This document discusses various data transformations that can be used to satisfy assumptions of normality, homogeneity of variance, and linearity. It describes how to compute logarithmic, square root, inverse, and square transformations in SPSS. For each transformation, it provides details on specifying the target variable name, selecting the appropriate function or formula, and adding any necessary constants to the values. The purpose of these transformations is to modify non-normal, non-linear, or heteroscedastic data to better meet the assumptions of parametric statistical tests.

1. Computing Transformations Transforming variables Transformations for normality Transformations for linearity

3. Transformations change the measurement scale In the diagram to the right, the values of 5 through 20 are plotted on the different scales used in the transformations. These scales would be used in plotting the horizontal axis of the histogram depicting the distribution. When comparing values measured on the decimal scale to which we are accustomed, we see that each transformation changes the distance between the benchmark measurements. All of the transformations increase the distance between small values and decrease the distance between large values. This has the effect of moving the positively skewed values to the left, reducing the effect of the skewing and producing a distribution that more closely resembles a normal distribution.

13. Transformations: Transformations for normality Both the histogram and the normality plot for Total Time Spent on the Internet (netime) indicate that the variable is not normally distributed.

14. Transformations: Determine whether reflection is required Skewness, in the table of Descriptive Statistics, indicates whether or not reflection (reversing the values) is required in the transformation. If Skewness is positive, as it is in this problem, reflection is not required. If Skewness is negative, reflection is required.

15. Transformations: Compute the adjustment to the argument In this problem, the minimum value is 0, so 1 will be added to each value in the formula, i.e. the argument to the SPSS functions and formula for the inverse will be: netime + 1.

16. Transformations: Computing the logarithmic transformation To compute the transformation, select the Compute … command from the Transform menu.

17. Transformations: Specifying the transform variable name and function First , in the Target Variable text box, type a name for the log transformation variable, e.g. “lgnetime“. Second , scroll down the list of functions to find LG10, which calculates logarithmic values use a base of 10. (The logarithmic values are the power to which 10 is raised to produce the original number.) Third , click on the up arrow button to move the highlighted function to the Numeric Expression text box.

18. Transformations: Adding the variable name to the function First , scroll down the list of variables to locate the variable we want to transform. Click on its name so that it is highlighted. Second , click on the right arrow button. SPSS will replace the highlighted text in the function (?) with the name of the variable.

19. Transformations: Adding the constant to the function Following the rules stated for determining the constant that needs to be included in the function either to prevent mathematical errors, or to do reflection, we include the constant in the function argument. In this case, we add 1 to the netime variable. Click on the OK button to complete the compute request.

20. Transformations: The transformed variable The transformed variable which we requested SPSS compute is shown in the data editor in a column to the right of the other variables in the dataset.

21. Transformations: Computing the square root transformation To compute the transformation, select the Compute … command from the Transform menu.

22. Transformations: Specifying the transform variable name and function First , in the Target Variable text box, type a name for the square root transformation variable, e.g. “sqnetime“. Second , scroll down the list of functions to find SQRT, which calculates the square root of a variable. Third , click on the up arrow button to move the highlighted function to the Numeric Expression text box.

23. Transformations: Adding the variable name to the function Second , click on the right arrow button. SPSS will replace the highlighted text in the function (?) with the name of the variable. First , scroll down the list of variables to locate the variable we want to transform. Click on its name so that it is highlighted.

24. Transformations: Adding the constant to the function Following the rules stated for determining the constant that needs to be included in the function either to prevent mathematical errors, or to do reflection, we include the constant in the function argument. In this case, we add 1 to the netime variable. Click on the OK button to complete the compute request.

25. Transformations: The transformed variable The transformed variable which we requested SPSS compute is shown in the data editor in a column to the right of the other variables in the dataset.

26. Transformations: Computing the inverse transformation To compute the transformation, select the Compute … command from the Transform menu.

27. Transformations: Specifying the transform variable name and formula First , in the Target Variable text box, type a name for the inverse transformation variable, e.g. “innetime“. Second , there is not a function for computing the inverse, so we type the formula directly into the Numeric Expression text box. Third , click on the OK button to complete the compute request.

28. Transformations: The transformed variable The transformed variable which we requested SPSS compute is shown in the data editor in a column to the right of the other variables in the dataset.

29. Transformations: Adjustment to the argument for the square transformation In this problem, the minimum value is 0, no adjustment is needed for computing the square. If the minimum was a number less than zero, we would add the absolute value of the minimum (dropping the sign) as an adjustment to the variable. It is mathematically correct to square a value of zero, so the adjustment to the argument for the square transformation is different. What we need to avoid are negative numbers, since the square of a negative number produces the same value as the square of a positive number.

30. Transformations: Computing the square transformation To compute the transformation, select the Compute … command from the Transform menu.

31. Transformations: Specifying the transform variable name and formula First , in the Target Variable text box, type a name for the inverse transformation variable, e.g. “s2netime“. Second , there is not a function for computing the square, so we type the formula directly into the Numeric Expression text box. Third , click on the OK button to complete the compute request.

32. Transformations: The transformed variable The transformed variable which we requested SPSS compute is shown in the data editor in a column to the right of the other variables in the dataset.

33. Using the script to compute transformations When the script tests assumptions, it will create the transformations that are checked. If you want to retain the transformed variable to use in an analysis, clear the checkbox that tells the script to delete the transformed variables it created.

34. The transformed variables The transformed variables are added to the data editor. The variable names attempt to identify the transformation in the variable name. The variable labels fully identify the transformation, including the function and formula used to compute it.

35. Which transformation to use The recommendation of which transform to use is often summarized in a pictorial chart like the above. In practice, it is difficult to determine which distribution is most like your variable. It is often more efficient to compute all transformations and examine the statistical properties of each.