Regression

class: center, middle, inverse, title-slide

.title[
# Regression
]
.subtitle[
## Part 1b
]
.author[
### Prof. Weldzius
]
.institute[
### Villanova University
]

---

# Agenda

1. Introducing the Data

2. Demonstrating Regressions

---

# Regression

- Let's demonstrate what we learned last time with the `sc_debt` data

``` r
require(tidyverse)

debt <- read_rds('../data/sc_debt.Rds')

glimpse(debt)
```

```
## Rows: 2,546
## Columns: 16
## $ unitid         <int> 100654, 100663, 100690, 100706, 100…
## $ instnm         <chr> "Alabama A & M University", "Univer…
## $ stabbr         <chr> "AL", "AL", "AL", "AL", "AL", "AL",…
## $ grad_debt_mdn  <int> 33375, 22500, 27334, 21607, 32000, …
## $ control        <chr> "Public", "Public", "Private", "Pub…
## $ region         <chr> "Southeast", "Southeast", "Southeas…
## $ preddeg        <chr> "Bachelor's", "Bachelor's", "Associ…
## $ openadmp       <int> 2, 2, 1, 2, 2, 2, 1, NA, 2, 2, 2, 1…
## $ adm_rate       <dbl> 0.9175, 0.7366, NA, 0.8257, 0.9690,…
## $ ccbasic        <int> 18, 15, 20, 16, 19, 15, 2, 22, 18, …
## $ sat_avg        <int> 939, 1234, NA, 1319, 946, 1261, NA,…
## $ md_earn_wne_p6 <int> 25200, 35100, 30700, 36200, 22600, …
## $ ugds           <int> 5271, 13328, 365, 7785, 3750, 31900…
## $ costt4_a       <int> 23053, 24495, 14800, 23917, 21866, …
## $ selective      <dbl> 0, 0, NA, 0, 0, 0, NA, NA, 0, 0, 0,…
## $ research_u     <dbl> 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,…
```

---

# Research Camp

- .blue[Research Question]: What is the relationship between SAT scores and median future earnings?

- .blue[Theory]: Students with higher SAT scores work harder and have learned more. Employers reward these attributes with higher wages in the private market.

- .blue[Hypothesis]: The relationship between SAT scores and future earnings should be positive.

- **NB**: Important caveats to this simplistic theory!
  
--

- Socioeconomic status: predicts both higher SAT scores and higher wages
  
--

- **Correlation `$\neq$` Causation**
  
---

# Set Up

- Linking .blue[Theory] to .red[Data]

- Our SAT scores are theorized to explain future earnings

- Thus the SAT scores are the independent / explanatory / predictor variable `$X$`
  
  - And earnings are the dependent / outcome variable `$Y$`

---

# Regression

- There is a simple recipe to follow

- And it is exactly how the syllabus for the class is designed!

1. Look at your data to identify missingness (**Wrangling**)
  
  2. **Univariate** visualization of your variables 
  
  3. **Multivariate** visualization of your variables 
  
  4. **Regression** (today)
  
  5. Evaluation of **errors** (next lecture)

---

# Step 1: Look

- Why worry about **missingness**?

1. **Substantive:** external validity

2. **Technical:** cross validation won't work!

``` r
summary(debt %>% select(sat_avg,md_earn_wne_p6))
```

```
##     sat_avg     md_earn_wne_p6  
##  Min.   : 737   Min.   : 10600  
##  1st Qu.:1053   1st Qu.: 26100  
##  Median :1119   Median : 31500  
##  Mean   :1141   Mean   : 33028  
##  3rd Qu.:1205   3rd Qu.: 37400  
##  Max.   :1557   Max.   :120400  
##  NA's   :1317   NA's   :240
```

---

# Step 2: Univariate Viz

- Why visualize both `$Y$` and `$X$`?

1. **Substantive:** *See* which units you are talking about

2. **Technical:** Adjust for *skew*

---

# Step 2: Univariate Viz

- Why visualize both `$Y$` and `$X$`?

``` r
debt %>%
  ggplot(aes(x = sat_avg)) + 
  geom_histogram()
```

---

# Step 2: Univariate Viz

- Why visualize both `$Y$` and `$X$`?

``` r
debt %>%
  ggplot(aes(x = md_earn_wne_p6)) + 
  geom_histogram()
```

---

# Step 3: Multivariate

- Eyeball the relationship first!

``` r
debt %>%
  ggplot(aes(x = sat_avg,y = md_earn_wne_p6)) + 
  geom_point()
```

---

# Step 3: Multivariate Viz

- Adding regression line

``` r
debt %>%
  ggplot(aes(x = sat_avg,y = md_earn_wne_p6)) + 
  geom_point() + geom_smooth(method = 'lm',se = F)
```

---

# Step 3: Multivariate Viz

- Let's focus on two schools

``` r
toplot <- debt %>%
  mutate(hl = ifelse(unitid %in% c(100654,179265),'hl','none')) # Choosing two examples
p2 <- toplot %>%
  ggplot(aes(x = sat_avg, y = md_earn_wne_p6,color = hl,group = 1,alpha = hl)) + 
  geom_point(data = toplot %>% filter(hl == 'none')) + 
  geom_point(data = toplot %>% filter(hl == 'hl'),size =3) + 
  scale_alpha_manual(values = c(1,.3)) + 
  scale_color_manual(values = c('red','black')) + 
  geom_smooth(method = 'lm',se = F) + 
  theme(legend.position = 'none') + 
  labs(title = "Graduate Earnings and SAT Scores",
       subtitle = "By School",
       x = "Average SAT Score",
       y = "Median Earnings of Grads")
```

---

# Step 3: Multivariate Viz

- Adding regression line

``` r
p2
```

---

# Step 3: Multivariate Viz

- Defining `$\varepsilon$`

``` r
p3 <- toplot %>%
  ggplot(aes(x = sat_avg, y = md_earn_wne_p6,color = hl,group = 1,alpha = hl)) + 
  geom_point(data = toplot %>% filter(hl == 'none')) + 
  geom_point(data = toplot %>% filter(hl == 'hl'),size =3) + 
  scale_alpha_manual(values = c(1,.3)) + 
  scale_color_manual(values = c('red','black')) + 
  geom_smooth(method = 'lm',se = F) + 
  annotate(geom = 'segment',
           x = toplot %>% filter(hl == 'hl') %>% .$sat_avg,
           y = toplot %>% filter(hl == 'hl') %>% .$md_earn_wne_p6,
           xend = toplot %>% filter(hl == 'hl') %>% .$sat_avg,
           yend = c(27500,41000),color = 'red',lwd = 1.2) + 
  theme(legend.position = 'none') + 
  labs(title = "Graduate Earnings and SAT Scores",
       subtitle = "By School",
       x = "Average SAT Score",
       y = "Median Earnings of Grads")
```

---

# Step 3: Multivariate Viz

- Measuring errors

``` r
p3
```

<img src="9b_RegressionPart1_files/figure-html/unnamed-chunk-12-1.png" style="display: block; margin: auto;" />
---