><span class="cl-e3e5b932">1.0000</span></p></td></tr></tbody></table></div></template>
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---

### （案例）银行贷款：相关系数矩阵图

```r
#remotes::install_github("r-link/corrmorant")
require("corrmorant")
corrmorant::corrmorant(df_loan[,-1], style = "binned")+
  theme_dark() +
  theme(text = element_text(size = 14))
```

???

参考：[Correlation Analysis Different Types of Plots in R](https://finnstats.com/index.php/2021/05/13/correlation-analysis-plot/)

---

### （案例）银行贷款：相关系数矩阵图

```r
ggcorrm(data = df_loan[,-1]) +
  lotri(geom_point(alpha = 0.5)) +
  lotri(geom_smooth()) +
  utri_heatmap() +
  utri_corrtext() +
  dia_names(y_pos = 0.15, size = 3) +
  dia_histogram(lower = 0.3, fill = "grey80", color = 1) +
  scale_fill_corr() +
  theme(text = element_text(size = 14))
```

---

### （案例）银行贷款：偏相关系数

假定我们认为不良贷款（`loan.bad`）与贷款余额（`loan.surplus`）及贷款项目数（`loan.number`）存在相互关系。

前面我们已经计算出如下的简单相关系数：
`$$r_{12} = r_{_{bad},_{sur}}= 0.8436; \quad r_{13} = r_{_{bad},_{num}}= 0.7003; \quad r_{23} = r_{_{num},_{sur}}= 0.8484$$`

因此我们可以分别计算出**偏相关系数**

---

### （案例）银行贷款：偏相关系数

- 保持
`$X_{3i}$`不变，
`$Y_i$`和
`$X_{2i}$`之间的相关系数：

`$$\begin {align} 
r_{12 \cdot 3}=\frac{r_{12}-r_{13} r_{23}}{\sqrt{\left(1-r_{13}^{2}\right)\left(1-r_{23}^{2}\right)}}
=\frac{0.84-0.7\times 0.85}{\sqrt{\left(1-0.7^{2}\right)\left(1-0.85^{2}\right)}}
= 0.6601
\end {align}$$`

- 保持
`$X_{2i}$`不变，
`$Y_i$`和
`$X_{3i}$`之间的相关系数：

`$$\begin {align} 
r_{13.2}=\frac{r_{13}-r_{12} r_{23}}{\sqrt{\left(1-r_{12}^{2}\right)\left(1-r_{23}^{2}\right)}} 
=\frac{0.7-0.84 \times 0.85}{\sqrt{\left(1-0.84^{2}\right)\left(1-0.85^{2}\right)}} 
= -0.0542
\end {align}$$`

- 保持
`$Y_i$`不变，
`$X_{2i}$`和
`$X_{3i}$`之间的相关系数：

`$$\begin {align} 
r_{23.1}=\frac{r_{23}-r_{12} r_{13}}{\sqrt{\left(1-r_{12}^{2}\right)\left(1-r_{13}^{2}\right)}}
=\frac{0.85-0.84 \times  0.7}{\sqrt{\left(1-0.84^{2}\right)\left(1-0.7^{2}\right)}} 
= 0.6722
\end {align}$$`

---

### （案例）银行贷款：相关系数显著性检验(手算)

对于前述`loan.surplus`与`loan.bad`进行相关系数显著性检验（Pearson）：

- 1）提出假设：
`$H_0: \rho =0; H_1: \rho \neq 0$`

- 2）计算样本统计量：

`$$\begin{align}
T^{\ast} = |r|\sqrt{\frac{n-2}{1-r^2}} 
=0.84 \times \sqrt{\frac{25-2}{1-0.84^2}}
= 7.53
\end{align}$$`

- 3）给定显著性水平
`$\alpha=0.05$`，确定t理论分布值
`$t_{1-\alpha/2}(n-2)=t_{1-0.05/2}(25-2)=t_{0.975}(23)=2.07$`。

- 4）得到假设检验结论：因为t样本统计量大于t理论查表值，也即

`$$\left[T^{\ast}= 7.53\right] > \left[t_{0.975}(23) =2.07\right]$$`

因此拒绝原假设
`$H_0$`，认为变量`loan.surplus`（贷款余额）与`loan.bad`（不良贷款）显著存在相关关系。

---

### （案例）银行贷款：相关系数显著性检验(R软件)

我们可以使用R软件函数`cor.test()`对上述两个变量进行相关系数显著性检验：

```r
cor.test(df_rel1$loan.surplus, df_rel1$loan.bad,
         method = "pearson")
```

```

Pearson's product-moment correlation

data:  df_rel1$loan.surplus and df_rel1$loan.bad
t = 8, df = 23, p-value = 0.0000001
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.67 0.93
sample estimates:
 cor 
0.84 
```

---
layout:false
background-image: url("../pic/thank-you-gif-funny-little-yellow.gif")
class: inverse,center
# 本节结束