1. Chapter 10 Re-Expressing data: Get it Straight
AP Statistics
Chapter 10
Re-Expressing data:
Get it Straight

2. Objectives:
Re-expression of data
Ladder of powers

3. Straight to the Point
We cannot use a linear model unless the relationship between the two variables is linear. Often re-expression (transformation) can save the day, straightening bent relationships so that we can fit and use a simple linear model.
Two simple ways to re-express data are with logarithms and reciprocals.
Re-expressions can be seen in everyday life—everybody does it.

4. Straight to the Point
The relationship between fuel efficiency (in miles per gallon) and weight (in pounds) for late model cars looks fairly linear at first:

5. Straight to the Point
A look at the residuals plot shows a problem:

6. Straight to the Point
We can re-express fuel efficiency as gallons per hundred miles (a reciprocal) and eliminate the bend in the original scatterplot:

7. Straight to the Point
A look at the residuals plot for the new model seems more reasonable:

8. Goals of Re-expression
Goal 1: Make the distribution of a variable (as seen in its histogram, for example) more symmetric.
It’s easier to summarize the center of a symmetric distribution, we can use the mean and standard deviation.
If the distribution is unimodal also, we can analysis using the normal model.
Here taking the log of the explanatory variable.

9. Goals of Re-expression
Goal 2: Make the spread of several groups (as seen in side-by-side boxplots) more alike, even if their centers differ.
Groups that share a common spread are easier to compare.
Here taking the log makes the individual boxplots more symmetric and gives them spreads that are more nearly equal.

10. Goals of Re-expression
Goal 3: Make the form of a scatterplot more nearly linear.
Linear scatterplots are easier to model.
By re-expressing to straighten the scatterplot relationship we can fit a linear model and use linear techniques to analysis.
Here taking the log of the response variable.

11. Goals of Re-expression
Goal 4: Make the scatter in a scatterplot spread out evenly rather than thickening at one end.
Having an even scatter is a condition of many methods of Statistics, as we will see later.
This is closely related to goal 2, but often comes along with goal 3, as seen below. When taking the log to straighten the data, it also evened out the spread.

12. The Ladder of Powers
There is a family of simple re-expressions that move data toward our goals in a consistent way. This collection of re-expressions is called the Ladder of Powers.
The Ladder of Powers orders the effects that the re-expressions have on data.

13. The Ladder of Powers 2 1 ½ “0” –1/2 –1
Ratios of two quantities (e.g., mph) often benefit from a reciprocal.
The reciprocal of the data –1
An uncommon re-expression, but sometimes useful.
Reciprocal square root
–1/2
Measurements that cannot be negative often benefit from a log re-expression.
We’ll use logarithms here
“0”
Counts often benefit from a square root re-expression.
Square root of data values
Data with positive and negative values and no bounds are less likely to benefit from re-expression.
Raw data 1
Try with unimodal distributions that are skewed to the left.
Square of data values 2
Comment
Name
Power

14. The Ladder of Powers
The Ladder of Powers orders the effects that the re-expressions have on data.
How it works.
If you try taking the square root of all the values in a variable and it helps, but not enough, then move further down the ladder to the log or reciprocal root. Those re-expressions will have a similar, but even stronger, effect on your data.
If you go too far, you can always back up.
Remember, when you take a negative power, the direction of the relationship will change. This is OK, you can always change the sign of the response variable if you want to keep the same direction.

15. Plan B: Attack of the Logarithms
When none of the data values is zero or negative, logarithms can be a helpful ally in the search for a useful model.
Try taking the logs of both the x- and y-variable.
Then re-express the data using some combination of x or log(x) vs. y or log(y).

16. Plan B: Attack of the Logarithms

17. Multiple Benefits
We often choose a re-expression for one reason and then discover that it has helped other aspects of an analysis.
For example, a re-expression that makes a histogram more symmetric might also straighten a scatterplot or stabilize variance.

18. Why Not Just Use a Curve?
If there’s a curve in the scatterplot, why not just fit a curve to the data?

19. Why Not Just Use a Curve?
The mathematics and calculations for “curves of best fit” are considerably more difficult than “lines of best fit.”
Besides, straight lines are easy to understand.
We know how to think about the slope and the y-intercept.

20. More Plan B: Modeling Nonlinear Data - Logarithms
Two specific types of nonlinear growth.
Exponential function (form y = abx)
Power function (form y = axb)
Equations of both forms can be transformed into linear forms.
Can then use linear regression to model and analyze the transformed data.
Can also perform an inverse transformation to obtain a model of the original data.

21. Transforming or Re-Expressing Exponential Data

22. Linear vs. Exponential Growth
Linear Growth – A variable grows linearly over time if it adds a fixed increment in each equal time period.
Arthmetic Sequence – common difference
(yn-yn-1)
Exponential Growth – A variable grows exponentially if it is multiplied by a fixed number greater than 1 in each equal time period. Exponential decay occurs when the factor is less than 1.
Geometric Sequence – common ratio (yn/yn-1)

23. To Transform the exponential Function use its Inverse the Logarithmic Function
Properties of Logarithms

24. Using Logarithms to Transform Data
Logarithms can be useful in straightening a scatterplot whose data values are greater than zero.
Remember, you cannot take the logarithm of a nonpositive number.
When you use transformed data to create a linear model, your regression equation is not in terms of (x,y) but in terms of the transformed variable(s) (log ŷ or log x).

25. Logarithm Transformations

26. Test for Exponential Functions
View the scatterplot, does it look exponential?
Calculate the common ratio between successive response values – yn/yn-1.
Can only be used if the explanatory values (x) change in equal increments.

27. Example: Testing for Exponential Association
Data

28. View Scatterplot
Looks like it has a curved pattern, could possibly be an exponential relationship.

29. Verify Exponential Association
Density (y)
4.5
6.1
4.3
5.5
7.4
9.8
7.9
10.6
13.4
16.9
21.2
25.6
31.0
35.6
41.2
44.2
50.7
50.6
57.4
64.0
70.3
Common ratio = yn/yn-1
6.1/4.5 = 1.36
4.3/6.1 = .70
5.5/4.3 = 1.28
7.4/5.5 = 1.35
9.8/7.4 = 1.32
7.9/9.8 = .81
10.6/7.9 = 1.34
13.4/10.6 = 1.26
16.9/13.4 = 1.26
21.2/16.9 = 1.25
25.6/21.2 = 1.21
31.0/25.6 = 1.21
35.6/31.0 = 1.15
41.2/35.6 = 1.16
44.2/41.2 = 1.07
50.7/44.2 = 1.15
50.6/50.7 = 1.00
57.4/50.6 = 1.13
64.0/57.4 = 1.11
70.3/64.0 = 1.10
Is there a common ratio?
YES, common ratio≈1.2

30. Your Turn: Is the following data exponential & if so, what is the common ratio?

31. Your Turn: Is the following data exponential & if so, what is common ratio?

32. Exponential Regression Procedure
Verify data is exponential.
Graph scatterplot & calculate common ratio
Transform data to linear by taking the log of the response variable.
Calculate the LSRL for the transformed data; log ŷ =b0+b1x (linear model).
Analyze using linear techniques, LSRL, r, r2, and residuals.
Make predictions using the LSRL (linear model). Calculate ŷ from the log ŷ by exponentiating to base 10 (example; log ŷ = 2.41, then ŷ = = ).

33. Example: Data Year Mbbl 1880 1890 1900 1910 1920 1930 1940 1950 1960
Annual crude oil production from 1880 to 1970
Year
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
Mbbl 30 77
149
328
689
1,412
2,150
3,803
7,674
16,690

34. What to do:
Graph scatterplot.
scatterplot shows curved pattern

35. What to do: continued Calculate common ratio. 77/30 = 2.567
149/77 = 1.935
328/149 = 2.201
689/328 = 2.101
1412/689 = 2.049
2150/1412 = 1.523
3803/2150 = 1.769
7674/3803 = 2.018
16690/7674 = 2.175
There is a common ratio of ≈ 2, so the data are exponential.
Year
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
Mbbl 30 77
149
328
689
1,412
2,150
3,803
7,674
16,690

36. What to do: continued
Transform data to linear (take the log of y).

37. What to do: continued
Graph transformed data and calculate LSRL of transformed data & graph.
LSRL: log 𝑀𝑏𝑏𝑙 =− (𝑦𝑒𝑎𝑟)

38. What to do: continued Analyze transformed data (r, r2, residual plot).
r, R2, and residual plot support that the transformed data are linear.

39. What to do: continued
Make predictions using the LSRL (linear model, LSRL: log 𝑀𝑏𝑏𝑙 =− (𝑦𝑒𝑎𝑟) ).
Predict Mbbl for 1955.
log 𝑀𝑏𝑏𝑙 =− (1955)
log 𝑀𝑏𝑏𝑙 =3.8
𝑀𝑏𝑏𝑙 = =6309.6

40. Your Turn: Exponential Regression

41. Transforming or Re-Expression Power Data

42. Power Function Model Power Function general form: y = axb
When we apply the log transformation to the response variable y in an exponential growth model, we produce a linear relationship. To produce a linear relationship from a power function model, we apply the log transformation to both variables (x & y).
Here is how it is done.
Power function model: y = axb
Take the log of both sides of the equation:
log y = log (axb)
Using the product and power properties of logs, this results in a linear relationship between log y and log x.
log y = log a + log xb
log y = log a + b log x
The power b in the power function model becomes the slope of the straight line that links log y to log x.

43. Power Function Procedure
Graph scatterplot.
Determine it is a power function, calculate common ratio (ie. not exponential – no common ratio, ratio shows increasing or decreasing pattern).
Transform data to linear (take the log of y & x).
Calculate LSRL of transformed data & graph.
Analyze transformed data (r, r2, residual plot).
Make predictions based on the transformed LSRL.

44. Example:
The table shows the temperature of an instrument measured as its distance from a heat source is varied. Find a suitable model for Dist. vs Temp.

45. Solution:
Graph scatterplot.
Scatterplot shows a curved pattern.

46. Solution: continued
Determine it is a power function, calculate common ratio (ie. not exponential – no common ratio, ratio shows increasing or decreasing pattern).
105/130 = .808
95/105 = .905
87/95 = .916
83/87 = .954
80/83 = .964
78/80 = .975
77/78 = .987
No common ratio, ratio shows increasing pattern – therefore power function.

47. Solution: continued
Transform data to linear (take the log of y & x) and graph.
Transformed data is now approximately linear.

48. Solution: continued Calculate LSRL of transformed data & graph.
LRSL: log 𝑇𝑒𝑚𝑝 =2.104− log 𝐷𝑖𝑠𝑡

49. Solution: continued Analyze transformed data (r, r2, residual plot).
r, R2, and residual plot support that the transformed data are linear.

50. Solution: continued Make predictions based on the transformed LSRL.
Predict the temp for a distance of 10 cm.
LRSL: log 𝑇𝑒𝑚𝑝 =2.104− log 𝐷𝑖𝑠𝑡
log 𝑇𝑒𝑚𝑝 =2.104− log 10
log 𝑇𝑒𝑚𝑝 =2.104− log 𝐷𝑖𝑠𝑡
log 𝑇𝑒𝑚𝑝 =1.8489
𝑇𝑒𝑚𝑝 =70.6 0F

51. Your Turn: Power Regression
The owner of a Video Game Store records the business costs and revenue for different years with the results listed. Find the best model.

52. What Can Go Wrong? Don’t expect your model to be perfect.
Don’t stray too far from the ladder.
Don’t choose a model based on R2 alone:

53. What Can Go Wrong? Beware of multiple modes.
Re-expression cannot pull separate modes together.
Watch out for scatterplots that turn around.
Re-expression can straighten many bent relationships, but not those that go up then down, or down then up.

54. What Can Go Wrong? Watch out for negative data values.
It’s impossible to re-express negative values by any power that is not a whole number on the Ladder of Powers or to re-express values that are zero for negative powers.
Watch for data far from 1.
Data values that are all very far from 1 may not be much affected by re-expression unless the range is very large. If all the data values are large (e.g., years), consider subtracting a constant to bring them back near 1.

55. What have we learned?
When the conditions for regression are not met, a simple re-expression of the data may help.
A re-expression may make the:
Distribution of a variable more symmetric.
Spread across different groups more similar.
Form of a scatterplot straighter.
Scatter around the line in a scatterplot more consistent.

56. What have we learned?
Taking logs is often a good, simple starting point.
To search further, the Ladder of Powers or the log-log approach can help us find a good re-expression.
Our models won’t be perfect, but re-expression can lead us to a useful model.