Hi, thanks everyone!
pchisqsum() in the "survey" package does exactly what I was looking for! Best wishes, Klaus -------- Original-Nachricht -------- Datum: Thu, 29 Mar 2007 07:45:15 -0700 (PDT) Von: Thomas Lumley <[EMAIL PROTECTED]> An: S Ellison <[EMAIL PROTECTED]> CC: [EMAIL PROTECTED], [EMAIL PROTECTED], [email protected], [EMAIL PROTECTED] Betreff: Re: [R] Tail area of sum of Chi-square variables > > The Satterthwaite approximation is surprisingly good, especially in the > most interesting range in the right tail (say 0.9 to 0.999). There is also > another, better, approximation with a power of a chi-squared distribution > that has been used in the survey literature. > > However, since it is easy to write down the characteristic function and > perfectly feasible to invert it by numerical integration, we might as well > use the right answer. > > -thomas > > On Thu, 29 Mar 2007, S Ellison wrote: > >> I was wondering if there are any R functions that give the tail area > >> of a sum of chisquare distributions of the type: > >> a_1 X_1 + a_2 X_2 > >> where a_1 and a_2 are constants and X_1 and X_2 are independent > >> chi-square variables with different degrees of freedom. > > > > You might also check out Welch and Satterthwaite's (separate) papers on > effective degrees of freedom for compound estimates of variance, which led > to a thing called the welch-satterthwaite equation by one (more or less > notorious, but widely used) document called the ISO Guide to Expression of > Uncertainty in Measurement (ISO, 1995). The original papers are > > B. L. Welch, J. Royal Stat. Soc. Suppl.(1936) 3 29-48 > > B. L. Welch, Biometrika, (1938) 29 350-362 > > B. L. Welch, Biometrika, (1947) 34 28-35 > > > > F. E. Satterthwaite, Psychometrika (1941) 6 309-316 > > F. E. Satterthwaite, Biometrics Bulletin, (1946) 2 part 6 110-114 > > > > The W-S equation - which I believe is a special case of Welch's somewhat > more general treatment - says that if you have multiple independent > estimated variances v[i] (could be more or less equivalent to your a_i X_i?) > with > degrees of freedom nu[i], the distribution of their sum is approximately a > scaled chi-squared distribution with effective degrees of freedom > nu.effective given by > > > > nu.effective = sum(v[i])^2 / sum( (v[i]^2)/nu[i] ) > > > > If I recall correctly, with an observed variance s^2 (corresponding to > the sum(v[i] above if those are observed varianes), nu*(s^2 /sigma^2) is > distributed as chi-squared with degrees of freedom nu, so the scaling factor > for quantiles would come out of there (depending whether you're after the > tail areas for s^2 given sigma^2 or for a confidence interval for sigma^2 > given s^2) > > > > However, I will be most interested to see what a more exact calculation > provides! > > > > Steve Ellison > > > > > > ******************************************************************* > > This email and any attachments are confidential. Any use, > co...{{dropped}} > > > > ______________________________________________ > > [email protected] mailing list > > https://stat.ethz.ch/mailman/listinfo/r-help > > PLEASE do read the posting guide > http://www.R-project.org/posting-guide.html > > and provide commented, minimal, self-contained, reproducible code. > > > > Thomas Lumley Assoc. Professor, Biostatistics > [EMAIL PROTECTED] University of Washington, Seattle -- "Feel free" - 10 GB Mailbox, 100 FreeSMS/Monat ... ______________________________________________ [email protected] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
