+ - 0:00:00
Notes for current slide
Notes for next slide

Simple Linear Regression

Conditions

Prof. Maria Tackett

1

Topics

3

Topics

  • List the conditions for simple linear regression
3

Topics

  • List the conditions for simple linear regression

  • Use plots of the residuals to check the conditions

3

Movie ratings data

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

4

The model

^audience=32.316+0.519×critics

term estimate std.error statistic p.value
(Intercept) 32.316 2.343 13.795 0
critics 0.519 0.035 15.028 0

5

Y|XN(β0+β1X,σ2ϵ)

6

Model conditions

7

Model conditions

  1. Linearity: There is a linear relationship between the response and predictor variable.
7

Model conditions

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

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

7

Model conditions

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

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

  3. Normality: The errors follow a normal distribution.

7

Model conditions

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

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

  3. Normality: The errors follow a normal distribution.

  4. Independence: The errors are independent from each other.

7

residuali=ei=yiˆyi

8

Residuals vs. fitted values

9

Checking linearity

10

Checking linearity

10

Checking linearity

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

10

❌ Violation: distinguishable pattern

11

Checking constant variance

12

Checking constant variance

12

Checking constant variance

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

12

❌ Violation: non-constant variance

13

Normal quantile plot

14

Checking normality

15

Checking normality

15

Checking normality

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

15

Checking independence

16

Checking independence

  • We can often check the independence assumption based on the context of the data and how the observations were collected.
16

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.

16

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.

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

16

In practice

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

17

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.

17

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.

17

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.

17

Recap

18

Recap

  • Used plots of the residuals to check conditions for simple linear regression:
    • Linearity
    • Constant Variance
    • Normality
    • Independence
18
Paused

Help

Keyboard shortcuts

, , Pg Up, k Go to previous slide
, , Pg Dn, Space, j Go to next slide
Home Go to first slide
End Go to last slide
Number + Return Go to specific slide
b / m / f Toggle blackout / mirrored / fullscreen mode
c Clone slideshow
p Toggle presenter mode
t Restart the presentation timer
?, h Toggle this help
Esc Back to slideshow