boxcoxlm {AID}

Box-Cox Transformation for Linear Models

Description

boxcoxlm performs Box-Cox transformation for linear models and provides graphical analysis of residuals after transformation.

Usage

boxcoxlm(x, y, method = "lse", lambda = seq(-3,3,0.01), lambda2 = NULL, plot = TRUE, 
  alpha = 0.05, verbose = TRUE)

Arguments

x

a nxp matrix, n is the number of observations and p is the number of variables.

y

a vector of response variable.

method

a character string to select the desired method to be used to estimate Box-Cox transformation parameter. To use Shapiro-Wilk test method should be set to "sw". For method = "ad", boxcoxnc function uses Anderson-Darling test to estimate Box-Cox transformation parameter. Similarly, method should be set to "cvm", "pt", "sf", "lt", "jb", "mle", "lse" to use Cramer-von Mises, Pearson Chi-square, Shapiro-Francia, Lilliefors and Jarque-Bera tests, maximum likelihood estimation and least square estimation, respectively. Default is set to method = "lse".

lambda

a vector which includes the sequence of candidate lambda values. Default is set to (-3,3) with increment 0.01.

lambda2

a numeric for an additional shifting parameter. Default is set to lambda2 = 0.

plot

a logical to plot histogram with its density line and qqplot of residuals before and after transformation. Defaults plot = TRUE.

alpha

the level of significance to assess the normality of residuals after transformation. Default is set to alpha = 0.05.

verbose

a logical for printing output to R console.

Details

Denote y the variable at the original scale and y' the transformed variable. The Box-Cox power transformation is defined by:

y' = \left\{ \begin{array}{ll} \frac{y^\lambda - 1}{\lambda} = \beta_0 + \beta_1x_1 + ... + \epsilon \mbox{ , if $\lambda \neq 0$} \cr log(y) = \beta_0 + \beta_1x_1 + ... + \epsilon \mbox{ , if $\lambda = 0$} \end{array} \right.

If the data include any nonpositive observations, a shifting parameter \lambda_2 can be included in the transformation given by:

y' = \left\{ \begin{array}{ll} \frac{(y + \lambda_2)^\lambda - 1}{\lambda} = \beta_0 + \beta_1x_1 + ... + \epsilon \mbox{ , if $\lambda \neq 0$} \cr log(y + \lambda_2) = \beta_0 + \beta_1x_1 + ... + \epsilon \mbox{ , if $\lambda = 0$} \end{array} \right.

Maximum likelihood estimation and least square estimation are equivalent while estimating Box-Cox power transformation parameter (Kutner et al., 2005). Therefore, these two methods return the same result.

Value

A list with class "boxcoxlm" containing the following elements:

method

method preferred to estimate Box-Cox transformation parameter

lambda.hat

estimate of Box-Cox Power transformation parameter based on corresponding method

lambda2

additional shifting parameter

statistic

statistic of normality test for residuals after transformation based on specified normality test in method. For mle and lse, statistic is obtained by Shapiro-Wilk test for residuals after transformation

p.value

p.value of normality test for residuals after transformation based on specified normality test in method. For mle and lse, p.value is obtained by Shapiro-Wilk test for residuals after transformation

alpha

the level of significance to assess normality of residuals

tf.y

transformed response variable

tf.residuals

residuals after transformation

y.name

response name

x.name

x matrix name

Author(s)

Osman Dag, Ozlem Ilk

References

Asar, O., Ilk, O., Dag, O. (2017). Estimating Box-Cox Power Transformation Parameter via Goodness of Fit Tests. Communications in Statistics - Simulation and Computation, 46:1, 91–105.

Kutner, M. H., Nachtsheim, C., Neter, J., Li, W. (2005). Applied Linear Statistical Models. (5th ed.). New York: McGraw-Hill Irwin.

Examples

library(AID)

trees=as.matrix(trees)
boxcoxlm(x = trees[,1:2], y = trees[,3])