James Ankeny wrote:
> I have two questions regarding simple linear regression that I was hoping
> someone could help me with.
>
> 1) According to what I have learned so far, the levels of X are "fixed," so
> that only Y is the random variable ( error is random as well). My question
> is, what if X is a random variable as well? It seems like this could be the
> case with some of my textbook examples. Does simple model of y=a+bx+e still
> hold? Are assumptions the same, such as conditional distributions of Y are
> normal with same variance, E(Y) is a straight line function of X, and
> independence/normality of error terms? Also, in repeated sampling the sample
> slope is normal because Y is normal. However, if X also varies from sample
> to sample, is the sample slope still normally distributed (sampling
> distribution)?
The regression in which both Y and X are random variables is called
model II regression ("usual" model with X fixed is model I).
http://www.mbari.org/~etp3/regress/history.html
gives a short introduction to the subject and many references.
http://biology.queensu.ca/courses/bio243/regress.html
lists assumptions of both models and explains how to calculate model II
from the equation of model I using the reduced major axis method.
If your software does not indicate which model it calculates, most
likely you get model I which is treated as a "default". Both intercept
and slope needs to be recalculated if you want equation of the model II.
Regards,
k
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