Transforming
Relationships
AP Statistics
Practice of Statistics
Section 4.1
What You’ll Learn
 Recognize when the relationship
between two variables is either an
exponential relationship or a power
relationship
 Perform the appropriate transformation
to “linearize” the data, find the LSRL on
the transformed points, “untransform”
to find a model for the original data
Not everything in Linear!
 We’ve looked at several sets of data in which
the relationships are linear in nature
 What about those relationships that exhibit a
different “nonlinear” pattern?
 Consider for a moment gypsy moths.
 An outbreak of gypsy moths in Massachusetts
from 1978 to 1981 resulted in many acres of
defoliated land. The acreages are listed in the
following table.
Gypsy Moths
 The data and
graph depict
the number of
acres
defoliated by
gypsy moths in
Massachusetts
between 1978
and 1981.
Years
1978
1979
1980
1981
Acres of
Defoliated
land
63042
226260
907075
2826095
 So, this doesn’t look too bad! Let’s try a linear
regression on the data, remembering to check
both the correlation coefficient and the
residual plot.
Simple Linear Regression
Simple linear regression results:
Dependent Variable: Acres
Independent Variable: Year
Acres = -1.7746007E9 + 896997.4 (Year)
Sample size: 4
R (correlation coefficient) = 0.9136
R-sq = 0.8347045
Estimate of error standard deviation: 631139.44
Well a visual of the line doesn’t look
too bad, and that’s a great
correlation coefficient.
(remember though, sometimes “r” is
deceptive---be sure to check the
residuals!)
The Residuals
 A check of the residuals indicates that a linear model
is not appropriate! (Notice the parabolic pattern in
the plot that even with only 4 data points can be
seen!)
So, what type of relationship is
this?
 Remember from linear regression that when the
relationship is linear, the response variable
increases (or decreases) by a constant amount.
Years Since 1977
Acres of defoliated land
Difference in Acres
1
2
3
4
63042
226260
907075
2826095
163218
680815
1919020
•Notice that the difference between number of acres is not constant
•With this in mind and the problem with the residual plot, let’s consider
another type of relationship.
Exponential Relationships
 In an exponential relationship, the response variable increases by a
fixed percentage of the previous total. In other words, we should be
able to multiply the previous value by some constant to get the next
one.
 So, let’s check out this possibility (we will again disregard the increase
from 1990-1993 and only look at the increases for 1-year intervals.
Years Since 1977
Acres of defoliated land
Ratio (Next/Prev)
1
2
3
4
63042
226260
907075
2826095
3.5890
4.0090
3.1156
•Notice that although the ratio is not exactly the same (we wouldn’t
expect it to be exact with “real” data) that there does appear to be a
pretty consistent ratio value.
So How Do We Create the
Model?
 If the relationship is an exponential
one, we can use a mathematical
transformation to “linearize” the data,
find the LSRL of the transformed data,
then “untransform” to find the model
that will fit the original data.
 Ok, so let’s take all of that step by step
Finding the Model
 Step 1: Use a mathematical model to “linearize” (create a new
data set whose relationship is linear)
 If the original data is exponential, find the logarithm (either
common log or natural log) of each of the response values.
 When working with years it is also helpful to “code” the year data
so our calculators can handle the values (most computer programs
are capable of creating models using the full year) To do this we will
take each year and subtract 1977 (this way all of our values are > 0)
Years
1978
1979
1980
1981
Acres of Defoliated
land
63042
226260
907075
2826095
1
2
3
4
4.7996
5.3546
5.9576
6.4512
Years Since 1977
Log10 (acres)
Finding the Model
 Now, let’s check a scatterplot of the
transformed data
Notice the change in the pattern from our original data to the
transformed data. The logarithm transformation really “straightened
our data”. (Using the natural logarithm would have had the same
effect, our values would have just been different)
Finding the Model
 Step 2: Find the LSRL for the
transformed data (remember to check
the “r” and the residuals!)
Simple Linear Regression
Simple linear regression results:
Dependent Variable: log10(Acres)
Independent Variable: Year-1977
log 10(Acres) = 4.2513404 + 0.5557706 (Year-1977)
Sample size: 4
R (correlation coefficient) = 0.9993
R-sq = 0.9985874
Estimate of error standard deviation: 0.033050213
This model looks promising, but remember to check the residuals.
Finding the Model
A check of the residuals confirms that an
exponential model is appropriate.
“Untransforming” to find the
model for our original data
 Remember that our goal was to find a model that we
could use for prediction of the number of defoliated
acres of land for a given year.
 The linear model we have would predict the common
logarithm of acres. In order for our model to be
useful, we need to reverse the transformation to
create the model that fits the original data.
 Although many transformations are easier to
“untransform” after evaluating, we can use the
properties of logarithms with both exponential and
power (we’ll look at those next) to find the model for
our original data.
Properties of Logarithms
 Before we try to “untransform”, let’s
review the properties of logarithms you
learned in Algebra (yes, you really did
learn these!)
 Logb xy = logb x + logb y
(Addition rule)
 Logb xm = mlogb x
(Power rule)
 Logb bn = n
(Same base)
 Logb(x/y) = logb x – logb y (Subtraction rule)
 Since any subtraction can be changed to an addition
equation, we will not use this last rule much!
“Untransforming” exponential
expressions
 An exponential function takes the form:
 y = abx, where a, b are constants
 (This is the form we want to end up with)
 So, let’s get started
log10 (Acres) = 4.2513404 + 0.5557706 (Year-1977)
Linear regression of the transformed data
10log10(Acres) = 10
Raise both sides using power of 10 (same
base)
4.2513404 + 0.5557706 (Year-1977)
Acres = 10 4.2513404 (10.5557706(Year-1977))
Same base law and multiplication law for
exponents.
Acres = 17837.7634 (3.5956(Year-1977))
Simplify the constants
This is now in the form of y=abx, where a=17837.7634 and b = 3.5956
Notice that “b” is approximately the average of the ratios (next/prev)
we calculated when we began looking for a model.
So, does it fit our original data?
 Since our original goal was to find a model that would
allow us to predict the number of acres of defoliated
land if we knew the year, we need to check to see if
our model actually fits the data.
Scatter Plot
Gypsy Moth Outbreak
3000000
2500000
2000000
Acres
The model looks pretty
good, but as with any model
we need to use caution when
predicting outside our
original data range.
1500000
1000000
500000
0
1.0
Acres =
1.5
2.0
2.5
3.0
YearsSince1977
Y earsSince1977
3.5
4.0
4.5
Power Models
 Another important transformation used in
modeling is the power model.
 Power models have the form
 Y = axb where a and b are constants
 We can find an appropriate power model by
taking the logarithms for both the response
and explanatory variables, finding the linear
regression for the transformed data, then
using the laws of logarithms and exponents to
“untransform”
 Let’s look at an example
Fishing Tournament
 In a fishing tournament that you are in charge of you
need to find a way to record the weight of each fish
caught without destroying or killing the fish.
 Since it is easier to measure the length of the fish
rather than it’s weight, we must find a way to convert
the length to weight.
 The local marine research lab has been gracious
enough to provide you with the data for the average
length and weight at different ages for Atlantic
Ocean rockfish which model most fish species
growing under normal feeding conditions.
The Data
Age (yr)
Length
(cm)
Weight
(g)
1
5.2
2
2
8.5
8
3
11.5
21
4
14.3
38
5
16.8
69
6
19.2
117
7
21.3
148
8
23.3
190
9
25.0
264
10
26.7
293
11
28.2
318
12
29.6
371
13
30.8
455
14
32.0
504
15
33.0
518
16
34.0
537
17
34.9
651
18
36.4
719
19
37.1
726
20
37.7
810
•Since length is one dimensional and weight is
three dimensional we should be able to find a
reasonable model using power model (the residuals
for a regression on the original data confirms
that the variables are NOT linearly related—but
we already knew that!)
•As before we need to first transform our data
but we have to perform transformations on both
length and weight
Transforming the Data
Age (yr)
Length
(cm)
Log 10
(length)
Weight
(g)
Log10
(weight)
1
5.2
.7160
2
.3010
2
8.5
.9294
8
.9031
3
11.5
1.0607
21
1.3222
4
14.3
1.1553
38
1.5798
5
16.8
1.2253
69
1.8388
6
19.2
1.2833
117
2.0682
7
21.3
1.3284
148
2.1703
8
23.3
1.3674
190
2.2788
9
25.0
1.3979
264
2.4216
10
26.7
1.4265
293
2.4669
11
28.2
1.4502
318
2.5024
12
29.6
1.4713
371
2.5694
13
30.8
1.4886
455
2.6580
14
32.0
1.5052
504
2.7024
15
33.0
1.5315
518
2.7143
16
34.0
1.5428
537
2.7300
17
34.9
1.5611
651
2.8136
18
36.4
1.5694
719
2.8567
19
37.1
1.5763
726
2.8609
20
37.7
1.5763
810
2.9085
This scatterplot indicates that a
linear regression on the
logarithms of both variables is
certainly one to consider.
Linear Regression on the
transformed data
Simple linear regression results:
Dependent Variable: log10(Weight(g))
Independent Variable: log10(Length(cm))
log10 (Weight(g)) = -1.8993973 + 3.049418 log10 (Length(cm))
Sample size: 20
R (correlation coefficient) = 0.9993
R-sq = 0.9985228
A check of the correlation coefficient is
certainly promising (r=.9993), the
scatterplot of the transformed data
indicates the line fits very well, and most
importantly-----look at those residuals!!!
Yes, statisticians get very excited when
they see residuals that look that good!
“Untransforming” a power model
log10 (Weight(g)) = -1.8993973 + 3.049418 log10 (Length(cm))
Linear equation of the transformed data
10log10(Weight(g)) = 10-1.8993973 + 3.049418 log10(length(cm))
Raise both sides using a base of 10
Same base and Multiplication law for
exponents
Weight = 10-1.8993973 (103.049418log10(length(cm)))
Weight = 10-1.8993973(10log10(length(cm))
3.049418
)
Power rule for logarithms
Weight = 10-1.8993973(length(cm))3.049418)
Same base
Weight = .01261 (length(cm))3.049418
Simplify constants
Looks like we’ve got a model
that will be very useful for
estimating the weight of a
fish if we know its length!
Weight
Last check: plot the new
model on the original data.
Scatter Plot
Atlantic Ocean Rockfish
900
800
700
600
500
400
300
200
100
0
5
Weight =
10
15
20
25
Length
Length
30
35
40
Are there Other Possibilities?
 There are many other possibilities to
transform data in order to find a model.
 If either an exponential or power model
is not appropriate you may try:
 Square the response or explanatory variable
 Take the square root of either variable
 Take the reciprocal of either variable
 The possibilities are endless, but for now we
will concentrate mostly on either an
exponential or power model.
Transforming on the TI
 There are a couple of different ways to
find both an exponential and power
regression model on your TI-calculator
 Using lists to transform
 Using the built in regression models
Using lists to transform
 We’ll use the Gypsy Moth data first.
Enter in lists 1 & 2
L1: years since 1977
L2: acres of defoliated land
Take the common log of the
values in list 2 and put the new
values in list 3
L3: log (L2)
Now do a linear regression on
lists 1 & 3
You can check residuals just like
we did before to verify this
regression.
Now “untransform” as we did
before to get the exponential
Note: for a power model create
another list for the logarithm of the
explanatory variable and do the linear
regression on these two lists.
Using the Regression Models
 The TI family of calculators has both an exponential and power
model built into the stat calc menus.
 Create a list for the explanatory variable and one for the
response variable
 From the home screen
 STAT
 CALC
 0:ExpReg
(A:PwrReg)
L1, L2
 The model does not need
untransforming
 The residuals created are
the residuals from the
linear transformation on
the transformed data
(yes, your calculator
actually transforms the
data, does a linear
regression, then
untransforms
How to decide which model
 Creating mathematical models for real data
involves a lot of trial and error.
 One strategy:
 Try a linear model first (
residuals)
 Then try an exponential model (
residuals)
 Then try a power model (
residuals)
 If all residuals show a pattern, you can
continue to try different transformations or
choose the one with the best correlation
 Remember, no model is perfect, some models
are useful…..we wish to find a useful model.