Engineers tend to use a version of the residual sum of squares (RSS) called PRESS, for predictive RSS. The idea is that RSS describes how well a linear model fits the data to which it was fitted, but PRESS tells you how well the model will predict new data.

Let's make up some data and fit a model,

n <- 10
x <- rnorm(n)
y <- rnorm(n)
(m <- lm(y ~ x))

##
## Call:
## lm(formula = y ~ x)
##
## Coefficients:
## (Intercept)            x
##      -1.037       -0.551


Here are the residuals,

(r <- resid(m))

##        1        2        3        4        5        6        7        8
## -0.51730 -0.19795  0.61276 -1.76696  1.45084  0.06266  1.01057 -0.29810
##        9       10
## -0.35962  0.00310


And here are the predictively adjusted residuals,

(pr <- resid(m)/(1 - lm.influence(m)\$hat))

##         1         2         3         4         5         6         7
## -0.742199 -0.220755  0.689584 -2.066720  1.614070  0.115833  1.231767
##         8         9        10
## -0.337332 -0.530903  0.003685


There is some theoretical magic that makes this equal to the cross-validated residuals. So the regular RSS is,

sum(r^2)

## [1] 7.153


and the PRESS is,

sum(pr^2)

## [1] 9.878


which is bigger because predicting is harder than fitting.