naive "collinear" weighted linear regression

Peter Dalgaard <p.dalgaard <at> biostat.ku.dk> writes:

The point is that R (as well as almost all other mainstream statistical 
software) assumes that a "weight" means that the variance of the 
corresponding observation is the general variance divided by the weight 
factor. The general variance is still determined from the residuals, and 
if they are zero to machine precision, well, there you go. I suspect you 
  get closer to the mark with glm, which allows you to assume that the 
dispersion is known:

 > summary(glm(y~x,family="gaussian"),dispersion=0.3^2)

or

 > summary(glm(y~x,family="gaussian",weights=1/error^2),dispersion=1)

Excellent; any of these commands provide Std. Errors which now coincide with my
naive expectation: though the data fall perfectly in a straight line, since they
have some associated "uncertainties" (only) in the response variables
(homoskedasticity), the estimated coefficients should have some kind of
nonvanishing "uncertainties" as well, should they not??

Now, forgive me, but I did not get the explanation for the distinct meanings of
Std. Error when calling simply summary(lm(y~x,weights=1/error^2), which I had
done before, and your suggested calls; could you rephrase and dwell a little bit
more upon this point. What does the option dispersion exactly mean?

Also, could you suggest some specific reference for me to read about this? I
have your excellent book "Introductory statistics with R", 1st edition, but was
not able (perhaps I have missed some point) to find this kind of distinction
there... Does this theme is specifically what statisticians call really
generalized linear models (glm) as opposed to (ordinary) linear models? If so,
which good references could you please suggest?? I thought of the following
books and would feel much obliged should you give me your impressions about
them, if any, or about any other relevant references at all:

1) Faraway, "Linear models with R"
2) Faraway, "Extending the linear model with R: generalized linear..."
3) Fox, "An R and S-Plus companion.."
4) Uusipaikka, "Confidence intervals in generalized linear regression models"

Thank you very much!!