1. Transforming Relationships
AP Statistics
Practice of Statistics
Section 4.1

2. 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

3. 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.

4. 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

5. So, this doesn’t look too bad
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 = E (Year) Sample size: 4 R (correlation coefficient) = R-sq = Estimate of error standard deviation:
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!)

6. 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!)

7. 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 1 2 3 4
Acres of defoliated land
63042
226260
907075
Difference in Acres
163218
680815
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.

8. 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 and only look at the increases for 1-year intervals.
Years Since 1977 1 2 3 4
Acres of defoliated land
63042
226260
907075
Ratio (Next/Prev)
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.

9. 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

10. 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
Years Since 1977 1 2 3 4
Log10 (acres)
4.7996
5.3546
5.9576
6.4512

11. 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)

12. 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 log 10(Acres) = (Year-1977) Sample size: 4 R (correlation coefficient) = R-sq = Estimate of error standard deviation:
This model looks promising, but remember to check the residuals.

13. Finding the Model
A check of the residuals confirms that an exponential model is appropriate.

14. “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.

15. 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!

16. “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) = (Year-1977)
10log10(Acres) = (Year-1977)
Acres = ( (Year-1977))
Acres = (3.5956(Year-1977))
Linear regression of the transformed data
Raise both sides using power of 10 (same base)
Same base law and multiplication law for exponents.
Simplify the constants
This is now in the form of y=abx, where a= and b =
Notice that “b” is approximately the average of the ratios (next/prev) we calculated when we began looking for a model.

17. 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.
The model looks pretty good, but as with any model we need to use caution when predicting outside our original data range.

18. 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

19. 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.

20. The Data
Age (yr)
Length (cm)
Weight (g) 1
5.2 2
8.5 8 3
11.5 21 4
14.3 38 5
16.8 69 6
19.2
117 7
21.3
148
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

21. Transforming the Data
Age (yr)
Length (cm)
Log 10 (length)
Weight (g)
Log10
(weight) 1
5.2
.7160 2
.3010
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
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
810
2.9085
This scatterplot indicates that a linear regression on the logarithms of both variables is certainly one to consider.

22. Linear Regression on the transformed data
Simple linear regression results: Dependent Variable: log10(Weight(g)) Independent Variable: log10(Length(cm)) log10 (Weight(g)) = log10 (Length(cm)) Sample size: 20 R (correlation coefficient) = R-sq =
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!

23. “Untransforming” a power model
log10 (Weight(g)) = log10 (Length(cm))
10log10(Weight(g)) = log10(length(cm))
Weight = ( log10(length(cm)))
Weight = (10log10(length(cm)) )
Weight = (length(cm)) )
Weight = (length(cm))
Linear equation of the transformed data
Raise both sides using a base of 10
Same base and Multiplication law for exponents
Power rule for logarithms
Same base
Simplify constants
Last check: plot the new model on the original data.
Looks like we’ve got a model that will be very useful for estimating the weight of a fish if we know its length!

24. 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.

25. 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

26. 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.

27. 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

28. 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.