chapter 10 re-expressing the data math2200. if the relationship is nonlinear we may re-express the...
TRANSCRIPT
Chapter 10 Re-expressing the data
math2200
If the relationship is nonlinear
• We may re-express the data to straighten the bent relationship.
• Common ways to re-express data – Logarithms of the response variable or the ex
planatory variable, or both.– Square root of the response variable.– Reciprocals of the response variable.
Straight to the Point (cont.)
• The relationship between fuel efficiency (in miles per gallon) and weight (in pounds) for late model cars looks fairly linear at first
• Using the linear model, we get the intercept 40.65, slope -0.0057 and R-squared 82%.
Straight to the Point (cont.)• The residual plot shows a problem:
Straight to the Point (cont.)• We can re-express fuel efficiency as gallons per mile (a
reciprocal):
scatter plots: before and after re-expression
Straight to the Point (cont.)
• Improved the residuals plot.• R-squared = 88% (was 82%)• 7,200 lbs Lincoln Navigator (mpg) has predicted
9.8 mpg (reported 11.0) c.f. -0.32 mpg before the re-expression.
When do we need to re-express data?
1. To make skewed distributions more symmetric
• Why?– Easier to summarize symmetric distributions
• We can use mean and sd
– We can use a normal model
Assets of 77 large companies
After log transformation
When do we need to re-express data?
2. Make the spread of several groups more alike
• Why?– Easier to compare groups that share a comm
on spread– Some statistical methods require the assumpti
on that all groups have a common sd
Assets by market sectors
After log transformation
When do we need to re-expressing data?
3. Make the relationship more linear• Why?
– To apply linear regression
4. Make the points in a scatterplot spread out evenly rather than thicken at one end.
• Why?– Make the dataset easier to model.
After log transformation
Assets vs. Sales
The Ladder of Powers
• There is a family of simple re-expressions that move data toward our goals in a consistent way. We call this collection of re-expressions the Ladder of Powers.
• The Ladder of Powers orders the effects that the re-expressions have on data.
The Ladder of Powers
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. For counted data, start here.
Square root of data values½
Data with positive and negative values and no bounds are less likely to benefit from re-expression.
Raw data1
Try with unimodal distributions that are skewed to the left.
Square of data values2
CommentNamePower
Straighten the scatterplot
0 50 100 150 200 250 300
05
00
10
00
15
00
x
y
0 50 100 150 200 250 300
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
x
log
(y)
0 50 100 150 200 250 300
10
15
20
25
30
35
40
x
sqrt
(y)
0 50 100 150 200 250 300
-0.0
15
-0.0
10
-0.0
05
0.0
00
x
-1/y
0 50 100 150 200 250 300
-0.1
2-0
.10
-0.0
8-0
.06
-0.0
4-0
.02
x
-1/s
qrt
(y)
Raw Data Square RootLogarithm
-reciprocal - 1/square root
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).
Plan B: Attack of the Logarithms (cont.)
More generally
• We can transform both x and y
model X-axis Y-axis
Y = exp(a+bx) x Log(y)
Y = a+b log(x) Log(x) y
Y = a xb Log(x) Log(y)
Plan B: Attack of the Logarithms (cont.)
Scatterplots : GDP and log(GDP) vs. Year
Logarithms: Fishing Line Length and Strength
length vs. strength
log (length) vs. log (strength)
1/length vs. strength
1/sqrt(length) vs. strength
Multiple Benefits
• We often choose a re-expression for one reason and then discover that it has helped other aspects of the analysis.
• For example, a re-expression that makes a histogram more symmetric might also straighten a scatterplot or stabilize variance.
Why Not Just a Curve?
• If there’s a curve in the scatterplot, why not just fit a curve to the data?
-Computationally more difficult
-Straight lines are easy to understand and interpret
A general principle
• Occam’s Razor
entia non sunt multiplicanda praeter necessitatem
entities should not be multiplied beyond necessity.
When multiple competing theories are equal in other respects, the principle recommends selecting the theory that introduces the fewest assumptions and postulates the fewest hypothetical entities
What Can Go Wrong?
• Don’t expect your model to be perfect.
• Don’t choose a model based on R2 alone:– Example: large R2, but
residual plot shows curvature
What Can Go Wrong? (cont.)
• 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 and down.
What Can Go Wrong? (cont.)
• Watch out for negative data values.– Can NOT use log or square root transformations
• 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.
• Don’t stray too far from the ladder– Too difficult to interpret
What have we learned?
• When the conditions for linear 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? (cont.)
• Taking logs is often a good and 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.