Computing TransformationsTransformations: Transforming variables to satisfy assumptionsTransformations change the measurement scaleTransformations: Computing transformations in SPSSTransformations: Two forms for computing transformationsTransformations: Functions and formulas for transformationsTransformations: Transformation of positively skewed variablesTransformations: Example of positively skewed variableTransformations: Transformation of negatively skewed variablesTransformations: Example of negatively skewed variableTransformations: The Square Transformation for LinearityTransformations: Example of the square transformationTransformations: Transformations for normalityTransformations: Determine whether reflection is requiredTransformations: Compute the adjustment to the argumentTransformations: Computing the logarithmic transformationTransformations: Specifying the transform variable name and functionTransformations: Adding the variable name to the functionTransformations: Adding the constant to the functionTransformations: The transformed variableTransformations: Computing the square root transformationSlide 22Slide 23Slide 24Slide 25Transformations: Computing the inverse transformationTransformations: Specifying the transform variable name and formulaSlide 28Transformations: Adjustment to the argument for the square transformationTransformations: Computing the square transformationSlide 31Slide 32Using the script to compute transformationsThe transformed variablesWhich transformation to useSW388R7Data Analysis & Computers IISlide 1Computing TransformationsTransforming variablesTransformations for normalityTransformations for linearitySW388R7Data Analysis & Computers IISlide 2Transformations:Transforming variables to satisfy assumptionsWhen a metric variable fails to satisfy the assumption of normality, homogeneity of variance, or linearity, we may be able to correct the deficiency by using a transformation.We will consider three transformations for normality, homogeneity of variance, and linearity:the logarithmic transformationthe square root transformation, and the inverse transformationplus a fourth that may be useful for problems of linearity:the square transformationSW388R7Data Analysis & Computers IISlide 3Transformations change the measurement scaleIn 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.SW388R7Data Analysis & Computers IISlide 4Transformations:Computing transformations in SPSSIn SPSS, transformations are obtained by computing a new variable. SPSS functions are available for the logarithmic (LG10) and square root (SQRT) transformations. The inverse transformation uses a formula which divides one by the original value for each case.For each of these calculations, there may be data values which are not mathematically permissible. For example, the log of zero is not defined mathematically, division by zero is not permitted, and the square root of a negative number results in an “imaginary” value. We will usually adjust the values passed to the function to make certain that these illegal operations do not occur.SW388R7Data Analysis & Computers IISlide 5Transformations:Two forms for computing transformationsThere are two forms for each of the transformations to induce normality, depending on whether the distribution is skewed negatively to the left or skewed positively to the right. Both forms use the same SPSS functions and formula to calculate the transformations. The two forms differ in the value or argument passed to the functions and formula. The argument to the functions is an adjustment to the original value of the variable to make certain that all of the calculations are mathematically correct.SW388R7Data Analysis & Computers IISlide 6Transformations:Functions and formulas for transformationsSymbolically, if we let x stand for the argument passes to the function or formula, the calculations for the transformations are:Logarithmic transformation: compute log = LG10(x)Square root transformation: compute sqrt = SQRT(x)Inverse transformation: compute inv = -1 / (x)Square transformation: compute s2 = x * xFor all transformations, the argument must be greater than zero to guarantee that the calculations are mathematically legitimate.SW388R7Data Analysis & Computers IISlide 7Transformations:Transformation of positively skewed variablesFor positively skewed variables, the argument is an adjustment to the original value based on the minimum value for the variable.If the minimum value for a variable is zero, the adjustment requires that we add one to each value, e.g. x + 1.If the minimum value for a variable is a negative number (e.g., –6), the adjustment requires that we add the absolute value of the minimum value (e.g. 6) plus one (e.g. x + 6 + 1, which equals x +7).SW388R7Data Analysis & Computers IISlide 8Transformations:Example of positively skewed variableSuppose our dataset contains the number of books read (books) for 5 subjects: 1, 3, 0, 5, and 2, and the distribution is positively skewed.The minimum value for the variable books is 0. The adjustment for each case is books + 1.The transformations would be calculated as follows:Compute logBooks = LG10(books + 1)Compute sqrBooks = SQRT(books + 1)Compute invBooks = -1 / (books + 1)SW388R7Data Analysis & Computers IISlide 9Transformations:Transformation of negatively skewed variablesIf the distribution of a variable is negatively skewed, the adjustment of the values reverses, or reflects, the distribution so that it becomes positively skewed. The transformations are then computed on the values in the positively skewed distribution.Reflection is computed by subtracting all of the values for a variable from one plus the absolute value of maximum value for the variable. This