stat question - not R question so ignore if not interested

If do a scattrplot of data ( x and y ) and there are two clouds of
points. One cloud is in the left
bottom corner of the plot and the other cloud is in the upper right.

If I fit a regression line to this data ( or equivalently ,  
calculate a
correlation ), then obviously, it is going to seem like
x and y are related because a line has to be connected between the 2
clouds. But, there must be a regression assumption that
is violated here because if the regressions are done separately on  
each
cloud, then there really isn't
a relationship between x and y. I was just wondering 1) what  
assumption
in regression is being violated in
the first case or 2) possibly if the regression is valid and the  
results
just have some different interpreation ?

One needs only to look at diagnostic plots:

Suppose
set.seed(2)
xy <- data.frame(y = c(rnorm(300), rnorm(300, 5)), x = c(rnorm(300),  
rnorm(300, 5)))
op <- par(mfrow = c(2,2))
plot(lm(y ~ x, xy))
par(op)

The model does not fit well because the residuals aren't flat as a  
function of fit and because homoscedasticity is violated.

When this happens we might try a different approach:
require(sm)
xy.sm <- sm.regression(xy$x, xy$y)

Whenever there's a big discrepancy between an OLS fit and a robust  
one, we should not pursue the OLS one w/o reinterpretation, which  
others have discussed in their replies.
_____________________________
Professor Michael Kubovy
University of Virginia
Department of Psychology
USPS:     P.O.Box 400400    Charlottesville, VA 22904-4400
Parcels:    Room 102        Gilmer Hall
         McCormick Road    Charlottesville, VA 22903
Office:    B011    +1-434-982-4729
Lab:        B019    +1-434-982-4751
Fax:        +1-434-982-4766
WWW:    http://www.people.virginia.edu/~mk9y/