# bcPower: Box-Cox, Box-Cox with Negatives Allowed, Yeo-Johnson and... In car: Companion to Applied Regression

 bcPower R Documentation

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

### Description

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.

### Usage

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

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

bcnPowerInverse(z, lambda, gamma)

basicPower(U,lambda, gamma=NULL)
```

### Arguments

 `U` A vector, matrix or data.frame of values to be transformed `lambda` Power transformation parameter with one element for each column of U, usuallly in the range from -2 to 2. `jacobian.adjusted` If `TRUE`, the transformation is normalized to have Jacobian equal to one. The default `FALSE` is almost always appropriate. `gamma` 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. `z` a numeric vector the result of a call to `bcnPower` with `jacobian.adjusted=FALSE`

.

### Details

The Box-Cox family of scaled power transformations equals (x^(lambda)-1)/lambda for lambda not equal to 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 + √[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`.

### Value

Returns a vector or matrix of transformed values.

### Author(s)

Sanford Weisberg, <sandy@umn.edu>

### References

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.

`powerTransform`, `testTransform`

### Examples

```U <- c(NA, (-3:3))
## Not run: bcPower(U, 0)  # produces an error as U has negative values
bcPower(U, 0, gamma=4)