mvlinks: Multivariate link functions in mvord

Description

Different link functions are available in mvord:

Usage

mvprobit()
mvlogit(df = 8L)

Value

The functions mvlogit() and mvprobit() returns an object of class

'mvlink'. An object of class

'mvlink' is a list containing the following components:

name

name of the multivariate link function

df

degrees of freedom of the t copula; returned only for mvlogit()

F_uni

a function corresponding to the univariate margins of the multivariate distribution \(F\) of the subject errors; the function returns \(Pr(X \leq x) = F_1(x)\)

F_biv

a function corresponding to the bivariate distribution of the multivariate distribution \(F\) of the subject errors \(Pr(X \leq x, Y\leq y|r) = F_2(x, y, r)\);

F_biv_rect

the function computes the rectangle probabilities from based on F_biv; the function has the matrices U (upper bounds) and L (lower bounds) as well as vector r containing the correlation coefficients corresponding to the bivariate distribution as arguments; the matrices U and L both have two columns, first corresponding to the bounds of x, second to the bounds of y; the number of rows corresponds to the number of observations; the rectangle probabilities are defined as \(Pr(L[,1]\leq X\leq U[,1], L[,2]\leq Y \leq U[,2]|r) = F_2(U[,1], U[,2],r) - F_2(U[,1], L[,2],r)- F_2(L[,1], U[,2],r) + F_2(L[,1], L[,2],r)\)

F_multi

the function computes the multivariate probabilities for distribution function \(F\); the function has the matrices U (upper bounds) and L (lower bounds) as well as the list list_R containing for each observation the correlation matrix; F is needed for the computation of the fitted/predicted joint probabilities. If NULL only marginal probabilities can be computed.

deriv.fun

(needed for computation of analytic standard errors) a list containing the following gradient functions:

dF1dx: derivative \(dF_1(x)/dx\) function,
dF2dx: derivative \(dF_2(x,y,r)/dx\) function,
dF2dr: derivative \(dF_2(x,y,r)/dr \) function.

If deriv.fun = NULL numeric standard errors will be computed.

Arguments

df: integer specifying the degrees of freedom of the t copula

Details

We allow for two different link functions, the multivariate probit link and the multivariate logit link. For the multivariate probit link a multivariate normal distribution for the errors is applied. The normal bivariate probabilities which enter the pairwise log-likelihood are computed with the package pbivnorm.

For the multivariate logit link a \(t\) copula based multivariate distribution with logistic margins is used. The mvlogit() function has an optional integer valued argument df which specifies the degrees of freedom to be used for the \(t\) copula. The default value of the degrees of freedom parameter is 8. We restrict the degrees of freedom to be integer valued because the most efficient routines for computing bivariate \(t\) probabilities do not support non-integer degrees of freedom. For further details see vignette.