Description Usage Arguments Details Value Author(s) References See Also Examples
Test statistics of normal distributionbased orderrelated likelihood ratio tests are often distributed as mixtures of chisquare or betadistributions with different parameters. These functions determine the mixing weights and the cumulative distribution functions based on these. They can be directly used and are called by function ic.test.
1 2 3  ic.weights(corr, ...)
pchibar(x, df, wt)
pbetabar(x, df1, df2, wt)

corr 

... 
... contains further arguments to be given to function

x 

df 
is the vector of the degrees of freedom for the chisquare
distributions that are mixed into the chibarsquaredistribution
with the proportions given in 
wt 
each element of 
df1 
vector of first parameters of the betadistributions to be mixed into the betabardistribution 
df2 
second parameter of the betadistributions to be
mixed into the betabardistribution; error degrees of freedom
in the tests implemented for linear models in summary.orlm; 
Function ic.weights
uses results by Kudo (1963)
regarding the calculation of the weights. The weights are the probabilities that
the projection along its covariance onto the nonnegative orthant
of a multivariate normal random vector with expectation 0 and
correlation corr
lies in faces of dimensions nrow(corr):1
(in this order). It is known that these probabilities coincide with
various other useful probabilities related to orderrelated hypothesis testing,
cf. e.g. Shapiro (1988). Calculation of the weights involves various calls
to function pmvnorm
from package mvtnorm
.
Functions pchibar
(taken from package ibdreg) and pbetabar
calculate cumulative probabilities from mixtures of chisquare and
betadistributions, respectively.
IMPORTANT: Contrary to likelihood ratio theory in linear models, the beta
distributions mixed always use the error sum of squares from the unrestricted model,
i.e. the smallest possible error sum of squares with a fixed no. of df. Therefore,
the second df entry is not increased when decreasing the first!
This is appropriate for the test statistics calculated by functions ic.test
or summary.orlm
, but not necessarily for test statistics obtained elsewhere.
ic.weights
returns the vector of weights,
pchibar
and pchibar
return the cumulative probability of the
respective distribution.
Function ic.weights
relies on package mvtnorm for determining
multivariate normal rectangle probabilities. Note that these calculations
involve Monte Carlo steps so that these weights are not completely repeatable.
Ulrike Groemping, BHT Berlin
Kudo, A. (1963) A multivariate analogue of the onesided test. Biometrika 50, 403–418
Shapiro, A. (1988) Towards a unified theory of inequalityconstrained testing in multivariate analysis. International Statistical Review 56, 49–62
Silvapulle, M.J. and Sen, P.K. (2004) Constrained Statistical Inference. Wiley, New York
ic.test
, orlm
, pmvnorm
,
GenzBretz
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  z < 0.5
corr < matrix(c(1,0.9,0.9,1),2,2)
print(wt.plus < ic.weights(corr))
T < c(z,z)%*%solve(corr,c(z,z))
1pchibar(T,2:0,wt.plus)
1pbetabar(T/(T+10),2:0,10,wt.plus)
corr < matrix(c(1,0,0,1),2,2)
print(wt.0 < ic.weights(corr))
T < c(z,z)%*%solve(corr,c(z,z))
1pchibar(T,2:0,wt.0)
1pbetabar(T/(T+10),2:0,10,wt.0)
corr < matrix(c(1,0.9,0.9,1),2,2)
print(wt.minus < ic.weights(corr))
T < c(z,z)%*%solve(corr,c(z,z))
1pchibar(T,2:0,wt.minus)
1pbetabar(T/(T+10),2:0,10,wt.minus)

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