\name{areg}
\alias{areg}
\alias{print.areg}
\alias{predict.areg}
\alias{plot.areg}
\title{Additive Regression with Optimal Transformations on Both Sides using
Canonical Variates}
\description{
Expands continuous variables into restricted cubic spline bases and
categorical variables into dummy variables and fits a multivariate
equation using canonical variates.  This finds optimum transformations
that maximize \eqn{R^2}.  Optionally, the bootstrap is used to estimate
the covariance matrix of both left- and right-hand-side transformation parameters.
}
\usage{
areg(x, y, xtype = NULL, ytype = NULL, nk = 4,
     linear.predictors = FALSE, B = 0, na.rm = TRUE,
     tolerance = NULL)

\method{print}{areg}(x, \dots)

\method{plot}{areg}(x, whichx = 1:ncol(x$x), \dots)

\method{predict}{areg}(object, x, \dots)
}
\arguments{
  \item{x}{
	A single predictor or a matrix of predictors.  Categorical
	predictors are required to be coded as integers (as \code{factor}
	does internally).
	For \code{predict}, \code{x} is a data matrix with the same integer
	codes that were originally used for categorical variables.
	}
  \item{y}{a \code{factor}, categorical, character, or numeric response
	variable}
  \item{xtype}{
	a vector of one-letter character codes specifying how each predictor
	is to be modeled, in order of columns of \code{x}.  The codes are
	\code{"s"} for smooth function (using restricted cubic splines),
	\code{"l"} for no transformation (linear), or \code{"c"} for
	categorical (to cause expansion into dummy variables).  Default is
	\code{"s"} if \code{nk > 0} and \code{"l"} if \code{nk=0}.
  }
  \item{ytype}{same coding as for \code{xtype}.  Default is \code{"s"}
	for a numeric variable with more than two unique values, \code{"l"}
	for a binary numeric variable, and \code{"c"} for a factor,
	categorical, or character variable.}
  \item{nk}{number of knots, 0 for linear, or 3 or more.  Default is 4
	which will fit 3 parameters to continuous variables (one linear term
  and two nonlinear terms)}
  \item{linear.predictors}{set to \code{TRUE} to store predicted
	transformed \code{y} in the result}
  \item{B}{number of bootstrap resamples used to estimate covariance
	matrices of transformation parameters.  Default is no bootstrapping.}
  \item{na.rm}{set to \code{FALSE} if you are sure that observations
	with \code{NA}s have already been removed}
  \item{tolerance}{singularity tolerance.  List source code for
	\code{lm.fit.qr.bare} for details.}
  \item{object}{an object created by \code{areg}}
  \item{whichx}{integer or character vector specifying which predictors
	are to have their transformations plotted (default is all).  The
	\code{y} transformation is always plotted.}
  \item{\dots}{arguments passed to the plot function.}
}
\details{
\code{areg} is a competitor of \code{ace} in the \code{acepack}
package.  Transformations from \code{ace} are seldom smooth enough and
are often overfitted.  With \code{areg} the complexity can be controlled
with the \code{nk} parameter, and predicted values are easy to obtain
because parametric functions are fitted.

If one side of the equation has a categorical variable with more than
two categories and the other side has a continuous variable not assumed
to act linearly, larger sample sizes are needed to reliably estimate
transformations, as it is difficult to optimally score categorical
variables to maximize \eqn{R^2} against a simultaneously optimally
transformed continuous variable.
}
\value{
  a list of class \code{"areg"} containing many objects
}
\references{Breiman and Friedman, Journal of the American Statistical
     Association (September, 1985).} 
\author{
Frank Harrell
\cr
Department of Biostatistics
\cr
Vanderbilt University
\cr
\email{f.harrell@vanderbilt.edu}
}
\seealso{\code{\link{cancor}},\code{\link[acepack]{ace}}, \code{\link{transcan}}}
\examples{
set.seed(1)

ns <- c(30,300,3000)
for(n in ns) {
  y <- sample(1:5,n,TRUE)
  x <- abs(y-3) + runif(n)
  par(mfrow=c(3,4))
  for(k in c(0,3:5)) {
    z <- areg(x,y,ytype='c',nk=k)
    plot(x, z$tx)
	title(paste('R2=',format(z$rsquared)))
    tapply(z$ty, y, range)
    a <- tapply(x,y,mean)
    b <- tapply(z$ty,y,mean)
    plot(a,b)
	abline(lsfit(a,b))
    # Should get same result to within linear transformation if reverse x and y
    w <- areg(y,x,xtype='c',nk=k)
    plot(z$ty, w$tx)
    title(paste('R2=',format(w$rsquared)))
    abline(lsfit(z$ty, w$tx))
 }
}

par(mfrow=c(2,2))
# Example where one category in y differs from others but only in variance of x
n <- 50
y <- sample(1:5,n,TRUE)
x <- rnorm(n)
x[y==1] <- rnorm(sum(y==1), 0, 5)
z <- areg(x,y,xtype='l',ytype='c')
z
plot(z)
z <- areg(x,y,ytype='c')
z
plot(z)
		
# Examine overfitting when true transformations are linear
par(mfrow=c(2,2))
for(n in c(200,2000)) {
  x <- rnorm(n); y <- rnorm(n) + x
  z <- areg(x,y,nk=5)
  plot(z)
  title(paste('n=',n))
}

# True transformation of x1 is quadratic, y is linear
n <- 200
x1 <- rnorm(n); x2 <- rnorm(n); y <- rnorm(n) + x1^2
z <- areg(cbind(x1,x2),y,xtype=c('s','l'),nk=3)
par(mfrow=c(2,2))
plot(z)

# y transformation is inverse quadratic but areg gets the same answer by
# making x1 quadratic
n <- 5000
x1 <- rnorm(n); x2 <- rnorm(n); y <- (x1 + rnorm(n))^2
z <- areg(cbind(x1,x2),y,nk=5)
par(mfrow=c(2,2))
plot(z)

# Overfit 20 predictors when no true relationships exist
n <- 1000
x <- matrix(runif(n*20),n,20)
y <- rnorm(n)
z <- areg(x,y,nk=5)

# Test predict function
n <- 50
x <- rnorm(n)
y <- rnorm(n) + x
g <- sample(1:3, n, TRUE)
z <- areg(cbind(x,g),y,xtype=c('s','c'),linear.predictors=TRUE)
range(predict(z, cbind(x,g)) - z$linear.predictors)
}
\keyword{smooth}
\keyword{regression}
\keyword{multivariate}
\keyword{models}
\concept{bootstrap}