### Linear Regression

#### Introduction

#### Simple Regression

- Linear Regression
- Regression Example
- Residual Analysis
- Transformations
- Influential Points
- Slope Estimate
- Slope Test

#### Multiple Regression

### Linear Regession: Table of Contents

#### Introduction

#### Simple Regression

- Linear Regression
- Regression Example
- Residual Analysis
- Transformations
- Influential Points
- Slope Estimate
- Slope Test

#### Multiple Regression

# Residual Analysis in Regression

Because a linear regression model is not always appropriate for the data, you should assess the appropriateness of the model by defining residuals and examining residual plots.

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## Residuals

The difference between the observed value of the dependent variable
(*y*) and the predicted value (*ŷ*) is called the
**residual** (*e*). Each data point has one
residual.

Residual = Observed value - Predicted value

*e* = *y* - *ŷ*

Both the sum and the mean of the residuals are equal to zero. That is,
Σ *e* = 0 and e = 0.

## Residual Plots

A **residual plot** is a graph that shows the
residuals on the vertical axis and the independent variable
on the horizontal axis. If the points in a residual plot
are randomly dispersed
around the horizontal axis, a linear regression model is
appropriate for the data; otherwise, a nonlinear model is more
appropriate.

The table below shows inputs and outputs from a simple linear regression analysis.

x | y | ŷ | e |
---|---|---|---|

60 | 70 | 65.411 | 4.589 |

70 | 65 | 71.849 | -6.849 |

80 | 70 | 78.288 | -8.288 |

85 | 95 | 81.507 | 13.493 |

95 | 85 | 87.945 | -2.945 |

And the chart below displays the residual (e) and independent variable (X) as a residual plot.

The residual plot shows a fairly random pattern - the first residual is positive, the next two are negative, the fourth is positive, and the last residual is negative. This random pattern indicates that a linear model provides a decent fit to the data.

Below, the residual plots show three typical patterns. The first plot shows a random pattern, indicating a good fit for a linear model.

Random pattern

Non-random: U-shaped

Non-random: Inverted U

The other plot patterns are non-random (U-shaped and inverted U), suggesting a better fit for a nonlinear model.

In the next lesson, we will work on a problem, where the residual plot shows a non-random pattern. And we will show how to "transform" the data to use a linear model with nonlinear data.

## Test Your Understanding

In the context of regression analysis, which of the following statements are true?

I. When the sum of the residuals is greater than zero, the data set is
nonlinear.

II. A random pattern of residuals supports a linear model.

III. A random pattern of residuals supports a nonlinear model.

(A) I only

(B) II only

(C) III only

(D) I and II

(E) I and III

**Solution**

The correct answer is (B). A random pattern of residuals supports a linear model; a non-random pattern supports a nonlinear model. The sum of the residuals is always zero, whether the data set is linear or nonlinear.

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