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powerTransform computes members of families of transformations indexed by one
parameter, the Box-Cox power family, or the Yeo and Johnson (2000) family, or the
basic power family, interpreting zero power as logarithmic.
The family can be modified to have Jacobian one, or not, except for the basic
power family.powerTransform(object,...)
## S3 method for class 'default':
powerTransform(object,...)
## S3 method for class 'lm':
powerTransform(object, ...)
## S3 method for class 'formula':
powerTransform(object, data, subset, weights, na.action,
...)lm, a formula, or a matrix or vector; see below.lm.lm.lm.estimateTransform,
which does the actual computing, or the optim
function, which does the maximizationpowerTransform is an object of class powerTransform
that gives the estimates of the the transformation parameters and related
statistics. The print method for the object will display the estimates only;
the summary method provides both the estimates, standard errors, marginal Wald
confidence intervals and relevant likelihood ratio tests.
Several helper functions are available. The coef method returns
the estimated transformation parameters, while coef(object,round=TRUE)
will
return the transformations rounded to nearby convenient values within 1.96
standard errors of the mle.
The vcov function returns the estimated covariance matrix of the
estimated
transformation parameters. A print method is used to
print the objects and summary to provide more information. By default
the summary method calls testTransform and provides likelihood ratio type
tests that all transformation parameters equal one and that all transformation
parameters equal zero, for log transformations, and for a convenient rounded value
not far from the mle. The function can be called
directly to test any other value for $\lambda$.z~x1+x2+x3 will find estimate a transformation for the response
z from the family of transformations indexed by the parameter lambda
that makes the residuals from the regression of the transformed z on the predictors
as closed to normally distributed as possible. This generalizes the Box and
Cox (1964) transformations to normality only by allowing for families other than the
power transformations used in that paper.
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. 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 (2005) 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
powerTransform(cbind(LBM, SSF) ~ 1, ais).
Two families of power transformations are available.
The bcPower family of scaled power transformations,
family="bctrans",
equals $(U^{\lambda}-1)/\lambda$
for $\lambda$ $\neq$ 0, and
$\log(U)$ if $\lambda =0$.
If 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 $U$
negative.
Other families can be added by writing a function whose first argument is a
matrix or vector to be transformed, and whose second argument is the value of the
transformation parameter. The function must return modified transformations
so that the Jacobian of the transformation is equal to one; see Cook and
Weisberg
(1982).
The function powerTransform is a front-end for
estimateTransform.
The function testTransform is used to obtain likelihood ratio
tests for
any specified value for the transformation parameters. It is used by the
summary method for powerTransform objects.estimateTransform, testTransform,
optim, bcPower, transform.# 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$roundlam) ~ len + amp + load, Wool))
# Multivariate Box Cox
summary(powerTransform(cbind(len, ADT, trks, sigs1) ~ 1, Highway1))
# Multivariate transformation to normality within levels of 'hwy'
summary(a3 <- powerTransform(cbind(len, ADT, trks, sigs1) ~ hwy, Highway1))
# test lambda = (0 0 0 -1)
testTransform(a3, c(0, 0, 0, -1))
# save the rounded transformed values, plot them with a separate
# color for males and females
transformedY <- bcPower(with(Highway1, cbind(len, ADT, trks, sigs1)),
coef(a3, round=TRUE))
pairs(transformedY, col=as.numeric(Highway1$hwy))Run the code above in your browser using DataLab