[R] Weighted least squares

John Fox jfox at mcmaster.ca
Tue May 8 17:19:16 CEST 2007

Dear Hadley,

I think that the problem is that the term "weights" has different meanings,
which, although they are related, are not quite the same. 

The weights used by lm() are (inverse-)"variance weights," reflecting the
variances of the errors, with observations that have low-variance errors
therefore being accorded greater weight in the resulting WLS regression.
What you have are sometimes called "case weights," and I'm unaware of a
general way of handling them in R, although you could regenerate the
unaggregated data. As you discovered, you get the same coefficients with
case weights as with variance weights, but different standard errors.
Finally, there are "sampling weights," which are inversely proportional to
the probability of selection; these are accommodated by the survey package. 

To complicate matters, this terminology isn't entirely standard.

I hope this helps,

John Fox, Professor
Department of Sociology
McMaster University
Hamilton, Ontario
Canada L8S 4M4

> -----Original Message-----
> From: r-help-bounces at stat.math.ethz.ch 
> [mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of hadley wickham
> Sent: Tuesday, May 08, 2007 5:09 AM
> To: R Help
> Subject: [R] Weighted least squares
> Dear all,
> I'm struggling with weighted least squares, where something 
> that I had assumed to be true appears not to be the case.  
> Take the following data set as an example:
> df <- data.frame(x = runif(100, 0, 100)) df$y <- df$x + 1 + 
> rnorm(100, sd=15)
> I had expected that:
> summary(lm(y ~ x, data=df, weights=rep(2, 100))) summary(lm(y 
> ~ x, data=rbind(df,df)))
> would be equivalent, but they are not.  I suspect the 
> difference is how the degrees of freedom is calculated - I 
> had expected it to be sum(weights), but seems to be 
> sum(weights > 0).  This seems unintuitive to me:
> summary(lm(y ~ x, data=df, weights=rep(c(0,2), each=50))) 
> summary(lm(y ~ x, data=df, weights=rep(c(0.01,2), each=50)))
> What am I missing?  And what is the usual way to do a linear 
> regression when you have aggregated data?
> Thanks,
> Hadley
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