Additive Regression and Transformations using ace or avas

Description

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).

Usage

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, ...)

## 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='', frame=0, where=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, ...)

Arguments

Details

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 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.

Value

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.

Author(s)

Frank Harrell
Department of Biostatistics
Vanderbilt University School of Medicine
f.harrell@vanderbilt.edu

References

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.

See avas, ace for primary references.

See Also

avas, ace, ols, validate, predab.resample, label, nomogram

Examples

# 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)

## 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-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)
## 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=-.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 value