Residual Analysis

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# Residual Analysis
### STAT 021 with Prof Suzy
### Swarthmore College

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## Multiple linear regression
### Assumptions

- Relationship between the response and the set of predictor variables (the regressors) is approximately linear

- The error term, `\(\epsilon\)`, has zero mean

- The error term has constant variance `\(\sigma^2\)`

- The errors are uncorrelated

- The errors are Normally distributed

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## Multiple linear regression
### Assumptions

- Relationship between the response and the set of predictor variables (the regressors) is approximately linear `\(\leftarrow\)` .red[Use residual plots!]

- ~~The error term, `\(\epsilon\)`, has zero mean~~

- The error term has constant variance `\(\sigma^2\)` `\(\leftarrow\)` .red[Use residual plots!]

- The errors are uncorrelated

- The errors are Normally distributed `\(\leftarrow\)` .red[Use Normal probability plots of the residuals!]

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## Multiple linear regression
### (Studentized) Residuals

Residuals measure the deviation between the observed data, `\(y_{obs}\)`, and the fitted response, `\(\hat{y}\)`. Some properties of residuals are

1. The mean of the residuals is always zero.

2. The variance of the residuals is, on average, equal to the `\(MSE = \frac{SSE}{n-p}\)`.

3. The residuals are correlated. (This is mainly a problem when the number of parameters `\(p\)` is close to - or larger than - your sample size, `\(n\)`.)

- This is one reason we want to use studentized residuals.
  
  - This is also why we plot the residuals, `\(e_i\)`, vs the fitted values, `\(\hat{y}_i\)`, (rather than the observed `\(y\)` values)

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From now on, we are going to use only use plots of the *studentized residuals*

```r
library("MASS")
studres(mod)  ## this is the function to calculate the studentized residuals of "mod" 
```

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## Normal probability plots

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## Residual plots 
### Residuals vs fitted values

- Outliers

- Non-constant variance

- Non-linearity

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## Residual plots 
### Residuals vs regressors

- Transformations of predictors (non-linear relationships)

- Extrapolation

- Potentially useful additional predictors

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### Reading along in your textbook

Chapter 4 Sections 1 and 2