This function corrects the final correlation of simulated variables to be within a precision value (epsilon) of the
target correlation. It updates the pairwise intermediate MVN correlation iteratively in a loop until either the maximum error
is less than epsilon or the number of iterations exceeds the maximum number set by the user (maxit). It uses
error_vars to simulate all variables and calculate the correlation of all variables in each
iteration. This function would not ordinarily be called directly by the user. The function is a
modification of Barbiero & Ferrari's ordcont function in GenOrd-package.
The ordcont has been modified in the following ways:
1) It works for continuous, ordinal (r >= 2 categories), and count variables.
2) The initial correlation check has been removed because this intermediate correlation
Sigma from rcorrvar or rcorrvar2 has already been
checked for positive-definiteness and used to generate variables.
3) Eigenvalue decomposition is done on Sigma to impose the correct interemdiate correlations on the normal variables.
If Sigma is not positive-definite, the negative eigen values are replaced with 0.
3) The final positive-definite check has been removed.
4) The intermediate correlation update function was changed to accomodate more situations.
5) A final "fail-safe" check was added at the end of the iteration loop where if the absolute
error between the final and target pairwise correlation is still > 0.1, the intermediate correlation is set
equal to the target correlation (if extra_correct = "TRUE").
6) Allowing specifications for the sample size and the seed for reproducibility.
error_loop(k_cat, k_cont, k_pois, k_nb, Y_cat, Y, Yb, Y_pois, Y_nb, marginal,
support, method, means, vars, constants, lam, size, prob, mu, n, seed,
epsilon, maxit, rho0, Sigma, rho_calc, extra_correct)the number of ordinal (r >= 2 categories) variables
the number of continuous variables
the number of Poisson variables
the number of Negative Binomial variables
the continuous (mean 0, variance 1) variables
the continuous variables with desired mean and variance
the Poisson variables
the Negative Binomial variables
a list of length equal k_cat; 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)
a list of length equal k_cat; the i-th element is a vector of containing the r
ordered support values; if not provided, the default is for the i-th element to be the vector 1, ..., r
the method used to generate the continuous variables. "Fleishman" uses a third-order polynomial transformation and "Polynomial" uses Headrick's fifth-order transformation.
a vector of means for the continuous variables
a vector of variances
a matrix with k_cont rows, each a vector of constants c0, c1, c2, c3 (if method = "Fleishman") or
c0, c1, c2, c3, c4, c5 (if method = "Polynomial"), like that returned by
find_constants
a vector of lambda (> 0) constants for the Poisson variables (see dpois)
a vector of size parameters for the Negative Binomial variables (see dnbinom)
a vector of success probability parameters
a vector of mean parameters (*Note: either prob or mu should be supplied for all Negative Binomial variables,
not a mixture)
the sample size
the seed value for random number generation
the maximum acceptable error between the final and target correlation matrices; smaller epsilons take more time
the maximum number of iterations to use to find the intermediate correlation; the
correction loop stops when either the iteration number passes maxit or epsilon is reached
the target correlation matrix
if "TRUE", a final "fail-safe" check is used at the end of the iteration loop where if the absolute error between the final and target pairwise correlation is still > 0.1, the intermediate correlation is set equal to the target correlation
A list with the following components:
Sigma the intermediate MVN correlation matrix resulting from the error loop
rho_calc the calculated final correlation matrix generated from Sigma
Y_cat the ordinal variables
Y the continuous (mean 0, variance 1) variables
Yb the continuous variables with desired mean and variance
Y_pois the Poisson variables
Y_nb the Negative Binomial variables
niter a matrix containing the number of iterations required for each variable pair
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.
Fleishman AI (1978). A Method for Simulating Non-normal Distributions. Psychometrika, 43, 521-532. 10.1007/BF02293811.
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Headrick TC (2004). On Polynomial Transformations for Simulating Multivariate Nonnormal Distributions. Journal of Modern Applied Statistical Methods, 3(1), 65-71. 10.22237/jmasm/1083370080.
Headrick TC, Kowalchuk RK (2007). The Power Method Transformation: Its Probability Density Function, Distribution Function, and Its Further Use for Fitting Data. Journal of Statistical Computation and Simulation, 77, 229-249. 10.1080/10629360600605065.
Headrick TC, Sawilowsky SS (1999). Simulating Correlated Non-normal Distributions: Extending the Fleishman Power Method. Psychometrika, 64, 25-35. 10.1007/BF02294317.
Headrick TC, Sheng Y, & Hodis FA (2007). Numerical Computing and Graphics for the Power Method Transformation Using Mathematica. Journal of Statistical Software, 19(3), 1 - 17. 10.18637/jss.v019.i03.
Higham N (2002). Computing the nearest correlation matrix - a problem from finance; IMA Journal of Numerical Analysis 22: 329-343.