Transformations

Transformation (re-expression) of a Variable

• A very useful transformation is the natural log transformation

• Transformation of a variable can change its distribution from a skewed distribution to a normal distribution (bell-shaped, symmetric about its centre

ln( )newx transformed x x • For any value of x, ln(x) can be:

• Looked up in tables• Calculated by most calculators• Calculated by most statistical packages

Graph of ln(x)

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ln( )newx x

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The effect of the transformation ln( )newx x

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

ln(x)newx

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

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

Intrinsically Linear (Linearizable) Curves 1 Hyperbolas

y = x/(ax-b)

Linear form: 1/y = a -b (1/x) or Y = 0 + 1 X

Transformations: Y = 1/y, X=1/x, 0 = a, 1 = -b

b/a

1/a

positive curvature b>0

y=x/(ax-b)

y=x/(ax-b)

negative curvature b< 0

1/a

b/a

2. Exponential

y = ex = x

Linear form: ln y = ln + x = ln + ln x or Y = 0 + 1 X

Transformations: Y = ln y, X = x, 0 = ln, 1 = = ln

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Exponential (B > 1)

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Exponential (B < 1)

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3. Power Functionsy = a xb

Linear from: ln y = lna + blnx or Y = 0 + 1 X

Transformations: Y = ln y, X = ln x, 0 = lna, 1 = b

Power functionsb>0

b > 1

b = 1

0 < b < 1

Power functionsb < 0

b < -1b = -1

-1 < b < 0

Summary

Transformations can be useful for:1. Changing data from a skewed distribution to a

Normal (bell- shaped) distribution

2. Straightening out Non-linear data

3. A common transformation is the natural log transformation ln(x)

Example – Motor Vehicle Data

The data is in an Excel file – MtrVeh.xlsDependent = mpg

Independent = Engine size, horsepower and weight

The data in an SPSS file

We will try to fit a model predicting mpg with Engine (engine size).

First a scatter plot:

The dialog box selecting the variables:

The scatter-plot

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ENGINE

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MPG

Similar to:2. Exponentialy = ex = x

Linear form: ln y = ln + x = ln + ln x or Y = 0 + 1 XTransformations: Y = ln y, X = x, 0 = ln, 1 = = ln

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Exponential (B < 1)

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• To perform a ln transformation in SPSS

• Go to the menu Transform->Compute

• In this dialogue box you define the tansformation

• Press OK and the trasformation will be performed

• The new variable has been added to the SPSS spreadsheet

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

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ENGINE

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lnmpg

Transformationssummary

• 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.

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Probability