Finding Univariate or Multivariate Power Transformations
powerTransform uses the maximum likelihood-like approach of Box and Cox (1964) to select a transformatiion of a univariate or multivariate response for normality, linearity and/or constant variance. Available families are the default Box-Cox power family, and the Yeo-Johnson and skew power familes that may be useful when a response is not strictly positive.
powerTransform passes arguments to
estimateTransform, so you may need to include arguments to estimateTransform in a call to powerTransform.
# S3 method for default powerTransform(object, family="bcPower", ...)
# S3 method for lm powerTransform(object, family="bcPower", ...)
# S3 method for formula powerTransform(object, data, subset, weights, na.action, family="bcPower", ...) # S3 method for lmerMod powerTransform(object, family="bcPower", lambda=c(-3, 3), gamma=NULL, ...) estimateTransform(X, Y, weights=NULL, family="bcPower", start=NULL, method="L-BFGS-B", ...) # S3 method for default estimateTransform(X, Y, weights=NULL, family="bcPower", start=NULL, method="L-BFGS-B", ...) # S3 method for skewPower estimateTransform(X, Y, weights=NULL, lambda=c(-3, 3), gamma=NULL, ...)
# S3 method for lmerMod estimateTransform(object, family="bcPower", lambda=c(-3, 3), start=NULL, method="L-BFGS-B", ...)
# S3 method for skewPowerlmer estimateTransform(object, lambda=c(-3, +3), gamma=NULL, ...)
This can either be an object of class
lmerMod, a formula, or a matrix or vector; see below.
A data frame or environment, as in
Case indices to be used, as in
Weights as in
Missing value action, as in ‘lm’.
The quoted name of a family of transformations. The available options are
"bcPower"the default for the Box-Cox power family;
"yjpower"for the Yeo-Johnson family, and
"skewPower"for the two-parameter skew power family. The families are documented at
The range to be considered for the estimate of the power parameter lambda, equal to
c(-3, +3)by default. Values of lambda outside the default range is unlikely to be useful in practice.
The skewPower family has two parameters, adding a location parameter gamma to the power parameter lambda present in most other transformation families. If
gamma=NULLthen the location parameter will be estimated; if
gammais set to a numeric value, or a numeric vector of positive values equal in length to the number of responses,
gammawill be fixed and only the power will be estimated.
Additional arguments that are passed to
estimateTransformwhich does the actual computing, or to the
optimfunction, which does the maximization for all the methods except for
lmerModmodels with the
skewPowerfamily. For this case, computing is done using the
neldermeadfunction in the
noptrpackage is used.
A matrix or data.frame giving the “right-side variables”, including a column of ones if the intercept is present.
A vector or matrix or data.frame giving the “left-side variables.”
Starting values for the computations. The default value of NULL is usually adequate.
The computing alogrithm used by
optimfor the maximization. The default
"L-BFGS-B"appears to work well.
powerTransform is used to estimate normalizing/linearizing/variance stabilizing transformations
of a univariate or a multivariate response in a linear regression. For a univariate response,
a formula like
z~x1+x2+x3 will estimate a transformation for the response
z from a family of transformations indexed by one parameter for Box-Cox and Yeo-Johnson transformations,
or two parameters for the skew power family,
that makes the residuals from the regression of the transformed
z on the predictors
as closed to normally distributed as possible.
For a formula like
cbind(y1,y2,y3)~x1+x2+x3, the three variables on
the left-side are all transformed, generally with different transformations
to make all the residuals as close to
normally distributed as possible. This is not the same as three univariate transformations becuase the variables transformed are allowed to be correlated.
cbind(y1,y2,y3)~1 would specify transformations
to multivariate normality with no predictors. This generalizes the multivariate
power transformations suggested by Velilla (1993) by allowing for different
families of transformations, and by allowing for predictors. Cook and Weisberg (1999)
and Weisberg (2014) suggest the usefulness of transforming
a set of predictors
z1, z2, z3 for multivariate normality and for transforming
for multivariate normality conditional on levels of a factor, which is equivalent
to setting the predictors to be indicator variables for that factor.
Specifying the first argument as a vector, for example
powerTransform(ais$LBM), is equivalent to
powerTransform(LBM ~ 1, ais). Similarly,
powerTransform(cbind(ais$LBM, ais$SSF)), where the first argument is a matrix
rather than a formula is equivalent to specification of a mulitvariate linear model
powerTransform(cbind(LBM, SSF) ~ 1, ais).
Three families of power transformations are available.
The Box-Cox pwoer family of power transformations,
for \(\lambda\) \(\neq\) 0, and
\(\log(U)\) if \(\lambda =0\). A scaled version of this transformation is used in computing with all the families to make the Jacobian of the transformation equal to 1.
family="yjPower" then the Yeo-Johnson transformations are used.
This is is Box-Cox transformation of \(U+1\) for nonnegative values,
and of \(|U|+1\) with parameter \(2-\lambda\) for \(U\)
family="skewPower" then the skew power family of transformations suggested
by Hawkins and Weisberg (2015) is used. This is a two-parameter family that would
generally be applied with a response with occasional negative values; see
for the details and examples. This family has a power parameter \(\lambda\) and a non-negative start parameter \(\gamma\), with \(\gamma = 0\) equal to the Box-Cox transformation.
The same generally methodology can be applied for linear mixed models fit with the
lmer function in the
lme4 package. A multivariate response is not permitted.
testTransform is used to obtain likelihood ratio
any specified value for the transformation parameter(s).
An object of class
powerTransform or class
powerTransfrom is returned, including the components listed below.
Several methods are available for use with
powerTransform objects. The
coef method returns
the estimated transformation parameters, while
return the transformations rounded to nearby convenient values within 1.96
standard errors of the mle, if any exist.
vcov function returns the estimated covariance matrix of the
transformation parameters. A
summary method provides more information including likelihood ratio type
tests that all power parameters equal one and that all transformation
parameters equal zero, for log transformations, and for a convenient rounded value
not far from the mle. In the case of the skew power family, these tests are based on the profile log-likelihood obtained by maximizing over the start parameter, thus treating the start as a nusiance parameter of lesser interest than the pwoer parameter.
testTransform can be called
directly to test any other value for \(\lambda\) or for skew power \(\lambda\) and \(\gamma\). There is a
plot.powerTransform method for plotting the transformed values, and also a
contour.skewpowerTransform method to obtain a contour plot of the two-dimensional log-likelihood for the skew power parameters when the response in univariate. Finally, the
boxCox method can be used to plot the univariate log-likleihood for the Box-Cox or Yeo-Johnson power families, or the profile log-likelihood of each of the parameters in the skew power family.
The components of the returned object are
The value of the loglikelihood at the mle.
The hessian matrix.
Starting values for the computations.
The ml estimate for the power parameter
The ml estimate for the start parameter for the skew power family
Convenient rounded values for the estimates. These rounded values will often be the desirable transformations.
The transformation family
QR decomposition of the predictor matrix.
The responses to be transformed
The weights if weighted least squares.
Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations. Journal of the Royal Statisistical Society, Series B. 26 211-46.
Cook, R. D. and Weisberg, S. (1999) Applied Regression Including Computing and Graphics. Wiley.
Fox, J. and Weisberg, S. (2011) An R Companion to Applied Regression, Second Edition, Sage.
Hawkins, D. and Weisberg, S. (2015) Combining the Box-Cox Power and Genralized Log Transformations to Accomodate Negative Responses, submitted for publication.
Velilla, S. (1993) A note on the multivariate Box-Cox transformation to normality. Statistics and Probability Letters, 17, 259-263.
Weisberg, S. (2014) Applied Linear Regression, Fourth Edition, Wiley.
Yeo, I. and Johnson, R. (2000) A new family of power transformations to improve normality or symmetry. Biometrika, 87, 954-959.
# Box Cox Method, univariate summary(p1 <- powerTransform(cycles ~ len + amp + load, Wool)) # fit linear model with transformed response: coef(p1, round=TRUE) summary(m1 <- lm(bcPower(cycles, p1$roundlam) ~ len + amp + load, Wool)) # Multivariate Box Cox uses Highway1 data summary(powerTransform(cbind(len, adt, trks, sigs1) ~ 1, Highway1)) # Multivariate transformation to normality within levels of 'htype' summary(a3 <- powerTransform(cbind(len, adt, trks, sigs1) ~ htype, Highway1)) # test lambda = (0 0 0 -1) testTransform(a3, c(0, 0, 0, -1)) # save the rounded transformed values, plot them with a separate # color for each highway type transformedY <- bcPower(with(Highway1, cbind(len, adt, trks, sigs1)), coef(a3, round=TRUE)) scatterplotMatrix( ~ transformedY|htype, Highway1)