chapter 4: more on two-variable (bivariate) data
TRANSCRIPT
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)