car (version 3.0-10)

bcPower: 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


Returns a vector or matrix of transformed values.


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.


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))
# }
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))
# }