1. HS 67 (Intro Health Stat) Regression
Friday, January 18, 2019
Chapter 5
Regression
1/18/2019
Chapter 5
Chapter 5 (Regression)

2. Regression
Like correlation, regression addresses the relationship between a quantitative explanatory variable (X) and quantitative response variable (Y)
The objective of regression is to describe the best fitting line through the data
As with correlation, start by looking at the data with a scatterplot
1/18/2019
Chapter 5

3. Same data as last week Per Capita GDP X Life Expectancy Y Country
Austria
21.4
77.48
Belgium
23.2
77.53
Finland
20.0
77.32
France
22.7
78.63
Germany
20.8
77.17
Ireland
18.6
76.39
Italy
21.5
78.51
Netherlands
22.0
78.15
Switzerland
23.8
78.99 UK
21.2
77.37
1/18/2019
Chapter 5

4. Inspect scatterplot for linearity
1/18/2019
Chapter 5

5. The Regression Line
The regression line predicts values of Y with this equation (the “regression model”):
ŷ = a + b∙X
where:
ŷ ≡ predicted value of Y at given X
a ≡ intercept
b = slope
a and b are called regression coefficients
1/18/2019
Chapter 5

6. Calculation of slope & intercept
1/18/2019
Chapter 5

7. Example: calculation of regression coefficients
Last week we calculated:
Therefore:
ŷ = a + b∙X = ∙X
1/18/2019
Chapter 5

8. Regression Coefficients by Calculator
This course supports the TI-30IIS. Other calculators are acceptable but are not supported by the instructor.
BEWARE!
The TI-30XIIS mislabels the slope & intercept. The slope is mislabeled as a and the intercept is mislabeled as b.
It should be the other way around!
1/18/2019
Chapter 5

9. Interpretation of Slope b
The slope predicts the increase in Y per unit X.
Example: ŷ = ∙X
The slope = Each unit increase in X (GDP) is associated with a increase in Y (life expectancy)
1/18/2019
Chapter 5

10. Interpretation: Intercept a
The intercept is where the line would pass through the Y-axis (when X = 0).
Example: ŷ = ∙X
The intercept = 68.7.
We do NOT normally interpolate the intercept
1/18/2019
Chapter 5

11. Regression Line for Prediction
Use regression equation to predict Y given X
Example ŷ = (0.420)X
What is the predicted life expectancy in a country with a GDP of 20.0?
ŷ = a + bX
= 68.7+(0.420)(20.0) = 77.12
1/18/2019
Chapter 5

12. Coefficient of Determination
Denoted r2 (the square r)
Interpretation: fraction of the Y “explained” by X
Illustration: Our example showed r =.809.
Therefore, r2 = = 0.66.
Interpretation: 66% of the variation in Y (life expectancy) is mathematically “explained” by X (GDP)
1/18/2019
Chapter 5

13. Cautions about regression
Linear relationships only (see prior lecture)
Influenced by outliers
Cannot be extrapolated
Association is not equal to causation! (Beware of lurking variables.)
1/18/2019
Chapter 5

14. Outliers and Influential Points
An outlier is an observation that lies far from the regression line
Outliers in the Y direction have large residuals
Outliers in the X direction are influential
1/18/2019
Chapter 5

15. Example: Influential Outlier Gesell Adaptive Score and “First Word”
After removing child 18
Line for all data
1/18/2019
Chapter 5

16. Extrapolation
Extrapolation is the use of the regression equation for predictions outside the range of explanatory variable X
Do NOT extrapolate!
See next slide
1/18/2019
Chapter 5

17. Example: extrapolation (Sarah’s height)
HS 67 (Intro Health Stat)
Friday, January 18, 2019
Example: extrapolation (Sarah’s height)
Figure: Sarah’s height from age 36 to 60 months (3 to 5 years)
Regression model: ŷ = (X)
To predict Sarah’s height at 42 months: ŷ = (42) = 88.8 cm ≈ 35” (~ 3’)
1/18/2019
Chapter 5
Chapter 5 (Regression)

18. Example: Extrapolation
Do NOT use the regression model to predict Sarah’s height at age 360 months (30 years)!
ŷ = (X)
= (360) = 216 cm = more than 7’ tall (clearly ridiculous)
1/18/2019
Chapter 5

19. Association does not imply causation
HS 67 (Intro Health Stat)
Friday, January 18, 2019
Association does not imply causation
Even strong correlations may be non-causal
See pp. 144 – 145 for examples!
1/18/2019
Chapter 5
Chapter 5 (Regression)

20. Association does not imply causation
HS 67 (Intro Health Stat)
Friday, January 18, 2019
Association does not imply causation
Criteria to establish causation (pp. 144 – 146):
Strength of relationship
Experimentation
Consistency
Dose-response
Temporality
Plausibility
1/18/2019
Chapter 5
Chapter 5 (Regression)