In appliedstatistics, avariance-stabilizing transformation is adata transformation that is specifically chosen either to simplify considerations in graphicalexploratory data analysis or to allow the application of simple regression-based oranalysis of variance techniques.^[1]

The aim behind the choice of a variance-stabilizing transformation is to find a simple functionƒ to apply to valuesx in a data set to create new valuesy =ƒ(x) such that the variability of the valuesy is not related to their mean value. For example, suppose that the values x are realizations from differentPoisson distributions: i.e. the distributions each have different mean valuesμ. Then, because for the Poisson distribution the variance is identical to the mean, the variance varies with the mean. However, if the simple variance-stabilizing transformation

${\displaystyle y=\sqrt{x}}$

is applied, the sampling variance associated with observation will be nearly constant: seeAnscombe transform for details and some alternative transformations.

While variance-stabilizing transformations are well known for certain parametric families of distributions, such as the Poisson and thebinomial distribution, some types of data analysis proceed more empirically: for example by searching amongpower transformations to find a suitable fixed transformation. Alternatively, if data analysis suggests a functional form for the relation between variance and mean, this can be used to deduce a variance-stabilizing transformation.^[2] Thus if, for a meanμ,

${\displaystyle \mathrm{var}(X)=h(\mu ),}$

a suitable basis for a variance stabilizing transformation would be

${\displaystyle y\propto {\int }^x\frac{1}{\sqrt{h(\mu )}}d\mu ,}$

where the arbitrary constant of integration and an arbitrary scaling factor can be chosen for convenience.

IfX is a positiverandom variable and for some constant, s, the variance is given ash(μ) =s²μ² then the standard deviation is proportional to the mean, which is called fixedrelative error. In this case, the variance-stabilizing transformation is

${\displaystyle y={\int }^x\frac{d\mu }{\sqrt{s^2{\mu }^2}}=\frac{1}{s}\ln (x)\propto \log (x).}$

That is, the variance-stabilizing transformation is the logarithmic transformation.

If the variance is given ash(μ) =σ² +s²μ² then the variance is dominated by a fixed varianceσ² when|μ| is small enough and is dominated by the relative variances²μ² when|μ| is large enough. In this case, the variance-stabilizing transformation is

${\displaystyle y={\int }^x\frac{d\mu }{\sqrt{{\sigma }^2+s^2{\mu }^2}}=\frac{1}{s}\mathrm{asinh}\frac{x}{\sigma /s}\propto \mathrm{asinh}\frac{x}{\lambda }.}$

That is, the variance-stabilizing transformation is theinverse hyperbolic sine of the scaled valuex /λ forλ =σ /s.

TheFisher transformation is a variance stabilizing transformation for thepearson correlation coefficient.

Here the delta method is presented informally to show the link to variance-stabilizing transformations. For a more formal statement of the delta method, seeDelta method.

Let${\displaystyle X}$ be a random variable, with${\displaystyle E[X]=\mu }$ and${\displaystyle \mathrm{Var}(X)={\sigma }^2}$.Define${\displaystyle Y=g(X)}$, where${\displaystyle g}$ is a regular function. A first order Taylor approximation for${\displaystyle Y=g(x)}$ is:

${\displaystyle Y=g(X)\approx g(\mu )+g^{\prime }(\mu )(X-\mu )}$

From the equation above, we obtain:

${\displaystyle E[Y]\approx g(\mu )}$ and${\displaystyle \mathrm{Var}[Y]\approx {\sigma }^2{g}^{\prime }(\mu {)}^2}$

This approximation method is called delta method.

Consider now a random variable${\displaystyle X}$ such that${\displaystyle E[X]=\mu }$ and${\displaystyle \mathrm{Var}[X]=h(\mu )}$.Notice the relation between the variance and the mean, which implies, for example,heteroscedasticity in a linear model. Therefore, the goal is to find a function${\displaystyle g}$ such that${\displaystyle Y=g(X)}$ has a variance independent (at least approximately) of its expectation.

Imposing the condition${\displaystyle \mathrm{Var}[Y]\approx h(\mu )g^{\prime }(\mu {)}^2=\text{constant}}$, this equality implies the differential equation:

${\displaystyle \frac{dg}{d\mu }=\frac{C}{\sqrt{h(\mu )}}}$

This ordinary differential equation has, byseparation of variables, the following solution:

${\displaystyle g(\mu )=\int \frac{Cd\mu }{\sqrt{h(\mu )}}}$

This last expression appeared for the first time in aM. S. Bartlett paper.^[3]

• Statistical data transformation

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