Week 4 - Part 2

class: center, middle, inverse, title-slide

.title[
# Week 4 - Part 2
]
.subtitle[
## Estimating the Error Variance
]
.author[
### Suzanne Thornton
]
.institute[
### Swarthmore College
]
.date[
### For class in Week 9 (updated: 2023-07-31)
]

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## What does the rest of the R output mean? 
### Residual standard error and it's degrees of freedom

Recall that `$\sigma$` is also a model parameter that we don't know and must estimate.

`$$\hat{\sigma} = \text{Residual standard error} =  \sqrt{\frac{SSres}{n-2}} = \sqrt{MSE}$$`

--
The *residual degrees of freedom* are `$n-2$`, where we are subtracting `$2$` because *SSres* is computed after we have estimated two model parameters, `$\beta_0$` and `$\beta_1$`.

```r
df.residual(SLR_hc)
```

```
## [1] 49
```

```r
dim(hc_employer_2013)
```

```
## [1] 51  6
```

---
## What does the rest of the R output mean? 
### Residual standard error and it's degrees of freedom

To extract this estimate for `$\sigma$` from the model use:

```r
SLR_hc_summary <- SLR_hc %>% summary
SLR_hc_summary$sigma
```

```
## [1] 0.02945572
```

---
## What does the rest of the R output mean? 
### Residual standard error and it's degrees of freedom

To see that `$\hat{\sigma} = \text{Residual standard error} =  \sqrt{\frac{SSres}{n-2}}$`,
look at the following output:

```r
anova(SLR_hc)
```

```
## Analysis of Variance Table
## 
## Response: prop_uninsured
##                 Df   Sum Sq   Mean Sq F value    Pr(>F)    
## spending_capita  1 0.023730 0.0237300   27.35 3.503e-06 ***
## Residuals       49 0.042514 0.0008676                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

.footnote[<a href="https://campus.datacamp.com/courses/correlation-and-regression/model-fit?ex=3">Here</a> is a useful R tutorial that helps to explain what *residual standard error* is.]

---
## What does the rest of the R output mean? 
### Residual standard error and it's degrees of freedom

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```r
hc_resid_data <- hc_employer_2013 %>% 
                        mutate(residuals = SLR_hc$residuals, 
                               fitted_vals = SLR_hc$fitted.values)

ggplot(hc_resid_data) +
         geom_point(aes(x=fitted_vals, y=residuals)) + 
         labs(title="Residual plot", subtitle="Cost of health care a predictor of proportion of people uninsured",
         x="Fitted values", y="Residuals") + 
  geom_hline(yintercept=0)
```

<img src="Figs/class14-2-1.png" style="display: block; margin: auto;" />
]