On Wed, 25 Sep 2002 [EMAIL PROTECTED] wrote in part:
> When two lines cross, where they cross they will be equal and error
> will be zero. As the nonparallel lines extend, they grow further
> apart. I see a potential problem with this model... there is only one
> line, the regression of y on x.
But measured with uncertainty: y = b_0 + b_1*x + e.
The uncertainty in b_0 adds to the uncertainty in e, implying a kind of
probabilistic sheaf of parallel lines that are most dense at b_0.
The uncertainty in b_1 implies a probabilistic fan of lines of slightly
different slopes centered on (and most dense at) the slope b_1, and all
intersecting at (x_bar, b_0 + b_1*x_bar).
Combining both of these uncertainties leads to a confidence band of
hyperbolic shape (for a given confidence level) that is narrowest (the
two arcs of the hyperbola are closest together) at the mean of x.
> "Gus Gassmann" <[EMAIL PROTECTED]> had written in part:
> >
> > If you have two lines, y = beta_0 + beta_1 x and y = b_0 + b_1 x,
> > where b_i is close to beta_i, then the lines diverge and the distance
> > between them increases as you move away from the center (where they
> > typically intersect). So your confidence limits widen as you move
> > out into the tails.
Gus was describing the difference between the "true" regression line
(using population values beta_0 and beta_1) and the regression line
estimated from the sample (using the sample values b_0 and b_1).
Same shape of result as I've described for the uncertainty about the
estimated line, of course.
-- Don.
-----------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
[Old address: 184 Nashua Road, Bedford, NH 03110 (603) 471-7128]
.
.
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