Indexes of Absolute Prediction Error for Linear Models

Computes the mean and median of various absolute errors related to ordinary multiple regression models. The mean and median absolute errors correspond to the mean square due to regression, error, and total. The absolute errors computed are derived from \(\hat{Y} - \mbox{median($\hat{Y}$)}\), \(\hat{Y} - Y\), and \(Y - \mbox{median($Y$)}\). The function also computes ratios that correspond to \(R^2\) and \(1 - R^2\) (but these ratios do not add to 1.0); the \(R^2\) measure is the ratio of mean or median absolute \(\hat{Y} - \mbox{median($\hat{Y}$)}\) to the mean or median absolute \(Y - \mbox{median($Y$)}\). The \(1 - R^2\) or SSE/SST measure is the mean or median absolute \(\hat{Y} - Y\) divided by the mean or median absolute \(\hat{Y} - \mbox{median($Y$)}\).

Usage

abs.error.pred(fit, lp=NULL, y=NULL)

# S3 method for class 'abs.error.pred'
print(x, ...)

Arguments

fit

a fit object typically from lm or ols that contains a y vector (i.e., you should have specified y=TRUE to the fitting function) unless the y argument is given to abs.error.pred. If you do not specify the lp argument, fit must contain fitted.values or linear.predictors. You must specify fit or both of lp and y.

lp

a vector of predicted values (Y hat above) if fit is not given

y

a vector of response variable values if fit (with y=TRUE in effect) is not given

x

an object created by abs.error.pred

...

unused

Value

a list of class abs.error.pred (used by print.abs.error.pred) containing two matrices: differences and ratios.

Author

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

See also

lm, ols, cor, validate.ols

References

Schemper M (2003): Stat in Med 22:2299-2308.

Tian L, Cai T, Goetghebeur E, Wei LJ (2007): Biometrika 94:297-311.

Examples

set.seed(1)         # so can regenerate results
x1 <- rnorm(100)
x2 <- rnorm(100)
y  <- exp(x1+x2+rnorm(100))
f <- lm(log(y) ~ x1 + poly(x2,3), y=TRUE)
abs.error.pred(lp=exp(fitted(f)), y=y)
#> 
#> Mean/Median |Differences|
#> 
#>                              Mean    Median
#> |Yi hat - median(Y hat)| 1.983447 0.8651185
#> |Yi hat - Yi|            2.184563 0.5436367
#> |Yi - median(Y)|         2.976277 1.0091661
#> 
#> 
#> Ratios of Mean/Median |Differences|
#> 
#>                                                Mean    Median
#> |Yi hat - median(Y hat)|/|Yi - median(Y)| 0.6664189 0.8572607
#> |Yi hat - Yi|/|Yi - median(Y)|            0.7339920 0.5386989
rm(x1,x2,y,f)