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Transformations. 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.
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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
The effect of the ln transformation • It spreads out values that are close to zero • Compacts values that are large
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 1Hyperbolas 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
2.Exponential y = aebx = 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
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
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)
Example – Motor Vehicle Data The data is in an Excel file – MtrVeh.xls Dependent = mpg Independent = Engine size, horsepower and weight
We will try to fit a model predicting mpg with Engine (engine size). First a scatter plot: The dialog box selecting the variables:
Similar to: 2.Exponential y = aebx = 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
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 scatterplot showing a better fit to a straight line using the new variable 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.
Next topic Probability