Transformations to Achieve Linearity

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.

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 Need Help? Click here. A + BX This is a linear model because ln(a) and ln(b) are constants

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

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!!

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.

Example: Starbucks Growth This table represents the number of Starbucks from 1984- 2004. Put the data in your calculator Year in L1 Stores in L2 Construct scatter plot. Forgotten how to make scatterplots? Click here.

Note that the data appear to be non-linear.

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 Forgotten how to determine the LSRL? Click here.

Original

Exponential (x, ln y)

Original

Power (ln x, ln y)

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. Forgotten how to make residual plots? Click here.

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). Forgotten how to find r-squared? Click here.

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.

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

Transformed Linear Model: a + bx Conversion to Power LSRL for transformed data (ln x, ln y) ln y = -102.4 + 23.6 ln x eln y = e-102.4 + 23.6 ln x eln y = e-102.4 (e23.6 ln x) y = e-102.4 x23.6 Transformed Linear Model: a + bx Exponential Model: a + bx

View non-linear model with non-linear data