9: Correlation and Regression

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.title[
# 9: Correlation and Regression
]
.subtitle[
## Empirical Methods
]
.author[
### <large>J. Seawright</large>
]
.institute[
### <small>Northwestern Political Science</small>
]
.date[
### Oct. 14, 2025
]

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We have two problems:

1. What if our independent variable is quantitative?

2. What if we have multiple control variables?

---
### Regression Analysis

-   Regression analysis is a tool that fits a line to a scatter plot.

---

Remember from your studies that the algebraic formula for a line is: `$Y = mX + b$`.

---
### Regression Analysis

When we pivot from geometry to regression, two important changes!

1.  Instead of `$m$` usually call the slope `$\beta$` or `$b$`, and instead of
    `$b$` we usually call the intercept `$\alpha$`, `$a$`, or sometimes
    `$\beta_0$`.

2.  We add a term at the end of the equation for error/noise above and
    below the line. This term is called `$e_i$` or `$\epsilon_i$`.

---

-   How do we get numbers for `$\alpha$` and `$\beta$` in regression analysis?

-   Data plus calculus!

---

``` r
anes2024 <- read.csv("https://raw.githubusercontent.com/jnseawright/ps210/refs/heads/main/Data/anes2024.csv")

anes2024$transbathroom <- ifelse(anes2024$V241372x < 0, NA, anes2024$V241372x)

table(anes2024$transbathroom)
```

```
## 
##    1    2    3    4    5    6    7 
##  779  459  131 1317  102  383 2059
```

---

``` r
anes2024$carbonregulations <- ifelse(anes2024$V242324x < 0, NA, anes2024$V242324x)

table(anes2024$carbonregulations)
```

```
## 
##    1    2    3    4    5    6    7 
## 1738  928  200 1182   89  263  306
```

---

``` r
library(ggplot2)
library(hrbrthemes)

transcarbonplot <- anes2024 %>% ggplot(aes(x=transbathroom,y=carbonregulations)) + 
  geom_point( color="#69b3a2") +
  theme_ipsum()
```

---

``` r
transcarbonplot
```

```
## Warning: Removed 845 rows containing missing values or values outside the scale range
## (`geom_point()`).
```

---

``` r
transcarbonjitterplot <- anes2024 %>% ggplot(aes(x=transbathroom,y=carbonregulations)) + 
  geom_point( color="#69b3a2") +
  geom_jitter() +
  theme_ipsum()
```

---

``` r
transcarbonjitterplot
```

```
## Warning: Removed 845 rows containing missing values or values outside the scale range
## (`geom_point()`).
## Removed 845 rows containing missing values or values outside the scale range
## (`geom_point()`).
```

---

`$CarbonRegulations_{i} = \alpha + \beta TransBathroom_{i} + \epsilon_{i}$`

---

``` r
transcarbonlm <- lm(carbonregulations ~ transbathroom, data=anes2024)

summary(transcarbonlm)
```

```
## 
## Call:
## lm(formula = carbonregulations ~ transbathroom, data = anes2024)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.6524 -1.5278 -0.4033  0.7225  5.5967 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    1.02846    0.05579   18.43   <2e-16 ***
## transbathroom  0.37484    0.01074   34.89   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.665 on 4674 degrees of freedom
##   (845 observations deleted due to missingness)
## Multiple R-squared:  0.2067,	Adjusted R-squared:  0.2065 
## F-statistic:  1218 on 1 and 4674 DF,  p-value: < 2.2e-16
```

---

``` r
transcarbonlmplot <- anes2024 %>% ggplot(aes(x=transbathroom,y=carbonregulations)) + 
  geom_point( color="#69b3a2") +
  geom_jitter() +
  geom_smooth(method=lm , color="red", se=TRUE) +  
  theme_ipsum()
```

---

``` r
transcarbonlmplot
```

---

What are some possible *confounding variables* for this relationship?

---

`$\tiny CarbonRegulations_{i} = \alpha + \beta_{1} TransBathroom_{i} + \beta_{2} Partisanship_{i} + \epsilon_{i}$`

---

``` r
anes2024$partyid <- ifelse(anes2024$V241227x < 0, NA, anes2024$V241227x)

table(anes2024$partyid)
```

```
## 
##    1    2    3    4    5    6    7 
## 1314  616  714  380  716  577 1166
```

---

``` r
transcarbonlm2 <- lm(carbonregulations ~ transbathroom + partyid, data=anes2024)

summary(transcarbonlm2)
```

```
## 
## Call:
## lm(formula = carbonregulations ~ transbathroom + partyid, data = anes2024)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.1742 -0.9916 -0.1910  0.8090  5.8090 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    0.69378    0.05454   12.72   <2e-16 ***
## transbathroom  0.20156    0.01245   16.19   <2e-16 ***
## partyid        0.29565    0.01234   23.97   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.571 on 4658 degrees of freedom
##   (860 observations deleted due to missingness)
## Multiple R-squared:  0.2932,	Adjusted R-squared:  0.2929 
## F-statistic: 966.3 on 2 and 4658 DF,  p-value: < 2.2e-16
```

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