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Transformations

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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Transformations

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  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 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

  9. 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

  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 = 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

  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. 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.

  22. Next topic Probability

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