emdbook (version 1.3.2.1)

dchibarsq: Mixed chi-squared distributions

Description

Calculates "mixed" chi-squared distributions (mixtures of chi-square(n) and chi-square(n-1)); useful for Likelihood Ratio Tests when parameters are on the boundary

Usage

dchibarsq(x, df = 1, mix = 0.5, log = FALSE)
pchibarsq(p, df = 1, mix = 0.5, lower.tail=TRUE, log.p = FALSE)
qchibarsq(q, df = 1, mix = 0.5)
rchibarsq(n, df = 1, mix = 0.5)

Arguments

x

numeric vector of positive values

p

numeric vector of positive values

q

numeric vector of quantiles (0-1)

n

integer: number of random deviates to pick

df

degrees of freedom (positive integer)

mix

mixture parameter: fraction of distribution that is chi-square(n-1) distributed

log

return log densities?

log.p

return log probabilities?

lower.tail

return lower tail values?

Value

• Vectors of probability densities (dchibarsq), cumulative probabilities (pchibarsq), quantiles (qchibarsq), or random deviates (rchibarsq) from Goldman and Whelan's "chi-bar-squared" distribution. qchibarsq uses simple algebra for df=1 and uniroot for df>1.

References

N. Goldman and S. Whelan (2000) "Statistical Tests of Gamma-Distributed Rate Heterogeneity in Models of Sequence Evolution in Phylogenetics", Mol. Biol. Evol. 17:975-978. D. O. Stram and J. W. Lee (1994) "Variance Components Testing in the Longitudinal Fixed Effects Model", Biometrics 50:1171-1177.

Examples

x <- rchibarsq(100)
plot(density(x,from=0))
curve(dchibarsq(x),add=TRUE,col=2,from=0)
library(lattice)
print(qqmath(~ simdist,
       distribution=qchibarsq,
       panel = function(x, ...) {
         panel.qqmathline(x, ...)
         panel.qqmath(x, ...)
            }))
## create first line of table in Goldman and Whelan 2000
round(qchibarsq(c(0.01,0.05,0.9,0.95,0.975,0.99,0.995),df=1),2)
## check second line of table
round(pchibarsq(c(3.81,5.14,6.48,8.27,9.63),df=2),3)
## create middle column
round(qchibarsq(0.95,df=1:10))