# areg: Additive Regression with Optimal Transformations on Both... In Hmisc: Harrell Miscellaneous

 areg R Documentation

## 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 R^2. Optionally, the bootstrap is used to estimate the covariance matrix of both left- and right-hand-side transformation parameters, and to estimate the bias in the R^2 due to overfitting and compute the bootstrap optimism-corrected R^2. Cross-validation can also be used to get an unbiased estimate of R^2 but this is not as precise as the bootstrap estimate. The bootstrap and cross-validation may also used to get estimates of mean and median absolute error in predicted values on the original `y` scale. These two estimates are perhaps the best ones for gauging the accuracy of a flexible model, because it is difficult to compare R^2 under different y-transformations, and because R^2 allows for an out-of-sample recalibration (i.e., it only measures relative errors).

Note that uncertainty about the proper transformation of `y` causes an enormous amount of model uncertainty. When the transformation for `y` is estimated from the data a high variance in predicted values on the original `y` scale may result, especially if the true transformation is linear. Comparing bootstrap or cross-validated mean absolute errors with and without restricted the `y` transform to be linear (`ytype='l'`) may help the analyst choose the proper model complexity.

### Usage

```areg(x, y, xtype = NULL, ytype = NULL, nk = 4,
B = 0, na.rm = TRUE, tolerance = NULL, crossval = NULL)

## S3 method for class 'areg'
print(x, digits=4, ...)

## S3 method for class 'areg'
plot(x, whichx = 1:ncol(x\$x), ...)

## S3 method for class 'areg'
predict(object, x, type=c('lp','fitted','x'),
what=c('all','sample'), ...)
```

### Arguments

 `x` A single predictor or a matrix of predictors. Categorical predictors are required to be coded as integers (as `factor` does internally). For `predict`, `x` is a data matrix with the same integer codes that were originally used for categorical variables. `y` a `factor`, categorical, character, or numeric response variable `xtype` a vector of one-letter character codes specifying how each predictor is to be modeled, in order of columns of `x`. The codes are `"s"` for smooth function (using restricted cubic splines), `"l"` for no transformation (linear), or `"c"` for categorical (to cause expansion into dummy variables). Default is `"s"` if `nk > 0` and `"l"` if `nk=0`. `ytype` same coding as for `xtype`. Default is `"s"` for a numeric variable with more than two unique values, `"l"` for a binary numeric variable, and `"c"` for a factor, categorical, or character variable. `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) `B` number of bootstrap resamples used to estimate covariance matrices of transformation parameters. Default is no bootstrapping. `na.rm` set to `FALSE` if you are sure that observations with `NA`s have already been removed `tolerance` singularity tolerance. List source code for `lm.fit.qr.bare` for details. `crossval` set to a positive integer k to compute k-fold cross-validated R-squared (square of first canonical correlation) and mean and median absolute error of predictions on the original scale `digits` number of digits to use in formatting for printing `object` an object created by `areg` `whichx` integer or character vector specifying which predictors are to have their transformations plotted (default is all). The `y` transformation is always plotted. `type` tells `predict` whether to obtain predicted untransformed `y` (`type='lp'`, the default) or predicted `y` on the original scale (`type='fitted'`), or the design matrix for the right-hand side (`type='x'`). `what` When the `y`-transform is non-monotonic you may specify `what='sample'` to `predict` to obtain a random sample of `y` values on the original scale instead of a matrix of all `y`-inverses. See `inverseFunction`. `...` arguments passed to the plot function.

### Details

`areg` is a competitor of `ace` in the `acepack` package. Transformations from `ace` are seldom smooth enough and are often overfitted. With `areg` the complexity can be controlled with the `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 R^2 against a simultaneously optimally transformed continuous variable.

### Value

a list of class `"areg"` containing many objects

### Author(s)

Frank Harrell
Department of Biostatistics
Vanderbilt University
fh@fharrell.com

### References

Breiman and Friedman, Journal of the American Statistical Association (September, 1985).

`cancor`,`ace`, `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)

## Not run:
# Examine overfitting when true transformations are linear
par(mfrow=c(4,3))
for(n in c(200,2000)) {
x <- rnorm(n); y <- rnorm(n) + x
for(nk in c(0,3,5)) {
z <- areg(x, y, nk=nk, crossval=10, B=100)
print(z)
plot(z)
title(paste('n=',n))
}
}
par(mfrow=c(1,1))

# Underfitting when true transformation is quadratic but overfitting
# when y is allowed to be transformed
set.seed(49)
n <- 200
x <- rnorm(n); y <- rnorm(n) + .5*x^2
#areg(x, y, nk=0, crossval=10, B=100)
#areg(x, y, nk=4, ytype='l', crossval=10, B=100)
z <- areg(x, y, nk=4) #, crossval=10, B=100)
z
# Plot x vs. predicted value on original scale.  Since y-transform is
# not monotonic, there are multiple y-inverses
xx <- seq(-3.5,3.5,length=1000)
yhat <- predict(z, xx, type='fitted')
plot(x, y, xlim=c(-3.5,3.5))
for(j in 1:ncol(yhat)) lines(xx, yhat[,j], col=j)
# Plot a random sample of possible y inverses
yhats <- predict(z, xx, type='fitted', what='sample')
points(xx, yhats, pch=2)

## End(Not run)

# 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
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)  # add crossval=4 to expose the problem

# 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'))
range(predict(z, cbind(x,g)) - z\$linear.predictors)
```

Hmisc documentation built on April 19, 2022, 9:05 a.m.