Noncentral t & F distributions
Alan Arnholt wrote:
Dear List:
The square of the noncentral t-statistic with noncentrality parameter
\delta is a noncentral F with noncentrality parameter \lambda=\delta^2.
So, t^2_{\nu,\delta} = F_{1,\nu,\lambda=\delta^2}. Consequently, it
should follow that t^2_{1-\alpha/2,\nu,\delta} =
f_{1-alpha,1,\vu,\lambda=\delta^2}. However, this is not what is
happening with the following code. The central distributions agree as
they should but the noncentral distributions do not. Am I missing
something or is there a bug in the code?
You're missing something: The noncentral t is not symmetric, so there's no obvious relation between qf(1-alpha), qt(alpha/2) and qt(1-alpha/2)
alpha <- 0.05
nu <- 10
NCP <- c(0,1,2,3)
TV <- (qt(1-alpha/2,nu,NCP))^2
FV <- qf(1-alpha,1,nu,NCP^2)
rbind(TV,FV)
[,1] [,2] [,3] [,4] TV 4.964603 12.535179 24.58013 41.71937 FV 4.964603 9.285829 18.98771 32.97855
TV <- (qt(1-alpha/2,nu,NCP))^2
FV <- qf(1-alpha/2,1,nu,NCP^2)
rbind(TV,FV)
[,1] [,2] [,3] [,4] TV 4.964603 12.53518 24.58013 41.71937 FV 6.936728 12.56450 24.58023 41.71937
rbind(pt(sqrt(FV),nu,NCP,lower=F), pt(-sqrt(FV),nu,NCP))
[,1] [,2] [,3] [,4] [1,] 0.025 0.0496216546 4.999922e-02 5.0000e-02 [2,] 0.025 0.0003783451 7.776214e-07 7.8153e-10
colSums(rbind(pt(sqrt(FV),nu,NCP,lower=F), pt(-sqrt(FV),nu,NCP)))
[1] 0.05 0.05 0.05 0.05
Thanks, Alan-
version
_ platform i386-pc-mingw32 arch i386 os mingw32 system i386, mingw32 status major 2 minor 4.0 year 2006 month 10 day 03 svn rev 39566 language R version.string R version 2.4.0 (2006-10-03) Alan T. Arnholt Associate Professor Dept. of Mathematical Sciences Appalachian State University TEL: 828 262 2863 FAX: 828 265 8617
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