ic.weights: functions for calculating the distributions of normal...

In ic.infer: Inequality Constrained Inference in Linear Normal Situations

functions for calculating the distributions of normal distribution order-related likelihood ratio tests

Description

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.

Usage

ic.weights(corr, ...)
pchibar(x, df, wt)
pbetabar(x, df1, df2, wt)

Arguments

corr	corr is the correlation or covariance matrix (or any multiple thereof) of the data or coefficients for which weights are to be calculated
...	... contains further arguments to be given to function pmvnorm of package mvtnorm for calculating multivariate normal rectangle probabilities; it is possible to select an algorithm (default in current version of mvtnorm: algorithm = GenzBretz()) and to tune weight accuracy by modifying including additional parameters into the algorithm specification, cf. help for GenzBretz
x	x is the quantile for which the distribution function is to be calculated
df	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 wt
wt	each element of wt is the mixing weight of the chi-square distribution with df as in the corresponding element of df; such weights can be calculated with function ic.weights
df1	vector of first parameters of the beta-distributions to be mixed into the betabar-distribution
df2	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; NOTE: see details for the (perhaps unexpected) constancy of df2

Details

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 non-negative 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 order-related 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 chi-square and beta-distributions, 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.

Value

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.

Author(s)

Ulrike Groemping, BHT Berlin

References

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

See Also

ic.test, orlm, pmvnorm, GenzBretz

Examples

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)

ic.infer documentation built on Oct. 5, 2023, 5:09 p.m.