AP Statistics
Chapter 10
Re-Expressing data:
Get it Straight
Objectives:
• Re-expression of data
• Ladder of powers
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.
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:
Straight to the Point
• A look at the residuals plot shows a problem:
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:
Straight to the Point
• A look at the residuals plot for the new model
seems more reasonable:
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.
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.
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.
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.
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.
The Ladder of Powers
Power Name
Comment
2
Square of
data values
1
Raw data
½
Square root of
data values
“0”
Measurements that cannot be negative
We’ll use
logarithms here often benefit from a log re-expression.
–1/2
–1
Reciprocal
square root
The reciprocal
of the data
Try with unimodal distributions that are
skewed to the left.
Data with positive and negative values
and no bounds are less likely to
benefit from re-expression.
Counts often benefit from a square
root re-expression.
An uncommon re-expression, but
sometimes useful.
Ratios of two quantities (e.g., mph)
often benefit from a reciprocal.
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.
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 yvariable.
• Then re-express the data using some
combination of x or log(x) vs. y or log(y).
Plan B: Attack of the Logarithms
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.
Why Not Just Use a Curve?
• If there’s a curve in the scatterplot, why not just
fit a curve to the data?
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 yintercept.
More Plan B: Modeling Nonlinear
Data - Logarithms
• Two specific types of nonlinear growth.
1. Exponential function (form y = abx)
2. 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.
Transforming or Re-Expressing
Exponential Data
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)
To Transform the exponential
Function use its Inverse the
Logarithmic Function
• Properties of Logarithms




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).
Logarithm Transformations
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.
Example: Testing for Exponential
Association
• Data
View Scatterplot
• Looks like it has a curved pattern, could
possibly be an exponential relationship.
Verify Exponential Association
• Density (y) • r = yn/yn-1
Is there a common ratio?
YES, r≈1.2
(mean r=1.16 and median r=1.19)
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
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
Your Turn: Is the following data
exponential & if so, what is r?
• Yes, it is exponential and r ≈ 1.45
Your Turn: Is the following data
(Hours vs. Number) exponential
& if so, what is r?
• No, it is not exponential.
Exponential Regression Procedure
1. Verify data is exponential.
 Graph scatterplot & calculate common ratio
2. Transform data to linear by taking the log of
the response variable.
3. Calculate the LSRL for the transformed data;
log ŷ =b0+b1x (linear model). Analyze using
linear techniques, LSRL, r, r2, and residuals.
4. Find exponential model for the original data by
inverse transformation of the LSRL,
exponentiating both sides of the LSRL
equation to base 10; ŷ = C • 10kx (exponential
model).
Example: Data
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
What to do:
1. Graph scatterplot.
2. Calculate common ratio.
3. Transform data to linear (take the log of y).
4. Calculate LSRL of transformed data & graph.
5. Analyze transformed data (r, r2, residual plot).
6. Perform inverse transformation (exponentiate
LSRL to base 10).
7. Graph exponential model.
Back to the Data
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
Models of Data
• Data is exponential (scatterplot curved pattern
and constant common ratio ≈ 2.1)
• Linear model
 log ŷ=-53.7+.0294x
• Exponential model
 ŷ=(10-53.7) • 10.0294x
 Use model on the calculate to make predictions,
not the exponential model equation.
 Predict oil production for 1956.
• 6564 Mbbl
 Predict oil production for 1992.
• 75027 Mbbl – extrapolation, be careful.
Your Turn: Exponential
Regression
Models of Data
• Data is exponential (scatterplot curved pattern
and constant common ratio ≈ 1.5)
• Linear Model
 Log ŷ = -24.11 + .0157x
• Exponential Model
 ŷ = (10-24.11) • (10.0157x)
Your Turn: Age vs Height
Models for Data
• Data is exponential (scatterplot curved pattern
and constant common ratio ≈ 1.04)
• Linear Model
 Log ŷ = 1.89 + .0241x
• Exponential Model
 ŷ = (101.89) • (10.0214x)
• If comparing Height vs Weight, could a common
ratio be calculated?
 NO, because the explanatory variable Height
does not in crease in equal increments.
 Have to calculate different models and see which
best fits the data.
Transforming or Re-Expression
Power Data
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.
Inverse Transformation
• Obtaining a power function model for the
original data from the LSRL on the transformed
data.
• LSRL will have the form:
 log ŷ = a + b log x
• Inverse transform the LSRL by exponentiating
both sides of the equation to base 10.
 10log ŷ = 10(a + b log x)
 ŷ = (10a)(10b log x)
 ŷ = (10a)(10log x)b
 ŷ = (10a)(xb) which is in the form y = C · xb
• A Power Function (can not be done on the
calulator, must be done by hand).
Power Function Procedure
1. Graph scatterplot.
2. Determine it is a power function (ie. not
exponential).
3. Transform data to linear (take the log of y & x).
4. Calculate LSRL of transformed data & graph.
5. Analyze transformed data (r, r2, residual plot).
6. Perform inverse transformation (exponentiate
LSRL to base 10).
7. Graph power model.
8. Make predictions based on the power model.
Example 1
• 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.
• LSRL: log(Temp.) = 4.84 - .255 log(Dist.)
log ŷ = 4.84 - .255 log x
• Power model: Temp. = (104.84)·(Dist.)-.255
ŷ = 104.84 · x-.255
Your Turn:
• The owner of a Video Game Store records the
business costs and revenue for different years
with the results listed. Find the best model.
• LSRL: log ŷ = 3.3 + .4 log x
• Power model: ŷ = 103.3 · x.4 or ŷ = (1995)x.4
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:
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.
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.
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.
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 reexpression.
• Our models won’t be perfect, but re-expression
can lead us to a useful model.