Troubleshooting
Problem
I performed an arcsine transformation on a variable which was expressed as a proportion, but the new values I got do not match the results given in the table in my statistics book. What should I do?
Resolving The Problem
There are two common approaches to this transformation which yield numbers that look very different. However, in practice, the two techniques result in identical statistical conclusions.
One technique, often used in the social sciences (e.g., Cohen & Cohen, 1983, pp. 266-268), is to let the transformed variable be equal to "twice the angle (measured in radians) whose trigonometric sine equals the square root of the proportion being transformed." Given a variable, var1, expressed as a proportion, the following command language would generate the desired transformation.
COMPUTE newvar1 = 2 * ARSIN(SQRT(var1)).
The other technique, often used in the biomedical sciences (e.g., Sokal & Rohlf, 1981, pp. 427), is to let the transformed variable be equal to the angle (measured in degrees) whose sine equals the square root of the proportion.
COMPUTE newvar2 = ARSIN(SQRT(var1)) * 45 / ARTAN(1).
On the right hand side of the above expression, the multiplication of ARSIN(SQRT(var1)) by 180 / Pi
yields the angle in degrees. To compute Pi in SPSS, we can use the fact that tan(Pi/4) = 1 and
SPSS will return a value equal to Pi/4 for ARTAN(1), so 45/ARTAN(1) equals 180/(4*ARTAN(1)), which equals 180/Pi.
These transformations are most often employed in the analysis of a dependent variable in general linear modeling (e.g., regression or ANOVA), when the raw values are proportions (or percents). No matter which of the above formulas is used, the test statistics (F- and t-values) and their associated probabilities will be identical. This will be true because each expression is a linear combination of the other.
The input to the ARSIN function should be a proportion, rather than a percentage. You can divide a variable with percentages by 100 within the arcsine transformation command without creating a new variable with proportions. For example, if the values of VAR2 are percentages :
COMPUTE arcpct = 2 * ARSIN(SQRT(var2/100)).
using the first method above, or
COMPUTE arcpct = ARSIN(SQRT(var2/100)) * 45 / ARTAN(1) .
using the second method.
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References
Cohen, J. & Cohen, P. (1983). Applied multiple regression/correlation
analysis for the behavioral sciences, 2nd Ed. Hillsdale, NJ:
Lawrence Erlbaum.
Sokal, R.R. & Rohlf, F.J. (1981). Biometry, 2nd Ed. New York: W.H.
Freeman
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Modified date:
16 April 2020
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