Chapter 4:More on

Two-Variable (Bivariate) Data

4.1 Transforming Relationships Animal’s Brain Weight vs. Weight of BodyAnimal’s Brain Weight vs. Weight of Body

Outliers

r=.86

Logarithm

r=.50

Drop Outliers

Plot Logarithm vs. Logarithm

r=.96 The vertical spread about the LSRL is similar everywhere, so the predictions of brain weight from body weight will be pretty precise (high r2) – in LOG SCALE

Working with a function of our original measurements can greatly simplify statistical analysis.

Transforming-

How?

Recall…Recall… Chapter 1 we did Linear Transformations

Took a set of data and transformed it linearly Called:

SHIFTING

C to F

Meters to Miles

4 669

15

A Linear Transformation CANNOT make a curved relationship between 2 variables “straight”

Resort to common non-linear functions like the logarithm, positive & negative powers

We can transform either one of the explanatory/response variables OR BOTH when we do we will call the variable “t”

Real World Example:Real World Example:We measure fuel consumption of a car in

miles per gallon

Engineers measure it in gallons per mile (how many gallons of fuel the car needs to travel 1 mile)

Reciprocal Transformation: 1/f(t)

My Car- 25 miles per gallon

1/25=.04 gallons per mile

Monotonic Function A A monotonic functionmonotonic function f(t) moves in one f(t) moves in one

direction as its argument “t” increases direction as its argument “t” increases Monotonic IncreasingMonotonic Increasing

Monotonic DecreasingMonotonic Decreasing

Monotonic Increasing:

2t

Positive “t”

a + bt

slope b>0tlog2t

Monotonic Decreasing:

Positive “t”

a + bt

slope b<0 2

11

tt

11 tt

Nonlinear monotonic transformations change data enough to change form or relations between 2 variables, yet preserve order and allow recovery of original data.

Strategy:1.1. If the variable that you want to If the variable that you want to

transform has values that are 0 or transform has values that are 0 or negative apply linear transformation negative apply linear transformation (add a constant) to get all positive.(add a constant) to get all positive.

2.2. Choose power or logarithmic Choose power or logarithmic transformation that approximately transformation that approximately straightens the scatterplot.straightens the scatterplot.

Ladder of Power Transformations:

Power Function: tP

Power Functions:

Monotonic Power Function

For t > 0….

1. Positive p – are monotonic increasing

2. Negative p – are monotonic decreasing

2t

2

11

tt

Monotonic Decreasing- Hard to interpret because reversed order of original data point

We want to make all tP therefore monotonic increasing.

We can apply a

LINEAR TRANSFORMATION p

t p 1

Original Data (t) Original Data (t) Power FunctionPower Function Linear Trans:Linear Trans:

00 undefinedundefined UndefinedUndefined

11 11 00

22

33

44

Linear Transformation: Linear Transformation:

1

11

t1t

p

t p 1

11 tt

11 tt

11 tt

11 ttp

t p 1

p

t p 1

This is log t

This is a line

Concavity of Power Functions:

P is greater than 1 =

- Push out right tail & pull in left tail

- Gets stronger as power p moves up away from 1

P is less than 1 =

- Push out left tail & pull in right tail

- Gets stronger as power p moves down away from 1

Country’s GDB vs. Life Expectancy

P=

P=

P=

Use

x

1

How do you know what transformation will make the scatterplot straight?

** DO NOT just push buttons!! ** We will develop methods of selection

1. Logarithmic Transformation

2. Power Transformation

1. Logarithmic 1. Logarithmic TransformationsTransformations

Exponential GrowthExponential Growth

A variable grows…Linearly:

Exponentially:

The King’s Chess Board…The King’s Chess Board…

xbay King’s Offer: 1,000,000 grains - 30 days

bxay

Wise Man: 1 grain per day and double for 30 days

Cell Phone GrowthCell Phone Growth

Suspect Exponential Suspect Exponential Growth…Growth…1.1. Calculate Ratios of Consecutive TermsCalculate Ratios of Consecutive Terms

- IF approximately the same… continue- IF approximately the same… continue

Suspect Exponential Suspect Exponential Growth…Growth…

2. Apply a Transformation that: 2. Apply a Transformation that:

a. Transforms exponential growth into a. Transforms exponential growth into linear growth linear growth

b. Transforms non-exponential growth b. Transforms non-exponential growth into non-linear growthinto non-linear growth

xbay

Logarithm Review…Logarithm Review…

xbifonlyandifyx yb log

1.log(AB)=

2.log(A/B)=

3.logXp =

The Transformation…The Transformation…We hypothesize an exponential model of the

form y=abx

To gain linearity, use the (x, log(y)) To gain linearity, use the (x, log(y)) transformationtransformation

)log(log xaby ylogylog

Form? –

When our data is growing exponentially… if When our data is growing exponentially… if we plot the log of y versus x, we should we plot the log of y versus x, we should observe a straight line for the observe a straight line for the transformed data!transformed data!

xbay )(logloglog

LOG (Y) = -263 + 0.134 (year)

R-sq = 98.2%

Eliminate first 4 years & perform regressionEliminate first 4 years & perform regression

LOG (New Y) = -189 + 0.0970(New X)

R-sq = 99.99%

Predictions in Predictions in Exponential Growth ModelExponential Growth Model

Regression is often used for predictionsRegression is often used for predictions In exponential growth, ________ rather In exponential growth, ________ rather

than actual values follow a linear patternthan actual values follow a linear pattern To make a prediction of Exp. Growth we To make a prediction of Exp. Growth we

must thus “undo” the logarithmic must thus “undo” the logarithmic transformation.transformation.

The inverse operation of a logarithm is _____________________

LOG (New Y) = -189 + 0.0970(New X)

R-sq = 99.99%

Predict the number of cell phone users in 2000.

)(0970.189)log( 1010 NewXNewY

y

y

ˆ

ˆ

If a variable grows exponentially… its ___________ grow linearly!

In other words… if (x, y) is exponential, then (x, log(y)) is linear!

Read and do Technology Toolbox- Page 210-211 on your own!!

2. Power 2. Power TransformationsTransformations

Example:

Pizza Shop- order pizza by diameter10 inch 12 inch 14 inch

Amount you get depends on the area of the pizza

Area circle = pi times the square of the radius

222

2

442x

xxrarea

Power Law Model

Power Law ModelPower Law Model

Power LawsPower Laws We expect area to go up with the square of We expect area to go up with the square of

dimensiondimension We expect volume to go up with the cube of a We expect volume to go up with the cube of a

dimensiondimension

Real Examples: Many Characteristics of Living Real Examples: Many Characteristics of Living ThingsThings

Kleiber’s Law-Kleiber’s Law- The rate at which animals use The rate at which animals use energy goes up as the ¾ power of their body energy goes up as the ¾ power of their body weight (works from bacteria to whales).weight (works from bacteria to whales).

Power Laws Become LinearPower Laws Become Linear Exponential growth becomes linear when Exponential growth becomes linear when

we apply the logarithm to the response we apply the logarithm to the response variable (y).variable (y).

Power Laws become linear when we Power Laws become linear when we apply the logarithm transformation to apply the logarithm transformation to BOTH variables.BOTH variables.

To Achieve Linearity…To Achieve Linearity…

1. The power law model is

2. Take the logarithm of both sides of equation (this straightens scatterplot)

3. Power p in the power law becomes the slope of the straight line that links log(y) to log(x)

4. Undo transformation to make prediction

pxay

Fish Example…Fish Example…Read Example 4.9 page 216Read Example 4.9 page 216

Model: weight = a x length3

Log (weight) = log a + [3x log(length)]

Yes appears very linear- perform LSRL on [log(length), log(weight)]

LSRL:

log(weight)= -1.8994 + 3.0494log(length)

r = .9926 r2 = .9985

log(weight)= -1.8994 + 3.0494log(length)

= -1.894 + log(length)3.0494

This is the final power equation for the original data (note- look at p-value)!

Prediction…Prediction…

Why did we do this?

weight = 10-1.8994 x length3.0494

Predict the weight of a 36cm fish

Summary- Order of Summary- Order of Checking…Checking… 1. Look to see if there is a 1. Look to see if there is a

___________________if so use LSRL___________________if so use LSRL 2. If points are ____________plot (x, log 2. If points are ____________plot (x, log

y) or (x, ln y) to gain linearityy) or (x, ln y) to gain linearity 3. If there is a _________ relationship 3. If there is a _________ relationship

(power model) plot (logx, logy)(power model) plot (logx, logy) 4. If the scatterplot looks ________ plot 4. If the scatterplot looks ________ plot

(logx, y)(logx, y)