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transace is ace packaged for easily automatically
transforming all variables in a matrix. transace is a fast
one-iteration version of transcan without imputation of NAs. areg.boot uses avas or ace to fit additive regression models
allowing all variables in the model (including the right-hand-side) to
be transformed, with transformations chosen so as to optimize
certain criteria. The default method uses avas, which
explicity tries to transform the response variable so as to stabilize
the variance of the residuals. All-variables-transformed models tend
to inflate R^2 and it can be difficult to get confidence limits for
each transformation. areg.boot solves both of these problems using
the bootstrap. As with the validate function in the Design library,
the Efron bootstrap is used to estimate the optimism in the apparent R^2,
and this optimism is subtracted from the apparent R^2 to optain a
bias-corrected R^2. This is done however on the transformed response
variable scale.
Tests with 3 predictors show that the AVAS and ACE estimates used by
areg.boot are unstable unless the sample size exceeds 350. Apparent
R^2 with low sample sizes can be very inflated, and bootstrap
estimates of R^2 can be even more unstable in such cases,
resulting in optimism-corrected R^2 that are much lower even than the
actual R^2. The situation can be improved a little by restricting
predictor transformations to be monotonic.
For method="avas" the response transformation is restricted to be
monotonic. You can specify restrictions for transformations of
predictors (and linearity for the response, or its monotonicity if
using ace). When the first argument is a formula, the function
automatically determines which variables are categorical (i.e., factor,
category, or character vectors). Specify linear transformations by
enclosing variables by the identify function (I()), and specify
monotonicity by using monotone(variable).
The summary method for areg.boot computes bootstrap estimates of
standard errors of differences in predicted responses (usually
on the original scale)
for selected levels of each predictor against the lowest level of the
predictor. The smearing estimator (see below) can be used here to
estimate differences in predicted means, medians, or many other
statistics. By default, quartiles are used for continuous predictors
and all levels are used for categorical ones. See DETAILS below.
There is also a plot method for plotting transformation estimates,
transformations for individual bootstrap re--samples, and pointwise
confidence limits for transformations. Unless you already have a
par(mfrow=) in effect with more than one row or column, plot will
try to fit the plots on one page. A predict method computes
predicted values on the original or transformed response scale, or a
matrix of transformed predictors. There is a Function method
for producing a list of S-PLUS functions that perform the final fitted
transformations. There is also a print method for areg.boot
objects.
When estimated means (or medians or other statistical parameters) are
requested for models fitted with areg.boot (by summary.areg.boot
or predict.areg.boot), the "smearing" estimator of Duan (1983) is
used. Here we estimate the mean of the untransformed response by
computing the arithmetic mean of ginverse(lp + residuals), where
ginverse is the inverse of the nonparametric transformation of the
response (obtained by reverse linear interpolation), lp is the
linear predictor for an individual observation on the transformed
scale, and residuals is the entire vector of residuals estimated
from the fitted model, on the transformed scales (n residuals for n
original observations). The smearingEst function computes the
general smearing estimate. For efficiency smearingEst recognizes
that quantiles are transformation-preserving, i.e., when one wishes to
estimate a quantile of the untransformed distribution one just needs
to compute the inverse transformation of the transformed estimate
after the chosen quantile of the vector of residuals is added to it.
When the median is desired, the estimate is ginverse(lp +
median(residuals)). See the last example for how smearingEst can be
used outside of areg.boot.
Mean is a generic function that returns an S function to compute
the estimate of the mean of a variable. Its input is typically some
kind of model fit object. Likewise, Quantile is a generic
quantile function-producing function. Mean.areg.boot and
Quantile.areg.boot create functions of a vector of linear
predictors that transform them into the smearing estimates of the mean
or quantile of the response variable, respectively.
Quantile.areg.boot produces exactly the same value as
predict.areg.boot or smearingEst. Mean
approximates the mapping of linear predictors to means over an evenly
spaced grid of by default 200 points. Linear interpolation is used
between these points. This approximate method is much faster than the
full smearing estimator once Mean creates the function. These
functions are especially useful in nomogram.Design (see the
example on hypothetical data).
transace(x, monotonic=NULL, categorical=NULL, binary=NULL, pl=TRUE)areg.boot(x, y, data, weights, subset, na.action=na.delete,
B=100, method=c("avas", "ace"), evaluation=100, valrsq=TRUE,
probs=c(.25,.5,.75), ...)
## S3 method for class 'areg.boot':
print(x, \dots)
## S3 method for class 'areg.boot':
plot(x, ylim, boot=TRUE, col.boot=2, lwd.boot=.15,
conf.int=.95, \dots)
smearingEst(transEst, inverseTrans, res,
statistic=c('median','quantile','mean','fitted','lp'),
q)
## S3 method for class 'areg.boot':
summary(object, conf.int=.95, values, adj.to,
statistic='median', q, \dots)
## S3 method for class 'summary.areg.boot':
print(x, \dots)
## S3 method for class 'areg.boot':
predict(object, newdata,
statistic=c("lp", "median",
"quantile", "mean", "fitted", "terms"),
q=NULL, ...)
## S3 method for class 'areg.boot':
Function(object, type=c('list','individual'),
ytype=c('transformed','inverse'),
prefix='.', suffix='', frame=0, where=1, ...)
Mean(object, ...)
Quantile(object, ...)
## S3 method for class 'areg.boot':
Mean(object, evaluation=200, \dots)
## S3 method for class 'areg.boot':
Quantile(object, q=.5, \dots)
transace a numeric matrix. For areg.boot x may
be a numeric matrix or a formula. For print or plot, an
object created by areg.boot. For print.summary.areg.bootareg.boot, or a model fit object suitable
for Mean or Quantile.transEst to the original untransformed scale. inverseTrans may
also be a 2-element list defining a mapping from the transformed
values to untransformed values. Lx for transace. Binary variables are not
transformed, of course.pl=FALSE to prevent transace from plotting each fitted transformationx is a formula.x is a formula and variables are not already in
the search listx is a formulana.delete (in Hmisc)."avas" (the default) or aceavas or ace. Default is
100. For Mean.areg.boot, evaluation is the number of points at
whichTRUE to more quickly do bootstrapping without validating R^2avas or ace (useful if x is not a
formula)statistic="lp" or statistic="fitted".smearingEst,
the default results in computation of the sample median of the model
residuals, then smearingEst adds the median residual and
back-transforms to get estimated mstatistic="quantile", or for Quantile.areg.boot.FALSE to not plot any bootstrapped transformations. Set it to an
integer k to plot the first k bootstrap estimates.summary.areg.boot. The
latter assumes normality of the estimated effects.probs. The list
must have named components, with names corresponding to the
predictors. Example: values=list(x1=c(2summary. The more nonlinear is the transformation of y the more
the adjustment settings will matter. Defactor predictors the levels
attribute do not need to be in the same order as those used in the
original fit, and not all leFunction is to return the series of functions that
define the transformations of all variables. By default a list is
created, with the names of the list elements being the names of the
variables. Specify type="individualFunction is the
y-transformation. Specify ytype="inverse" to instead create the
inverse of the transformation, to be able to obtain originally scaled
y-values.type="individual". By default, the function specifying the
transformation for variable x will be named .x.assign). The default
is frame 0, the session database, which disappears at the end of the
S-Plus session.assign). If where is
specified (e.g., where=1 to store functions in search position one),
frame is ignored. For R, the value of where is passed totransace returns a matrix like x but containing transformed
values. This matrix has attributes rsq (vector of R^2 with which
each variable can be predicted from the others) and omitted (row
numbers of x that were deleted due to NAs).
areg.boot returns a list of class "areg.boot" containing many
elements, including (if valrsq is TRUE) rsquare.app and
rsquare.val. summary.areg.boot returns a list of class
"summary.areg.boot" containing a matrix of results for each
predictor and a vector of adjust-to settings. It also contains the
call and a label for the statistic that was computed. A print method for
these objects handles the printing. predict.areg.boot returns a
vector unless statistic="terms", in which case it returns a matrix.
Function.areg.boot returns by default a list of functions whose
argument is one of the variables (on the original scale) and whose
returned values are the corresponding transformed values. The names
of the list of functions correspond to the names of the original
variables. When type="individual", Function.areg.boot invisibly
returns the vector of names of the created function objects.
Mean.areg.boot and Quantile.areg.boot also return functions.
smearingEst returns a vector of estimates of distribution parameters
of class "labelled" so that print.labelled wil print a label
documenting the estimate that was used (see label). This label can
be retrieved for other purposes by using e.g. label(obj), where
obj was the vector returned by smearingEst.
transace only does one iteration over the predictors, it may not
find optimal transformations and it will be dependent on the order of
the predictors in x.
ace and avas standardize transformed variables to have mean zero
and variance one for each bootstrap sample, so if a predictor is not
important it will still consistently have a positive regression
coefficient. Therefore using the bootstrap to estimate standard
errors of the additive least squares regression coefficients would not
help in drawing inferences about the importance of the predictors. To
do this, summary.areg.boot computes estimates of, e.g., the
inter-quartile range effects of predictors in predicting the
response variable (after untransforming it). As an example, at each
bootstrap repetition the estimated transformed value of one of the
predictors is computed at the lower quartile, median, and upper
quartile of the raw value of the predictor. These transformed x
values are then multipled by the least squares estimate of the
partial regression coefficient for that transformed predictor in
predicting transformed y. Then these weighted transformed x values have the
weighted transformed x value corresponding to the lower quartile
subtracted from them, to estimate an x effect accounting for
nonlinearity. The last difference computed is then the standardized
effect of raising x from its lowest to its highest quartile. Before
computing differences, predicted values are back-transformed to be on
the original y scale in a way depending on statistic and q.
The sample standard deviation of these effects
(differences) is taken over the bootstrap samples, and this is used to
compute approximate confidence intervals for effects and approximate
P-values, both assuming normality.
predict does not re-insert NAs corresponding to observations that
were dropped before the fit, when newdata is omitted.
statistic="fitted" estimates the same quantity as statistic="median" if the
residuals on the transformed response have a symmetric distribution.
The two provide identical estimates when the sample median of the
residuals is exactly zero. The sample mean of the residuals is
constrained to be exactly zero although this does not simplify anything.
Duan N (1983): Smearing estimate: A nonparametric retransformation method. JASA 78:605--610.
Wang N, Ruppert D (1995): Nonparametric estimation of the transformation in the transform-both-sides regression model. JASA 90:522--534.
See avas, ace for primary references.
avas, ace, ols, validate, predab.resample, label,
nomogram# xtrans <- transace(cbind(age,sex,blood.pressure,race.code),
# binary='sex', monotonic='age',
# categorical='race.code')
# Generate random data from the model y = exp(x1 + epsilon/3) where
# x1 and epsilon are Gaussian(0,1)
set.seed(171) # to be able to reproduce example
x1 <- rnorm(200)
x2 <- runif(200) # a variable that is really unrelated to y]
x3 <- factor(sample(c('cat','dog','cow'), 200,TRUE)) # also unrelated to y
y <- exp(x1 + rnorm(200)/3)
f <- areg.boot(y ~ x1 + x2 + x3, B=40)
f
plot(f)
# Note that the fitted transformation of y is very nearly log(y)
# (the appropriate one), the transformation of x1 is nearly linear,
# and the transformations of x2 and x3 are essentially flat
# (specifying monotone(x2) would have resulted in a smaller
# confidence band for x2)
summary(f)
# use summary(f, values=list(x2=c(.2,.5,.8))) for example if you
# want to use nice round values for judging effects
# Plot Y hat vs. Y (this doesn't work if there were NAs)
plot(fitted(f), y) # or: plot(predict(f,statistic='fitted'), y)
# Show fit of model by varying x1 on the x-axis and creating separate
# panels for x2 and x3. For x2 using only a few discrete values
newdat <- expand.grid(x1=seq(-2,2,length=100),x2=c(.25,.75),
x3=c('cat','dog','cow'))
yhat <- predict(f, newdat, statistic='fitted')
# statistic='mean' to get estimated mean rather than simple inverse trans.
xYplot(yhat ~ x1 | x2, groups=x3, type='l', data=newdat)
# Another example, on hypothetical data
f <- areg.boot(response ~ I(age) + monotone(blood.pressure) + race)
# use I(response) to not transform the response variable
plot(f, conf.int=.9)
# Check distribution of residuals
plot(fitted(f), resid(f))
qqnorm(resid(f))
# Refit this model using ols so that we can draw a nomogram of it.
# The nomogram will show the linear predictor, median, mean.
# The last two are smearing estimators.
Function(f, type='individual') # create transformation functions
f.ols <- ols(.response(response) ~ age +
.blood.pressure(blood.pressure) + .race(race))
# Note: This model is almost exactly the same as f but there
# will be very small differences due to interpolation of
# transformations
meanr <- Mean(f) # create function of lp computing mean response
medr <- Quantile(f) # default quantile is .5
nomogram(f.ols, fun=list(Mean=meanr,Median=medr))
# Create S-PLUS functions that will do the transformations
# This is a table look-up with linear interpolation
g <- Function(f)
plot(blood.pressure, g$blood.pressure(blood.pressure))
# produces the central curve in the last plot done by plot(f)
# Another simulated example, where y has a log-normal distribution
# with mean x and variance 1. Untransformed y thus has median
# exp(x) and mean exp(x + .5sigma^2) = exp(x + .5)
# First generate data from the model y = exp(x + epsilon),
# epsilon ~ Gaussian(0, 1)
set.seed(139)
n <- 1000
x <- rnorm(n)
y <- exp(x + rnorm(n))
f <- areg.boot(y ~ x, B=20)
plot(f) # note log shape for y, linear for x. Good!
xs <- c(-2, 0, 2)
d <- data.frame(x=xs)
predict(f, d, 'fitted')
predict(f, d, 'median') # almost same; median residual=-.003
exp(xs) # population medians
predict(f, d, 'mean')
exp(xs + .5) # population means
# Show how smearingEst works
res <- c(-1,0,1) # define residuals
y <- 1:5
ytrans <- log(y)
ys <- seq(.1,15,length=50)
trans.approx <- list(x=log(ys), y=ys)
options(digits=4)
smearingEst(ytrans, exp, res, 'fitted') # ignores res
smearingEst(ytrans, trans.approx, res, 'fitted') # ignores res
smearingEst(ytrans, exp, res, 'median') # median res=0
smearingEst(ytrans, exp, res+.1, 'median') # median res=.1
smearingEst(ytrans, trans.approx, res, 'median')
smearingEst(ytrans, exp, res, 'mean')
mean(exp(ytrans[2] + res)) # should equal 2nd # above
smearingEst(ytrans, trans.approx, res, 'mean')
smearingEst(ytrans, trans.approx, res, mean)
# Last argument can be any statistical function operating
# on a vector that returns a single valueRun the code above in your browser using DataLab