1. Chapter 10 Re-expressing Data: Get it Straight!!
*Straightening Relationships
*Goals of Re-Expression
*Ladder of Powers

2. Straightening Relationships
To use a linear model, the scatterplot must be straight enough
Check scatterplot AND residual plot
We have the ability to straighten data so that we can use a linear model for scatterplots that do not satisfy the straight enough condition

3. MPG and Weight

4. A Hummer weighs about 6000 pounds. What is the predicted MPG?

5. MPG vs Gallons/100 Miles
Change 25 mpg into gallons/100 miles

6. Scatterplot: gal/100 miles and weight

7. Revisit the Hummer
What is the predicted fuel efficiency for a Hummer? (6000 lbs)
The new model predicts that a 6000 lbs Hummer would get 9.7 gallons/100 miles
Convert that back into MPG

8. Not Sold?? You regularly use re-expression
What units do you use to talk about how fast you went on a bike?
What units do you use to talk about how fast you run?

9. Goals of Re-Expressing
1) Make the distribution of a variable more symmetric
easier to compare centers
if its unimodal you could perhaps use the Normal Model

10. Goals Make the spread of several groups more alike
groups with similar spreads are easier to compare
centers may be different

11. Goals Make the form of a scatterplot more nearly linear
linear models are easier to describe

12. Goals
Make the scatter in a scatterplot spread out evenly rather than following a fan shape
having an even scatter is a condition of many methods in Stats (we will see later)

13. Ladder of Powers Use to systematically re-express data
The farther you move from 1 (original data) the greater the effect on the data
Certain re-expressions work better for different types of data

14. Ladder of Powers 2 y2 1 y 1/2 logy -1/2 -1/ -1 -1/y Power Name Comment
unimodal distributions that are skewed to the left 1 y
data that can be both positive and negative and continue without bond; less likely for re-expression
1/2
counted data
logy
measurements that can NOT be negative; values that grow by percentages (salaries, populations); if the data has zeros add a small constant to each value
-1/2
-1/
uncommon; changing the sign to take the negative of the reciprocal square root preserves the direction -1
-1/y
ratios of two quantities (mpg); change the sign if you want to preserve the direction; if there are zeros, add a small constant to all values

15. Plan B: Attack of the Logs
Try taking the logs of BOTH the x values and the y values.
Model Name
x-axis
y-axis
Comment
Exponential x
log(y)
“0” power from the ladder
Logarithmic
log (x) y
wide range of x-values, scatterplot descending rapidly at the left and trailing to the right
Power
log (y)
when you are in between powers on the ladder

16. Example
Let’s try to predict the shutter speed based off the f/stop of a cameras lens.
Enter data
Shutter Speed
1/1000
1/500
1/250
1/125
1/60
1/30
1/15
1/8
f/stop
2.8 4
5.6 8 11 16 22 32

17. What Can Go Wrong? Don’t expect your model to be perfect
Don’t choose a model based on R2 alone
always check the residual plot
Watch out for scatterplots that change direction
Watch out for negative values
Rescale years
Don’t stray too far from the ladder