This function calculates the intermediate MVN correlation needed to generate a variable described by
a discrete marginal distribution and associated finite support. This includes ordinal (\(r \ge 2\) categories) variables
or variables that are treated as ordinal (i.e. count variables in the Barbiero & Ferrari, 2015 method used in
corrvar2, 10.1002/asmb.2072). The function is a modification of Barbiero & Ferrari's
ordcont function in GenOrd-package.
It works by setting the intermediate MVN correlation equal to the target correlation and updating each intermediate pairwise
correlation until the final pairwise correlation is within epsilon of the target correlation or the maximum number of
iterations has been reached. This function uses norm_ord to calculate the ordinal correlation obtained
from discretizing the normal variables generated from the intermediate correlation matrix. The ordcont has been modified in the following ways:
1) the initial correlation check has been removed because this is done within the simulation functions
2) the final positive-definite check has been removed
3) the intermediate correlation update function was changed to accommodate more situations
This function would not ordinarily be called by the user. Note that this will return a matrix that is NOT positive-definite
because this is corrected for in the simulation functions corrvar and corrvar2
using the method of Higham (2002) and the nearPD function.
ord_norm(marginal = list(), rho = NULL, support = list(),
epsilon = 0.001, maxit = 1000, Spearman = FALSE)a list of length equal to the number of variables; the i-th element is a vector of the cumulative probabilities defining the marginal distribution of the i-th variable; if the variable can take r values, the vector will contain r - 1 probabilities (the r-th is assumed to be 1)
the target correlation matrix
a list of length equal to the number of variables; the i-th element is a vector of containing the r ordered support values; if not provided (i.e. support = list()), the default is for the i-th element to be the vector 1, ..., r
the maximum acceptable error between the final and target pairwise correlations (default = 0.001); smaller values take more time
the maximum number of iterations to use (default = 1000) to find the intermediate correlation; the
correction loop stops when either the iteration number passes maxit or epsilon is reached
if TRUE, Spearman's correlations are used (and support is not required); if FALSE (default) Pearson's correlations are used
A list with the following components:
SigmaC the intermediate MVN correlation matrix
rho0 the calculated final correlation matrix generated from SigmaC
rho the target final correlation matrix
niter a matrix containing the number of iterations required for each variable pair
maxerr the maximum final error between the final and target correlation matrices
Barbiero A, Ferrari PA (2015). Simulation of correlated Poisson variables. Applied Stochastic Models in Business and Industry, 31:669-80. 10.1002/asmb.2072.
Barbiero A, Ferrari PA (2015). GenOrd: Simulation of Discrete Random Variables with Given Correlation Matrix and Marginal Distributions. R package version 1.4.0. https://CRAN.R-project.org/package=GenOrd
Ferrari PA, Barbiero A (2012). Simulating ordinal data, Multivariate Behavioral Research, 47(4):566-589. 10.1080/00273171.2012.692630.