Test statistics of normal distribution-based order-related likelihood ratio tests are often distributed as mixtures of chi-square or beta-distributions 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.
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... contains further arguments to be given to function
is the vector of the degrees of freedom for the chi-square
distributions that are mixed into the chibar-square-distribution
with the proportions given in
each element of
vector of first parameters of the beta-distributions to be mixed into the betabar-distribution
second parameter of the beta-distributions to be
mixed into the betabar-distribution; error degrees of freedom
in the tests implemented for linear models in summary.orlm;
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 non-negative orthant
of a multivariate normal random vector with expectation 0 and
corr lies in faces of dimensions
(in this order). It is known that these probabilities coincide with
various other useful probabilities related to order-related hypothesis testing,
cf. e.g. Shapiro (1988). Calculation of the weights involves various calls
pmvnorm from package
pchibar (taken from package ibdreg) and
calculate cumulative probabilities from mixtures of chi-square and
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
summary.orlm, but not necessarily for test statistics obtained elsewhere.
ic.weights returns the vector of weights,
pchibar return the cumulative probability of the
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 one-sided test. Biometrika 50, 403–418
Shapiro, A. (1988) Towards a unified theory of inequality-constrained testing in multivariate analysis. International Statistical Review 56, 49–62
Silvapulle, M.J. and Sen, P.K. (2004) Constrained Statistical Inference. Wiley, New York
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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)) 1-pchibar(T,2:0,wt.plus) 1-pbetabar(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)) 1-pchibar(T,2:0,wt.0) 1-pbetabar(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)) 1-pchibar(T,2:0,wt.minus) 1-pbetabar(T/(T+10),2:0,10,wt.minus)
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