Additive Regression and Transformations using ace or avas
areg.boot uses areg or
avas to fit additive regression models allowing
all variables in the model (including the left-hand-side) to be
transformed, with transformations chosen so as to optimize certain
criteria. The default method uses areg whose goal it is
to maximize R^2. method="avas" 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
rms 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 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. On the other hand, the areg approach allows one to
control overfitting by specifying the number of knots to use for each
continuous variable in a restricted cubic spline function.
For method="avas" the response transformation is restricted to
be monotonic. You can specify restrictions for transformations of
predictors (and linearity for the response). 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). Monotonicity restrictions are not
allowed with method="areg".
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 R 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 \var{ginverse}(\var{lp} + \var{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
\var{ginverse}(\var{lp} + median(\var{residuals})).
See the last example for how smearingEst can be used outside of
areg.boot.
Mean is a generic function that returns an R 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 (see the
example on hypothetical data).
transace(x, monotonic=NULL, categorical=NULL, binary=NULL, pl=TRUE)
areg.boot(x, data, weights, subset, na.action=na.delete,
B=100, method=c("areg","avas"), nk=4, evaluation=100, valrsq=TRUE,
probs=c(.25,.5,.75), tolerance=NULL)
## S3 method for class 'areg.boot'
print(x, ...)
## S3 method for class 'areg.boot'
plot(x, ylim, boot=TRUE, col.boot=2, lwd.boot=.15,
conf.int=.95, ...)
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, ...)
## S3 method for class 'summary.areg.boot'
print(x, ...)
## 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='', pos=-1, ...)
Mean(object, ...)
Quantile(object, ...)
## S3 method for class 'areg.boot'
Mean(object, evaluation=200, ...)
## S3 method for class 'areg.boot'
Quantile(object, q=.5, ...)x |
for |
object |
an object created by |
transEst |
a vector of transformed values. In log-normal regression these could be predicted log(Y) for example. |
inverseTrans |
a function specifying the inverse transformation needed to change
|
binary, categorical, monotonic |
These are vectors of variable names specifying what to assume about
each column of |
pl |
set |
data |
data frame to use if |
weights |
a numeric vector of observation weights. By default, all observations are weighted equally. |
subset |
an expression to subset data if |
na.action |
a function specifying how to handle |
B |
number of bootstrap samples (default=100) |
method |
|
nk |
number of knots for continuous variables not restricted to be
linear. Default is 4. One or two is not allowed. |
evaluation |
number of equally-spaced points at which to evaluate (and save) the
nonparametric transformations derived by |
valrsq |
set to |
probs |
vector probabilities denoting the quantiles of continuous predictors to use in estimating effects of those predictors |
tolerance |
singularity criterion; list source code for the
|
res |
a vector of residuals from the transformed model. Not required when
|
statistic |
statistic to estimate with the smearing estimator. For
|
q |
a single quantile of the original response scale to estimate, when
|
ylim |
2-vector of y-axis limits |
boot |
set to |
col.boot |
color for bootstrapped transformations |
lwd.boot |
line width for bootstrapped transformations |
conf.int |
confidence level (0-1) for pointwise bootstrap confidence limits and
for estimated effects of predictors in |
values |
a list of vectors of settings of the predictors, for predictors for
which you want to overide settings determined from |
adj.to |
a named vector of adjustment constants, for setting all other
predictors when examining the effect of a single predictor in
|
newdata |
a data frame or list containing the same number of values of all of
the predictors used in the fit. For |
type |
specifies how |
ytype |
By default the first function created by |
prefix |
character string defining the prefix for function names created when
|
suffix |
character string defining the suffix for the function names |
pos |
See |
... |
arguments passed to other functions |
As 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 andapproximate 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.
transace 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.
Frank Harrell
Department of Biostatistics
Vanderbilt University School of Medicine
fh@fharrell.com
Harrell FE, Lee KL, Mark DB (1996): Stat in Med 15:361–387.
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.
# 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) if method='avas' 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)
## Not run:
# 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 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)
## End(Not run)
# 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=-.001
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 valuePlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.