Boxcox Power Transformation
Currently, there is a default method and a method for objects of class "lm".
boxcox(x, ...)
## Default S3 method:
boxcox(x,
lambda = {if (optimize) c(-2, 2) else seq(-2, 2, by = 0.5)},
optimize = FALSE, objective.name = "PPCC",
eps = .Machine$double.eps, include.x = TRUE, ...)
## S3 method for class 'lm'
boxcox(x,
lambda = {if (optimize) c(-2, 2) else seq(-2, 2, by = 0.5)},
optimize = FALSE, objective.name = "PPCC",
eps = .Machine$double.eps, include.x = TRUE, ...)x |
an object of class |
lambda |
numeric vector of finite values indicating what powers to use for the
Box-Cox transformation. When |
optimize |
logical scalar indicating whether to simply evalute the objective function at the
given values of |
objective.name |
character string indicating what objective to use. The possible values are
|
eps |
finite, positive numeric scalar. When the absolute value of |
include.x |
logical scalar indicating whether to include the finite, non-missing values of
the argument |
... |
optional arguments for possible future methods. Currently not used. |
Two common assumptions for several standard parametric hypothesis tests are:
The observations all come from a normal distribution.
The observations all come from distributions with the same variance.
For example, the standard one-sample t-test assumes all the observations come from the same normal distribution, and the standard two-sample t-test assumes that all the observations come from a normal distribution with the same variance, although the mean may differ between the two groups.
When the original data do not satisfy the above assumptions, data transformations
are often used to attempt to satisfy these assumptions. The rest of this section
is divided into two parts: one that discusses Box-Cox transformations in the
context of the original observations, and one that discusses Box-Cox
transformations in the context of linear models.
Box-Cox Transformations Based on the Original Observations
Box and Cox (1964) presented a formalized method for deciding on a data
transformation. Given a random variable X from some distribution with
only positive values, the Box-Cox family of power transformations is defined as:
| Y | = | \frac{X^λ - 1}{λ} | λ \ne 0 |
| log(X) | λ = 0 \;\;\;\;\;\; (1) |
where Y is assumed to come from a normal distribution. This transformation is
continuous in λ. Note that this transformation also preserves ordering.
See the help file for boxcoxTransform for more information on data
transformations.
Let \underline{x} = x_1, x_2, …, x_n denote a random sample of n observations from some distribution and assume that there exists some value of λ such that the transformed observations
| y_i | = | \frac{x_i^λ - 1}{λ} | λ \ne 0 |
| log(x_i) | λ = 0 \;\;\;\;\;\; (2) |
(i = 1, 2, …, n) form a random sample from a normal distribution.
Box and Cox (1964) proposed choosing the appropriate value of λ based on maximizing the likelihood function. Alternatively, an appropriate value of λ can be chosen based on another objective, such as maximizing the probability plot correlation coefficient or the Shapiro-Wilk goodness-of-fit statistic.
In the case when optimize=TRUE, the function boxcox calls the
R function nlminb to minimize the negative value of the
objective (i.e., maximize the objective) over the range of possible values of
λ specified in the argument lambda. The starting value for
the optimization is always λ=1 (i.e., no transformation).
The rest of this sub-section explains how the objective is computed for the
various options for objective.name.
Objective Based on Probability Plot Correlation Coefficient (objective.name="PPCC")
When objective.name="PPCC", the objective is computed as the value of the
normal probability plot correlation coefficient based on the transformed data
(see the description of the Probability Plot Correlation Coefficient (PPCC)
goodness-of-fit test in the help file for gofTest). That is,
the objective is the correlation coefficient for the normal
quantile-quantile plot for the transformed data.
Large values of the PPCC tend to indicate a good fit to a normal distribution.
Objective Based on Shapiro-Wilk Goodness-of-Fit Statistic (objective.name="Shapiro-Wilk")
When objective.name="Shapiro-Wilk", the objective is computed as the value of
the Shapiro-Wilk goodness-of-fit statistic based on the transformed data
(see the description of the Shapiro-Wilk test in the help file for
gofTest). Large values of the Shapiro-Wilk statistic tend to
indicate a good fit to a normal distribution.
Objective Based on Log-Likelihood Function (objective.name="Log-Likelihood")
When objective.name="Log-Likelihood", the objective is computed as the value
of the log-likelihood function. Assuming the transformed observations in
Equation (2) above come from a normal distribution with mean μ and
standard deviation σ, we can use the change of variable formula to
write the log-likelihood function as:
log[L(λ, μ, σ)] = \frac{-n}{2}log(2π) - \frac{n}{2}log(σ^2) - \frac{1}{2σ^2} ∑_{i=1}^n (y_i - μ)^2 + (λ - 1) ∑_{i=1}^n log(x_i) \;\;\;\;\;\; (3)
where y_i is defined in Equation (2) above (Box and Cox, 1964). For a fixed value of λ, the log-likelihood function is maximized by replacing μ and σ with their maximum likelihood estimators:
\hat{μ} = \frac{1}{n} ∑_{i=1}^n y_i \;\;\;\;\;\; (4)
\hat{σ} = [\frac{1}{n} ∑_{i=1}^n (y_i - \bar{y})^2]^{1/2} \;\;\;\;\;\; (5)
Thus, when optimize=TRUE, Equation (3) is maximized by iteratively solving for
λ using the values for μ and σ given in
Equations (4) and (5). When optimize=FALSE, the value of the objective is
computed by using Equation (3), using the values of λ specified in the
argument lambda, and using the values for μ and σ given
in Equations (4) and (5).
Box-Cox Transformation for Linear Models
In the case of a standard linear regression model with n observations and
p predictors:
Y_i = β_0 + β_1 X_{i1} + … + β_p X_{ip} + ε_i, \; i=1,2,…,n \;\;\;\;\;\; (6)
the standard assumptions are:
The error terms ε_i come from a normal distribution with mean 0.
The variance is the same for all of the error terms and does not depend on the predictor variables.
Assuming Y is a random variable from some distribution that may depend on the predictor variables and Y takes on only positive values, the Box-Cox family of power transformations is defined as:
| Y^* | = | \frac{Y^λ - 1}{λ} | λ \ne 0 |
| log(Y) | λ = 0 \;\;\;\;\;\; (7) |
where Y^* becomes the new response variable and the errors are now assumed to come from a normal distribution with a mean of 0 and a constant variance.
In this case, the objective is computed as described above, but it is based on the residuals from the fitted linear model in which the response variable is now Y^* instead of Y.
When x is an object of class "lm", boxcox returns
a list of class "boxcoxLm" containing the results. See
the help file for boxcoxLm.object for details.
When x is simply a numeric vector of positive numbers,
boxcox returns a list of class "boxcox" containing the
results. See the help file for boxcox.object for details.
Data transformations are often used to induce normality, homoscedasticity, and/or linearity, common assumptions of parametric statistical tests and estimation procedures. Transformations are not “tricks” used by the data analyst to hide what is going on, but rather useful tools for understanding and dealing with data (Berthouex and Brown, 2002, p.61). Hoaglin (1988) discusses “hidden” transformations that are used everyday, such as the pH scale for measuring acidity. Johnson and Wichern (2007, p.192) note that "Transformations are nothing more than a reexpression of the data in different units."
In the case of a linear model, there are at least two approaches to improving a model fit: transform the Y and/or X variable(s), and/or use more predictor variables. Often in environmental data analysis, we assume the observations come from a lognormal distribution and automatically take logarithms of the data. For a simple linear regression (i.e., one predictor variable), if regression diagnostic plots indicate that a straight line fit is not adequate, but that the variance of the errors appears to be fairly constant, you may only need to transform the predictor variable X or perhaps use a quadratic or cubic model in X. On the other hand, if the diagnostic plots indicate that the constant variance and/or normality assumptions are suspect, you probably need to consider transforming the response variable Y. Data transformations for linear regression models are discussed in Draper and Smith (1998, Chapter 13) and Helsel and Hirsch (1992, pp. 228-229).
One problem with data transformations is that translating results on the
transformed scale back to the original scale is not always straightforward.
Estimating quantities such as means, variances, and confidence limits in the
transformed scale and then transforming them back to the original scale
usually leads to biased and inconsistent estimates (Gilbert, 1987, p.149;
van Belle et al., 2004, p.400). For example, exponentiating the confidence
limits for a mean based on log-transformed data does not yield a
confidence interval for the mean on the original scale. Instead, this yields
a confidence interval for the median (see the help file for elnormAlt).
It should be noted, however, that quantiles (percentiles) and rank-based
procedures are invariant to monotonic transformations
(Helsel and Hirsch, 1992, p.12).
Finally, there is no guarantee that a Box-Cox tranformation based on the “optimal” value of λ will provide an adequate transformation to allow the assumption of approximate normality and constant variance. Any set of transformed data should be inspected relative to the assumptions you want to make about it (Johnson and Wichern, 2007, p.194).
Steven P. Millard (EnvStats@ProbStatInfo.com)
Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers, Second Edition. Lewis Publishers, Boca Raton, FL.
Box, G.E.P., and D.R. Cox. (1964). An Analysis of Transformations (with Discussion). Journal of the Royal Statistical Society, Series B 26(2), 211–252.
Draper, N., and H. Smith. (1998). Applied Regression Analysis. Third Edition. John Wiley and Sons, New York, pp.47-53.
Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, NY.
Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, New York, NY.
Hinkley, D.V., and G. Runger. (1984). The Analysis of Transformed Data (with Discussion). Journal of the American Statistical Association 79, 302–320.
Hoaglin, D.C., F.M. Mosteller, and J.W. Tukey, eds. (1983). Understanding Robust and Exploratory Data Analysis. John Wiley and Sons, New York, Chapter 4.
Hoaglin, D.C. (1988). Transformations in Everyday Experience. Chance 1, 40–45.
Johnson, N. L., S. Kotz, and A.W. Kemp. (1992). Univariate Discrete Distributions, Second Edition. John Wiley and Sons, New York, p.163.
Johnson, R.A., and D.W. Wichern. (2007). Applied Multivariate Statistical Analysis, Sixth Edition. Pearson Prentice Hall, Upper Saddle River, NJ, pp.192–195.
Shumway, R.H., A.S. Azari, and P. Johnson. (1989). Estimating Mean Concentrations Under Transformations for Environmental Data With Detection Limits. Technometrics 31(3), 347–356.
Stoline, M.R. (1991). An Examination of the Lognormal and Box and Cox Family of Transformations in Fitting Environmental Data. Environmetrics 2(1), 85–106.
van Belle, G., L.D. Fisher, Heagerty, P.J., and Lumley, T. (2004). Biostatistics: A Methodology for the Health Sciences, 2nd Edition. John Wiley & Sons, New York.
Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ, Chapter 13.
# Generate 30 observations from a lognormal distribution with
# mean=10 and cv=2. Look at some values of various objectives
# for various transformations. Note that for both the PPCC and
# the Log-Likelihood objective, the optimal value of lambda is
# about 0, indicating that a log transformation is appropriate.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
x <- rlnormAlt(30, mean = 10, cv = 2)
dev.new()
hist(x, col = "cyan")
# Using the PPCC objective:
#--------------------------
boxcox(x)
#Results of Box-Cox Transformation
#---------------------------------
#
#Objective Name: PPCC
#
#Data: x
#
#Sample Size: 30
#
# lambda PPCC
# -2.0 0.5423739
# -1.5 0.6402782
# -1.0 0.7818160
# -0.5 0.9272219
# 0.0 0.9921702
# 0.5 0.9581178
# 1.0 0.8749611
# 1.5 0.7827009
# 2.0 0.7004547
boxcox(x, optimize = TRUE)
#Results of Box-Cox Transformation
#---------------------------------
#
#Objective Name: PPCC
#
#Data: x
#
#Sample Size: 30
#
#Bounds for Optimization: lower = -2
# upper = 2
#
#Optimal Value: lambda = 0.04530789
#
#Value of Objective: PPCC = 0.9925919
# Using the Log-Likelihodd objective
#-----------------------------------
boxcox(x, objective.name = "Log-Likelihood")
#Results of Box-Cox Transformation
#---------------------------------
#
#Objective Name: Log-Likelihood
#
#Data: x
#
#Sample Size: 30
#
# lambda Log-Likelihood
# -2.0 -154.94255
# -1.5 -128.59988
# -1.0 -106.23882
# -0.5 -90.84800
# 0.0 -85.10204
# 0.5 -88.69825
# 1.0 -99.42630
# 1.5 -115.23701
# 2.0 -134.54125
boxcox(x, objective.name = "Log-Likelihood", optimize = TRUE)
#Results of Box-Cox Transformation
#---------------------------------
#
#Objective Name: Log-Likelihood
#
#Data: x
#
#Sample Size: 30
#
#Bounds for Optimization: lower = -2
# upper = 2
#
#Optimal Value: lambda = 0.0405156
#
#Value of Objective: Log-Likelihood = -85.07123
#----------
# Plot the results based on the PPCC objective
#---------------------------------------------
boxcox.list <- boxcox(x)
dev.new()
plot(boxcox.list)
#Look at QQ-Plots for the candidate values of lambda
#---------------------------------------------------
plot(boxcox.list, plot.type = "Q-Q Plots", same.window = FALSE)
#==========
# The data frame Environmental.df contains daily measurements of
# ozone concentration, wind speed, temperature, and solar radiation
# in New York City for 153 consecutive days between May 1 and
# September 30, 1973. In this example, we'll plot ozone vs.
# temperature and look at the Q-Q plot of the residuals. Then
# we'll look at possible Box-Cox transformations. The "optimal" one
# based on the PPCC looks close to a log-transformation
# (i.e., lambda=0). The power that produces the largest PPCC is
# about 0.2, so a cube root (lambda=1/3) transformation might work too.
head(Environmental.df)
# ozone radiation temperature wind
#05/01/1973 41 190 67 7.4
#05/02/1973 36 118 72 8.0
#05/03/1973 12 149 74 12.6
#05/04/1973 18 313 62 11.5
#05/05/1973 NA NA 56 14.3
#05/06/1973 28 NA 66 14.9
tail(Environmental.df)
# ozone radiation temperature wind
#09/25/1973 14 20 63 16.6
#09/26/1973 30 193 70 6.9
#09/27/1973 NA 145 77 13.2
#09/28/1973 14 191 75 14.3
#09/29/1973 18 131 76 8.0
#09/30/1973 20 223 68 11.5
# Fit the model with the raw Ozone data
#--------------------------------------
ozone.fit <- lm(ozone ~ temperature, data = Environmental.df)
# Plot Ozone vs. Temperature, with fitted line
#---------------------------------------------
dev.new()
with(Environmental.df,
plot(temperature, ozone, xlab = "Temperature (degrees F)",
ylab = "Ozone (ppb)", main = "Ozone vs. Temperature"))
abline(ozone.fit)
# Look at the Q-Q Plot for the residuals
#---------------------------------------
dev.new()
qqPlot(ozone.fit$residuals, add.line = TRUE)
# Look at Box-Cox transformations of Ozone
#-----------------------------------------
boxcox.list <- boxcox(ozone.fit)
boxcox.list
#Results of Box-Cox Transformation
#---------------------------------
#
#Objective Name: PPCC
#
#Linear Model: ozone.fit
#
#Sample Size: 116
#
# lambda PPCC
# -2.0 0.4286781
# -1.5 0.4673544
# -1.0 0.5896132
# -0.5 0.8301458
# 0.0 0.9871519
# 0.5 0.9819825
# 1.0 0.9408694
# 1.5 0.8840770
# 2.0 0.8213675
# Plot PPCC vs. lambda based on Q-Q plots of residuals
#-----------------------------------------------------
dev.new()
plot(boxcox.list)
# Look at Q-Q plots of residuals for the various transformation
#--------------------------------------------------------------
plot(boxcox.list, plot.type = "Q-Q Plots", same.window = FALSE)
# Compute the "optimal" transformation
#-------------------------------------
boxcox(ozone.fit, optimize = TRUE)
#Results of Box-Cox Transformation
#---------------------------------
#
#Objective Name: PPCC
#
#Linear Model: ozone.fit
#
#Sample Size: 116
#
#Bounds for Optimization: lower = -2
# upper = 2
#
#Optimal Value: lambda = 0.2004305
#
#Value of Objective: PPCC = 0.9940222
#==========
# Clean up
#---------
rm(x, boxcox.list, ozone.fit)
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