2: Starting with Regression

class: center, middle, inverse, title-slide

.title[
# 2: Starting with Regression
]
.subtitle[
## Linear Models
]
.author[
### <large>Jaye Seawright</large>
]
.institute[
### <small>Northwestern Political Science</small>
]
.date[
### Jan. 7, 2026
]

---

class: center, middle

pre[class] {
  max-height: 200px;
}
</style>

Imagine someone interested in predicting U.S. presidential election results based on economic performance.

---

``` r
library(tidyverse)
```

```
## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ forcats   1.0.1     ✔ stringr   1.6.0
## ✔ lubridate 1.9.4     ✔ tibble    3.3.0
## ✔ purrr     1.2.0     ✔ tidyr     1.3.1
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ nlme::collapse() masks dplyr::collapse()
## ✖ mice::filter()   masks dplyr::filter(), stats::filter()
## ✖ dplyr::lag()     masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
```

``` r
library(rosdata)
data("hibbs")
```

---

``` r
econvoteplot <- ggplot(hibbs, aes(x = growth, y = vote, label = year)) +
     geom_text(size = 3) +
     scale_x_continuous(labels = function(x) paste0(x, "%")) +
     scale_y_continuous(labels = function(x) paste0(x, "%")) +
     labs(title = "Forecasting the Election from the Economy",
          x = "Average recent growth in personal income",
          y = "Incumbent party's vote share") +
     theme_minimal()
```

---

---

``` r
econvoteplot.line <- ggplot(hibbs, aes(x = growth, y = vote, label = year)) +
     geom_text(size = 3) +
     scale_x_continuous(labels = function(x) paste0(x, "%")) +
     scale_y_continuous(labels = function(x) paste0(x, "%")) +
     labs(title = "Forecasting the Election from the Economy",
          x = "Average recent growth in personal income",
          y = "Incumbent party's vote share") +
     geom_smooth(method = "lm") +
     theme_minimal()
```

---

```
## `geom_smooth()` using formula = 'y ~ x'
```

---

``` r
econvote.lm <- lm(vote ~ growth, data = hibbs)
```
---

``` r
library(rstanarm)
```

```
## Loading required package: Rcpp
```

```
## This is rstanarm version 2.32.2
```

```
## - See https://mc-stan.org/rstanarm/articles/priors for changes to default priors!
```

```
## - Default priors may change, so it's safest to specify priors, even if equivalent to the defaults.
```

```
## - For execution on a local, multicore CPU with excess RAM we recommend calling
```

```
##   options(mc.cores = parallel::detectCores())
```

```
## 
## Attaching package: 'rstanarm'
```

```
## The following objects are masked from 'package:rosdata':
## 
##     kidiq, roaches, wells
```

``` r
econvote.lm.bayes <- stan_glm(vote ~ growth, data = hibbs)
```

```
## 
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```

---

``` r
library(modelsummary)
library(kableExtra)
```

```
## 
## Attaching package: 'kableExtra'
```

```
## The following object is masked from 'package:dplyr':
## 
##     group_rows
```

``` r
econvote.summary <- modelsummary( list("Economic Voting Model" = econvote.lm),
             output = "kableExtra",
             stars = TRUE,
             statistic = "std.error",  
             coef_rename = TRUE,
             escape = FALSE,
             gof_map = "all") %>%      
  kable_styling(
    font_size = 24,                    # Even larger font
    bootstrap_options = c("striped", "hover", "condensed"),
    full_width = FALSE
  )
```

---

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto; font-size: 24px; width: auto !important; margin-left: auto; margin-right: auto;" class="table table table-striped table-hover table-condensed">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;">  Economic Voting Model </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 46.248*** </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (1.622) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> growth </td>
   <td style="text-align:center;"> 3.061*** </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1.5px">  </td>
   <td style="text-align:center;box-shadow: 0px 1.5px"> (0.696) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 16 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 </td>
   <td style="text-align:center;"> 0.580 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 Adj. </td>
   <td style="text-align:center;"> 0.550 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> AIC </td>
   <td style="text-align:center;"> 91.7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> BIC </td>
   <td style="text-align:center;"> 94.0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Log.Lik. </td>
   <td style="text-align:center;"> −42.839 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> F </td>
   <td style="text-align:center;"> 19.321 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> RMSE </td>
   <td style="text-align:center;"> 3.52 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

---

``` r
econvote.bayes.summary <- modelsummary( list("Bayesian Voting Model" = econvote.lm.bayes),
             output = "kableExtra",
             stars = TRUE,
             statistic = "conf.int",  
             coef_rename = TRUE,
             escape = FALSE,
             gof_map = "all") %>%      
  kable_styling(
    font_size = 24,                    # Even larger font
    bootstrap_options = c("striped", "hover", "condensed"),
    full_width = FALSE
  )
```

---

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto; font-size: 24px; width: auto !important; margin-left: auto; margin-right: auto;" class="table table table-striped table-hover table-condensed">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;">  Bayesian Voting Model </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 46.240 </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> [42.698, 49.570] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> growth </td>
   <td style="text-align:center;"> 3.069 </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1.5px">  </td>
   <td style="text-align:center;box-shadow: 0px 1.5px"> [1.617, 4.539] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 16 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 </td>
   <td style="text-align:center;"> 0.551 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 Adj. </td>
   <td style="text-align:center;"> 0.522 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Log.Lik. </td>
   <td style="text-align:center;"> −43.766 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ELPD </td>
   <td style="text-align:center;"> −46.4 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ELPD s.e. </td>
   <td style="text-align:center;"> 3.9 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> LOOIC </td>
   <td style="text-align:center;"> 92.8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> LOOIC s.e. </td>
   <td style="text-align:center;"> 7.7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> WAIC </td>
   <td style="text-align:center;"> 92.2 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> RMSE </td>
   <td style="text-align:center;"> 3.52 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sigma </td>
   <td style="text-align:center;"> 3.875 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

---

A large literature in political science worries that Republicans and Democrats are treated differently in terms of economic voting.

---

``` r
hibbs$inc_party_candidate
```

```
##  [1] "Stevenson"  "Eisenhower" "Nixon"      "Johnson"    "Humphrey"  
##  [6] "Nixon"      "Ford"       "Carter"     "Reagan"     "Bush, Sr." 
## [11] "Bush, Sr."  "Clinton"    "Gore"       "Bush, Jr."  "McCain"    
## [16] "Obama"
```

---

``` r
hibbs$inc_party <- 1*(hibbs$inc_party_candidate %in% c("Johnson", "Humphrey", "Carter", "Clinton", "Gore", "Obama"))
```

---

``` r
econvoteplotparty <- ggplot(hibbs, aes(x = growth, y = vote, label = year)) +
     geom_text(aes(color = as.factor(inc_party)), size = 3) +
     scale_x_continuous(labels = function(x) paste0(x, "%")) +
     scale_y_continuous(labels = function(x) paste0(x, "%")) +
     labs(title = "Forecasting the Election from the Economy",
          x = "Average recent growth in personal income",
          y = "Incumbent party's vote share") +
     scale_color_manual(
       values = c("1" = "#0015BC",  # Classic Democratic blue
                "0" = "#DE0100") # Classic Republican red
       ) +
     theme_minimal()
```

---

---

``` r
econvoteplotparty.line <- ggplot(hibbs, aes(x = growth, y = vote, color = as.factor(inc_party))) +
     geom_point() +
     scale_x_continuous(labels = function(x) paste0(x, "%")) +
     scale_y_continuous(labels = function(x) paste0(x, "%")) +
     labs(title = "Forecasting the Election from the Economy",
          x = "Average recent growth in personal income",
          y = "Incumbent party's vote share") +
     scale_color_manual(
       values = c("1" = "#0015BC",  # Classic Democratic blue
                "0" = "#DE0100") # Classic Republican red
       ) +
     geom_smooth(method = "lm") +
     theme_minimal()
```

---

```
## `geom_smooth()` using formula = 'y ~ x'
```

---

``` r
econvoteparty.lm <- lm(vote ~ growth + inc_party, data = hibbs)
```

---

``` r
econparty.summary <- modelsummary( list("Economic Voting Model" = econvoteparty.lm),
             output = "kableExtra",
             stars = TRUE,
             statistic = "std.error",  
             coef_rename = TRUE,
             escape = FALSE,
             gof_map = "all") %>%      
  kable_styling(
    font_size = 24,                    # Even larger font
    bootstrap_options = c("striped", "hover", "condensed"),
    full_width = FALSE
  )
```

---

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto; font-size: 24px; width: auto !important; margin-left: auto; margin-right: auto;" class="table table table-striped table-hover table-condensed">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;">  Economic Voting Model </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 46.153*** </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (1.855) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> growth </td>
   <td style="text-align:center;"> 3.062*** </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (0.722) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> inc_party </td>
   <td style="text-align:center;"> 0.245 </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1.5px">  </td>
   <td style="text-align:center;box-shadow: 0px 1.5px"> (2.016) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 16 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 </td>
   <td style="text-align:center;"> 0.580 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 Adj. </td>
   <td style="text-align:center;"> 0.516 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> AIC </td>
   <td style="text-align:center;"> 93.7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> BIC </td>
   <td style="text-align:center;"> 96.8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Log.Lik. </td>
   <td style="text-align:center;"> −42.830 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> F </td>
   <td style="text-align:center;"> 8.988 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> RMSE </td>
   <td style="text-align:center;"> 3.52 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>
---

``` r
econvotepartyinteract.lm <- lm(vote ~ growth + inc_party + growth:inc_party, data = hibbs)
```

---

``` r
econinteract.summary <- modelsummary( list("Economic Voting Model" = econvotepartyinteract.lm),
             output = "kableExtra",
             stars = TRUE,
             statistic = "std.error",  
             coef_rename = TRUE,
             escape = FALSE,
             gof_map = "all") %>%      
  kable_styling(
    font_size = 24,                    # Even larger font
    bootstrap_options = c("striped", "hover", "condensed"),
    full_width = FALSE
  )
```

---

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto; font-size: 24px; width: auto !important; margin-left: auto; margin-right: auto;" class="table table table-striped table-hover table-condensed">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;">  Economic Voting Model </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 44.990*** </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (2.284) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> growth </td>
   <td style="text-align:center;"> 3.669** </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (0.999) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> inc_party </td>
   <td style="text-align:center;"> 2.692 </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (3.426) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> growth × inc_party </td>
   <td style="text-align:center;"> −1.295 </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1.5px">  </td>
   <td style="text-align:center;box-shadow: 0px 1.5px"> (1.459) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 16 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 </td>
   <td style="text-align:center;"> 0.606 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 Adj. </td>
   <td style="text-align:center;"> 0.508 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> AIC </td>
   <td style="text-align:center;"> 94.6 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> BIC </td>
   <td style="text-align:center;"> 98.5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Log.Lik. </td>
   <td style="text-align:center;"> −42.322 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> F </td>
   <td style="text-align:center;"> 6.157 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> RMSE </td>
   <td style="text-align:center;"> 3.41 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

---

``` r
library(interplot)
```

```
## Loading required package: abind
```

```
## Loading required package: arm
```

```
## Loading required package: MASS
```

```
## 
## Attaching package: 'MASS'
```

```
## The following object is masked from 'package:rosdata':
## 
##     newcomb
```

```
## The following object is masked from 'package:dplyr':
## 
##     select
```

```
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---

``` r
econ.interplot <- interplot(m = econvotepartyinteract.lm, var1 = "growth", var2 = "inc_party") + 
  xlab('Party (1 for Democrats)') +
  ylab('Marginal Effect of Income Growth') +
  ggtitle('Interaction between Party and Economics as Predictors of Elections') +
  theme(plot.title = element_text(face='bold'))
```

---

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###Goals

People can and do pursue a lot of different goals with regression.

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###Goals

1. Summarizing relationships
2. Prediction
3. Theory testing
4. Causal inference