[R] weight in lm

[David Winsemius]

> On Aug 14, 2017, at 5:17 AM, peter dalgaard <pdalgd at gmail.com> wrote:
> 
> 
>> On 14 Aug 2017, at 13:43 , Spencer Graves <spencer.graves at effectivedefense.org> wrote:
>> 
>> 
>> 
>> On 2017-08-14 5:53 AM, peter dalgaard wrote:
>>>> On 14 Aug 2017, at 10:13 , Troels Ring <tring at gvdnet.dk> wrote:
>>>> 
>>>> Dear friends - I hope you will accept a naive question on lm: R version 3.4.1, Windows 10
>>>> 
>>>> I have 204 "baskets" of three types corresponding to factor F, each of size from 2 to 33 containing measurements, and need to know if the standard deviation on the measurements  in each basket,sdd, is different across types, F. Plotting the observed sdd  versus the sizes from 2 to 33, called "k" , does show a decreasing spread as k increases towards 33.
>>>> 
>>>> I tried lm(sdd ~ F,weight=k) and got different results if omitting the weight argument but would it be the correct way to use sqrt(k) as weight instead?
>>>> 
>>> I doubt that there is a "correct" way, but theory says that if the baskets have the same SD and data are normally distributed, then the variance of the sample VARIANCE is proportional to 1/f = 1/(k-1). Weights in lm are inverse-variance, so the "natural" thing to do would seem to be to regress the square of sdd with weights (k-1).
>>> 
>>> (If the distribution is not normal, the variance of the sample variance is complicated by a term that involves both n and the excess kurtosis, whereas the variance of the sample SD is complicated in any case. All according to the gospel of St.Google.)
>> 
>> 
>>     The Wikipedia article on "standard deviation" gives the more general formula.  (That article does NOT give a citation for that formula.  I you know one, please add it -- or post it here, to make it easier for someone else to add it.)
>> 
> 
> Er, I don't see that (i.e. var(S) etc.) in there? 
> 
> My sources were
> 
> https://math.stackexchange.com/questions/72975/variance-of-sample-variance
> https://stats.stackexchange.com/questions/631/standard-deviation-of-standard-deviation
> 
> which contains further links, but no references to publications. I suspect that this stuff is easy enough to do ab initio that people don't bother to fire up a literature search.

I don't see why that page doesn't cite: https://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation

... which had several citations including to Johnson, Kotz and Balakrishnan, v 1, ch 13 sect 8.2. I dug out my copy from the bottom of a large pile of tomes that I had not reshelved and can confirm that the formula is almost (but not quite) the same as appears in print.

JK&M give a formula (p 127) with no derivation or citation:

E[S] = sigma*( 2/n )^(1/2)*Gamma(n/2)/Gamma[ (n-1)/2 ]

Whereas the Wikipedia page citing a 1968 TAS article gives:

E[S] = sigma*( 2/(n-1) )^(1/2)*Gamma(n/2)/Gamma[ (n-1)/2 ]

I looked up the Bloch note online:

http://www.tandfonline.com/doi/abs/10.1080/00031305.1968.10480476?journalCode=utas20

And it does not have the formula. It was a note on an earlier article by Cureton, who in turn cited an American Journal of Psychology article by Holtxman(1950, v63, 615-617).
http://amstat.tandfonline.com/doi/abs/10.1080/00031305.1968.10480435?src=recsys

Searching on that article I see the first hit is a citation to some R documentation for hte MBESS::s.u function, which does implement it as recommended by Holtzman.

If I were voting on this I would put greater weight on the JK&M but that's just because it is incredibly likely that I could do the math.

Best;
David.





> 
> -pd
> 
> 
>> 
>>     Thanks, Peter.
>>     Spencer Graves
>>> 
>>> -pd
>>> 

David Winsemius
Alameda, CA, USA

'Any technology distinguishable from magic is insufficiently advanced.'   -Gehm's Corollary to Clarke's Third Law