# boxCox

##### Graph the profile log-likelihood for Box-Cox transformations in 1D, or in 2D with the bcnPower family.

Computes and optionally plots profile log-likelihoods for the parameter of the
Box-Cox power family, the Yeo-Johnson power family, or for either of the parameters in a bcnPower family. This is a slight generalization of the
`boxcox`

function in the MASS package that allows for families of
transformations other than the Box-Cox power family. the `boxCox2d`

function
produces a contour
plot of the two-dimensional likelihood profile for the bcnPower family.

- Keywords
- regression

##### Usage

`boxCox(object, ...)`# S3 method for default
boxCox(object,
lambda = seq(-2, 2, 1/10), plotit = TRUE,
interp = plotit, eps = 1/50,
xlab=NULL, ylab=NULL,
family="bcPower",
param=c("lambda", "gamma"), gamma=NULL,
grid=TRUE, ...)

# S3 method for formula
boxCox(object, lambda = seq(-2, 2, 1/10), plotit = TRUE, family = "bcPower",
param = c("lambda", "gamma"), gamma = NULL, grid = TRUE,
...)

# S3 method for lm
boxCox(object, lambda = seq(-2, 2, 1/10), plotit = TRUE, ...)

boxCox2d(x, ksds = 4, levels = c(0.5, 0.95, 0.99, 0.999),
main = "bcnPower Log-likelihood", grid=TRUE, ...)

##### Arguments

- object
a formula or fitted model object of class

`lm`

or`aov`

.- lambda
vector of values of \(\lambda\), with default (-2, 2) in steps of 0.1, where the profile log-likelihood will be evaluated.

- plotit
logical which controls whether the result should be plotted; default

`TRUE`

.- interp
logical which controls whether spline interpolation is used. Default to

`TRUE`

if plotting with lambda of length less than 100.- eps
Tolerance for lambda = 0; defaults to 0.02.

- xlab
defaults to

`"lambda"`

or`"gamma"`

.- ylab
defaults to

`"log-Likelihood"`

or for bcnPower family to the appropriate label.- family
Defaults to

`"bcPower"`

for the Box-Cox power family of transformations. If set to`"yjPower"`

the Yeo-Johnson family, which permits negative responses, is used. If set to`bcnPower`

the function gives the profile log-likelihood for the parameter selected via`param`

.- param
Relevant only to

`family="bcnPower"`

, produces a profile log-likelihood for the parameter selected, maximizing over the remaining parameter.- gamma
For use when the

`family="bcnPower", param="gamma"`

. If this is a vector of positive values, then the profile log-likelihood for the location (or start) parameter in the`bcnPower`

family is evaluated at these values of gamma. If gamma is`NULL`

, then evaulation is done at 100 equally spaced points between`min(.01, gmax - 3*sd)`

and`gmax + 3*sd`

, where`gmax`

is the maximimum likelihood estimate of gamma, and`sd`

is the sd of the response. See`bcnPower`

for the definition of`gamma`

.- grid
If TRUE, the default, a light-gray background grid is put on the graph.

- …
additional arguments passed to the

`lm`

method with`boxCox.formula`

or passed to`contour`

with`boxCox2d`

.- x
An object created by a call to

`powerTransform`

using`family="bcnPower"`

.- ksds
Contour plotting of the log-likelihood surface will cover plus of minus

`ksds`

standard deviations on each axis.- levels
Contours will be drawn at the values of levels. For example,

`levels=c(.5, .99)`

would display two contours, at the 50% level and at the 99% level.- main
Title for the contour plot

##### Details

The `boxCox`

function is an elaboration of the `boxcox`

function in the
MASS package. The first 7 arguments are the same as in `boxcox`

, and if the argument `family="bcPower"`

is used, the result is essentially identical to the function in MASS. Two additional families are the `yjPower`

and `bcnPower`

families that allow a few values of the response to be non-positive.
The bcnPower family has two parameters: a power \(\lambda\) and a start or location parameter \(\gamma\), and the `boxCox`

function can be used to obtain a profile log-likelihood for either parameter with \(\lambda\) as the default. Alternatively, the `boxCox2d`

function can be used to get a contour plot of the profile log-likelihood.

##### Value

Both functions ae designed for their side effects of drawing a graph. The `boxCox`

function returns a list of the lambda (or possibly, gamma) vector and the computed profile log-likelihood vector,
invisibly if the result is plotted. If `plotit=TRUE`

plots log-likelihood vs
lambda and indicates a 95% confidence interval about the maximum observed value of
lambda. If `interp=TRUE`

, spline interpolation is used to give a smoother plot.

##### 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. (2016)
*Applied Regression Analysis and Generalized Linear Models*,
Third Edition. Sage.

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.

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

##### See Also

##### Examples

```
# NOT RUN {
with(trees, boxCox(Volume ~ log(Height) + log(Girth), data = trees,
lambda = seq(-0.25, 0.25, length = 10)))
data("quine", package = "MASS")
with(quine, boxCox(Days ~ Eth*Sex*Age*Lrn, data = quine,
lambda = seq(-0.05, 0.45, len = 20), family="yjPower"))
# }
```

*Documentation reproduced from package car, version 3.0-0, License: GPL (>= 2)*