PADP 7120 Data Applications in PA

class: center, middle, inverse, title-slide

.title[
# PADP 7120 Data Applications in PA
]
.subtitle[
## Regression with Interactions
]
.author[
### Alex Combs
]
.institute[
### UGA | SPIA | PADP
]
.date[
### Last updated: September 28, 2026
]

---

# Outline

- Interact numerical and categorical variables

---
# Parallel Slopes

---
# Example: Parallel Slopes

`$$Salary=\beta_0+\beta_1YrsEmployed+\beta_2Sex + \epsilon$$`

``` r
sal_mod <- lm(salary ~ yrs.service + sex, data = Salaries)
```

``` r
get_regression_table(sal_mod)
```
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std_error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p_value </th>
   <th style="text-align:right;"> lower_ci </th>
   <th style="text-align:right;"> upper_ci </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> intercept </td>
   <td style="text-align:right;"> 92356.947 </td>
   <td style="text-align:right;"> 4740.188 </td>
   <td style="text-align:right;"> 19.484 </td>
   <td style="text-align:right;"> 0.000 </td>
   <td style="text-align:right;"> 83037.722 </td>
   <td style="text-align:right;"> 101676.171 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> yrs.service </td>
   <td style="text-align:right;"> 747.612 </td>
   <td style="text-align:right;"> 111.396 </td>
   <td style="text-align:right;"> 6.711 </td>
   <td style="text-align:right;"> 0.000 </td>
   <td style="text-align:right;"> 528.607 </td>
   <td style="text-align:right;"> 966.617 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> sex: Male </td>
   <td style="text-align:right;"> 9071.800 </td>
   <td style="text-align:right;"> 4861.644 </td>
   <td style="text-align:right;"> 1.866 </td>
   <td style="text-align:right;"> 0.063 </td>
   <td style="text-align:right;"> -486.208 </td>
   <td style="text-align:right;"> 18629.808 </td>
  </tr>
</tbody>
</table>

`$$\hat{Salary}=92357+748 \times YrsEmployed+9072 \times Sex$$`

- Note that Female is the reference group, Sex = 1 for Male and 0 for Female

---
class: inverse, middle, center

# We aren't required to model the slopes to be parallel. Could allow the effect of years employed to differ between male and female professors.

---
# Visualizing Interaction Model

.pull-left[

``` r
ggplot(Salaries, 
  aes(x = yrs.service, 
  y = salary, 
* color = sex)) +
  geom_point(alpha = 0.6, 
             size = 5) +
  geom_smooth(method = 'lm', 
              se = FALSE, 
              linewidth = 2) +
  theme_minimal() +
  theme(text = element_text(size = 20))
```
]

.pull-right[
<img src="Regression-with-Interactions_files/figure-html/unnamed-chunk-7-1.png" alt="" width="99%" />
]

---
# Interaction model implication

`$$y = \beta_0 + \beta_1x + \beta_2d + \beta_3 xd + \epsilon$$`
`$$\hat{y} = b_0 + b_1x + b_2d + b_3 xd$$`
- Now `$x$` and `$d$` are interacted (multiplied together), allowing the effect of `$x$` to differ between groups of `$d$`

---
# Interaction model equations

`$$\hat{y} = b_0 + b_1x + b_2d + b_3 xd$$`
--

- For the `$d=0$` group:

`$$\hat{y} = b_0 + b_1x$$`

- For the `$d=1$` group:

`$$\hat{y} = b_0 + b_1x + b_2 + b_3x$$`

`$$\hat{y} = (b_0 + b_2) + (b_1+b_3)x$$`

- Intercept *and* slope *can* differ between groups

- Intercept differs by `$b_2$` and slope (affect of `$x$`) differs by `$b_3$`

---
# Interaction

- An interaction allows the effect of one variable to differ depending on the value of another variable

`$$Salary=\beta_0+\beta_1YrsEmp+\beta_2Sex + \beta_3YrsEmp*Sex + \epsilon$$`

- `$\beta_3$` tells us how the slopes compare
  - For Sex = 1 (male), the association between years and salary is `$\beta_1+\beta_3$`
  - For Sex = 0 (female), it is `$\beta_1$`

- `$\beta_2$` tells us how the intercepts compare
  - For Sex = 1 (male), intercept is `$\beta_0+\beta_2$`
  - For Sex = 0 (female), it is `$\beta_0$`

---
# Running an interaction model

``` r
sal_mod2 <- lm(salary ~ yrs.service + sex + 
*                yrs.service*sex,
               data = Salaries)
```

``` r
get_regression_table(sal_mod2)
```
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std_error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p_value </th>
   <th style="text-align:right;"> lower_ci </th>
   <th style="text-align:right;"> upper_ci </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> intercept </td>
   <td style="text-align:right;"> 82068.51 </td>
   <td style="text-align:right;"> 7568.72 </td>
   <td style="text-align:right;"> 10.84 </td>
   <td style="text-align:right;"> 0.00 </td>
   <td style="text-align:right;"> 67188.26 </td>
   <td style="text-align:right;"> 96948.76 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> yrs.service </td>
   <td style="text-align:right;"> 1637.30 </td>
   <td style="text-align:right;"> 523.03 </td>
   <td style="text-align:right;"> 3.13 </td>
   <td style="text-align:right;"> 0.00 </td>
   <td style="text-align:right;"> 609.02 </td>
   <td style="text-align:right;"> 2665.58 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> sex: Male </td>
   <td style="text-align:right;"> 20128.63 </td>
   <td style="text-align:right;"> 7991.14 </td>
   <td style="text-align:right;"> 2.52 </td>
   <td style="text-align:right;"> 0.01 </td>
   <td style="text-align:right;"> 4417.90 </td>
   <td style="text-align:right;"> 35839.35 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> yrs.service:sexMale </td>
   <td style="text-align:right;"> -931.74 </td>
   <td style="text-align:right;"> 535.24 </td>
   <td style="text-align:right;"> -1.74 </td>
   <td style="text-align:right;"> 0.08 </td>
   <td style="text-align:right;"> -1984.04 </td>
   <td style="text-align:right;"> 120.56 </td>
  </tr>
</tbody>
</table>

`$$\hat{Sal}=82069+1637*YrsEmp+20129*Sex - 932*YrsEmp*Sex$$`

---
# Interpreting an interaction model

`$$\hat{Sal}=82069+1637*YrsEmp+20129*Sex - 932*YrsEmp*Sex$$`
- Recall that `Sex` = 1 for male, 0 for female

- What is the predicted change in salary for a male professor given one more year of employment?

- What is the predicted change in salary for a female professor given one more year of employment?

---
# Interpreting an interaction model

`$$\hat{Sal}=82069+1637*YrsEmp+20129*Sex - 932*YrsEmp*Sex$$`

- What is the predicted salary for a female professor employed for 10 years?

- What is the predicted salary for a male professor employed for 10 years?