Finding Univariate or Multivariate Power Transformations

Description

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.

Usage

powerTransform(object,...)

## Default S3 method:
powerTransform(object,...)

## S3 method for class 'lm'
powerTransform(object, ...)

## S3 method for class 'formula'
powerTransform(object, data, subset, weights, na.action,
  ...)

Arguments

Details

The function powerTransform is used to estimate normalizing transformations of a univariate or a multivariate random variable. For a univariate transformation, a formula like 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 not equal to 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.

Value

The result of powerTransform 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.

Author(s)

Sanford Weisberg, <sandy@stat.umn.edu>

References

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.

Velilla, S. (1993) A note on the multivariate Box-Cox transformation to normality. Statistics and Probability Letters, 17, 259-263.

Weisberg, S. (2005) Applied Linear Regression, Third Edition. Wiley.

Yeo, I. and Johnson, R. (2000) A new family of power transformations to improve normality or symmetry. Biometrika, 87, 954-959.

See Also

estimateTransform, testTransform, optim, bcPower, transform.

Examples

# 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))
## Not run: pairs(transformedY, col=as.numeric(Highway1$hwy))