On 2/6/03 2:29 PM, in article b1uk3q$9cm$[EMAIL PROTECTED], "hey"
<[EMAIL PROTECTED]> wrote:
> hi,
>
> i am reading on regression, and the presumptions it made were:
> 1. all the error terms Ei's have the same constant variance (sigma squared)
>
> 2. all of the Yi's have that same variance (sigma squared) but different
> means that depend on the unknown constants Xi's
>
> later it mentions that a common objective in a regression analysis is to
> estimate the mean for one or more probability distributins of Y. so if Xh
> denote the level of X we wish to estimate a mean response(Xh may be a valuse
> that occured in a sample, ir some other value within the scope of the
> model)E{Yh} is the mean response for that Xh.
>
> if you plot the regression line with the confidence interval limits for each
> Yh, youll see that if you were close to Xbar the confidence limits will be
> closer than if you went to an Xh farther from Xbar.
>
> that is because the variance for that Yh depends on the X....
>
> my question is: i thought that all Y's have the same variance. i know that
> this is predicted. is it differnet from the fitted Yhats that you get when
> you insert a regression line? arent the variances of the fitted Ybars equal?
> what is the differencem and why are the variances different in this case?
>
The confidence interval limits for Yhat are only partially a function of the
common variance of the error term. Also important it the precision with
which we can estimate each Yh. Yh's in the center (i.e., near the mean of
X) can be estimated with greater precision because those estimates are
essentially determined by the mean of Y and we use all the data to estimate
that mean. As we move from the center (i.e., more and more extreme values
of X) then are estimate of Yh is increasingly determined by the estimate of
the slope. Any error in our estimate of the slope will have no effect in
the center, but will have a larger effect as we move towards extremes. The
increasing dependence on having estimated the slope correctly implies that
our confidence in our estimate must decrease as we move away from Xbar.
gary
.
.
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