View source for Box-Cox transformation

Transformations of data designated to achieve a specified purpose, e.g., stability of variance, additivity of effects and symmetry of the density. If one is successful in finding a suitable transformation, the ordinary method for analysis will be available. Among the many parametric transformations, the family in [[#References|[a1]]] is commonly utilized.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b1107901.png" /> be a [[Random variable|random variable]] on the positive half-line. Then the Box–Cox transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b1107902.png" /> with power parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b1107903.png" /> is defined by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b1107904.png" /></td> </tr></table>

The formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b1107905.png" /> is chosen so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b1107906.png" /> is continuous as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b1107907.png" /> tends to zero and monotone increasing with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b1107908.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b1107909.png" />.

The power parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079010.png" /> is estimated by a graphical technique or by the [[Maximum-likelihood method|maximum-likelihood method]]. Unfortunately, a closed form for the estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079011.png" /> can be rarely found. Hence, the plot of the maximum likelihood against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079012.png" /> is helpful. The value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079013.png" /> obtained in this way is treated as if it were a true value, and then one fits the model to the transformed data. Such an approach may be easily carried out, and an asymptotic theory associated with other parameters is useful. See [[#References|[a1]]] and [[#References|[a3]]].

This treatment has, however, some difficulties because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079014.png" /> has a variability and depends on the given data itself. It is known that estimation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079015.png" /> by maximum likelihood and related likelihood-ratio tests can be heavily influenced by outliers (cf. also [[Outlier|Outlier]]). Further, in certain situations, the usual limiting theory based on knowing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079016.png" /> does not hold in the unknown case. Therefore, several robust estimation procedures have been proposed (see [[Robust statistics|Robust statistics]]; and [[#References|[a5]]] and references therein).

In the literature, Box–Cox transformations are applied to basic distributions, e.g., the cubic root transformation of chi-squared variates is used for acceleration to normality (cf. also [[Normal distribution|Normal distribution]]), and the square-root transformation stabilizes variances of Poisson distributions (cf. also [[Poisson distribution|Poisson distribution]]). These results are unified by appealing to features of the following family of distributions.

Consider a collection of densities of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079017.png" /></td> </tr></table>

satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079018.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079019.png" />. This family is called an exponential dispersion model with power variance function (EDM-PVF) of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079020.png" />. The existence of such a model was shown in [[#References|[a2]]] unless <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079021.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079022.png" />. It is a flexible family, including the normal, Poisson, gamma-, inverse Gaussian, etc., distributions.

It is known that both of the normalizing and the variance-stabilizing transformations of the exponential dispersion model with power variance function are given by Box–Cox transformations, see [[#References|[a4]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079023.png" /> follows the exponential dispersion model with power variance function and with index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079024.png" />, the normalizing and variance-stabilizing transformations are given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079025.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079027.png" /> (the power for normalization) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079028.png" /> (the power for variance-stabilization) are summarized in the Table below (recall that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079029.png" />). The similar characteristics of familiar distributions are also tabulated there. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079030.png" />, it has been proved in [[#References|[a4]]] that the density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079031.png" /> has a uniformly convergent Gram–Charlier expansion (cf. also [[Gram–Charlier series|Gram–Charlier series]]). This implies that the normalizing transformation which is obtained by reducing the third-order cumulant reduces all higher-order cumulants as a result (cf. also [[Cumulant|Cumulant]]).''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">Distribution</td> <td colname="2" style="background-color:white;" colspan="1">index</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079032.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079033.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079034.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Normal</td> <td colname="2" style="background-color:white;" colspan="1">2</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079035.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079036.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079037.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Poisson</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079038.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079039.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079040.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079041.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Gamma</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079042.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079043.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079044.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079045.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Inverse Gaussian</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079046.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079047.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079048.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079049.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">EDM-PVF</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079050.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079051.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079052.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110790/b11079053.png" /></td> </tr> </tbody> </table>

</td></tr> </table>

Box–Cox transformations are also applied to link functions in generalized linear models. The transformations mainly aim to get the linearity of effects of covariates. See [[#References|[a3]]] for further detail. Generalized Box–Cox transformations for random variables and link functions can be found in [[#References|[a5]]].

See also [[Exponential distribution|Exponential distribution]]; [[Regression|Regression]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E.P. Box,   D.R. Cox,   "An analysis of transformations"  ''J. Roy. Statist. Soc. B'' , '''26'''  (1964)  pp. 211–252</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Jørgensen,   "Exponential dispersion models"  ''J. Roy. Statist. Soc. B'' , '''49'''  (1987)  pp. 127–162</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. McCullagh,   J.A. Nelder,   "Generalized linear models" , Chapman and Hall  (1990)  (Edition: Second)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Nishii,   "Convergence of the Gram–Charlier expansion after the normalizing Box–Cox transformation"  ''Ann. Inst. Statist. Math.'' , '''45''' :  1  (1993)  pp. 173–186</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G.A.F. Seber,   C.J. Wild,   "Nonlinear regression" , Wiley  (1989)</TD></TR></table>