Description Vignettes Functions References See Also

Generate correlated systems of statistical equations which represent **repeated measurements** or clustered data.
These systems contain either: *a)* continuous normal, non-normal, and mixture variables based on the techniques of Headrick and Beasley (2004,
doi: 10.1081/SAC-120028431) or *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 SimMultiCorrData and 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
`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.

There are vignettes which accompany this package that may help the user understand the simulation and analysis methods.

1) **Theory and Equations for Correlated Systems of Continuous Variables** describes the system of continuous variables generated with `nonnormsys`

and
derives the equations used in `calc_betas`

, `calc_corr_y`

, `calc_corr_ye`

,
and `calc_corr_yx`

.

2) **Correlated Systems of Statistical Equations with Non-Mixture and Mixture Continuous Variables** provides examples of using
`nonnormsys`

.

3) **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 `corrsys`

and `corrsys2`

.

4) **Correlated Systems of Statistical Equations with Multiple Variable Types** provides examples of using `corrsys`

and
`corrsys2`

.

This package contains 3 *simulation* functions:

`nonnormsys`

, `corrsys`

, `corrsys2`

4 *support* functions for `nonnormsys`

:

`calc_betas`

, `calc_corr_y`

, `calc_corr_ye`

, `calc_corr_yx`

1 *parameter check* function:

`checkpar`

1 *summary* function:

`summary_sys`

1 *correction* function for non-PD correlation matrices:

`adj_grad`

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.

https://CRAN.R-project.org/package=GenOrd

Berend H (2017). nleqslv: Solve Systems of Nonlinear Equations. R package version 3.2. https://CRAN.R-project.org/package=nleqslv

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.

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. 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. 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.

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. (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.

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). *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. 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.

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.

Useful link: https://github.com/AFialkowski/SimMultiCorrData, https://github.com/AFialkowski/SimCorrMix, https://github.com/AFialkowski/SimRepeat

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.