mvtnorm (version 0.3-0)

pmvt: Multivariate t Distribution

Description

Computes the the distribution function of the multivariate t distribution for arbitrary limits, degrees of freedom and correlation matrices based on algorithms by Genz and Bretz.

Usage

pmvt(lower, upper, df, corr, delta, maxpts=25000, abseps=0.001, releps=0)

Arguments

lower

the vector of lower limits of length n.

upper

the vector of upper limits of length n.

df

degree of freedom as integer.

corr

the correlation matrix of dimension n.

delta

the vector of noncentrality parameters of length n.

maxpts

maximum number of function values as integer.

abseps

absolute error tolerance as double.

releps

relative error tolerance as double.

Value

• A list with the following components:
• value
  estimated integral value.
• error
  estimated absolute error.
• msg
  status messages.

Details

This program involves the computation of central and noncentral multivariate t-probabilities with arbitrary correlation matrices. It involves both the computation of singular and nonsingular probabilities. The methodology is described in Genz and Bretz (1999, 2001).

For a given correlation matrix corr, for short $A$, say, (which has to be positive semi-definite) and degrees of freedom df the following values are numerically evaluated

$$I = K \int s^{df-1} \exp(-s^2/2) \Phi(s \cdot lower/\sqrt{df}-delta, s \cdot upper/\sqrt{df}-delta) ds$$

where $\Phi(a,b) = K^\prime \int_a^b \exp(-x^\prime Ax/2) dx$ is the multivariate normal distribution, $K^\prime = 1/\sqrt{det(A)(2\pi)^m}$ and $K = 2^{1-df/2} / Gamma(df/2)$ are constants and the (single) integral of $I$ goes from 0 to +Inf.

Note that both -Inf and +Inf may be specified in the lower and upper integral limits. Randomized quasi-Monte Carlo methods are used for the computations.

Univariate problems are passed to pt.

Further information can be obtained from the quoted articles, which can be downloaded (together with additional material and additional codes) from the websites http://www.bioinf.uni-hannover.de/~bretz/ and http://www.sci.wsu.edu/math/faculty/genz/homepage.

References

Genz, A. and Bretz, F. (1999), Numerical computation of multivariate t-probabilities with application to power calculation of multiple contrasts. Journal of Statistical Computation and Simulation, 63, 361--378.

Genz, A. and Bretz, F. (2001), Methods for the computation of multivariate t-probabilities. (submitted)

Edwards D. and Berry, Jack J. (1987), The efficiency of simulation-based multiple comparisons. Biometrics, 43, 913--928

Examples

n <- 5
lower <- rep(-1, 5)
upper <- rep(3, 5)
df <- 4
corr <- diag(5)
corr[lower.tri(corr)] <- 0.5
delta <- rep(0, 5)
prob <- pmvt(lower, upper, df, corr , delta)
print(prob)

pmvt(-Inf, 3, df = 3, corr = 0)$value == pt(3, 3)

# Example from R News paper (original by Edwards and Berry, 1987)

n <- c(26, 24, 20, 33, 32)
V <- diag(1/n)
df <- 130
C <- c(1,1,1,0,0,-1,0,0,1,0,0,-1,0,0,1,0,0,0,-1,-1,0,0,-1,0,0)
C <- matrix(C, ncol=5)
cv <- C %*% V %*% t(C)
cr <- matrix(rep(0, ncol(cv)^2), ncol=ncol(cv))
for (i in 1:5) {
  for (j in 1:5) {
    cr[i,j] <- cv[i,j]/sqrt(cv[i,i]*cv[j,j] )
  }
}
delta <- rep(0,5)

myfct <- function(q, alpha) {
  lower <- rep(-q, ncol(cv))
  upper <- rep(q, ncol(cv))
  pmvt(lower, upper, df, cr, delta, abseps=0.0001)$value - alpha
}

round(uniroot(myfct, lower=1, upper=5, alpha=0.95)$root, 3)