geoR (version 1.9-4)

boxcox: The Box-Cox Transformed Normal Distribution

Description

Functions related with the Box-Cox family of transformations. Density and random generation for the Box-Cox transformed normal distribution with mean equal to mean and standard deviation equal to sd, in the normal scale.

Usage

rboxcox(n, lambda, lambda2 = NULL, mean = 0, sd = 1)
dboxcox(x, lambda, lambda2 = NULL, mean = 0, sd = 1)

Value

The functions returns the following results:

rboxcox

a vector of random deviates.

dboxcox

a vector of densities.

Arguments

lambda

numerical value(s) for the transformation parameter \(\lambda\).

lambda2

logical or numerical value(s) of the additional transformation (see DETAILS below). Defaults to NULL.

n

number of observations to be generated.

x

a vector of quantiles (dboxcox) or an output of boxcoxfit (print, plot, lines).

mean

a vector of mean values at the normal scale.

sd

a vector of standard deviations at the normal scale.

Author

Paulo Justiniano Ribeiro Jr. ,
Peter J. Diggle .

Details

Denote \(Y\) the variable at the original scale and \(Y'\) the transformed variable. The Box-Cox transformation is defined by:

$$Y' = \left\{ \begin{array}{ll} log(Y) \mbox{ , if $\lambda = 0$} \cr \frac{Y^\lambda - 1}{\lambda} \mbox{ , otherwise} \end{array} \right.$$.

An additional shifting parameter \(\lambda_2\) can be included in which case the transformation is given by:

$$Y' = \left\{ \begin{array}{ll} log(Y + \lambda_2) \mbox{ , $\lambda = 0$ } \cr \frac{(Y + \lambda_2)^\lambda - 1}{\lambda} \mbox{ , otherwise} \end{array} \right.$$.

The function rboxcox samples \(Y'\) from the normal distribution using the function rnorm and backtransform the values according to the equations above to obtain values of \(Y\). If necessary the back-transformation truncates the values such that \(Y' \geq \frac{1}{\lambda}\) results in \(Y = 0\) in the original scale. Increasing the value of the mean and/or reducing the variance might help to avoid truncation.

References

Box, G.E.P. and Cox, D.R.(1964) An analysis of transformations. JRSS B 26:211--246.

See Also

The parameter estimation function is boxcoxfit. Other packages has BoxCox related functions such as boxcox in the package MASS and the function box.cox in the package car.

Examples

## Simulating data
simul <- rboxcox(100, lambda=0.5, mean=10, sd=2)
##
## Comparing models with different lambdas,
## zero  means and unit variances
curve(dboxcox(x, lambda=-1), 0, 8)
for(lambda in seq(-.5, 1.5, by=0.5))
  curve(dboxcox(x, lambda), 0, 8, add = TRUE)