bcPower {car}R Documentation

Box-Cox, Box-Cox with Negatives Allowed, Yeo-Johnson and Basic Power Transformations


Transform the elements of a vector or columns of a matrix using, the Box-Cox, Box-Cox with negatives allowed, Yeo-Johnson, or simple power transformations.


bcPower(U, lambda, jacobian.adjusted=FALSE, gamma=NULL)

bcnPower(U, lambda, jacobian.adjusted = FALSE, gamma)

bcnPowerInverse(z, lambda, gamma)

yjPower(U, lambda, jacobian.adjusted = FALSE)

basicPower(U,lambda, gamma=NULL)



A vector, matrix or data.frame of values to be transformed


Power transformation parameter with one element for each column of U, usuallly in the range from -2 to 2.


If TRUE, the transformation is normalized to have Jacobian equal to one. The default FALSE is almost always appropriate.


For bcPower or basicPower, the transformation is of U + gamma, where gamma is a positive number called a start that must be large enough so that U + gamma is strictly positive. For the bcnPower, Box-cox power with negatives allowed, see the details below.


a numeric vector the result of a call to bcnPower with jacobian.adjusted=FALSE



The Box-Cox family of scaled power transformations equals (x^{\lambda}-1)/\lambda for \lambda \neq 0, and \log(x) if \lambda =0. The bcPower function computes the scaled power transformation of x = U + \gamma, where \gamma is set by the user so U+\gamma is strictly positive for these transformations to make sense.

The Box-Cox family with negatives allowed was proposed by Hawkins and Weisberg (2017). It is the Box-Cox power transformation of

z = .5 (U + \sqrt{U^2 + \gamma^2)})

where for this family \gamma is either user selected or is estimated. gamma must be positive if U includes negative values and non-negative otherwise, ensuring that z is always positive. The bcnPower transformations behave similarly to the bcPower transformations, and introduce less bias than is introduced by setting the parameter \gamma to be non-zero in the Box-Cox family.

The function bcnPowerInverse computes the inverse of the bcnPower function, so U = bcnPowerInverse(bcnPower(U, lambda=lam, jacobian.adjusted=FALSE, gamma=gam), lambda=lam, gamma=gam) is true for any permitted value of gam and lam.

If family="yeo.johnson" then the Yeo-Johnson transformations are used. This is the Box-Cox transformation of U+1 for nonnegative values, and of |U|+1 with parameter 2-\lambda for U negative.

The basic power transformation returns U^{\lambda} if \lambda is not 0, and \log(\lambda) otherwise for U strictly positive.

If jacobian.adjusted is TRUE, then the scaled transformations are divided by the Jacobian, which is a function of the geometric mean of U for skewPower and yjPower and of U + gamma for bcPower. With this adjustment, the Jacobian of the transformation is always equal to 1. Jacobian adjustment facilitates computing the Box-Cox estimates of the transformation parameters.

Missing values are permitted, and return NA where ever U is equal to NA.


Returns a vector or matrix of transformed values.


Sanford Weisberg, <sandy@umn.edu>


Fox, J. and Weisberg, S. (2019) An R Companion to Applied Regression, Third Edition, Sage.

Hawkins, D. and Weisberg, S. (2017) Combining the Box-Cox Power and Generalized Log Transformations to Accomodate Nonpositive Responses In Linear and Mixed-Effects Linear Models South African Statistics Journal, 51, 317-328.

Weisberg, S. (2014) Applied Linear Regression, Fourth Edition, Wiley Wiley, Chapter 7.

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

See Also

powerTransform, testTransform


U <- c(NA, (-3:3))
## Not run: bcPower(U, 0)  # produces an error as U has negative values
bcPower(U, 0, gamma=4)
bcPower(U, .5, jacobian.adjusted=TRUE, gamma=4)
bcnPower(U, 0, gamma=2)
basicPower(U, lambda = 0, gamma=4)
yjPower(U, 0)
V <- matrix(1:10, ncol=2)
bcPower(V, c(0, 2))
basicPower(V, c(0,1))

[Package car version 3.1-0 Index]