R/SimRepeat.R
In AFialkowski/SimRepeat: Simulation of Correlated Systems of Equations with Multiple Variable Types
#' @title Simulation of Correlated Systems of Statistical Equations with Multiple Variable Types
#'
#' @description Generate correlated systems of statistical equations which represent \strong{repeated measurements} or clustered data.
#' These systems contain either: \emph{a)} continuous normal, non-normal, and mixture variables based on the techniques of Headrick and Beasley (2004,
#' \doi{10.1081/SAC-120028431}) or \emph{b)} continuous (normal, non-normal and mixture), ordinal, and count (regular or zero-inflated, Poisson and
#' Negative Binomial) variables based on the hierarchical linear models (HLM) approach. Headrick and Beasley's method for continuous variables calculates
#' the beta (slope) coefficients based on the target correlations between independent variables and between outcomes and independent variables.
#' The package provides functions to calculate the expected correlations between outcomes, between outcomes and error terms, and between outcomes and
#' independent variables, extending Headrick and Beasley's equations to include mixture variables. These theoretical values can be compared to the
#' simulated correlations. The HLM approach requires specification of the beta
#' coefficients, but permits group and subject-level independent variables, interactions among independent variables, and fixed and random effects,
#' providing more flexibility in the system of equations. Both methods permit simulation of data sets that mimic real-world clinical or genetic data sets (i.e. plasmodes, as in Vaughan et al.,
#' 2009, \doi{10.1016/j.csda.2008.02.032}).
#'
#' The techniques extend those found in the \pkg{SimMultiCorrData} and \pkg{SimCorrMix}
#' packages. Standard normal variables with an imposed intermediate correlation matrix are transformed to generate the desired distributions. Continuous
#' variables are simulated using either Fleishman's third-order (\doi{10.1007/BF02293811}) or Headrick's fifth-order (\doi{10.1016/S0167-9473(02)00072-5})
#' power method transformation (PMT). Simulation occurs at the component-level for continuous mixture distributions. These components are transformed into
#' the desired mixture variables using random multinomial variables based on the mixing probabilities. The target correlation matrices are specified in terms of
#' correlations with components of continuous mixture variables. Binary and ordinal variables are simulated by discretizing the normal variables at quantiles
#' defined by the marginal distributions. Count variables are simulated using the inverse CDF method.
#'
#' There are two simulation pathways for the multi-variable type systems which differ by intermediate correlations involving count variables. Correlation Method 1
#' adapts Yahav and Shmueli's 2012 method (\doi{10.1002/asmb.901}) and performs best with large count variable means and positive correlations or small means and
#' negative correlations. Correlation Method 2 adapts Barbiero and Ferrari's 2015 modification of
#' \code{\link[GenOrd]{GenOrd-package}} (\doi{10.1002/asmb.2072}) and performs best under the opposite scenarios.
#' There are three methods available for correcting non-positive definite correlation matrices. The optional error loop may be used to improve the accuracy of the final
#' correlation matrices. The package also provides function to check parameter inputs and summarize the generated systems of equations.
#'
#' @seealso Useful link: \url{https://github.com/AFialkowski/SimMultiCorrData}, \url{https://github.com/AFialkowski/SimCorrMix},
#' \url{https://github.com/AFialkowski/SimRepeat}
#' @section Vignettes:
#' There are vignettes which accompany this package that may help the user understand the simulation and analysis methods.
#'
#' 1) \bold{Theory and Equations for Correlated Systems of Continuous Variables} describes the system of continuous variables generated with \code{\link[SimRepeat]{nonnormsys}} and
#' derives the equations used in \code{\link[SimRepeat]{calc_betas}}, \code{\link[SimRepeat]{calc_corr_y}}, \code{\link[SimRepeat]{calc_corr_ye}},
#' and \code{\link[SimRepeat]{calc_corr_yx}}.
#'
#' 2) \bold{Correlated Systems of Statistical Equations with Non-Mixture and Mixture Continuous Variables} provides examples of using
#' \code{\link[SimRepeat]{nonnormsys}}.
#'
#' 3) \bold{The Hierarchical Linear Models Approach for a System of Correlated Equations with Multiple Variable Types} describes the system of ordinal,
#' continuous, and count variables generated with \code{\link[SimRepeat]{corrsys}} and \code{\link[SimRepeat]{corrsys2}}.
#'
#' 4) \bold{Correlated Systems of Statistical Equations with Multiple Variable Types} provides examples of using \code{\link[SimRepeat]{corrsys}} and
#' \code{\link[SimRepeat]{corrsys2}}.
#'
#' @section Functions:
#' This package contains 3 \emph{simulation} functions:
#'
#' \code{\link[SimRepeat]{nonnormsys}}, \code{\link[SimRepeat]{corrsys}}, \code{\link[SimRepeat]{corrsys2}}
#'
#' 4 \emph{support} functions for \code{\link[SimRepeat]{nonnormsys}}:
#'
#' \code{\link[SimRepeat]{calc_betas}}, \code{\link[SimRepeat]{calc_corr_y}}, \code{\link[SimRepeat]{calc_corr_ye}}, \code{\link[SimRepeat]{calc_corr_yx}}
#'
#' 1 \emph{parameter check} function:
#'
#' \code{\link[SimRepeat]{checkpar}}
#'
#' 1 \emph{summary} function:
#'
#' \code{\link[SimRepeat]{summary_sys}}
#'
#' 1 \emph{correction} function for non-PD correlation matrices:
#'
#' \code{\link[SimRepeat]{adj_grad}}
#'
#' @docType package
#' @name SimRepeat
#' @references
#' Amatya A & Demirtas H (2015). Simultaneous generation of multivariate mixed data with Poisson and normal marginals.
#' Journal of Statistical Computation and Simulation, 85(15):3129-39. \doi{10.1080/00949655.2014.953534}.
#'
#' Barbiero A & Ferrari PA (2015). Simulation of correlated Poisson variables. Applied Stochastic Models in
#' Business and Industry, 31:669-80. \doi{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. \cr \url{https://CRAN.R-project.org/package=GenOrd}
#'
#' Davenport JW, Bezder JC, & Hathaway RJ (1988). Parameter Estimation for Finite Mixture Distributions.
#' Computers & Mathematics with Applications, 15(10):819-28.
#'
#' Demirtas H (2006). A method for multivariate ordinal data generation given marginal distributions and correlations. Journal of Statistical
#' Computation and Simulation, 76(11):1017-1025. \cr \doi{10.1080/10629360600569246}.
#'
#' Demirtas H (2014). Joint Generation of Binary and Nonnormal Continuous Data. Biometrics & Biostatistics, S12.
#'
#' Demirtas H & Hedeker D (2011). A practical way for computing approximate lower and upper correlation bounds.
#' American Statistician, 65(2):104-109. \doi{10.1198/tast.2011.10090}.
#'
#' Demirtas H, Hedeker D, & Mermelstein RJ (2012). Simulation of massive public health data by power polynomials.
#' Statistics in Medicine, 31(27):3337-3346. \doi{10.1002/sim.5362}.
#'
#' Emrich LJ & Piedmonte MR (1991). A Method for Generating High-Dimensional Multivariate Binary Variables. The American Statistician, 45(4): 302-4.
#' \doi{10.1080/00031305.1991.10475828}.
#'
#' Everitt BS (1996). An Introduction to Finite Mixture Distributions. Statistical Methods in Medical Research, 5(2):107-127. \doi{10.1177/096228029600500202}.
#'
#' Ferrari PA & Barbiero A (2012). Simulating ordinal data. Multivariate Behavioral Research, 47(4): 566-589.
#' \doi{10.1080/00273171.2012.692630}.
#'
#' Fialkowski AC (2017). SimMultiCorrData: Simulation of Correlated Data with Multiple Variable Types. R package version 0.2.1.
#' \url{https://CRAN.R-project.org/package=SimMultiCorrData}.
#'
#' Fialkowski AC (2018). SimCorrMix: Simulation of Correlated Data of Multiple Variable Types including Continuous and Count
#' Mixture Distributions. R package version 0.1.0. \url{https://github.com/AFialkowski/SimCorrMix}
#'
#' Fleishman AI (1978). A Method for Simulating Non-normal Distributions. Psychometrika, 43:521-532. \doi{10.1007/BF02293811}.
#'
#' Frechet M (1951). Sur les tableaux de correlation dont les marges sont donnees. Ann. l'Univ. Lyon SectA, 14:53-77.
#'
#' Hasselman B (2017). nleqslv: Solve Systems of Nonlinear Equations. R package version 3.3.1.
#' \url{https://CRAN.R-project.org/package=nleqslv}
#'
#' Headrick TC (2002). Fast Fifth-order Polynomial Transforms for Generating Univariate and Multivariate
#' Non-normal Distributions. Computational Statistics & Data Analysis, 40(4):685-711. \doi{10.1016/S0167-9473(02)00072-5}.
#' (\href{http://www.sciencedirect.com/science/article/pii/S0167947302000725}{ScienceDirect})
#'
#' Headrick TC (2004). On Polynomial Transformations for Simulating Multivariate Nonnormal Distributions.
#' Journal of Modern Applied Statistical Methods, 3(1):65-71. \doi{10.22237/jmasm/1083370080}.
#'
#' Headrick TC, Beasley TM (2004). A Method for Simulating Correlated Non-Normal Systems of Linear Statistical Equations.
#' Communications in Statistics - Simulation and Computation, 33(1). \doi{10.1081/SAC-120028431}
#'
#' 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. \doi{10.1080/10629360600605065}.
#'
#' Headrick TC, Sawilowsky SS (1999). Simulating Correlated Non-normal Distributions: Extending the Fleishman Power
#' Method. Psychometrika, 64:25-35. \doi{10.1007/BF02294317}.
#'
#' Headrick TC, Sawilowsky SS (2002). Weighted Simplex Procedures for Determining Boundary Points and Constants for the
#' Univariate and Multivariate Power Methods. Journal of Educational and Behavioral Statistics, 25:417-436. \doi{10.3102/10769986025004417}.
#'
#' 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. \cr \doi{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.
#'
#' Hoeffding W. Scale-invariant correlation theory. In: Fisher NI, Sen PK, editors. The collected works of Wassily Hoeffding.
#' New York: Springer-Verlag; 1994. p. 57-107.
#'
#' Ismail N & Zamani H (2013). Estimation of Claim Count Data Using Negative Binomial, Generalized Poisson, Zero-Inflated Negative Binomial and
#' Zero-Inflated Generalized Poisson Regression Models. Casualty Actuarial Society E-Forum 41(20):1-28.
#'
#' Kincaid C (2005). Guidelines for Selecting the Covariance Structure in Mixed Model Analysis. Computational Statistics and
#' Data Analysis, 198(30):1-8.
#'
#' Lambert D (1992). Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing. Technometrics 34(1):1-14.
#'
#' Lininger M, Spybrook J, & Cheatham CC (2015). Hierarchical Linear Model: Thinking Outside the Traditional Repeated-Measures
#' Analysis-of-Variance Box. Journal of Athletic Training, 50(4):438-441. \doi{10.4085/1062-6050-49.5.09}.
#'
#' McCulloch CE, Searle SR, Neuhaus JM (2008). \emph{Generalized, Linear, and Mixed Models} (2nd ed.). Wiley Series in Probability and
#' Statistics. Hoboken, New Jersey: John Wiley & Sons, Inc.
#'
#' Olsson U, Drasgow F, & Dorans NJ (1982). The Polyserial Correlation Coefficient. Psychometrika, 47(3):337-47.
#' \doi{10.1007/BF02294164}.
#'
#' Pearson RK (2011). Exploring Data in Engineering, the Sciences, and Medicine. In. New York: Oxford University Press.
#'
#' Schork NJ, Allison DB, & Thiel B (1996). Mixture Distributions in Human Genetics Research. Statistical Methods in Medical Research,
#' 5:155-178. \doi{10.1177/096228029600500204}.
#'
#' Vale CD & Maurelli VA (1983). Simulating Multivariate Nonnormal Distributions. Psychometrika, 48:465-471. \doi{10.1007/BF02293687}.
#'
#' Van Der Leeden R (1998). Multilevel Analysis of Repeated Measures Data. Quality & Quantity, 32(1):15-29.
#'
#' Varadhan R, Gilbert PD (2009). BB: An R Package for Solving a Large System of Nonlinear Equations and for
#' Optimizing a High-Dimensional Nonlinear Objective Function, J. Statistical Software, 32(4). \doi{10.18637/jss.v032.i04}.
#' \url{http://www.jstatsoft.org/v32/i04/}
#'
#' Vaughan LK, Divers J, Padilla M, Redden DT, Tiwari HK, Pomp D, Allison DB (2009). The use of plasmodes as a supplement to simulations:
#' A simple example evaluating individual admixture estimation methodologies. Comput Stat Data Anal, 53(5):1755-66.
#' \doi{10.1016/j.csda.2008.02.032}.
#'
#' Yahav I & Shmueli G (2012). On Generating Multivariate Poisson Data in Management Science Applications. Applied Stochastic
#' Models in Business and Industry, 28(1):91-102. \doi{10.1002/asmb.901}.
#'
#' Yee TW (2017). VGAM: Vector Generalized Linear and Additive Models. \cr \url{https://CRAN.R-project.org/package=VGAM}.
#'
#' Zhang X, Mallick H, & Yi N (2016). Zero-Inflated Negative Binomial Regression for Differential Abundance Testing in Microbiome
#' Studies. Journal of Bioinformatics and Genomics 2(2):1-9. \doi{10.18454/jbg.2016.2.2.1}.
#'
#'
NULL
AFialkowski/SimRepeat documentation built on May 14, 2019, 6:12 p.m.
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Note that we can't provide technical support on individual packages. You should contact the package authors for that.
#' @title Simulation of Correlated Systems of Statistical Equations with Multiple Variable Types
#'
#' @description Generate correlated systems of statistical equations which represent \strong{repeated measurements} or clustered data.
#' These systems contain either: \emph{a)} continuous normal, non-normal, and mixture variables based on the techniques of Headrick and Beasley (2004,
#' \doi{10.1081/SAC-120028431}) or \emph{b)} continuous (normal, non-normal and mixture), ordinal, and count (regular or zero-inflated, Poisson and
#' Negative Binomial) variables based on the hierarchical linear models (HLM) approach. Headrick and Beasley's method for continuous variables calculates
#' the beta (slope) coefficients based on the target correlations between independent variables and between outcomes and independent variables.
#' The package provides functions to calculate the expected correlations between outcomes, between outcomes and error terms, and between outcomes and
#' independent variables, extending Headrick and Beasley's equations to include mixture variables. These theoretical values can be compared to the
#' simulated correlations. The HLM approach requires specification of the beta
#' coefficients, but permits group and subject-level independent variables, interactions among independent variables, and fixed and random effects,
#' providing more flexibility in the system of equations. Both methods permit simulation of data sets that mimic real-world clinical or genetic data sets (i.e. plasmodes, as in Vaughan et al.,
#' 2009, \doi{10.1016/j.csda.2008.02.032}).
#'
#' The techniques extend those found in the \pkg{SimMultiCorrData} and \pkg{SimCorrMix}
#' packages. Standard normal variables with an imposed intermediate correlation matrix are transformed to generate the desired distributions. Continuous
#' variables are simulated using either Fleishman's third-order (\doi{10.1007/BF02293811}) or Headrick's fifth-order (\doi{10.1016/S0167-9473(02)00072-5})
#' power method transformation (PMT). Simulation occurs at the component-level for continuous mixture distributions. These components are transformed into
#' the desired mixture variables using random multinomial variables based on the mixing probabilities. The target correlation matrices are specified in terms of
#' correlations with components of continuous mixture variables. Binary and ordinal variables are simulated by discretizing the normal variables at quantiles
#' defined by the marginal distributions. Count variables are simulated using the inverse CDF method.
#'
#' There are two simulation pathways for the multi-variable type systems which differ by intermediate correlations involving count variables. Correlation Method 1
#' adapts Yahav and Shmueli's 2012 method (\doi{10.1002/asmb.901}) and performs best with large count variable means and positive correlations or small means and
#' negative correlations. Correlation Method 2 adapts Barbiero and Ferrari's 2015 modification of
#' \code{\link[GenOrd]{GenOrd-package}} (\doi{10.1002/asmb.2072}) and performs best under the opposite scenarios.
#' There are three methods available for correcting non-positive definite correlation matrices. The optional error loop may be used to improve the accuracy of the final
#' correlation matrices. The package also provides function to check parameter inputs and summarize the generated systems of equations.
#'
#' @seealso Useful link: \url{https://github.com/AFialkowski/SimMultiCorrData}, \url{https://github.com/AFialkowski/SimCorrMix},
#' \url{https://github.com/AFialkowski/SimRepeat}
#' @section Vignettes:
#' There are vignettes which accompany this package that may help the user understand the simulation and analysis methods.
#'
#' 1) \bold{Theory and Equations for Correlated Systems of Continuous Variables} describes the system of continuous variables generated with \code{\link[SimRepeat]{nonnormsys}} and
#' derives the equations used in \code{\link[SimRepeat]{calc_betas}}, \code{\link[SimRepeat]{calc_corr_y}}, \code{\link[SimRepeat]{calc_corr_ye}},
#' and \code{\link[SimRepeat]{calc_corr_yx}}.
#'
#' 2) \bold{Correlated Systems of Statistical Equations with Non-Mixture and Mixture Continuous Variables} provides examples of using
#' \code{\link[SimRepeat]{nonnormsys}}.
#'
#' 3) \bold{The Hierarchical Linear Models Approach for a System of Correlated Equations with Multiple Variable Types} describes the system of ordinal,
#' continuous, and count variables generated with \code{\link[SimRepeat]{corrsys}} and \code{\link[SimRepeat]{corrsys2}}.
#'
#' 4) \bold{Correlated Systems of Statistical Equations with Multiple Variable Types} provides examples of using \code{\link[SimRepeat]{corrsys}} and
#' \code{\link[SimRepeat]{corrsys2}}.
#'
#' @section Functions:
#' This package contains 3 \emph{simulation} functions:
#'
#' \code{\link[SimRepeat]{nonnormsys}}, \code{\link[SimRepeat]{corrsys}}, \code{\link[SimRepeat]{corrsys2}}
#'
#' 4 \emph{support} functions for \code{\link[SimRepeat]{nonnormsys}}:
#'
#' \code{\link[SimRepeat]{calc_betas}}, \code{\link[SimRepeat]{calc_corr_y}}, \code{\link[SimRepeat]{calc_corr_ye}}, \code{\link[SimRepeat]{calc_corr_yx}}
#'
#' 1 \emph{parameter check} function:
#'
#' \code{\link[SimRepeat]{checkpar}}
#'
#' 1 \emph{summary} function:
#'
#' \code{\link[SimRepeat]{summary_sys}}
#'
#' 1 \emph{correction} function for non-PD correlation matrices:
#'
#' \code{\link[SimRepeat]{adj_grad}}
#'
#' @docType package
#' @name SimRepeat
#' @references
#' Amatya A & Demirtas H (2015). Simultaneous generation of multivariate mixed data with Poisson and normal marginals.
#' Journal of Statistical Computation and Simulation, 85(15):3129-39. \doi{10.1080/00949655.2014.953534}.
#'
#' Barbiero A & Ferrari PA (2015). Simulation of correlated Poisson variables. Applied Stochastic Models in
#' Business and Industry, 31:669-80. \doi{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. \cr \url{https://CRAN.R-project.org/package=GenOrd}
#'
#' Davenport JW, Bezder JC, & Hathaway RJ (1988). Parameter Estimation for Finite Mixture Distributions.
#' Computers & Mathematics with Applications, 15(10):819-28.
#'
#' Demirtas H (2006). A method for multivariate ordinal data generation given marginal distributions and correlations. Journal of Statistical
#' Computation and Simulation, 76(11):1017-1025. \cr \doi{10.1080/10629360600569246}.
#'
#' Demirtas H (2014). Joint Generation of Binary and Nonnormal Continuous Data. Biometrics & Biostatistics, S12.
#'
#' Demirtas H & Hedeker D (2011). A practical way for computing approximate lower and upper correlation bounds.
#' American Statistician, 65(2):104-109. \doi{10.1198/tast.2011.10090}.
#'
#' Demirtas H, Hedeker D, & Mermelstein RJ (2012). Simulation of massive public health data by power polynomials.
#' Statistics in Medicine, 31(27):3337-3346. \doi{10.1002/sim.5362}.
#'
#' Emrich LJ & Piedmonte MR (1991). A Method for Generating High-Dimensional Multivariate Binary Variables. The American Statistician, 45(4): 302-4.
#' \doi{10.1080/00031305.1991.10475828}.
#'
#' Everitt BS (1996). An Introduction to Finite Mixture Distributions. Statistical Methods in Medical Research, 5(2):107-127. \doi{10.1177/096228029600500202}.
#'
#' Ferrari PA & Barbiero A (2012). Simulating ordinal data. Multivariate Behavioral Research, 47(4): 566-589.
#' \doi{10.1080/00273171.2012.692630}.
#'
#' Fialkowski AC (2017). SimMultiCorrData: Simulation of Correlated Data with Multiple Variable Types. R package version 0.2.1.
#' \url{https://CRAN.R-project.org/package=SimMultiCorrData}.
#'
#' Fialkowski AC (2018). SimCorrMix: Simulation of Correlated Data of Multiple Variable Types including Continuous and Count
#' Mixture Distributions. R package version 0.1.0. \url{https://github.com/AFialkowski/SimCorrMix}
#'
#' Fleishman AI (1978). A Method for Simulating Non-normal Distributions. Psychometrika, 43:521-532. \doi{10.1007/BF02293811}.
#'
#' Frechet M (1951). Sur les tableaux de correlation dont les marges sont donnees. Ann. l'Univ. Lyon SectA, 14:53-77.
#'
#' Hasselman B (2017). nleqslv: Solve Systems of Nonlinear Equations. R package version 3.3.1.
#' \url{https://CRAN.R-project.org/package=nleqslv}
#'
#' Headrick TC (2002). Fast Fifth-order Polynomial Transforms for Generating Univariate and Multivariate
#' Non-normal Distributions. Computational Statistics & Data Analysis, 40(4):685-711. \doi{10.1016/S0167-9473(02)00072-5}.
#' (\href{http://www.sciencedirect.com/science/article/pii/S0167947302000725}{ScienceDirect})
#'
#' Headrick TC (2004). On Polynomial Transformations for Simulating Multivariate Nonnormal Distributions.
#' Journal of Modern Applied Statistical Methods, 3(1):65-71. \doi{10.22237/jmasm/1083370080}.
#'
#' Headrick TC, Beasley TM (2004). A Method for Simulating Correlated Non-Normal Systems of Linear Statistical Equations.
#' Communications in Statistics - Simulation and Computation, 33(1). \doi{10.1081/SAC-120028431}
#'
#' 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. \doi{10.1080/10629360600605065}.
#'
#' Headrick TC, Sawilowsky SS (1999). Simulating Correlated Non-normal Distributions: Extending the Fleishman Power
#' Method. Psychometrika, 64:25-35. \doi{10.1007/BF02294317}.
#'
#' Headrick TC, Sawilowsky SS (2002). Weighted Simplex Procedures for Determining Boundary Points and Constants for the
#' Univariate and Multivariate Power Methods. Journal of Educational and Behavioral Statistics, 25:417-436. \doi{10.3102/10769986025004417}.
#'
#' 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. \cr \doi{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.
#'
#' Hoeffding W. Scale-invariant correlation theory. In: Fisher NI, Sen PK, editors. The collected works of Wassily Hoeffding.
#' New York: Springer-Verlag; 1994. p. 57-107.
#'
#' Ismail N & Zamani H (2013). Estimation of Claim Count Data Using Negative Binomial, Generalized Poisson, Zero-Inflated Negative Binomial and
#' Zero-Inflated Generalized Poisson Regression Models. Casualty Actuarial Society E-Forum 41(20):1-28.
#'
#' Kincaid C (2005). Guidelines for Selecting the Covariance Structure in Mixed Model Analysis. Computational Statistics and
#' Data Analysis, 198(30):1-8.
#'
#' Lambert D (1992). Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing. Technometrics 34(1):1-14.
#'
#' Lininger M, Spybrook J, & Cheatham CC (2015). Hierarchical Linear Model: Thinking Outside the Traditional Repeated-Measures
#' Analysis-of-Variance Box. Journal of Athletic Training, 50(4):438-441. \doi{10.4085/1062-6050-49.5.09}.
#'
#' McCulloch CE, Searle SR, Neuhaus JM (2008). \emph{Generalized, Linear, and Mixed Models} (2nd ed.). Wiley Series in Probability and
#' Statistics. Hoboken, New Jersey: John Wiley & Sons, Inc.
#'
#' Olsson U, Drasgow F, & Dorans NJ (1982). The Polyserial Correlation Coefficient. Psychometrika, 47(3):337-47.
#' \doi{10.1007/BF02294164}.
#'
#' Pearson RK (2011). Exploring Data in Engineering, the Sciences, and Medicine. In. New York: Oxford University Press.
#'
#' Schork NJ, Allison DB, & Thiel B (1996). Mixture Distributions in Human Genetics Research. Statistical Methods in Medical Research,
#' 5:155-178. \doi{10.1177/096228029600500204}.
#'
#' Vale CD & Maurelli VA (1983). Simulating Multivariate Nonnormal Distributions. Psychometrika, 48:465-471. \doi{10.1007/BF02293687}.
#'
#' Van Der Leeden R (1998). Multilevel Analysis of Repeated Measures Data. Quality & Quantity, 32(1):15-29.
#'
#' Varadhan R, Gilbert PD (2009). BB: An R Package for Solving a Large System of Nonlinear Equations and for
#' Optimizing a High-Dimensional Nonlinear Objective Function, J. Statistical Software, 32(4). \doi{10.18637/jss.v032.i04}.
#' \url{http://www.jstatsoft.org/v32/i04/}
#'
#' Vaughan LK, Divers J, Padilla M, Redden DT, Tiwari HK, Pomp D, Allison DB (2009). The use of plasmodes as a supplement to simulations:
#' A simple example evaluating individual admixture estimation methodologies. Comput Stat Data Anal, 53(5):1755-66.
#' \doi{10.1016/j.csda.2008.02.032}.
#'
#' Yahav I & Shmueli G (2012). On Generating Multivariate Poisson Data in Management Science Applications. Applied Stochastic
#' Models in Business and Industry, 28(1):91-102. \doi{10.1002/asmb.901}.
#'
#' Yee TW (2017). VGAM: Vector Generalized Linear and Additive Models. \cr \url{https://CRAN.R-project.org/package=VGAM}.
#'
#' Zhang X, Mallick H, & Yi N (2016). Zero-Inflated Negative Binomial Regression for Differential Abundance Testing in Microbiome
#' Studies. Journal of Bioinformatics and Genomics 2(2):1-9. \doi{10.18454/jbg.2016.2.2.1}.
#'
#'
NULL
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