Freeman-Tukey (square root and arcsine) transforms

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Freeman-Tukey (square root and arcsine) transforms

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Freeman-Tukey (FT) transforms seek to adjust data to make the distribution more similar to a Normal distribution. The basic square root transform is a form of power transform, but the FT variant (see Freeman and Tukey, 1950 [FRE1]) was specifically designed for Poisson-like data, especially with a mean value >1. The FT angular or arcsine transform was developed for Binomial-like data, in particular, data representing proportions or percentages.

Square root transform

The basic square root transform is illustrated in the chart below and is simply:

Anscombe (1948, [ANS1]) proposed a form of the square root transform aimed at stabilizing the variance of the Poisson distribution to a value of approximately 1 with the transformed distribution being approximately Normal (especially for larger mean values, m>20). His transform is:

The Freeman-Tukey (FT) transform, which is widely used for variance stabilizing, is a variation of this transform:

If x is a Poisson variate with mean value m>1, the above transform is excellent at achieving variance stabilization, with the variance tending to 1. The adjusted transform:

where μ is the known or estimated mean value of {x}, has an approximate mean value of 0 and variance 1, with the transform thus providing a good approximation to a standard Normal distribution, i.e. z* ~N(0,1).

The back-transforms are for the unadjusted cases are:

Square root transform - basic form

Angular or ArcSin Transform

The angular or arcsin transform applies to source data in the range [-1,1] or more commonly in the range [0,1] and is designed to spread the set of values near the end of the range. Frequently data in the [0,1] range will represent proportions. However, these should be genuine measurements and not derived values that are essentially nominal, e.g. presence/absence data. Once transformed data are scaled to the range [-π/2,+π/2], as illustrated in the chart below. k is typically 0.5 and thus provides the square root. This transform is often used to correct S-shaped relationships between response and explanatory variables. The basic form of the transform is:

Letting p=x/n and using the value k=0.5 we have

The Freeman-Tukey (FT) version of this transform is:

or the same expression divided by 2. As with the square root transform (above) this is a variance-stabilizing transform. If x is a Binomial variate with parameters (n,p) and np>1 then the transform yields an approximately Normal variate with mean sin-1(p) and variance 1/(4n+2).

Arcsin function

References

[ANS1] Anscombe F J (1948) The transformation of Poisson, binomial and negative-binomial data. Biometrika, 35, 246-254

[FRE1] Freeman M F, Tukey J W (1950) Transformations related to the angular and the square root. Ann. Math. Statist., 21, 607-11