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<!DOCTYPE html>
<html lang="" xml:lang="">
  <head>
    <title>Estimating the Error Variance</title>
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    <meta name="author" content="STAT 021 with Prof Suzy" />
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class: center, middle, inverse, title-slide

# Estimating the Error Variance
### STAT 021 with Prof Suzy
### Swarthmore College

---


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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 &lt;- SLR_hc %&gt;% 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(&gt;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[&lt;a href="https://campus.datacamp.com/courses/correlation-and-regression/model-fit?ex=3"&gt;Here&lt;/a&gt; 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

.scroll-output[

```r
hc_resid_data &lt;- hc_employer_2013 %&gt;%
                        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)
```

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