Chapter 10 Re-expressing the data

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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

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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.