1. Transformations

2. Transformation (re-expression) of a Variable
Transformation of a variable can change its distribution from a skewed distribution to a normal distribution (bell-shaped, symmetric about its centre
A very useful transformation is the natural log transformation
For any value of x, ln(x) can be:
Looked up in tables
Calculated by most calculators
Calculated by most statistical packages

3. Graph of ln(x)

4. The effect of the transformation

5. The effect of the ln transformation
It spreads out values that are close to zero
Compacts values that are large

6. Transforming data to a normal distribution allows one to use powerful statistical procedures (discussed later on) that assumes the data is normally distributed.

7. Transformations to Linearity
Many non-linear curves can be put into a linear form by appropriate transformations of the either
the dependent variable Y or
the independent variable X
or both.
This leads to the wide utility of the Linear model.
Another use of trans

8. Intrinsically Linear (Linearizable) Curves
1 Hyperbolas
y = x/(ax-b)
Linear form: 1/y = a -b (1/x) or Y = b0 + b1 X
Transformations: Y = 1/y, X=1/x, b0 = a, b1 = -b

9. 2. Exponential
y = a ebx = aBx
Linear form: ln y = lna + b x = lna + lnB x or Y = b0 + b1 X
Transformations: Y = ln y, X = x, b0 = lna, b1 = b = lnB

10. 3. Power Functions
y = a xb
Linear from: ln y = lna + blnx or Y = b0 + b1 X
Transformations: Y = ln y, X = ln x, b0 = lna, b1 = b

11. Summary Transformations can be useful for:
Changing data from a skewed distribution to a Normal (bell- shaped) distribution
Straightening out Non-linear data
A common transformation is the natural log transformation ln(x)

12. Example – Motor Vehicle Data
The data is in an Excel file – MtrVeh.xls
Dependent = mpg
Independent = Engine size, horsepower and weight

13. The data in an SPSS file

14. We will try to fit a model predicting mpg with Engine (engine size).
First a scatter plot:
The dialog box selecting the variables:

15. The scatter-plot

16. Similar to:
2. Exponential
y = a ebx = aBx
Linear form: ln y = lna + b x = lna + lnB x or Y = b0 + b1 X
Transformations: Y = ln y, X = x, b0 = lna, b1 = b = lnB

17. To perform a ln transformation in SPSS
Go to the menu Transform->Compute

18. In this dialogue box you define the tansformation
Press OK and the trasformation will be performed

19. The new variable has been added to the SPSS spreadsheet

20. The scatterplot showing a better fit to a straight line using the new variable lnmpg.

21. Transformations summary
Transformations can be used to convert non-normal data to normally (bell-shaped) distributed data (allowing for the use of the more powerful techniques assuming normality)
Transformations can be used to convert non-linear data linear (straight line) data.

22. Next topic
Probability