Computing Transformations in SPSS

8/12/2019 Computing Transformations in SPSS

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Summary made by: Gernimo Maldonado-MartnezBiostatistician

Data Management & Statistical Research Support Unit

Universidad Central del Caribe


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Transforming variablesTransformations for normalityTransformations for linearity


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When a metric variable fails to satisfy theassumption of normality, homogeneity ofvariance, or linearity, we may be able to correctthe deficiency by using a transformation.

We will consider three transformations fornormality, homogeneity of variance, andlinearity:

the logarithmic transformation

the square root transformation the inverse transformation

plus a fourth that is useful for problems oflinearity:

the square transformation


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SPSS functions are available for the logarithmic(LG10) and square root (SQRT) transformations. The multiplicative inverse transformation uses a

formula which divides one by the original value foreach case. (1/x)

Sometimes there may be data values which arenot mathematically permissible. For example:

the log of zero is not defined mathematically division by zero is not permitted 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 illegaloperations do not occur.


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There are two forms for each of thetransformations to induce normality, dependingon whether the distribution is skewednegatively to the left or skewed positively to theright.

Both forms use the same SPSS functions andformula to calculate the transformations.

The two forms differ in the value or argumentpassed to the functions and formula. The argument to the functions is an adjustment

to the original value of the variable to makecertain that all of the calculations aremathematically correct.


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Symbolically, if we let stand for the argumentpasses to the function or formula, thecalculations 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 begreater than zero to guarantee that thecalculations are mathematically legitimate .


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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, (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, whichequals x +7)

Elongated tail @ the R.More data in the R tail

than would be expectedin a Gaussian distribution


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Suppose our dataset contains the number ofbooks read (books) for 5 subjects: 1, 3, 0, 5,and 2, and the distribution is positivelyskewed.

The minimum value for the variable books is0. The adjustment for each case is books + 1.

The transformations would be calculated asfollows:

Compute logBooks = LG10(books + 1) Compute sqrBooks = SQRT(books + 1) Compute invBooks = 1 / (books + 1)


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For negatively skewed variables: The adjustment of the values reverses, or reflects, thedistribution so that it becomes positively skewed.

The transformations are then computed on the values in thepositively skewed distribution.

Reflection is computed by subtracting all of the values fora variable from one plus the absolute value of maximumvalue for the variable.

This results in a positively skewed distribution with allvalues larger than zero.

Elongated tail @ the L.More data in the L tail

than would be expectedin a Gaussian distribution


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Suppose our dataset contains the number ofbooks read (books) for 5 subjects: 1, 3, 0, 5,and 2, and the distribution is negativelyskewed.

The maximum value for the variable books is5. The adjustment for each case is 6 - books.

The transformations would be calculated asfollows:

Compute logBooks = LG10(6 - books) Compute sqrBooks = SQRT(6 - books) Compute invBooks = 1 / (6 - books)


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The square transformation is computed bymultiplying the value for the variable byitself.

It does not matter whether the distribution ispositively or negatively skewed.

It does matter if the variable has negativevalues, since we would not be able todistinguish their squares from the square of a

comparable positive value (e.g. the square of-4 is equal to the square of +4). If the variable has negative values, we add the

absolute value of the minimum value to eachscore before squaring it.


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Suppose our dataset contains change scores(chg) for 5 subjects that indicate thedifference between test scores at the end ofa semester and test scores at mid-term: -10,0, 10, 20, and 30.

The minimum score is -10. The absolutevalue of the minimum score is 10.

The transformation would be calculated asfollows: Compute squarChg = (chg + 10) * (chg + 10)


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Normal Q-Q Plot of TOTAL TIME SPENT ON TH

Observed Value

120100806040200-20-40

3

2

1

0

-1

-2

-3

TOTAL TIME SPENT ON THE INTERNET

100.090.0

80.070.0

60.050.0

40.030.0

20.010.0

0.0

Histogram50

40

30

20

10

0

Std. Dev = 15.35

Mean = 10.7

N = 93.00

Both the histogram andthe normality plot for

Total Time Spent on theInternet (netime) indicate

that the variable is notnormally distributed.


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Skewness, in the table of Descriptive Statistics,

indicates whether or not reflection (reversing thevalues) 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 .


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In this problem, the minimum value is 0, so 1will be added to each value in the formula, i.e.

the argument to the SPSS functions and formulafor the inverse will be: netime + 1.


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To compute thetransformation, select theCompute command from

the Transform menu.


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First , in the Target Variable

text box, type a name forthe log transformation

variable, e.g. lgnetime .

Second , scroll down the list of functions to findLG10, which calculates logarithmic values use a

base of 10. (The logarithmic values are thepower to which 10 is raised to produce the

original number.)

Third , click onthe up arrow

button tomove the

highlightedfunction to

the NumericExpressiontext box.


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First , scroll down the list ofvariables to locate the

variable we want totransform. Click on its name

so that it is highlighted.

Second , click on the right arrowbutton. SPSS will replace the

highlighted text in the function(?) with the name of the variable.


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Following the rules stated for determiningthe constant that needs to be included in the

function either to prevent mathematicalerrors, or to do reflection, we include theconstant in the function argument. In this

case, we add 1 to the netime variable.

Click on the OKbutton to

complete thecompute request.


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The transformed variable whichwe requested SPSS compute isshown in the data editor in a

column to the right of theother variables in the dataset.


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To compute thetransformation, select theCompute command from

the Transform menu.


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First , in the TargetVariable text box, type a

name for the squareroot transformation

variable, e.g.sqnetime.

Second , scroll down thelist of functions to findSQRT, which calculates

the square root of avariable.

Third , click onthe up arrow

button to movethe highlightedfunction to the

Numeric

Expression textbox.


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Second , click on theright arrow button.

SPSS will replace thehighlighted text in thefunction (?) with thename of the variable.

First , scroll down the listof variables to locate

the variable we want totransform. Click on its

name so that it ishighlighted.


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Following the rules stated for determining the

constant that needs to be included in the functioneither to prevent mathematical errors, or to do

reflection, we include the constant in the functionargument. In this case, we add 1 to the netime

variable.

Click on theOK button tocomplete the

computerequest.


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To compute thetransformation,

select theCompute

command fromthe Transform

menu.


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First , in theTarget Variable text box, type a

name for theinverse

transformationvariable, e.g.innetime.

Second , there is not afunction for computing the

inverse, so we type theformula directly into the

Numeric Expression text box.

Third , click on

the OK buttonto completethe compute

request.


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The transformed variable which werequested SPSS compute is shown in

the data editor in a column to theright of the other variables in the

dataset.


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In this problem, the minimum value is 0,no adjustment is needed for computing

the square. If the minimum was a numberless than zero, we would add the absolutevalue of the minimum (dropping the sign)

as an adjustment to the variable.

It is mathematically correct to square a value of zero, so theadjustment 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.


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To compute thetransformation, select

the Compute commandfrom the Transform

menu.


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First , in the Target Variable text box, type a name for

the inverse transformationvariable, e.g. s2netime.

Second , there is not a functionfor computing the square, sowe type the formula directlyinto the Numeric Expression

text box.

Third , click onthe OK buttonto completethe compute

request.


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The transformed variablewhich we requested SPSS

compute is shown in the dataeditor in a column to the rightof the other variables in the

dataset.

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Computing Transformations in SPSS

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