Simple Linear Regression

# Simple Linear Regression
## Conditions
### Prof. Maria Tackett

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## [Click here for PDF of slides](05-slr-conditions.pdf)
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## Topics

- List the conditions for simple linear regression
  
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- Use plots of the residuals to check the conditions

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## Movie ratings data

The data set contains the "Tomatometer" score (.term[critics]) and audience score (.term[audience]) for 146 movies rated on rottentomatoes.com.

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## The model

`$$\color{red}{\hat{\text{audience}} = 32.316 + 0.519 \times \text{critics}}$$`

<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;"> 32.316 </td>
   <td style="text-align:right;"> 2.343 </td>
   <td style="text-align:right;"> 13.795 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> critics </td>
   <td style="text-align:right;"> 0.519 </td>
   <td style="text-align:right;"> 0.035 </td>
   <td style="text-align:right;"> 15.028 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>

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class: middle

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## Model conditions

1. .vocab[Linearity: ]There is a linear relationship between the response and predictor variable.

2. .vocab[Constant Variance: ]The variability of the errors is equal for all values of the predictor variable.

3. .vocab[Normality: ]The errors follow a normal distribution.

4. .vocab[Independence: ]The errors are independent from each other.
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## Residuals vs. fitted values

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## Checking linearity

✅ There is no distinguishable pattern or structure. The residuals are randomly scattered.

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## ❌ Violation: distinguishable pattern

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## Checking constant variance

✅ The vertical spread of the residuals is relatively constant across the plot.

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## ❌ Violation: non-constant variance

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## Normal quantile plot

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## Checking normality

✅ Points fall along a straight diagonal line on the normal quantile plot.

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## Checking independence

- We can often check the independence assumption based on the context of the data and how the observations were collected.

- If the data were collected in a particular order, examine a scatterplot of the residuals versus order in which the data were collected. 
<br>

✅ Based on available information, the error for one movie does not tell us anything about the error for another movie.

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## In practice

As you check the model conditions, ask if any observed deviation from the model conditions are so great that

1️⃣ a different model should be proposed.

2️⃣ conclusions drawn from the model should be used with caution.

✅ If not, the conditions are sufficiently met and we can proceed with the current model.

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## Recap

- Used plots of the residuals to check conditions for simple linear regression: 
  - .vocab[Linearity]
  - .vocab[Constant Variance]
  - .vocab[Normality]
  - .vocab[Independence]