1. Transformations to Achieve Linearity

2. Common Models for Curved Data
Exponential Model
y = a bx
Power Model
y = a xb
The variable is in the exponent.
The variable is the base and b is its power.

3. Linearizing Exponential Data
Accomplished by taking ln(y)
To illustrate, we can take the logarithm of both sides of the model.
y = a bx
ln(y) = ln (a bx)
ln(y) = ln(a) + ln(bx)
ln(y) = ln(a) + ln(b)x
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A + BX
This is a linear model because ln(a) and ln(b) are constants

4. Linearizing the Power Model
Accomplished by taking the logarithm of both x and y.
Again, we can take the logarithm of both sides of the model.
y = a xb
ln(y) = ln (a xb)
ln(y) = ln(a) + ln(xb)
ln(y) = ln(a) + b ln(x)
Note that this time the logarithm remains attached to both y AND x.
A + BX

5. Why Should We Linearize Data?
Much of bivariate data analysis is built on linear models. By linearizing non-linear data, we can assess the fit of non-linear models using linear tactics.
In other words, we don’t have to invent new procedures for non-linear data.
HOORAY!!

6. Procedures for testing models
Step 1
Inspect Data.
If it is non-linear, you should test both exponential and power models.
Step 2
Transform Data.
Try exponential model first, since it requires taking only one logarithm.
Step 3
Inspect Transformed Data.
Pay close attention to residual plot and linear correlation coefficient.
Step 4
Repeat with the other model.

7. Example: Starbucks Growth
This table represents the number of Starbucks from
Put the data in your calculator
Year in L1
Stores in L2
Construct scatter plot.
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8. Note that the data appear to be non-linear.

9. Forgotten how to determine the LSRL? Click here.
Transformation time
Transform the data
Let L3 = ln (L1)
Let L4 = ln (L2)
Redraw scatterplot
Determine new LSRL
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10. Original

11. Exponential (x, ln y)

12. Original

13. Power (ln x, ln y)

14. Remember: Inspect Residual Plots!!
Exponential
Power
NOTE: Since both residual plots show curved patterns, neither model is completely appropriate, but both are improvements over the basic linear model.
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15. R-squared (A.K.A. Tiebreaker)
If plots are similar, the decision should be based on the value of r-squared.
Power has the highest value (r2 = .94), so it is the most appropriate model for this data (given your choices of models in this course).
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16. Writing equation for model
Once a model has been chosen, the LSRL must be converted to the non-linear model.
This is done using inverses.
In practice, you would only need to convert the best fit model.

17. Conversion to Exponential
LSRL for transformed data (x, ln y)
ln y = x
eln y = e x
eln y = e-20.7 (e.2707x)
y = e-20.7 (e.2707)x
Transformed Linear Model: a + bx
Exponential Model:
a + bx

18. Transformed Linear Model: a + bx
Conversion to Power
LSRL for transformed data (ln x, ln y)
ln y = ln x
eln y = e ln x
eln y = e (e23.6 ln x)
y = e x23.6
Transformed Linear Model: a + bx
Exponential Model:
a + bx

19. View non-linear model with non-linear data