# [R] Weighted least squares

John Fox jfox at mcmaster.ca
Wed May 9 13:10:54 CEST 2007

```Dear Adai,

> -----Original Message-----
> From: Adaikalavan Ramasamy [mailto:ramasamy at cancer.org.uk]
> Sent: Tuesday, May 08, 2007 8:38 PM
> To: S Ellison
> Cc: h.wickham at gmail.com; jfox at mcmaster.ca; R-help at stat.math.ethz.ch
> Subject: Re: [R] Weighted least squares
>
> http://en.wikipedia.org/wiki/Weighted_least_squares gives a
> formulaic description of what you have said.
>
> I believe the original poster has converted something like this
>
> 	y	  x
> 	0	1.1
> 	0	2.2
> 	0	2.2
> 	0	2.2
> 	1	3.3
> 	1	3.3
> 	2	4.4
>          ...
>
> into something like the following
>
> 	y	  x	freq
> 	0	1.1	   1
> 	0	2.2	   3
> 	1	3.3        2
> 	2	4.4        1
>          ...
>
> Now, the variance of means of each row in table above is ZERO
> because the individual elements that comprise each row are
> identical. Therefore your method of using inverse-variance
> will not work here.
>
> Then is it valid then to use lm( y ~ x, weights=freq ) ?

No, because the weights argument gives inverse-variance weights not case
weights.

Regards,
John

>
>
>
>
> S Ellison wrote:
> >
> >> .. what is the usual way to do a linear regression when you have
> >> aggregated data?
> >
> > Least squares generally uses inverse variance weighting.
> For aggregated data fitted as mean values, you just need the
> variances for the _means_.
> >
> > So if you have individual means x_i and sd's s_i that arise
> from aggregated data with n_i observations in group i, the
> natural weighting is by inverse squared standard error of the
> mean. The appropriate weight for x_i would then be
> n_i/(s_i^2). In R, that's n/(s^2), as n and s would be
> vectors with the same length as x. If all the groups had the
> same variance, or nearly so, s is a scalar; if they have the
> same number of observations, n is a scalar.
> >
> > Of course, if they have the same variance and same number
> of observations, they all have the same weight and you
> needn't weight them at all: see previous posting!
> >
> > Steve E
> >
> >
> >
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