POL90: Statistics Chapter 7: Regression by Calculation

class: center, middle, inverse, title-slide

# POL90: Statistics <br/> Chapter 7: Regression by Calculation
### Prof Wasow <br/> Assistant Professor, Politics <br/> Pomona College
### 2022-03-12

---

.regression12 table {
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.regression14 table {
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# Announcements

.large[
* Assignments

+ PS06

]

.large[
* Statistical Sleuth

+ Read Chapter 7

+ Supplement
      - http://appliedstats.org/chapter7.html

]

---
# Schedule

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> Week </th>
   <th style="text-align:left;"> Date </th>
   <th style="text-align:left;"> Day </th>
   <th style="text-align:left;"> Title </th>
   <th style="text-align:right;"> Chapter </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:left;"> Feb 23 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> Comparison Among Several Samples </td>
   <td style="text-align:right;"> 5 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:left;"> Feb 28 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Comparison Among Several Samples </td>
   <td style="text-align:right;"> 5 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:left;"> Mar 2 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> Simple Linear Regression </td>
   <td style="text-align:right;"> 7 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:left;"> Mar 7 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Simple Linear Regression </td>
   <td style="text-align:right;"> 7 </td>
  </tr>
  <tr>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 8 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Mar 9 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Wed </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Regression by Calculation </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 7 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:left;"> Mar 14 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Spring Recess </td>
   <td style="text-align:right;"> - </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:left;"> Mar 16 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> Spring Recess </td>
   <td style="text-align:right;"> - </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:left;"> Mar 21 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Null hypothesis, R-squared </td>
   <td style="text-align:right;"> 8 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:left;"> Mar 23 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> Multiple regression </td>
   <td style="text-align:right;"> 8 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 11 </td>
   <td style="text-align:left;"> Mar 28 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Interaction terms </td>
   <td style="text-align:right;"> 9 </td>
  </tr>
</tbody>
</table>

---
## Assignment schedule

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> Week </th>
   <th style="text-align:left;"> Date </th>
   <th style="text-align:left;"> Day </th>
   <th style="text-align:left;"> Assignment </th>
   <th style="text-align:right;"> Percent </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:left;"> Mar 4 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> Report1 </td>
   <td style="text-align:right;"> 6 </td>
  </tr>
  <tr>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 8 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Mar 11 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Fri </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> PS06 </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:left;"> Mar 18 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> Spring break </td>
   <td style="text-align:right;"> NA </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:left;"> Mar 25 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS07 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 11 </td>
   <td style="text-align:left;"> Apr 1 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS08 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 12 </td>
   <td style="text-align:left;"> Apr 8 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> Report2 </td>
   <td style="text-align:right;"> 8 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 13 </td>
   <td style="text-align:left;"> Apr 15 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS09 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 14 </td>
   <td style="text-align:left;"> Apr 22 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS10 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 15 </td>
   <td style="text-align:left;"> Apr 29 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> Report3 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 16 </td>
   <td style="text-align:left;"> May 6 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> NA </td>
   <td style="text-align:right;"> NA </td>
  </tr>
</tbody>
</table>

---
class: middle, center

# Report 1: Did Donald Trump 
# Drive Public Opinion
# on "Build the Wall"?

---
## Welch, Rodi, Shaw & Wonghirundacha (2022)

---
## Welch, Rodi, Shaw & Wonghirundacha (2022)
<br>
<img src="images/welch_rodi_shaw_wonghirundacha02.png" width="592" style="display: block; margin: auto;" />

---
## Welch, Rodi, Shaw & Wonghirundacha (2022)
<br>
<img src="images/welch_rodi_shaw_wonghirundacha03.png" width="621" style="display: block; margin: auto;" />

---
## Welch, Rodi, Shaw & Wonghirundacha (2022)
<br><br><br><br>
<img src="images/welch_rodi_shaw_wonghirundacha04.png" width="873" style="display: block; margin: auto;" />

---
## Welch, Rodi, Shaw & Wonghirundacha (2022)
<br><br>
<img src="images/welch_rodi_shaw_wonghirundacha05.png" width="860" style="display: block; margin: auto;" />

---
class: middle, center
# Mathematical Approach to 
# Linear Models

---
## Revisiting Meat Processing and PH Level

```r
meat <- Sleuth3::case0702
meat
```

```
   Time   pH
1     1 7.02
2     1 6.93
3     2 6.42
4     2 6.51
5     4 6.07
6     4 5.99
7     6 5.59
8     6 5.80
9     8 5.51
10    8 5.36
```
---

```r
meat %>%
ggplot() +
  aes(x = Time, y = pH) +
  geom_point() +
* geom_smooth(method = "loess", se = FALSE)
```

---

```r
meat <- meat %>% mutate(log_time = log(Time))
meat %>%
  ggplot() +
    aes(x = log_time, y = pH) +
    geom_point() +
*   geom_smooth(method = "lm", se = FALSE)
```

<img src="week08_02_files/figure-html/unnamed-chunk-11-1.png" width="792" style="display: block; margin: auto;" />
---
## Terminology

- Greek letters like `$\beta$`, are the unobserved truth

- Modified Greek letters like `$\hat{\beta}$` are our estimate. They are what we think the truth is based on our data

- English letters like `$\boldsymbol{X}$` are actual data from our sample

- Modified English letters like `$\bar{X}$` are calculations from our sample. They're what we do with our data

- We can say that our estimate of the truth is that calculation, e.g. `$\hat{\mu} = \bar{X}$`

$$ Data \longrightarrow Calculation \longrightarrow Estimate \overset{\text{hopefully}}\longrightarrow Truth $$

.footnote[Source: Nick Huntington-Klein, https://twitter.com/nickchk/status/1272993322395557888]

---
## Review notation for fitted values and residuals

.vertical-center[
.large[

- Hat notation denotes an estimate of the parameter

`\begin{eqnarray*}
\hat{\mu}\{Y|X\} & = &\hat{\beta_0} + \hat{\beta_1}X 
\end{eqnarray*}`

- Estimated mean is the <span style="color:blue">fitted value</span> or <span style="color:blue">predicted value<span>

- Difference between observed response and estimated mean is the <span style="color:blue">residual<span>
]
]

`\begin{eqnarray*}
fit_i =\hat{\mu}\{Y_i|X_i\} &= &\hat{\beta_0} + \hat{\beta_1}X_i \\
res_i & = & Y_i - fit_i
\end{eqnarray*}`

---
## Visualizing a single `$Y_i$` and `$\hat{Y_i}$`

.center[
<img src="week08_02_files/figure-html/unnamed-chunk-12-1.png" width="792" style="display: block; margin: auto;" />
]
---
## Visualizing a single `$Y_i$` and `$\hat{Y_i}$`

.center[
<img src="week08_02_files/figure-html/unnamed-chunk-13-1.png" width="792" style="display: block; margin: auto;" />
]
---
## Visualizing a single `$Y_i$` and `$\hat{Y_i}$`

.center[
<img src="week08_02_files/figure-html/unnamed-chunk-14-1.png" width="792" style="display: block; margin: auto;" />
]
---
## Visualizing a single `$Y_i$` and `$\hat{Y_i}$`

.center[
<img src="week08_02_files/figure-html/unnamed-chunk-15-1.png" width="792" style="display: block; margin: auto;" />
]
---
## Visualizing a single `$Y_i$` and `$\hat{Y_i}$`

.center[
<img src="week08_02_files/figure-html/unnamed-chunk-16-1.png" width="792" style="display: block; margin: auto;" />
]
---
## Visualizing a single `$Y_i$` and `$\hat{Y_i}$`

.center[
<img src="week08_02_files/figure-html/unnamed-chunk-17-1.png" width="792" style="display: block; margin: auto;" />
]
---
## Visualizing a single `$Y_i$` and `$\hat{Y_i}$` (zoomed in)

.center[
<img src="week08_02_files/figure-html/unnamed-chunk-18-1.png" width="792" style="display: block; margin: auto;" />
]

---

## Model assumptions

.vertical-center[
.large[
`\begin{eqnarray*}
Y_i & = & \beta_0 + \beta_1X_i + \epsilon_i,
\end{eqnarray*}`

- where `$\epsilon_i \sim N(0, \sigma^2)$`
]
]

---
## Ideal, normal, simple linear regression
.center[
<img src="images/ss_display_7_5.png" width="60%" style="display: block; margin: auto;" />
]
.footnote[Source: *Statistical Sleuth*, 3e, Display 7.5]

---
## Model assumptions

.large[
- <span style="color:blue">Linearity assumption</span>: the means of the sub-population of responses for each value of the explanatory variable

- <span style="color:blue">Equal spread assumption</span> (constant variance assumption): the sub-population standard deviations are all equal (to `$\sigma$`)

- <span style="color:blue">Normality assumption</span>: there is a normally distributed sub-population of responses for each value of the explanatory variable

- <span style="color:blue">Independence assumption</span>: each response is drawn independently of all other responses from the same sub-population and independently of all responses drawn from other sub-populations
]

---
## Least squares estimators

.vertical-center[
.large[

`\begin{eqnarray*}
\hat{\beta}_1  & = & \frac{\sum^{n}_{i=1}(X_i-\bar{X})(Y_i-\bar{Y}) }{\sum^{n}_{i=1} (X_i-\bar{X})^2}, \\
& & \\
\hat{\beta}_0 & = & \bar{Y} - \hat{\beta}_1\bar{X}
\end{eqnarray*}`
]
]

---
## Least squares estimators manually in R

```r
x_bar <- mean(meat$log_time)
x_bar
```

```
[1] 1.19
```

```r
y_bar <- mean(meat$pH)
y_bar
```

```
[1] 6.12
```

```r
x_deviation <- meat$log_time - x_bar
x_deviation 
```

```
 [1] -1.1901 -1.1901 -0.4970 -0.4970  0.1962  0.1962  0.6016  0.6016  0.8893
[10]  0.8893
```

```r
y_deviation <- meat$pH - y_bar
y_deviation 
```

```
 [1]  0.90  0.81  0.30  0.39 -0.05 -0.13 -0.53 -0.32 -0.61 -0.76
```

---
## Least squares estimators manually in R

```r
beta_hat_1 <- sum((x_deviation) * (y_deviation))/sum((x_deviation)^2)
beta_hat_1
```

```
[1] -0.7257
```

```r
beta_hat_0 <- y_bar - beta_hat_1 * x_bar
beta_hat_0
```

```
[1] 6.984
```

```r
# Compare to base R regression
lm(formula = pH ~ log_time, data = meat) %>%
    broom::tidy() %>% kable(digits = 4)
```

<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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 6.9836 </td>
   <td style="text-align:right;"> 0.0485 </td>
   <td style="text-align:right;"> 143.90 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> log_time </td>
   <td style="text-align:right;"> -0.7257 </td>
   <td style="text-align:right;"> 0.0344 </td>
   <td style="text-align:right;"> -21.08 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>

---
## For standard error, first estimate of &sigma; from residuals

.vertical-center[
.large[

`\begin{eqnarray*}
\hat{\sigma}  & = & \sqrt{\frac{\textrm{Sum of all squared residuals}}{\textrm{Degrees of freedom}} }
\end{eqnarray*}`

- Where d.f. = (Number of observations) - (Number of parameters in the model for the means)
]
]

---
## Recall residuals are `$Y_i$` - `$\hat{Y_i}$`

.center[
<img src="week08_02_files/figure-html/unnamed-chunk-22-1.png" width="792" style="display: block; margin: auto;" />
]

---
## Sampling distributions for OLS

.center[
<img src="images/ss_display_7_7.png" width="70%" style="display: block; margin: auto;" />
]
.footnote[Source: *Statistical Sleuth*, 3e, Display 7.7]

---
## Estimation of &sigma; from Residuals in R

```r
y_hat <- beta_hat_0 + beta_hat_1 * meat$log_time
y_hat
```

```
 [1] 6.984 6.984 6.481 6.481 5.978 5.978 5.683 5.683 5.475 5.475
```

```r
residuals <- meat$pH - y_hat
residuals
```

```
 [1]  0.03637 -0.05363 -0.06064  0.02936  0.09235  0.01235 -0.09342  0.11658
 [9]  0.03534 -0.11466
```

```r
sum_squared_residuals <- sum(residuals^2)
sum_squared_residuals
```

```
[1] 0.05413
```
---
## Estimation of &sigma; from Residuals in R

```r
n  <- nrow(meat)
n
```

```
[1] 10
```

```r
df <- n - 2
df
```

```
[1] 8
```

```r
sigma_hat <- sqrt(sum_squared_residuals / df)
sigma_hat
```

```
[1] 0.08226
```

---
## Standard errors for OLS

.center[
<img src="images/ss_display_7_7_formulas.png" width="100%" style="display: block; margin: auto;" />
]
.footnote[Source: *Statistical Sleuth*, 3e, Section 7.3.5]

---
## Estimation of standard errors manually in R

```r
s_x <- sd(meat$log_time)
s_x
```

```
[1] 0.7965
```

```r
s_x2 <- s_x^2
s_x2
```

```
[1] 0.6344
```

```r
# s_x2 is variance
var(meat$log_time)
```

```
[1] 0.6344
```

```r
se_beta_hat_1 <- sigma_hat * sqrt(1 / (( n - 1) * ( s_x2 )))
se_beta_hat_1
```

```
[1] 0.03443
```
---
## Estimation of standard errors manually in R

```r
s_x <- sd(meat$log_time)
s_x
```

```
[1] 0.7965
```

```r
s_x2 <- s_x^2
s_x2
```

```
[1] 0.6344
```

```r
# s_x2 is variance
var(meat$log_time)
```

```
[1] 0.6344
```

```r
se_beta_hat_0 <- sigma_hat * sqrt(1 / n + (x_bar^2 / (( n - 1) * ( s_x2 ))) )
se_beta_hat_0
```

```
[1] 0.04853
```
---
## Compare manual se calculations to `lm` se

```r
se_beta_hat_0
```

```
[1] 0.04853
```

```r
se_beta_hat_1
```

```
[1] 0.03443
```

```r
# Compare to base R regression
lm(formula = pH ~ log_time, data = meat) %>%
    broom::tidy() %>% kable(digits = 4)
```

---
class: middle, center

# Geometry of Regression

---
## Geometry of `$\hat{\beta}_1$` with `palmerpenguins`

- Data were collected and made available by Dr. Kristen Gorman and the Palmer Station, Antarctica LTER, a member of the Long Term Ecological Research Network.

```r
library(palmerpenguins)
penguins <- palmerpenguins::penguins

head(penguins)
```

```
# A tibble: 6 × 8
  species island bill_length_mm bill_depth_mm flipper_length_… body_mass_g sex  
  <fct>   <fct>           <dbl>         <dbl>            <int>       <int> <fct>
1 Adelie  Torge…           39.1          18.7              181        3750 male 
2 Adelie  Torge…           39.5          17.4              186        3800 fema…
3 Adelie  Torge…           40.3          18                195        3250 fema…
4 Adelie  Torge…           NA            NA                 NA          NA <NA> 
5 Adelie  Torge…           36.7          19.3              193        3450 fema…
6 Adelie  Torge…           39.3          20.6              190        3650 male 
# … with 1 more variable: year <int>
```

.footnote[Source: https://allisonhorst.github.io/palmerpenguins/index.html]

---
## Who are `palmerpenguins`?

---
## What are these measurements?

---
## Subsetting for Adelie

```r
# subset data
penguins_adelie <- penguins %>% 
  drop_na() %>% 
  filter(species == "Adelie")

dim(penguins_adelie)
```

```
[1] 146   8
```

---
## Visualizing bill length vs bill depth

---
## Modeling bill length vs bill depth

```r
# bill length vs bill depth
fit <- lm(formula = bill_length_mm ~ bill_depth_mm, data = penguins_adelie)

summary(fit)
```

```

Call:
lm(formula = bill_length_mm ~ bill_depth_mm, data = penguins_adelie)

Residuals:
   Min     1Q Median     3Q    Max 
-6.543 -1.837  0.016  1.718  6.510

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)     23.367      3.087    7.57  4.1e-12 ***
bill_depth_mm    0.842      0.168    5.02  1.5e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.46 on 144 degrees of freedom
Multiple R-squared:  0.149,	Adjusted R-squared:  0.143 
F-statistic: 25.2 on 1 and 144 DF,  p-value: 1.51e-06
```

---
## Modeling bill length vs bill depth

```r
penguins_adelie$predicted <- predict(fit)   # Save the predicted values
penguins_adelie$residuals <- residuals(fit) # Save the residual values

penguins_adelie %>% select(bill_depth_mm, bill_length_mm,  predicted, residuals) %>% head()
```

```
# A tibble: 6 × 4
  bill_depth_mm bill_length_mm predicted residuals
          <dbl>          <dbl>     <dbl>     <dbl>
1          18.7           39.1      39.1   -0.0211
2          17.4           39.5      38.0    1.47  
3          18             40.3      38.5    1.77  
4          19.3           36.7      39.6   -2.93  
5          20.6           39.3      40.7   -1.42  
6          17.8           38.9      38.4    0.537 
```

---
## Visualizing Residuals

---
## Visualizing Residuals (ratio of one to one)

---
## Recall Least Squares Estimators

.vertical-center[
.large[

---
## Visualizing `$\hat{\beta}_1$` Numerator Step-by-Step

---
## Visualizing `$\hat{\beta}_1$` Numerator *x*-mean

---
## Visualizing `$\hat{\beta}_1$` Numerator *y*-mean

---
## Visualizing `$\hat{\beta}_1$` Numerator *x*-difference

---
## Visualizing `$\hat{\beta}_1$` Numerator *y*-difference

---
## Visualizing `$\hat{\beta}_1$` Numerator `$x$`-diff `$\times y$`-diff

---
## Question: What do the Colors Represent?

---
## Question: What Explains Positive vs Negative Slope?

---
## Visualizing `$\hat{\beta}_1$` Denominator (ratio of one to one)

---
## Visualizing `$\hat{\beta}_1$` Denominator `$x$`-mean

---
## Visualizing `$\hat{\beta}_1$` Denominator `$x$`-difference

---
## Visualizing `$\hat{\beta}_1$` Denominator ( `$x$`-diff) `$^2$`

---
## Again, Recall Least Squares Estimators

.vertical-center[
.large[

---
## Calculating `$\hat{\beta}_1$` Algebraically

```r
x_bar <- mean(penguins_adelie$bill_depth_mm)
x_bar
```

```
[1] 18.35
```

```r
x_diff <- penguins_adelie$bill_depth_mm - x_bar
head(x_diff)
```

```
[1]  0.3527 -0.9473 -0.3473  0.9527  2.2527 -0.5473
```

```r
y_bar <- mean(penguins_adelie$bill_length_mm)
y_bar
```

```
[1] 38.82
```

```r
y_diff <- penguins_adelie$bill_length_mm - y_bar
head(y_diff)
```

```
[1]  0.27603  0.67603  1.47603 -2.12397  0.47603  0.07603
```

---
## Calculating `$\hat{\beta}_1$` Algebraically

```r
x_diff_y_diff <- x_diff * y_diff
head(x_diff_y_diff)
```

```
[1]  0.09737 -0.64037 -0.51257 -2.02359  1.07237 -0.04161
```

```r
# numerator
sum(x_diff_y_diff)
```

```
[1] 181.6
```

```r
# denominator
sum(x_diff^2)
```

```
[1] 215.6
```

```r
beta_hat_1 <- sum(x_diff_y_diff) / sum(x_diff^2)
beta_hat_1
```

```
[1] 0.8425
```

---
## Calculating `$\hat{\beta}_0$` Algebraically

```r
beta_hat_0 <- y_bar - beta_hat_1 * x_bar
beta_hat_0
```

```
[1] 23.37
```

---
## Calculating Coefficients with `lm()`

```r
lm(formula = bill_length_mm ~ bill_depth_mm, data = penguins_adelie) %>%
  summary()
```

```

Call:
lm(formula = bill_length_mm ~ bill_depth_mm, data = penguins_adelie)

Residuals:
   Min     1Q Median     3Q    Max 
-6.543 -1.837  0.016  1.718  6.510

Residual standard error: 2.46 on 144 degrees of freedom
Multiple R-squared:  0.149,	Adjusted R-squared:  0.143 
F-statistic: 25.2 on 1 and 144 DF,  p-value: 1.51e-06
```

---
## Recall "Rise over Run"
<br><br><br><br><br>
.large[

- Rise: Numerator is  `$\sum_i (x_i - \bar{x}) \times (y_i - \bar{y})$`

- Run: Denominator is `$\sum_i (x_i - \bar{x})^2$`
]

---
## Let's play

.vertical-center[
.large[

* Short: http://bit.ly/346ols

* Long: http://setosa.io/ev/ordinary-least-squares-regression/
]
]

---
class: center, middle

# Questions?