Simulation of Correlated Data with Multiple Variable Types Including Continuous and Count Mixture Distributions

Description

SimCorrMix generates continuous (normal, non-normal, or mixture distributions), binary, ordinal, and count (Poisson or Negative Binomial, regular or zero-inflated) variables with a specified correlation matrix, or one continuous variable with a mixture distribution. This package can be used to simulate 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 methods extend those found in the SimMultiCorrData package. 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). Non-mixture distributions require the user to specify mean, variance, skewness, standardized kurtosis, and standardized fifth and sixth cumulants. Mixture distributions require these inputs for the component distributions plus the mixing probabilities. Simulation occurs at the component-level for continuous mixture distributions. The target correlation matrix is specified in terms of correlations with components of continuous mixture variables. These components are transformed into the desired mixture variables using random multinomial variables based on the mixing probabilities. However, the package provides functions to approximate expected correlations with continuous mixture variables given target correlations with the components. Binary and ordinal variables are simulated using a modification of GenOrd-package's ordsample function. Count variables are simulated using the inverse CDF method. There are two simulation pathways which calculate intermediate correlations involving count variables differently. 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. The optional error loop may be used to improve the accuracy of the final correlation matrix. The package also provides functions to calculate the standardized cumulants of continuous mixture distributions, check parameter inputs, calculate feasible correlation boundaries, and summarize and plot simulated variables.

Vignettes

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

1) Comparison of Correlation Methods 1 and 2 describes the two simulation pathways that can be followed for generation of correlated data.

2) Continuous Mixture Distributions demonstrates how to simulate one continuous mixture variable using contmixvar1 and gives a step-by-step guideline for comparing a simulated distribution to the target distribution.

3) Expected Cumulants and Correlations for Continuous Mixture Variables derives the equations used by the function calc_mixmoments to find the mean, standard deviation, skew, standardized kurtosis, and standardized fifth and sixth cumulants for a continuous mixture variable. The vignette also explains how the functions rho_M1M2 and rho_M1Y approximate the expected correlations with continuous mixture variables based on the target correlations with the components.

4) Overall Workflow for Generation of Correlated Data gives a step-by-step guideline to follow with an example containing continuous non-mixture and mixture, ordinal, zero-inflated Poisson, and zero-inflated Negative Binomial variables. It executes both correlated data simulation functions with and without the error loop.

5) Variable Types describes the different types of variables that can be simulated in SimCorrMix, details the algorithm involved in the optional error loop that helps to minimize correlation errors, and explains how the feasible correlation boundaries are calculated for each of the two simulation pathways.

Functions

This package contains 3 simulation functions:

contmixvar1, corrvar, and corrvar2

4 data description (summary) function:

calc_mixmoments, summary_var, rho_M1M2, rho_M1Y

2 graphing functions:

plot_simpdf_theory, plot_simtheory

3 support functions:

validpar, validcorr, validcorr2

and 16 auxiliary functions (should not normally be called by the user, but are called by other functions):

corr_error, intercorr, intercorr2, intercorr_cat_nb, intercorr_cat_pois,
intercorr_cont_nb, intercorr_cont_nb2, intercorr_cont_pois, intercorr_cont_pois2,
intercorr_cont, intercorr_nb, intercorr_pois, intercorr_pois_nb, maxcount_support, ord_norm, norm_ord

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

Carnell R (2017). triangle: Provides the Standard Distribution Functions for the Triangle Distribution. R package version 0.11. https://CRAN.R-project.org/package=triangle.

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 (2018). SimMultiCorrData: Simulation of Correlated Data with Multiple Variable Types. R package version 0.2.2. https://CRAN.R-project.org/package=SimMultiCorrData.

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 (2018). nleqslv: Solve Systems of Nonlinear Equations. R package version 3.3.2. 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. (ScienceDirect)

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, 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.

Kendall M & Stuart A (1977). The Advanced Theory of Statistics, 4th Edition. Macmillan, New York.

Lambert D (1992). Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing. Technometrics 34(1):1-14.

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.

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 (2018). VGAM: Vector Generalized Linear and Additive Models. R package version 1.0-5. 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.

See Also

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

Find Standardized Cumulants of a Continuous Mixture Distribution by Method of Moments

Description

This function uses the method of moments to calculate the expected mean, standard deviation, skewness, standardized kurtosis, and standardized fifth and sixth cumulants for a continuous mixture variable based on the distributions of its components. The result can be used as input to find_constants or for comparison to a simulated mixture variable from contmixvar1, corrvar, or corrvar2. See the Expected Cumulants and Correlations for Continuous Mixture Variables vignette for equations of the cumulants.

Usage

calc_mixmoments(mix_pis = NULL, mix_mus = NULL, mix_sigmas = NULL,
  mix_skews = NULL, mix_skurts = NULL, mix_fifths = NULL,
  mix_sixths = NULL)

Arguments

Value

A vector of the mean, standard deviation, skewness, standardized kurtosis, and standardized fifth and sixth cumulants

References

Please see references for SimCorrMix.

Examples

# Mixture of Normal(-2, 1) and Normal(2, 1)
calc_mixmoments(mix_pis = c(0.4, 0.6), mix_mus = c(-2, 2),
  mix_sigmas = c(1, 1), mix_skews = c(0, 0), mix_skurts = c(0, 0),
  mix_fifths = c(0, 0), mix_sixths = c(0, 0))

Generation of One Continuous Variable with a Mixture Distribution Using the Power Method Transformation

Description

This function simulates one continuous mixture variable. Mixture distributions describe random variables that are drawn from more than one component distribution. For a random variable Y_{mix} from a finite continuous mixture distribution with k components, the probability density function (PDF) can be described by:

h_Y(y) = \sum_{i=1}^{k} \pi_i f_{Yi}(y), \sum_{i=1}^{k} \pi_i = 1.

The \pi_i are mixing parameters which determine the weight of each component distribution f_{Yi}(y) in the overall probability distribution. As long as each component has a valid PDF, the overall distribution h_Y(y) has a valid PDF. The main assumption is statistical independence between the process of randomly selecting the component distribution and the distributions themselves. Each component Y_i is generated using either Fleishman's third-order (method = "Fleishman", doi: 10.1007/BF02293811) or Headrick's fifth-order (method = "Polynomial", doi: 10.1016/S0167-9473(02)00072-5) power method transformation (PMT). It works by matching standardized cumulants – the first four (mean, variance, skew, and standardized kurtosis) for Fleishman's method, or the first six (mean, variance, skew, standardized kurtosis, and standardized fifth and sixth cumulants) for Headrick's method. The transformation is expressed as follows:

Y = c_0 + c_1 * Z + c_2 * Z^2 + c_3 * Z^3 + c_4 * Z^4 + c_5 * Z^5, Z \sim N(0,1),

where c_4 and c_5 both equal 0 for Fleishman's method. The real constants are calculated by
find_constants. These components are then transformed to the desired mixture variable using a random multinomial variable generated based on the mixing probabilities. There are no parameter input checks in order to decrease simulation time. All inputs should be checked prior to simulation with validpar. Summaries for the simulation results can be obtained with summary_var.

Mixture distributions provide a useful way for describing heterogeneity in a population, especially when an outcome is a composite response from multiple sources. The vignette Variable Types provides more information about simulation of mixture variables and the required parameters. The vignette Expected Cumulants and Correlations for Continuous Mixture Variables gives the equations for the expected cumulants of a mixture variable. In addition, Headrick & Kowalchuk (2007, doi: 10.1080/10629360600605065) outlined a general method for comparing a simulated distribution Y to a given theoretical distribution Y^*. These steps can be found in the Continuous Mixture Distributions vignette.

Usage

contmixvar1(n = 10000, method = c("Fleishman", "Polynomial"), means = 0,
  vars = 1, mix_pis = NULL, mix_mus = NULL, mix_sigmas = NULL,
  mix_skews = NULL, mix_skurts = NULL, mix_fifths = NULL,
  mix_sixths = NULL, mix_Six = list(), seed = 1234, cstart = list(),
  quiet = FALSE)

Arguments

Value

A list with the following components:

constants a data.frame of the constants

Y_comp a data.frame of the components of the mixture variable

Y_mix a data.frame of the generated mixture variable

sixth_correction the sixth cumulant correction values for Y_comp

valid.pdf "TRUE" if constants generate a valid PDF, else "FALSE"

Time the total simulation time in minutes

Overview of Simulation Process

1) A check is performed to see if any distributions are repeated within the parameter inputs, i.e. if the mixture variable contains 2 components with the same standardized cumulants. These are noted so that the constants are only calculated once.

2) The constants are calculated for each component variable using find_constants. If no solutions are found that generate a valid power method PDF, the function will return constants that produce an invalid PDF (or a stop error if no solutions can be found). Possible solutions include: 1) changing the seed, or 2) using a mix_Six list with vectors of sixth cumulant correction values (if method = "Polynomial"). Errors regarding constant calculation are the most probable cause of function failure.

3) A matrix X_cont of dim n x length(mix_pis) of standard normal variables is generated and singular-value decomposition is done to remove any correlation. The constants are applied to X_cont to create the component variables Y with the desired distributions.

4) A random multinomial variable M = rmultinom(n, size = 1, prob = mix_pis) is generated using stats::rmultinom. The continuous mixture variable Y_mix is created from the component variables Y based on this multinomial variable. That is, if M[i, k_i] = 1, then Y_mix[i] = Y[i, k_i]. A location-scale transformation is done on Y_mix to give it mean means and variance vars.

Reasons for Function Errors

1) The most likely cause for function errors is that no solutions to fleish or poly converged when using find_constants. If this happens, the simulation will stop. It may help to first use find_constants for each component variable to determine if a sixth cumulant correction value is needed. The solutions can be used as starting values (see cstart below). If the standardized cumulants are obtained from calc_theory, the user may need to use rounded values as inputs (i.e. skews = round(skews, 8)). For example, in order to ensure that skew is exactly 0 for symmetric distributions.

2) The kurtosis may be outside the region of possible values. There is an associated lower boundary for kurtosis associated with a given skew (for Fleishman's method) or skew and fifth and sixth cumulants (for Headrick's method). Use calc_lower_skurt to determine the boundary for a given set of cumulants.

References

See references for SimCorrMix.

See Also

find_constants, validpar, summary_var

Examples

# Mixture of Normal(-2, 1) and Normal(2, 1)
Nmix <- contmixvar1(n = 1000, "Polynomial", means = 0, vars = 1,
  mix_pis = c(0.4, 0.6), mix_mus = c(-2, 2), mix_sigmas = c(1, 1),
  mix_skews = c(0, 0), mix_skurts = c(0, 0), mix_fifths = c(0, 0),
  mix_sixths = c(0, 0))
## Not run: 
# Mixture of Beta(6, 3), Beta(4, 1.5), and Beta(10, 20)
Stcum1 <- calc_theory("Beta", c(6, 3))
Stcum2 <- calc_theory("Beta", c(4, 1.5))
Stcum3 <- calc_theory("Beta", c(10, 20))
mix_pis <- c(0.5, 0.2, 0.3)
mix_mus <- c(Stcum1[1], Stcum2[1], Stcum3[1])
mix_sigmas <- c(Stcum1[2], Stcum2[2], Stcum3[2])
mix_skews <- c(Stcum1[3], Stcum2[3], Stcum3[3])
mix_skurts <- c(Stcum1[4], Stcum2[4], Stcum3[4])
mix_fifths <- c(Stcum1[5], Stcum2[5], Stcum3[5])
mix_sixths <- c(Stcum1[6], Stcum2[6], Stcum3[6])
mix_Six <- list(seq(0.01, 10, 0.01), c(0.01, 0.02, 0.03),
  seq(0.01, 10, 0.01))
Bstcum <- calc_mixmoments(mix_pis, mix_mus, mix_sigmas, mix_skews,
  mix_skurts, mix_fifths, mix_sixths)
Bmix <- contmixvar1(n = 10000, "Polynomial", Bstcum[1], Bstcum[2]^2,
  mix_pis, mix_mus, mix_sigmas, mix_skews, mix_skurts, mix_fifths,
  mix_sixths, mix_Six)
Bsum <- summary_var(Y_comp = Bmix$Y_comp, Y_mix = Bmix$Y_mix, means = means,
  vars = vars, mix_pis = mix_pis, mix_mus = mix_mus,
  mix_sigmas = mix_sigmas, mix_skews = mix_skews, mix_skurts = mix_skurts,
  mix_fifths = mix_fifths, mix_sixths = mix_sixths)

## End(Not run)

Error Loop to Correct Final Correlation of Simulated Variables

Description

This function attempts to correct the final pairwise correlations of simulated variables to be within epsilon of the target correlations. It updates the intermediate normal correlation iteratively in a loop until either the maximum error is less than epsilon or the number of iterations exceeds maxit. 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 function has been modified in the following ways:

1) It works for continuous, ordinal (r >= 2 categories), and count (regular or zero-inflated, Poisson or Negative Binomial) variables.

2) The initial correlation check has been removed because the intermediate correlation matrix Sigma from corrvar or corrvar2 has already been checked for positive-definiteness and used to generate variables.

3) Eigenvalue decomposition is done on Sigma to impose the correct intermediate correlations on the normal variables. If Sigma is not positive-definite, the negative eigenvalues are replaced with 0.

4) The final positive-definite check has been removed.

5) The intermediate correlation update function was changed to accommodate more situations.

6) Allowing specifications for the sample size and the seed for reproducibility.

The vignette Variable Types describes the algorithm used in the error loop.

Usage

corr_error(n = 10000, k_cat = 0, k_cont = 0, k_pois = 0, k_nb = 0,
  method = c("Fleishman", "Polynomial"), means = NULL, vars = NULL,
  constants = NULL, marginal = list(), support = list(), lam = NULL,
  p_zip = 0, size = NULL, mu = NULL, p_zinb = 0, seed = 1234,
  epsilon = 0.001, maxit = 1000, rho0 = NULL, Sigma = NULL,
  rho_calc = NULL)

Arguments

Value

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

Y_cont 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

References

Please see references for SimCorrMix.

See Also

corrvar, corrvar2

Generation of Correlated Ordinal, Continuous (mixture and non-mixture), and/or Count (Poisson and Negative Binomial, regular and zero-inflated) Variables: Correlation Method 1

Description

This function simulates k_cat ordinal (r \ge 2 categories), k_cont continuous non-mixture, k_mix continuous mixture, k_pois Poisson (regular and zero-inflated), and/or k_nb Negative Binomial (regular and zero-inflated) variables with a specified correlation matrix rho. The variables are generated from multivariate normal variables with intermediate correlation matrix Sigma, calculated by intercorr, and then transformed. The intermediate correlations involving count variables are determined using correlation method 1. The ordering of the variables in rho must be 1st ordinal, 2nd continuous non-mixture, 3rd components of the continuous mixture, 4th regular Poisson, 5th zero-inflated Poisson, 6th regular NB, and 7th zero-inflated NB. Note that it is possible for k_cat, k_cont, k_mix, k_pois, and/or k_nb to be 0. Simulation occurs at the component-level for continuous mixture distributions. The target correlation matrix is specified in terms of correlations with components of continuous mixture variables. There are no parameter input checks in order to decrease simulation time. All inputs should be checked prior to simulation with validpar and validcorr. Summaries for the simulation results can be obtained with summary_var.

All continuous variables are simulated using either Fleishman's third-order (method = "Fleishman", doi: 10.1007/BF02293811) or Headrick's fifth-order (method = "Polynomial", doi: 10.1016/S0167-9473(02)00072-5) power method transformation. It works by matching standardized cumulants – the first four (mean, variance, skew, and standardized kurtosis) for Fleishman's method, or the first six (mean, variance, skew, standardized kurtosis, and standardized fifth and sixth cumulants) for Headrick's method. The transformation is expressed as follows:

Y = c_0 + c_1 * Z + c_2 * Z^2 + c_3 * Z^3 + c_4 * Z^4 + c_5 * Z^5, Z \sim N(0,1),

where c_4 and c_5 both equal 0 for Fleishman's method. The real constants are calculated by
find_constants. Continuous mixture variables are generated componentwise and then transformed to the desired mixture variables based on random multinomial variables generated from the mixing probabilities. Ordinal variables (r \ge 2 categories) are generated by discretizing the standard normal variables at quantiles. These quantiles are determined by evaluating the inverse standard normal CDF at the cumulative probabilities defined by each variable's marginal distribution. Count variables are generated using the inverse CDF method. The CDF of a standard normal variable has a uniform distribution. The appropriate quantile function (F_Y)^(-1) is applied to this uniform variable with the designated parameters to generate the count variable: Y = (F_Y)^(-1)(Phi(Z)). The Negative Binomial variable represents the number of failures which occur in a sequence of Bernoulli trials before the target number of successes is achieved. Zero-inflated Poisson or NB variables are obtained by setting the probability of a structural zero to be greater than 0. The optional error loop attempts to correct the final pairwise correlations to be within a user-specified precision value (epsilon) of the target correlations.

The vignette Variable Types discusses how each of the different variables are generated and describes the required parameters.

The vignette Overall Workflow for Generation of Correlated Data provides a detailed example discussing the step-by-step simulation process and comparing correlation methods 1 and 2.

Usage

corrvar(n = 10000, k_cat = 0, k_cont = 0, k_mix = 0, k_pois = 0,
  k_nb = 0, method = c("Fleishman", "Polynomial"), means = NULL,
  vars = NULL, skews = NULL, skurts = NULL, fifths = NULL,
  sixths = NULL, Six = list(), mix_pis = list(), mix_mus = list(),
  mix_sigmas = list(), mix_skews = list(), mix_skurts = list(),
  mix_fifths = list(), mix_sixths = list(), mix_Six = list(),
  marginal = list(), support = list(), lam = NULL, p_zip = 0,
  size = NULL, prob = NULL, mu = NULL, p_zinb = 0, rho = NULL,
  seed = 1234, errorloop = FALSE, epsilon = 0.001, maxit = 1000,
  use.nearPD = TRUE, nrand = 100000, Sigma = NULL, cstart = list(),
  quiet = FALSE)

Arguments

Value

A list whose components vary based on the type of simulated variables.

If ordinal variables are produced: Y_cat the ordinal variables,

If continuous variables are produced:

constants a data.frame of the constants,

Y_cont the continuous non-mixture variables,

Y_comp the components of the continuous mixture variables,

Y_mix the continuous mixture variables,

sixth_correction a list of sixth cumulant correction values,

valid.pdf a vector where the i-th element is "TRUE" if the constants for the i-th continuous variable generate a valid PDF, else "FALSE"

If Poisson variables are produced: Y_pois the regular and zero-inflated Poisson variables,

If Negative Binomial variables are produced: Y_nb the regular and zero-inflated Negative Binomial variables,

Additionally, the following elements:

Sigma the intermediate correlation matrix (after the error loop),

Error_Time the time in minutes required to use the error loop,

Time the total simulation time in minutes,

niter a matrix of the number of iterations used for each variable in the error loop,

Overview of Correlation Method 1

The intermediate correlations used in method 1 are more simulation based than those in method 2, which means that accuracy increases with sample size and the number of repetitions. In addition, specifying the seed allows for reproducibility. In addition, method 1 differs from method 2 in the following ways:

1) The intermediate correlation for count variables is based on the method of Yahav & Shmueli (2012, doi: 10.1002/asmb.901), which uses a simulation based, logarithmic transformation of the target correlation. This method becomes less accurate as the variable mean gets closer to zero.

2) The ordinal - count variable correlations are based on an extension of the method of Amatya & Demirtas (2015, doi: 10.1080/00949655.2014.953534), in which the correlation correction factor is the product of the upper Frechet-Hoeffding bound on the correlation between the count variable and the normal variable used to generate it and a simulated upper bound on the correlation between an ordinal variable and the normal variable used to generate it (see Demirtas & Hedeker, 2011, doi: 10.1198/tast.2011.10090).

3) The continuous - count variable correlations are based on an extension of the methods of Amatya & Demirtas (2015) and Demirtas et al. (2012, doi: 10.1002/sim.5362), in which the correlation correction factor is the product of the upper Frechet-Hoeffding bound on the correlation between the count variable and the normal variable used to generate it and the power method correlation between the continuous variable and the normal variable used to generate it (see Headrick & Kowalchuk, 2007, doi: 10.1080/10629360600605065). The intermediate correlations are the ratio of the target correlations to the correction factor.

Please see the Comparison of Correlation Methods 1 and 2 vignette for more information and a step-by-step overview of the simulation process.

Choice of Fleishman's third-order or Headrick's fifth-order method

Using the fifth-order approximation allows additional control over the fifth and sixth moments of the generated distribution, improving accuracy. In addition, the range of feasible standardized kurtosis (\gamma_2) values, given skew (\gamma_1) and standardized fifth (\gamma_3) and sixth (\gamma_4) cumulants, is larger than with Fleishman's method (see calc_lower_skurt). For example, the Fleishman method can not be used to generate a non-normal distribution with a ratio of \gamma_1^2/\gamma_2 > 9/14 (see Headrick & Kowalchuk, 2007). This eliminates the Chi-squared family of distributions, which has a constant ratio of \gamma_1^2/\gamma_2 = 2/3. The fifth-order method also generates more distributions with valid PDF's. However, if the fifth and sixth cumulants are unknown or do not exist, the Fleishman approximation should be used.

Reasons for Function Errors

1) The most likely cause for function errors is that no solutions to fleish or poly converged when using find_constants. If this happens, the simulation will stop. It may help to first use find_constants for each continuous variable to determine if a sixth cumulant correction value is needed. The solutions can be used as starting values (see cstart below). If the standardized cumulants are obtained from calc_theory, the user may need to use rounded values as inputs (i.e. skews = round(skews, 8)). For example, in order to ensure that skew is exactly 0 for symmetric distributions.

2) The kurtosis may be outside the region of possible values. There is an associated lower boundary for kurtosis associated with a given skew (for Fleishman's method) or skew and fifth and sixth cumulants (for Headrick's method). Use calc_lower_skurt to determine the boundary for a given set of cumulants.

3) The feasibility of the final correlation matrix rho, given the distribution parameters, should be checked first using validcorr. This function either checks if a given rho is plausible or returns the lower and upper final correlation limits. It should be noted that even if a target correlation matrix is within the "plausible range," it still may not be possible to achieve the desired matrix. This happens most frequently when generating ordinal variables or using negative correlations. The error loop frequently fixes these problems.

References

Please see references for SimCorrMix.

See Also

find_constants, validpar, validcorr, intercorr, corr_error, summary_var

Examples

Sim1 <- corrvar(n = 1000, k_cat = 1, k_cont = 1, method = "Polynomial",
  means = 0, vars = 1, skews = 0, skurts = 0, fifths = 0, sixths = 0,
  marginal = list(c(1/3, 2/3)), support = list(0:2),
  rho = matrix(c(1, 0.4, 0.4, 1), 2, 2), quiet = TRUE)

## Not run: 

# 2 continuous mixture, 1 binary, 1 zero-inflated Poisson, and
# 1 zero-inflated NB variable
n <- 10000
seed <- 1234

# Mixture variables: Normal mixture with 2 components;
# mixture of Logistic(0, 1), Chisq(4), Beta(4, 1.5)
# Find cumulants of components of 2nd mixture variable
L <- calc_theory("Logistic", c(0, 1))
C <- calc_theory("Chisq", 4)
B <- calc_theory("Beta", c(4, 1.5))

skews <- skurts <- fifths <- sixths <- NULL
Six <- list()
mix_pis <- list(c(0.4, 0.6), c(0.3, 0.2, 0.5))
mix_mus <- list(c(-2, 2), c(L[1], C[1], B[1]))
mix_sigmas <- list(c(1, 1), c(L[2], C[2], B[2]))
mix_skews <- list(rep(0, 2), c(L[3], C[3], B[3]))
mix_skurts <- list(rep(0, 2), c(L[4], C[4], B[4]))
mix_fifths <- list(rep(0, 2), c(L[5], C[5], B[5]))
mix_sixths <- list(rep(0, 2), c(L[6], C[6], B[6]))
mix_Six <- list(list(NULL, NULL), list(1.75, NULL, 0.03))
Nstcum <- calc_mixmoments(mix_pis[[1]], mix_mus[[1]], mix_sigmas[[1]],
  mix_skews[[1]], mix_skurts[[1]], mix_fifths[[1]], mix_sixths[[1]])
Mstcum <- calc_mixmoments(mix_pis[[2]], mix_mus[[2]], mix_sigmas[[2]],
  mix_skews[[2]], mix_skurts[[2]], mix_fifths[[2]], mix_sixths[[2]])
means <- c(Nstcum[1], Mstcum[1])
vars <- c(Nstcum[2]^2, Mstcum[2]^2)

marginal <- list(0.3)
support <- list(c(0, 1))
lam <- 0.5
p_zip <- 0.1
size <- 2
prob <- 0.75
p_zinb <- 0.2

k_cat <- k_pois <- k_nb <- 1
k_cont <- 0
k_mix <- 2
Rey <- matrix(0.39, 8, 8)
diag(Rey) <- 1
rownames(Rey) <- colnames(Rey) <- c("O1", "M1_1", "M1_2", "M2_1", "M2_2",
  "M2_3", "P1", "NB1")

# set correlation between components of the same mixture variable to 0
Rey["M1_1", "M1_2"] <- Rey["M1_2", "M1_1"] <- 0
Rey["M2_1", "M2_2"] <- Rey["M2_2", "M2_1"] <- Rey["M2_1", "M2_3"] <- 0
Rey["M2_3", "M2_1"] <- Rey["M2_2", "M2_3"] <- Rey["M2_3", "M2_2"] <- 0

# check parameter inputs
validpar(k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal, support,
  lam, p_zip, size, prob, mu = NULL, p_zinb, rho = Rey)

# check to make sure Rey is within the feasible correlation boundaries
validcorr(n, k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal,
  lam, p_zip, size, prob, mu = NULL, p_zinb, Rey, seed)

# simulate without the error loop
Sim2 <- corrvar(n, k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal, support,
  lam, p_zip, size, prob, mu = NULL, p_zinb, Rey, seed, epsilon = 0.01)

names(Sim2)

# simulate with the error loop
Sim2_EL <- corrvar(n, k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial",
  means, vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus,
  mix_sigmas, mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six,
  marginal, support, lam, p_zip, size, prob, mu = NULL, p_zinb, Rey,
  seed, errorloop = TRUE, epsilon = 0.01)

names(Sim2_EL)

## End(Not run)

Generation of Correlated Ordinal, Continuous (mixture and non-mixture), and/or Count (Poisson and Negative Binomial, regular and zero-inflated) Variables: Correlation Method 2

Description

This function simulates k_cat ordinal (r \ge 2 categories), k_cont continuous non-mixture, k_mix continuous mixture, k_pois Poisson (regular and zero-inflated), and/or k_nb Negative Binomial (regular and zero-inflated) variables with a specified correlation matrix rho. The variables are generated from multivariate normal variables with intermediate correlation matrix Sigma, calculated by intercorr2, and then transformed. The intermediate correlations involving count variables are determined using correlation method 2. The ordering of the variables in rho must be 1st ordinal, 2nd continuous non-mixture, 3rd components of the continuous mixture, 4th regular Poisson, 5th zero-inflated Poisson, 6th regular NB, and 7th zero-inflated NB. Note that it is possible for k_cat, k_cont, k_mix, k_pois, and/or k_nb to be 0. Simulation occurs at the component-level for continuous mixture distributions. The target correlation matrix is specified in terms of correlations with components of continuous mixture variables. There are no parameter input checks in order to decrease simulation time. All inputs should be checked prior to simulation with validpar and validcorr2. Summaries for the simulation results can be obtained with summary_var.

All continuous variables are simulated using either Fleishman's third-order (method = "Fleishman", doi: 10.1007/BF02293811) or Headrick's fifth-order (method = "Polynomial", doi: 10.1016/S0167-9473(02)00072-5) power method transformation. It works by matching standardized cumulants – the first four (mean, variance, skew, and standardized kurtosis) for Fleishman's method, or the first six (mean, variance, skew, standardized kurtosis, and standardized fifth and sixth cumulants) for Headrick's method. The transformation is expressed as follows:

Y = c_0 + c_1 * Z + c_2 * Z^2 + c_3 * Z^3 + c_4 * Z^4 + c_5 * Z^5, Z \sim N(0,1),

where c_4 and c_5 both equal 0 for Fleishman's method. The real constants are calculated by
find_constants. Continuous mixture variables are generated componentwise and then transformed to the desired mixture variables based on random multinomial variables generated from the mixing probabilities. Ordinal variables (r \ge 2 categories) are generated by discretizing the standard normal variables at quantiles. These quantiles are determined by evaluating the inverse standard normal CDF at the cumulative probabilities defined by each variable's marginal distribution. Count variables are generated using the inverse CDF method. The CDF of a standard normal variable has a uniform distribution. The appropriate quantile function (F_Y)^(-1) is applied to this uniform variable with the designated parameters to generate the count variable: Y = (F_Y)^(-1)(Phi(Z)). The Negative Binomial variable represents the number of failures which occur in a sequence of Bernoulli trials before the target number of successes is achieved. Zero-inflated Poisson or NB variables are obtained by setting the probability of a structural zero to be greater than 0. The optional error loop attempts to correct the final pairwise correlations to be within a user-specified precision value (epsilon) of the target correlations.

The vignette Variable Types discusses how each of the different variables are generated and describes the required parameters.

The vignette Overall Workflow for Generation of Correlated Data provides a detailed example discussing the step-by-step simulation process and comparing correlation methods 1 and 2.

Usage

corrvar2(n = 10000, k_cat = 0, k_cont = 0, k_mix = 0, k_pois = 0,
  k_nb = 0, method = c("Fleishman", "Polynomial"), means = NULL,
  vars = NULL, skews = NULL, skurts = NULL, fifths = NULL,
  sixths = NULL, Six = list(), mix_pis = list(), mix_mus = list(),
  mix_sigmas = list(), mix_skews = list(), mix_skurts = list(),
  mix_fifths = list(), mix_sixths = list(), mix_Six = list(),
  marginal = list(), support = list(), lam = NULL, p_zip = 0,
  size = NULL, prob = NULL, mu = NULL, p_zinb = 0, pois_eps = 0.0001,
  nb_eps = 0.0001, rho = NULL, seed = 1234, errorloop = FALSE,
  epsilon = 0.001, maxit = 1000, use.nearPD = TRUE, Sigma = NULL,
  cstart = list(), quiet = FALSE)

Arguments

Value

A list whose components vary based on the type of simulated variables.

If ordinal variables are produced: Y_cat the ordinal variables,

If continuous variables are produced:

constants a data.frame of the constants,

Y_cont the continuous non-mixture variables,

Y_comp the components of the continuous mixture variables,

Y_mix the continuous mixture variables,

sixth_correction a list of sixth cumulant correction values,

valid.pdf a vector where the i-th element is "TRUE" if the constants for the i-th continuous variable generate a valid PDF, else "FALSE"

If Poisson variables are produced: Y_pois the regular and zero-inflated Poisson variables,

If Negative Binomial variables are produced: Y_nb the regular and zero-inflated Negative Binomial variables,

Additionally, the following elements:

Sigma the intermediate correlation matrix (after the error loop),

Error_Time the time in minutes required to use the error loop,

Time the total simulation time in minutes,

niter a matrix of the number of iterations used for each variable in the error loop,

Overview of Method 2

The intermediate correlations used in method 2 are less simulation based than those in method 1, and no seed is needed. Their calculations involve greater utilization of correction loops which make iterative adjustments until a maximum error has been reached (if possible). In addition, method 2 differs from method 1 in the following ways:

1) The intermediate correlations involving count variables are based on the methods of Barbiero & Ferrari (2012, doi: 10.1080/00273171.2012.692630, 2015, doi: 10.1002/asmb.2072). The Poisson or Negative Binomial support is made finite by removing a small user-specified value (i.e. 1e-06) from the total cumulative probability. This truncation factor may differ for each count variable. The count variables are subsequently treated as ordinal and intermediate correlations are calculated using the correction loop of ord_norm.

2) The continuous - count variable correlations are based on an extension of the method of Demirtas et al. (2012, doi: 10.1002/sim.5362), and the count variables are treated as ordinal. The correction factor is the product of the power method correlation between the continuous variable and the normal variable used to generate it (see Headrick & Kowalchuk, 2007, doi: 10.1080/10629360600605065) and the point-polyserial correlation between the ordinalized count variable and the normal variable used to generate it (see Olsson et al., 1982, doi: 10.1007/BF02294164). The intermediate correlations are the ratio of the target correlations to the correction factor.

Please see the Comparison of Correlation Methods 1 and 2 vignette for more information and a step-by-step overview of the simulation process.

Choice of Fleishman's third-order or Headrick's fifth-order method

Using the fifth-order approximation allows additional control over the fifth and sixth moments of the generated distribution, improving accuracy. In addition, the range of feasible standardized kurtosis (\gamma_2) values, given skew (\gamma_1) and standardized fifth (\gamma_3) and sixth (\gamma_4) cumulants, is larger than with Fleishman's method (see calc_lower_skurt). For example, the Fleishman method can not be used to generate a non-normal distribution with a ratio of \gamma_1^2/\gamma_2 > 9/14 (see Headrick & Kowalchuk, 2007). This eliminates the Chi-squared family of distributions, which has a constant ratio of \gamma_1^2/\gamma_2 = 2/3. The fifth-order method also generates more distributions with valid PDF's. However, if the fifth and sixth cumulants are unknown or do not exist, the Fleishman approximation should be used.

Reasons for Function Errors

1) The most likely cause for function errors is that no solutions to fleish or poly converged when using find_constants. If this happens, the simulation will stop. It may help to first use find_constants for each continuous variable to determine if a sixth cumulant correction value is needed. The solutions can be used as starting values (see cstart below). If the standardized cumulants are obtained from calc_theory, the user may need to use rounded values as inputs (i.e. skews = round(skews, 8)). For example, in order to ensure that skew is exactly 0 for symmetric distributions.

2) The kurtosis may be outside the region of possible values. There is an associated lower boundary for kurtosis associated with a given skew (for Fleishman's method) or skew and fifth and sixth cumulants (for Headrick's method). Use calc_lower_skurt to determine the boundary for a given set of cumulants.

3) The feasibility of the final correlation matrix rho, given the distribution parameters, should be checked first using validcorr2. This function either checks if a given rho is plausible or returns the lower and upper final correlation limits. It should be noted that even if a target correlation matrix is within the "plausible range," it still may not be possible to achieve the desired matrix. This happens most frequently when generating ordinal variables or using negative correlations. The error loop frequently fixes these problems.

References

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

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, & Mermelstein RJ (2012). Simulation of massive public health data by power polynomials. Statistics in Medicine, 31(27):3337-3346. doi: 10.1002/sim.5362.

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 (2018). SimMultiCorrData: Simulation of Correlated Data with Multiple Variable Types. R package version 0.2.2. https://CRAN.R-project.org/package=SimMultiCorrData.

Fleishman AI (1978). A Method for Simulating Non-normal Distributions. Psychometrika, 43:521-532. doi: 10.1007/BF02293811.

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, 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, 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.

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.

Lambert D (1992). Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing. Technometrics 34(1):1-14.

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.

Yee TW (2018). VGAM: Vector Generalized Linear and Additive Models. R package version 1.0-5. 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.

See Also

find_constants, validpar, validcorr2, intercorr2, corr_error, summary_var

Examples

Sim1 <- corrvar2(n = 1000, k_cat = 1, k_cont = 1, method = "Polynomial",
  means = 0, vars = 1, skews = 0, skurts = 0, fifths = 0, sixths = 0,
  marginal = list(c(1/3, 2/3)), support = list(0:2),
  rho = matrix(c(1, 0.4, 0.4, 1), 2, 2), quiet = TRUE)

## Not run: 

# 2 continuous mixture, 1 binary, 1 zero-inflated Poisson, and
# 1 zero-inflated NB variable
n <- 10000
seed <- 1234

# Mixture variables: Normal mixture with 2 components;
# mixture of Logistic(0, 1), Chisq(4), Beta(4, 1.5)
# Find cumulants of components of 2nd mixture variable
L <- calc_theory("Logistic", c(0, 1))
C <- calc_theory("Chisq", 4)
B <- calc_theory("Beta", c(4, 1.5))

skews <- skurts <- fifths <- sixths <- NULL
Six <- list()
mix_pis <- list(c(0.4, 0.6), c(0.3, 0.2, 0.5))
mix_mus <- list(c(-2, 2), c(L[1], C[1], B[1]))
mix_sigmas <- list(c(1, 1), c(L[2], C[2], B[2]))
mix_skews <- list(rep(0, 2), c(L[3], C[3], B[3]))
mix_skurts <- list(rep(0, 2), c(L[4], C[4], B[4]))
mix_fifths <- list(rep(0, 2), c(L[5], C[5], B[5]))
mix_sixths <- list(rep(0, 2), c(L[6], C[6], B[6]))
mix_Six <- list(list(NULL, NULL), list(1.75, NULL, 0.03))
Nstcum <- calc_mixmoments(mix_pis[[1]], mix_mus[[1]], mix_sigmas[[1]],
  mix_skews[[1]], mix_skurts[[1]], mix_fifths[[1]], mix_sixths[[1]])
Mstcum <- calc_mixmoments(mix_pis[[2]], mix_mus[[2]], mix_sigmas[[2]],
  mix_skews[[2]], mix_skurts[[2]], mix_fifths[[2]], mix_sixths[[2]])
means <- c(Nstcum[1], Mstcum[1])
vars <- c(Nstcum[2]^2, Mstcum[2]^2)

marginal <- list(0.3)
support <- list(c(0, 1))
lam <- 0.5
p_zip <- 0.1
pois_eps <- 0.0001
size <- 2
prob <- 0.75
p_zinb <- 0.2
nb_eps <- 0.0001

k_cat <- k_pois <- k_nb <- 1
k_cont <- 0
k_mix <- 2
Rey <- matrix(0.39, 8, 8)
diag(Rey) <- 1
rownames(Rey) <- colnames(Rey) <- c("O1", "M1_1", "M1_2", "M2_1", "M2_2",
  "M2_3", "P1", "NB1")

# set correlation between components of the same mixture variable to 0
Rey["M1_1", "M1_2"] <- Rey["M1_2", "M1_1"] <- 0
Rey["M2_1", "M2_2"] <- Rey["M2_2", "M2_1"] <- Rey["M2_1", "M2_3"] <- 0
Rey["M2_3", "M2_1"] <- Rey["M2_2", "M2_3"] <- Rey["M2_3", "M2_2"] <- 0

# check parameter inputs
validpar(k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal, support,
  lam, p_zip, size, prob, mu = NULL, p_zinb, pois_eps, nb_eps, Rey)

# check to make sure Rey is within the feasible correlation boundaries
validcorr2(n, k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal,
  lam, p_zip, size, prob, mu = NULL, p_zinb, pois_eps, nb_eps, Rey, seed)

# simulate without the error loop
Sim2 <- corrvar2(n, k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal, support,
  lam, p_zip, size, prob, mu = NULL, p_zinb, pois_eps, nb_eps, Rey, seed,
  epsilon = 0.01)

names(Sim2)

# simulate with the error loop
Sim2_EL <- corrvar2(n, k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial",
  means, vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus,
  mix_sigmas, mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six,
  marginal, support, lam, p_zip, size, prob, mu = NULL, p_zinb, pois_eps,
  nb_eps, Rey, seed, errorloop = TRUE, epsilon = 0.01)

names(Sim2_EL)

## End(Not run)

Calculate Intermediate MVN Correlation for Ordinal, Continuous, Poisson, or Negative Binomial Variables: Correlation Method 1

Description

This function calculates a k x k intermediate matrix of correlations, where k = k_cat + k_cont + k_pois + k_nb, to be used in simulating variables with corrvar. The k_cont includes regular continuous variables and components of continuous mixture variables. The ordering of the variables must be ordinal, continuous non-mixture, components of continuous mixture variables, regular Poisson, zero-inflated Poisson, regular Negative Binomial (NB), and zero-inflated NB (note that it is possible for k_cat, k_cont, k_pois, and/or k_nb to be 0). There are no parameter input checks in order to decrease simulation time. All inputs should be checked prior to simulation with validpar. There is a message given if the calculated intermediate correlation matrix Sigma is not positive-definite because it may not be possible to find a MVN correlation matrix that will produce the desired marginal distributions. This function is called by the simulation function corrvar, and would only be used separately if the user wants to first find the intermediate correlation matrix. This matrix Sigma can be used as an input to corrvar.

Please see the Comparison of Correlation Methods 1 and 2 vignette for information about calculations by variable pair type and the differences between this function and intercorr2.

Usage

intercorr(k_cat = 0, k_cont = 0, k_pois = 0, k_nb = 0,
  method = c("Fleishman", "Polynomial"), constants = NULL,
  marginal = list(), support = list(), lam = NULL, p_zip = 0,
  size = NULL, prob = NULL, mu = NULL, p_zinb = 0, rho = NULL,
  seed = 1234, epsilon = 0.001, maxit = 1000, nrand = 100000,
  quiet = FALSE)

Arguments

Value

the intermediate MVN correlation matrix

References

Please see references for SimCorrMix.

See Also

corrvar

Examples

Sigma1 <- intercorr(k_cat = 1, k_cont = 1, method = "Polynomial",
  constants = matrix(c(0, 1, 0, 0, 0, 0), 1, 6), marginal = list(0.3),
  support = list(c(0, 1)), rho = matrix(c(1, 0.4, 0.4, 1), 2, 2),
  quiet = TRUE)
## Not run: 
# 1 continuous mixture, 1 binary, 1 zero-inflated Poisson, and
# 1 zero-inflated NB variable
seed <- 1234

# Mixture of N(-2, 1) and N(2, 1)
constants <- rbind(c(0, 1, 0, 0, 0, 0), c(0, 1, 0, 0, 0, 0))

marginal <- list(0.3)
support <- list(c(0, 1))
lam <- 0.5
p_zip <- 0.1
size <- 2
prob <- 0.75
p_zinb <- 0.2

k_cat <- k_pois <- k_nb <- 1
k_cont <- 2
Rey <- matrix(0.35, 5, 5)
diag(Rey) <- 1
rownames(Rey) <- colnames(Rey) <- c("O1", "M1_1", "M1_2", "P1", "NB1")

# set correlation between components of the same mixture variable to 0
Rey["M1_1", "M1_2"] <- Rey["M1_2", "M1_1"] <- 0

Sigma2 <- intercorr(k_cat, k_cont, k_pois, k_nb, "Polynomial", constants,
  marginal, support, lam, p_zip, size, prob, mu = NULL, p_zinb, Rey, seed)

## End(Not run)

Calculate Intermediate MVN Correlation for Ordinal, Continuous, Poisson, or Negative Binomial Variables: Correlation Method 2

Description

This function calculates a k x k intermediate matrix of correlations, where k = k_cat + k_cont + k_pois + k_nb, to be used in simulating variables with corrvar2. The k_cont includes regular continuous variables and components of continuous mixture variables. The ordering of the variables must be ordinal, continuous non-mixture, components of continuous mixture variables, regular Poisson, zero-inflated Poisson, regular Negative Binomial (NB), and zero-inflated NB (note that it is possible for k_cat, k_cont, k_pois, and/or k_nb to be 0). There are no parameter input checks in order to decrease simulation time. All inputs should be checked prior to simulation with validpar. There is a message given if the calculated intermediate correlation matrix Sigma is not positive-definite because it may not be possible to find a MVN correlation matrix that will produce the desired marginal distributions. This function is called by the simulation function corrvar2, and would only be used separately if the user wants to first find the intermediate correlation matrix. This matrix Sigma can be used as an input to corrvar2.

Please see the Comparison of Correlation Methods 1 and 2 vignette for information about calculations by variable pair type and the differences between this function and intercorr.

Usage

intercorr2(k_cat = 0, k_cont = 0, k_pois = 0, k_nb = 0,
  method = c("Fleishman", "Polynomial"), constants = NULL,
  marginal = list(), support = list(), lam = NULL, p_zip = 0,
  size = NULL, prob = NULL, mu = NULL, p_zinb = 0, pois_eps = 0.0001,
  nb_eps = 0.0001, rho = NULL, epsilon = 0.001, maxit = 1000,
  quiet = FALSE)

Arguments

Value

the intermediate MVN correlation matrix

References

Please see references for SimCorrMix.

See Also

corrvar2

Examples

Sigma1 <- intercorr2(k_cat = 1, k_cont = 1, method = "Polynomial",
  constants = matrix(c(0, 1, 0, 0, 0, 0), 1, 6), marginal = list(0.3),
  support = list(c(0, 1)), rho = matrix(c(1, 0.4, 0.4, 1), 2, 2),
  quiet = TRUE)
## Not run: 

# 1 continuous mixture, 1 binary, 1 zero-inflated Poisson, and
# 1 zero-inflated NB variable
# The defaults of pois_eps <- nb_eps <- 0.0001 are used.

# Mixture of N(-2, 1) and N(2, 1)
constants <- rbind(c(0, 1, 0, 0, 0, 0), c(0, 1, 0, 0, 0, 0))

marginal <- list(0.3)
support <- list(c(0, 1))
lam <- 0.5
p_zip <- 0.1
size <- 2
prob <- 0.75
p_zinb <- 0.2

k_cat <- k_pois <- k_nb <- 1
k_cont <- 2
Rey <- matrix(0.35, 5, 5)
diag(Rey) <- 1
rownames(Rey) <- colnames(Rey) <- c("O1", "M1_1", "M1_2", "P1", "NB1")

# set correlation between components of the same mixture variable to 0
Rey["M1_1", "M1_2"] <- Rey["M1_2", "M1_1"] <- 0

Sigma2 <- intercorr2(k_cat, k_cont, k_pois, k_nb, "Polynomial", constants,
  marginal, support, lam, p_zip, size, prob, mu = NULL, p_zinb, rho = Rey)

## End(Not run)

Calculate Intermediate MVN Correlation for Ordinal - Negative Binomial Variables: Correlation Method 1

Description

This function calculates the k_cat x k_nb intermediate matrix of correlations for the k_cat ordinal (r >= 2 categories) and k_nb Negative Binomial variables required to produce the target correlations in rho_cat_nb. It extends the method of Amatya & Demirtas (2015, doi: 10.1080/00949655.2014.953534) to ordinal - Negative Binomial pairs and allows for regular or zero-inflated NB variables. Here, the intermediate correlation between Z1 and Z2 (where Z1 is the standard normal variable discretized to produce an ordinal variable Y1, and Z2 is the standard normal variable used to generate a Negative Binomial variable via the inverse CDF method) is calculated by dividing the target correlation by a correction factor. The correction factor is the product of the upper Frechet-Hoeffding bound on the correlation between a Negative Binomial variable and the normal variable used to generate it and a simulated GSC upper bound on the correlation between an ordinal variable and the normal variable used to generate it (see Demirtas & Hedeker, 2011, doi: 10.1198/tast.2011.10090). The function is used in intercorr and corrvar. This function would not ordinarily be called by the user.

Usage

intercorr_cat_nb(rho_cat_nb = NULL, marginal = list(), size = NULL,
  mu = NULL, p_zinb = 0, nrand = 100000, seed = 1234)

Arguments

Value

a k_cat x k_nb matrix whose rows represent the k_cat ordinal variables and columns represent the k_nb Negative Binomial variables

References

Please see references for intercorr_cat_pois.

See Also

intercorr, corrvar

Calculate Intermediate MVN Correlation for Ordinal - Poisson Variables: Correlation Method 1

Description

This function calculates a k_cat x k_pois intermediate matrix of correlations for the k_cat ordinal (r >= 2 categories) and k_pois Poisson variables required to produce the target correlations in rho_cat_pois. It extends the method of Amatya & Demirtas (2015, doi: 10.1080/00949655.2014.953534) to ordinal - Poisson pairs and allows for regular or zero-inflated Poisson variables. Here, the intermediate correlation between Z1 and Z2 (where Z1 is the standard normal variable discretized to produce an ordinal variable Y1, and Z2 is the standard normal variable used to generate a Poisson variable via the inverse CDF method) is calculated by dividing the target correlation by a correction factor. The correction factor is the product of the upper Frechet-Hoeffding bound on the correlation between a Poisson variable and the normal variable used to generate it and a simulated GSC upper bound on the correlation between an ordinal variable and the normal variable used to generate it (see Demirtas & Hedeker, 2011, doi: 10.1198/tast.2011.10090). The function is used in intercorr and corrvar. This function would not ordinarily be called by the user.

Usage

intercorr_cat_pois(rho_cat_pois = NULL, marginal = list(), lam = NULL,
  p_zip = 0, nrand = 100000, seed = 1234)

Arguments

Value

a k_cat x k_pois matrix whose rows represent the k_cat ordinal variables and columns represent the k_pois Poisson variables

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.

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.

Frechet M (1951). Sur les tableaux de correlation dont les marges sont donnees. Ann. l'Univ. Lyon SectA, 14:53-77.

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.

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 (2018). VGAM: Vector Generalized Linear and Additive Models. R package version 1.0-5. https://CRAN.R-project.org/package=VGAM.

See Also

intercorr, corrvar

Calculate Intermediate MVN Correlation for Continuous Variables Generated by Polynomial Transformation Method

Description

This function finds the intermediate correlation for standard normal random variables which are used in Fleishman's third-order (doi: 10.1007/BF02293811) or Headrick's fifth-order (doi: 10.1016/S0167-9473(02)00072-5) polynomial transformation method (PMT) using nleqslv. It is used in intercorr and intercorr2 and would not ordinarily be called by the user. The correlations are found pairwise so that eigen-value or principal components decomposition should be done on the resulting Sigma matrix. The Comparison of Correlation Methods 1 and 2 vignette contains the equations which were derived by Headrick and Sawilowsky (doi: 10.1007/BF02294317) or Headrick (doi: 10.1016/S0167-9473(02)00072-5).

Usage

intercorr_cont(method = c("Fleishman", "Polynomial"), constants = NULL,
  rho_cont = NULL)

Arguments

Value

the intermediate matrix of correlations with the same dimensions as rho_cont

References

Please see additional references for SimCorrMix.

Fialkowski AC (2018). SimMultiCorrData: Simulation of Correlated Data with Multiple Variable Types. R package version 0.2.2. https://CRAN.R-project.org/package=SimMultiCorrData.

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, 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.

See Also

intercorr, intercorr2, nleqslv

Calculate Intermediate MVN Correlation for Continuous - Negative Binomial Variables: Correlation Method 1

Description

This function calculates a k_cont x k_nb intermediate matrix of correlations for the k_cont continuous and k_nb Negative Binomial variables. It extends the method of Amatya & Demirtas (2015, doi: 10.1080/00949655.2014.953534) to continuous variables generated using Headrick's fifth-order polynomial transformation and regular or zero-inflated NB variables. Here, the intermediate correlation between Z1 and Z2 (where Z1 is the standard normal variable transformed using Headrick's fifth-order or Fleishman's third-order method to produce a continuous variable Y1, and Z2 is the standard normal variable used to generate a Negative Binomial variable via the inverse CDF method) is calculated by dividing the target correlation by a correction factor. The correction factor is the product of the upper Frechet-Hoeffding bound on the correlation between a Negative Binomial variable and the normal variable used to generate it and the power method correlation (described in Headrick & Kowalchuk, 2007, doi: 10.1080/10629360600605065) between Y1 and Z1. The function is used in intercorr and corrvar. This function would not ordinarily be called by the user.

Usage

intercorr_cont_nb(method = c("Fleishman", "Polynomial"), constants = NULL,
  rho_cont_nb = NULL, size = NULL, mu = NULL, p_zinb = 0,
  nrand = 100000, seed = 1234)

Arguments

Value

a k_cont x k_nb matrix whose rows represent the k_cont continuous variables and columns represent the k_nb Negative Binomial variables

References

Please see references for intercorr_cont_pois.

See Also

find_constants, intercorr, corrvar

Calculate Intermediate MVN Correlation for Continuous - Negative Binomial Variables: Correlation Method 2

Description

This function calculates a k_cont x k_nb intermediate matrix of correlations for the k_cont continuous and k_nb Negative Binomial variables. It extends the methods of Demirtas et al. (2012, doi: 10.1002/sim.5362) and Barbiero & Ferrari (2015, doi: 10.1002/asmb.2072) by:

1) including non-normal continuous and regular or zero-inflated Negative Binomial variables

2) allowing the continuous variables to be generated via Fleishman's third-order or Headrick's fifth-order transformation, and

3) since the count variables are treated as ordinal, using the point-polyserial and polyserial correlations to calculate the intermediate correlations (similar to findintercorr_cont_cat in
SimMultiCorrData).

Here, the intermediate correlation between Z1 and Z2 (where Z1 is the standard normal variable transformed using Headrick's fifth-order or Fleishman's third-order method to produce a continuous variable Y1, and Z2 is the standard normal variable used to generate a Negative Binomial variable via the inverse CDF method) is calculated by dividing the target correlation by a correction factor. The correction factor is the product of the point-polyserial correlation between Y2 and Z2 (described in Olsson et al., 1982, doi: 10.1007/BF02294164) and the power method correlation (described in Headrick & Kowalchuk, 2007, doi: 10.1080/10629360600605065) between Y1 and Z1. After the maximum support value has been found using maxcount_support, the point-polyserial correlation is given by:

\rho_{Y2,Z2} = \frac{1}{\sigma_{Y2}} \sum_{j = 1}^{r-1} \phi(\tau_{j})(y2_{j+1} - y2_{j})

where

\phi(\tau) = (2\pi)^{-1/2} * exp(-0.5 \tau^2)

Here, y_{j} is the j-th support value and \tau_{j} is \Phi^{-1}(\sum_{i=1}^{j} Pr(Y = y_{i})). The power method correlation is given by:

\rho_{Y1, Z1} = c_1 + 3c_3 + 15c_5,

where c_5 = 0 if method = "Fleishman". The function is used in intercorr2 and corrvar2. This function would not ordinarily be called by the user.

Usage

intercorr_cont_nb2(method = c("Fleishman", "Polynomial"), constants = NULL,
  rho_cont_nb = NULL, nb_marg = list(), nb_support = list())

Arguments

Value

a k_cont x k_nb matrix whose rows represent the k_cont continuous variables and columns represent the k_nb Negative Binomial variables

References

Please see references in intercorr_cont_pois2.

See Also

find_constants, power_norm_corr, intercorr2, corrvar2

Calculate Intermediate MVN Correlation for Continuous - Poisson Variables: Correlation Method 1

Description

This function calculates a k_cont x k_pois intermediate matrix of correlations for the k_cont continuous and k_pois Poisson variables. It extends the method of Amatya & Demirtas (2015, doi: 10.1080/00949655.2014.953534) to continuous variables generated using Headrick's fifth-order polynomial transformation and zero-inflated Poisson variables. Here, the intermediate correlation between Z1 and Z2 (where Z1 is the standard normal variable transformed using Headrick's fifth-order or Fleishman's third-order method to produce a continuous variable Y1, and Z2 is the standard normal variable used to generate a Poisson variable via the inverse CDF method) is calculated by dividing the target correlation by a correction factor. The correction factor is the product of the upper Frechet-Hoeffding bound on the correlation between a Poisson variable and the normal variable used to generate it and the power method correlation (described in Headrick & Kowalchuk, 2007, doi: 10.1080/10629360600605065) between Y1 and Z1. The function is used in intercorr and corrvar. This function would not ordinarily be called by the user.

Usage

intercorr_cont_pois(method = c("Fleishman", "Polynomial"), constants = NULL,
  rho_cont_pois = NULL, lam = NULL, p_zip = 0, nrand = 100000,
  seed = 1234)

Arguments

Value

a k_cont x k_pois matrix whose rows represent the k_cont continuous variables and columns represent the k_pois Poisson variables

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.

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.

Frechet M (1951). Sur les tableaux de correlation dont les marges sont donnees. Ann. l'Univ. Lyon SectA, 14:53-77.

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.

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.

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 (2018). VGAM: Vector Generalized Linear and Additive Models. R package version 1.0-5. https://CRAN.R-project.org/package=VGAM.

See Also

power_norm_corr, find_constants, intercorr, corrvar

Calculate Intermediate MVN Correlation for Continuous - Poisson Variables: Correlation Method 2

Description

This function calculates a k_cont x k_pois intermediate matrix of correlations for the k_cont continuous and k_pois Poisson variables. It extends the methods of Demirtas et al. (2012, doi: 10.1002/sim.5362) and Barbiero & Ferrari (2015, doi: 10.1002/asmb.2072) by:

1) including non-normal continuous and regular or zero-inflated Poisson variables

2) allowing the continuous variables to be generated via Fleishman's third-order or Headrick's fifth-order transformation, and

3) since the count variables are treated as ordinal, using the point-polyserial and polyserial correlations to calculate the intermediate correlations (similar to findintercorr_cont_cat) in
SimMultiCorrData).

Here, the intermediate correlation between Z1 and Z2 (where Z1 is the standard normal variable transformed using Headrick's fifth-order or Fleishman's third-order method to produce a continuous variable Y1, and Z2 is the standard normal variable used to generate a Poisson variable via the inverse CDF method) is calculated by dividing the target correlation by a correction factor. The correction factor is the product of the point-polyserial correlation between Y2 and Z2 (described in Olsson et al., 1982, doi: 10.1007/BF02294164) and the power method correlation (described in Headrick & Kowalchuk, 2007, doi: 10.1080/10629360600605065) between Y1 and Z1. After the maximum support value has been found using maxcount_support, the point-polyserial correlation is given by:

\rho_{Y2,Z2} = \frac{1}{\sigma_{Y2}} \sum_{j = 1}^{r-1} \phi(\tau_{j})(y2_{j+1} - y2_{j})

where

\phi(\tau) = (2\pi)^{-1/2} * exp(-0.5 \tau^2)

Here, y_{j} is the j-th support value and \tau_{j} is \Phi^{-1}(\sum_{i=1}^{j} Pr(Y = y_{i})). The power method correlation is given by:

\rho_{Y1, Z1} = c_1 + 3c_3 + 15c_5,

where c_5 = 0 if method = "Fleishman". The function is used in intercorr2 and corrvar2. This function would not ordinarily be called by the user.

Usage

intercorr_cont_pois2(method = c("Fleishman", "Polynomial"),
  constants = NULL, rho_cont_pois = NULL, pois_marg = list(),
  pois_support = list())

Arguments

Value

a k_cont x k_pois matrix whose rows represent the k_cont continuous variables and columns represent the k_pois Poisson variables

References

Please see additional references in intercorr_cont_pois.

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.

See Also

find_constants, power_norm_corr, intercorr2, corrvar2

Calculate Intermediate MVN Correlation for Negative Binomial Variables: Correlation Method 1

Description

This function calculates a k_nb x k_nb intermediate matrix of correlations for the Negative Binomial variables by extending the method of Yahav & Shmueli (2012, doi: 10.1002/asmb.901). The intermediate correlation between Z1 and Z2 (the standard normal variables used to generate the Negative Binomial variables Y1 and Y2 via the inverse CDF method) is calculated using a logarithmic transformation of the target correlation. First, the upper and lower Frechet-Hoeffding bounds (mincor, maxcor) on \rho_{Y1, Y2} are simulated. Then the intermediate correlation is found as follows:

\rho_{Z1, Z2} = \frac{1}{b} * log(\frac{\rho_{Y1, Y2} - c}{a}),

where a = -(maxcor * mincor)/(maxcor + mincor), b = log((maxcor + a)/a), and c = -a. The function adapts code from Amatya & Demirtas' (2016) package PoisNor-package by:

1) allowing specifications for the number of random variates and the seed for reproducibility

2) providing the following checks: if Sigma_(Z1, Z2) > 1, Sigma_(Z1, Z2) is set to 1; if Sigma_(Z1, Z2) < -1, Sigma_(Z1, Z2) is set to -1

3) simulating regular and zero-inflated Negative Binomial variables.

The function is used in intercorr and corrvar and would not ordinarily be called by the user.

Usage

intercorr_nb(rho_nb = NULL, size = NULL, mu = NULL, p_zinb = 0,
  nrand = 100000, seed = 1234)

Arguments

Value

the k_nb x k_nb intermediate correlation matrix for the Negative Binomial variables

References

Please see references for intercorr_pois.

See Also

intercorr_pois, intercorr_pois_nb, intercorr, corrvar

Calculate Intermediate MVN Correlation for Poisson Variables: Correlation Method 1

Description

This function calculates a k_pois x k_pois intermediate matrix of correlations for the Poisson variables using the method of Yahav & Shmueli (2012, doi: 10.1002/asmb.901). The intermediate correlation between Z1 and Z2 (the standard normal variables used to generate the Poisson variables Y1 and Y2 via the inverse CDF method) is calculated using a logarithmic transformation of the target correlation. First, the upper and lower Frechet-Hoeffding bounds (mincor, maxcor) on \rho_{Y1, Y2} are simulated. Then the intermediate correlation is found as follows:

\rho_{Z1, Z2} = \frac{1}{b} * log(\frac{\rho_{Y1, Y2} - c}{a}),

where a = -(maxcor * mincor)/(maxcor + mincor), b = log((maxcor + a)/a), and c = -a. The function adapts code from Amatya & Demirtas' (2016) package PoisNor-package by:

1) allowing specifications for the number of random variates and the seed for reproducibility

2) providing the following checks: if Sigma_(Z1, Z2) > 1, Sigma_(Z1, Z2) is set to 1; if Sigma_(Z1, Z2) < -1, Sigma_(Z1, Z2) is set to -1

3) simulating regular and zero-inflated Poisson variables.

The function is used in intercorr and corrvar and would not ordinarily be called by the user.

Usage

intercorr_pois(rho_pois = NULL, lam = NULL, p_zip = 0, nrand = 100000,
  seed = 1234)

Arguments

Value

the k_pois x k_pois intermediate correlation matrix for the Poisson variables

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.

Demirtas H & Hedeker D (2011). A practical way for computing approximate lower and upper correlation bounds. American Statistician, 65(2):104-109.

Frechet M (1951). Sur les tableaux de correlation dont les marges sont donnees. Ann. l'Univ. Lyon SectA, 14:53-77.

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.

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 (2018). VGAM: Vector Generalized Linear and Additive Models. R package version 1.0-5. https://CRAN.R-project.org/package=VGAM.

See Also

intercorr_nb, intercorr_pois_nb, intercorr, corrvar

Calculate Intermediate MVN Correlation for Poisson - Negative Binomial Variables: Correlation Method 1

Description

This function calculates a k_pois x k_nb intermediate matrix of correlations for the Poisson and Negative Binomial variables by extending the method of Yahav & Shmueli (2012, doi: 10.1002/asmb.901). The intermediate correlation between Z1 and Z2 (the standard normal variables used to generate the Poisson and Negative Binomial variables Y1 and Y2 via the inverse CDF method) is calculated using a logarithmic transformation of the target correlation. First, the upper and lower Frechet-Hoeffding bounds (mincor, maxcor) on \rho_{Y1, Y2} are simulated. Then the intermediate correlation is found as follows:

\rho_{Z1, Z2} = \frac{1}{b} * log(\frac{\rho_{Y1, Y2} - c}{a}),

where a = -(maxcor * mincor)/(maxcor + mincor), b = log((maxcor + a)/a), and c = -a. The function adapts code from Amatya & Demirtas' (2016) package PoisNor-package by:

1) allowing specifications for the number of random variates and the seed for reproducibility

2) providing the following checks: if Sigma_(Z1, Z2) > 1, Sigma_(Z1, Z2) is set to 1; if Sigma_(Z1, Z2) < -1, Sigma_(Z1, Z2) is set to -1

3) simulating regular and zero-inflated Poisson and Negative Binomial variables.

The function is used in intercorr and corrvar and would not ordinarily be called by the user.

Usage

intercorr_pois_nb(rho_pois_nb = NULL, lam = NULL, p_zip = 0,
  size = NULL, mu = NULL, p_zinb = 0, nrand = 100000, seed = 1234)

Arguments

Value

the k_pois x k_nb intermediate correlation matrix whose rows represent the k_pois Poisson variables and columns represent the k_nb Negative Binomial variables

References

Please see references for intercorr_pois.

See Also

intercorr_pois, intercorr_nb, intercorr, corrvar

Calculate Maximum Support Value for Count Variables: Correlation Method 2

Description

This function calculates the maximum support value for count variables by extending the method of Barbiero & Ferrari (2015, doi: 10.1002/asmb.2072) to include regular and zero-inflated Poisson and Negative Binomial variables. In order for count variables to be treated as ordinal in the calculation of the intermediate MVN correlation matrix, their infinite support must be truncated (made finite). This is done by setting the total cumulative probability equal to 1 - a small user-specified value (pois_eps or nb_eps). The maximum support value equals the inverse CDF applied to this result. The truncation values may differ for each variable. The function is used in intercorr2 and corrvar2 and would not ordinarily be called by the user.

Usage

maxcount_support(k_pois = 0, k_nb = 0, lam = NULL, p_zip = 0,
  size = NULL, prob = NULL, mu = NULL, p_zinb = 0, pois_eps = NULL,
  nb_eps = NULL)

Arguments

Value

a data.frame with k_pois + k_nb rows; the column names are:

Distribution Poisson or Negative Binomial

Number the variable index

Max the maximum support value

References

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.

See Also

intercorr2, corrvar2

Calculate Correlations of Ordinal Variables Obtained from Discretizing Normal Variables

Description

This function calculates the correlation of ordinal variables (or variables treated as "ordinal"), with given marginal distributions, obtained from discretizing standard normal variables with a specified correlation matrix. The function modifies Barbiero & Ferrari's contord function in GenOrd-package. It uses pmvnorm function from the mvtnorm package to calculate multivariate normal cumulative probabilities defined by the normal quantiles obtained at marginal and the supplied correlation matrix Sigma. This function is used within ord_norm and would not ordinarily be called by the user.

Usage

norm_ord(marginal = list(), Sigma = NULL, support = list(),
  Spearman = FALSE)

Arguments

Value

the correlation matrix of the ordinal variables

References

Please see references in ord_norm.

Alan Genz, Frank Bretz, Tetsuhisa Miwa, Xuefei Mi, Friedrich Leisch, Fabian Scheipl, Torsten Hothorn (2018). mvtnorm: Multivariate Normal and t Distributions. R package version 1.0-8. https://CRAN.R-project.org/package=mvtnorm.

Alan Genz, Frank Bretz (2009), Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics, Vol. 195., Springer-Verlag, Heidelberg. ISBN 978-3-642-01688-2.

See Also

ord_norm

Calculate Intermediate MVN Correlation to Generate Variables Treated as Ordinal

Description

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, doi: 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.

Usage

ord_norm(marginal = list(), rho = NULL, support = list(),
  epsilon = 0.001, maxit = 1000, Spearman = FALSE)

Arguments

Value

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

References

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

Ferrari PA, Barbiero A (2012). Simulating ordinal data, Multivariate Behavioral Research, 47(4):566-589. doi: 10.1080/00273171.2012.692630.

See Also

corrvar, corrvar2, norm_ord, intercorr, intercorr2

Plot Simulated Probability Density Function and Target PDF by Distribution Name or Function for Continuous or Count Variables

Description

This plots the PDF of simulated continuous or count (regular or zero-inflated, Poisson or Negative Binomial) data and overlays the target PDF (if overlay = TRUE), which is specified by distribution name (plus up to 4 parameters) or PDF function fx (plus support bounds). If a continuous target distribution is provided (cont_var = TRUE), the simulated data y is scaled and then transformed (i.e. y = sigma * scale(y) + mu) so that it has the same mean (mu) and variance (sigma^2) as the target distribution. The PDF's of continuous variables are shown as lines (using geom_density and ggplot2::geom_line). It works for valid or invalid power method PDF's. The PMF's of count variables are shown as vertical bar graphs (using ggplot2::geom_col). The function returns a ggplot2-package object so the user can save it or modify it as necessary. The graph parameters (i.e. title, sim_color, sim_lty, sim_size, target_color, target_lty, target_size, legend.position, legend.justification, legend.text.size, title.text.size, axis.text.size, and axis.title.size) are inputs to the ggplot2-package functions so information about valid inputs can be obtained from that package's documentation.

Usage

plot_simpdf_theory(sim_y, title = "Simulated Probability Density Function",
  ylower = NULL, yupper = NULL, sim_color = "dark blue", sim_lty = 1,
  sim_size = 1, col_width = 0.5, overlay = TRUE, cont_var = TRUE,
  target_color = "dark green", target_lty = 2, target_size = 1,
  Dist = c("Benini", "Beta", "Beta-Normal", "Birnbaum-Saunders", "Chisq",
  "Dagum", "Exponential", "Exp-Geometric", "Exp-Logarithmic", "Exp-Poisson",
  "F", "Fisk", "Frechet", "Gamma", "Gaussian", "Gompertz", "Gumbel",
  "Kumaraswamy", "Laplace", "Lindley", "Logistic", "Loggamma", "Lognormal",
  "Lomax", "Makeham", "Maxwell", "Nakagami", "Paralogistic", "Pareto", "Perks",
  "Rayleigh", "Rice", "Singh-Maddala", "Skewnormal", "t", "Topp-Leone",
  "Triangular", "Uniform", "Weibull", "Poisson", "Negative_Binomial"),
  params = NULL, fx = NULL, lower = NULL, upper = NULL,
  legend.position = c(0.975, 0.9), legend.justification = c(1, 1),
  legend.text.size = 10, title.text.size = 15, axis.text.size = 10,
  axis.title.size = 13)

Arguments

Value

A ggplot2-package object.

References

Please see the references for plot_simtheory.

See Also

calc_theory, ggplot

Examples

# Using normal mixture variable from contmixvar1 example
Nmix <- contmixvar1(n = 1000, "Polynomial", means = 0, vars = 1,
  mix_pis = c(0.4, 0.6), mix_mus = c(-2, 2), mix_sigmas = c(1, 1),
  mix_skews = c(0, 0), mix_skurts = c(0, 0), mix_fifths = c(0, 0),
  mix_sixths = c(0, 0))
plot_simpdf_theory(Nmix$Y_mix[, 1],
  title = "Mixture of Normal Distributions",
  fx = function(x) 0.4 * dnorm(x, -2, 1) + 0.6 * dnorm(x, 2, 1),
  lower = -5, upper = 5)
## Not run: 
# Mixture of Beta(6, 3), Beta(4, 1.5), and Beta(10, 20)
Stcum1 <- calc_theory("Beta", c(6, 3))
Stcum2 <- calc_theory("Beta", c(4, 1.5))
Stcum3 <- calc_theory("Beta", c(10, 20))
mix_pis <- c(0.5, 0.2, 0.3)
mix_mus <- c(Stcum1[1], Stcum2[1], Stcum3[1])
mix_sigmas <- c(Stcum1[2], Stcum2[2], Stcum3[2])
mix_skews <- c(Stcum1[3], Stcum2[3], Stcum3[3])
mix_skurts <- c(Stcum1[4], Stcum2[4], Stcum3[4])
mix_fifths <- c(Stcum1[5], Stcum2[5], Stcum3[5])
mix_sixths <- c(Stcum1[6], Stcum2[6], Stcum3[6])
mix_Six <- list(seq(0.01, 10, 0.01), c(0.01, 0.02, 0.03),
  seq(0.01, 10, 0.01))
Bstcum <- calc_mixmoments(mix_pis, mix_mus, mix_sigmas, mix_skews,
  mix_skurts, mix_fifths, mix_sixths)
Bmix <- contmixvar1(n = 10000, "Polynomial", Bstcum[1], Bstcum[2]^2,
  mix_pis, mix_mus, mix_sigmas, mix_skews, mix_skurts, mix_fifths,
  mix_sixths, mix_Six)
plot_simpdf_theory(Bmix$Y_mix[, 1], title = "Mixture of Beta Distributions",
  fx = function(x) mix_pis[1] * dbeta(x, 6, 3) + mix_pis[2] *
    dbeta(x, 4, 1.5) + mix_pis[3] * dbeta(x, 10, 20), lower = 0, upper = 1)

## End(Not run)

Plot Simulated Data and Target Distribution Data by Name or Function for Continuous or Count Variables

Description

This plots simulated continuous or count (regular or zero-inflated, Poisson or Negative Binomial) data and overlays data (if overlay = TRUE) generated from the target distribution. The target is specified by name (plus up to 4 parameters) or PDF function fx (plus support bounds). Due to the integration involved in finding the CDF from the PDF supplied by fx, only continuous fx may be supplied. Both are plotted as histograms (using geom_histogram). If a continuous target distribution is specified (cont_var = TRUE), the simulated data y is scaled and then transformed (i.e. y = sigma * scale(y) + mu) so that it has the same mean (mu) and variance (sigma^2) as the target distribution. It works for valid or invalid power method PDF's. It returns a ggplot2-package object so the user can save it or modify it as necessary. The graph parameters (i.e. title, sim_color, target_color, legend.position, legend.justification, legend.text.size, title.text.size, axis.text.size, and axis.title.size) are inputs to the ggplot2-package functions so information about valid inputs can be obtained from that package's documentation.

Usage

plot_simtheory(sim_y, title = "Simulated Data Values", ylower = NULL,
  yupper = NULL, sim_color = "dark blue", overlay = TRUE,
  cont_var = TRUE, target_color = "dark green", binwidth = NULL,
  nbins = 100, Dist = c("Benini", "Beta", "Beta-Normal",
  "Birnbaum-Saunders", "Chisq", "Dagum", "Exponential", "Exp-Geometric",
  "Exp-Logarithmic", "Exp-Poisson", "F", "Fisk", "Frechet", "Gamma", "Gaussian",
  "Gompertz", "Gumbel", "Kumaraswamy", "Laplace", "Lindley", "Logistic",
  "Loggamma", "Lognormal", "Lomax", "Makeham", "Maxwell", "Nakagami",
  "Paralogistic", "Pareto", "Perks", "Rayleigh", "Rice", "Singh-Maddala",
  "Skewnormal", "t", "Topp-Leone", "Triangular", "Uniform", "Weibull",
  "Poisson", "Negative_Binomial"), params = NULL, fx = NULL, lower = NULL,
  upper = NULL, seed = 1234, sub = 1000, legend.position = c(0.975,
  0.9), legend.justification = c(1, 1), legend.text.size = 10,
  title.text.size = 15, axis.text.size = 10, axis.title.size = 13)

Arguments

Value

A ggplot2-package object.

References

Carnell R (2017). triangle: Provides the Standard Distribution Functions for the Triangle Distribution. R package version 0.11. https://CRAN.R-project.org/package=triangle.

Fialkowski AC (2018). SimMultiCorrData: Simulation of Correlated Data with Multiple Variable Types. R package version 0.2.2. https://CRAN.R-project.org/package=SimMultiCorrData.

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.

Wickham H. ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York, 2009.

Yee TW (2018). VGAM: Vector Generalized Linear and Additive Models. R package version 1.0-5. https://CRAN.R-project.org/package=VGAM.

See Also

calc_theory, ggplot, geom_histogram

Examples

# Using normal mixture variable from contmixvar1 example
Nmix <- contmixvar1(n = 1000, "Polynomial", means = 0, vars = 1,
  mix_pis = c(0.4, 0.6), mix_mus = c(-2, 2), mix_sigmas = c(1, 1),
  mix_skews = c(0, 0), mix_skurts = c(0, 0), mix_fifths = c(0, 0),
  mix_sixths = c(0, 0))
plot_simtheory(Nmix$Y_mix[, 1], title = "Mixture of Normal Distributions",
  fx = function(x) 0.4 * dnorm(x, -2, 1) + 0.6 * dnorm(x, 2, 1),
  lower = -5, upper = 5)
## Not run: 
# Mixture of Beta(6, 3), Beta(4, 1.5), and Beta(10, 20)
Stcum1 <- calc_theory("Beta", c(6, 3))
Stcum2 <- calc_theory("Beta", c(4, 1.5))
Stcum3 <- calc_theory("Beta", c(10, 20))
mix_pis <- c(0.5, 0.2, 0.3)
mix_mus <- c(Stcum1[1], Stcum2[1], Stcum3[1])
mix_sigmas <- c(Stcum1[2], Stcum2[2], Stcum3[2])
mix_skews <- c(Stcum1[3], Stcum2[3], Stcum3[3])
mix_skurts <- c(Stcum1[4], Stcum2[4], Stcum3[4])
mix_fifths <- c(Stcum1[5], Stcum2[5], Stcum3[5])
mix_sixths <- c(Stcum1[6], Stcum2[6], Stcum3[6])
mix_Six <- list(seq(0.01, 10, 0.01), c(0.01, 0.02, 0.03),
  seq(0.01, 10, 0.01))
Bstcum <- calc_mixmoments(mix_pis, mix_mus, mix_sigmas, mix_skews,
  mix_skurts, mix_fifths, mix_sixths)
Bmix <- contmixvar1(n = 10000, "Polynomial", Bstcum[1], Bstcum[2]^2,
  mix_pis, mix_mus, mix_sigmas, mix_skews, mix_skurts, mix_fifths,
  mix_sixths, mix_Six)
plot_simtheory(Bmix$Y_mix[, 1], title = "Mixture of Beta Distributions",
  fx = function(x) mix_pis[1] * dbeta(x, 6, 3) + mix_pis[2] *
    dbeta(x, 4, 1.5) + mix_pis[3] * dbeta(x, 10, 20), lower = 0, upper = 1)

## End(Not run)

Approximate Correlation between Two Continuous Mixture Variables M1 and M2

Description

This function approximates the expected correlation between two continuous mixture variables M1 and M2 based on their mixing proportions, component means, component standard deviations, and correlations between components across variables. The equations can be found in the Expected Cumulants and Correlations for Continuous Mixture Variables vignette. This function can be used to see what combination of component correlations gives a desired correlation between M1 and M2.

Usage

rho_M1M2(mix_pis = list(), mix_mus = list(), mix_sigmas = list(),
  p_M1M2 = NULL)

Arguments

Value

the expected correlation between M1 and M2

References

Davenport JW, Bezder JC, & Hathaway RJ (1988). Parameter Estimation for Finite Mixture Distributions. Computers & Mathematics with Applications, 15(10):819-28.

Pearson RK (2011). Exploring Data in Engineering, the Sciences, and Medicine. In. New York: Oxford University Press.

See Also

rho_M1Y

Examples

# M1 is mixture of N(-2, 1) and N(2, 1);
# M2 is mixture of Logistic(0, 1), Chisq(4), and Beta(4, 1.5)
# pairwise correlation between components across M1 and M2 set to 0.35
L <- calc_theory("Logistic", c(0, 1))
C <- calc_theory("Chisq", 4)
B <- calc_theory("Beta", c(4, 1.5))
rho_M1M2(mix_pis = list(c(0.4, 0.6), c(0.3, 0.2, 0.5)),
  mix_mus = list(c(-2, 2), c(L[1], C[1], B[1])),
  mix_sigmas = list(c(1, 1), c(L[2], C[2], B[2])),
  p_M1M2 = matrix(0.35, 2, 3))

Approximate Correlation between Continuous Mixture Variable M1 and Random Variable Y

Description

This function approximates the expected correlation between a continuous mixture variables M1 and another random variable Y based on the mixing proportions, component means, and component standard deviations of M1 and correlations between components of M1 and Y. The equations can be found in the Expected Cumulants and Correlations for Continuous Mixture Variables vignette. This function can be used to see what combination of correlations between components of M1 and Y gives a desired correlation between M1 and Y.

Usage

rho_M1Y(mix_pis = NULL, mix_mus = NULL, mix_sigmas = NULL, p_M1Y = NULL)

Arguments

Value

the expected correlation between M1 and Y

References

Please see references for rho_M1M2.

See Also

rho_M1Y

Examples

# M1 is mixture of N(-2, 1) and N(2, 1); pairwise correlation set to 0.35
rho_M1Y(mix_pis = c(0.4, 0.6), mix_mus = c(-2, 2), mix_sigmas = c(1, 1),
  p_M1Y = c(0.35, 0.35))

Summary of Simulated Variables

Description

This function summarizes the results of contmixvar1, corrvar, or corrvar2. The inputs are either the simulated variables or inputs for those functions. See their documentation for more information. If summarizing result from contmixvar1, mixture parameters may be entered as vectors instead of lists.

Usage

summary_var(Y_cat = NULL, Y_cont = NULL, Y_comp = NULL, Y_mix = NULL,
  Y_pois = NULL, Y_nb = NULL, means = NULL, vars = NULL, skews = NULL,
  skurts = NULL, fifths = NULL, sixths = NULL, mix_pis = list(),
  mix_mus = list(), mix_sigmas = list(), mix_skews = list(),
  mix_skurts = list(), mix_fifths = list(), mix_sixths = list(),
  marginal = list(), lam = NULL, p_zip = 0, size = NULL, prob = NULL,
  mu = NULL, p_zinb = 0, rho = NULL)

Arguments

Value

A list whose components vary based on the type of simulated variables.

If ordinal variables are produced:

ord_sum a list, where the i-th element contains a data.frame with target and simulated cumulative probabilities for ordinal variable Y_i

If continuous variables are produced:

cont_sum a data.frame summarizing Y_cont and Y_comp,

target_sum a data.frame with the target distributions for Y_cont and Y_comp,

mix_sum a data.frame summarizing Y_mix,

target_mix a data.frame with the target distributions for Y_mix,

If Poisson variables are produced:

pois_sum a data.frame summarizing Y_pois

If Negative Binomial variables are produced:

nb_sum a data.frame summarizing Y_nb

Additionally, the following elements:

rho_calc the final correlation matrix for Y_cat, Y_cont, Y_comp, Y_pois, and Y_nb

rho_mix the final correlation matrix for Y_cat, Y_cont, Y_mix, Y_pois, and Y_nb

maxerr the maximum final correlation error of rho_calc from the target rho.

References

See references for SimCorrMix.

See Also

contmixvar1, corrvar, corrvar2

Examples

# Using normal mixture variable from contmixvar1 example
Nmix <- contmixvar1(n = 1000, "Polynomial", means = 0, vars = 1,
  mix_pis = c(0.4, 0.6), mix_mus = c(-2, 2), mix_sigmas = c(1, 1),
  mix_skews = c(0, 0), mix_skurts = c(0, 0), mix_fifths = c(0, 0),
  mix_sixths = c(0, 0))
Nsum <- summary_var(Y_comp = Nmix$Y_comp, Y_mix = Nmix$Y_mix,
  means = 0, vars = 1, mix_pis = c(0.4, 0.6), mix_mus = c(-2, 2),
  mix_sigmas = c(1, 1), mix_skews = c(0, 0), mix_skurts = c(0, 0),
  mix_fifths = c(0, 0), mix_sixths = c(0, 0))

## Not run: 

# 2 continuous mixture, 1 binary, 1 zero-inflated Poisson, and
# 1 zero-inflated NB variable
n <- 10000
seed <- 1234

# Mixture variables: Normal mixture with 2 components;
# mixture of Logistic(0, 1), Chisq(4), Beta(4, 1.5)
# Find cumulants of components of 2nd mixture variable
L <- calc_theory("Logistic", c(0, 1))
C <- calc_theory("Chisq", 4)
B <- calc_theory("Beta", c(4, 1.5))

skews <- skurts <- fifths <- sixths <- NULL
Six <- list()
mix_pis <- list(c(0.4, 0.6), c(0.3, 0.2, 0.5))
mix_mus <- list(c(-2, 2), c(L[1], C[1], B[1]))
mix_sigmas <- list(c(1, 1), c(L[2], C[2], B[2]))
mix_skews <- list(rep(0, 2), c(L[3], C[3], B[3]))
mix_skurts <- list(rep(0, 2), c(L[4], C[4], B[4]))
mix_fifths <- list(rep(0, 2), c(L[5], C[5], B[5]))
mix_sixths <- list(rep(0, 2), c(L[6], C[6], B[6]))
mix_Six <- list(list(NULL, NULL), list(1.75, NULL, 0.03))
Nstcum <- calc_mixmoments(mix_pis[[1]], mix_mus[[1]], mix_sigmas[[1]],
  mix_skews[[1]], mix_skurts[[1]], mix_fifths[[1]], mix_sixths[[1]])
Mstcum <- calc_mixmoments(mix_pis[[2]], mix_mus[[2]], mix_sigmas[[2]],
  mix_skews[[2]], mix_skurts[[2]], mix_fifths[[2]], mix_sixths[[2]])
means <- c(Nstcum[1], Mstcum[1])
vars <- c(Nstcum[2]^2, Mstcum[2]^2)

marginal <- list(0.3)
support <- list(c(0, 1))
lam <- 0.5
p_zip <- 0.1
size <- 2
prob <- 0.75
p_zinb <- 0.2

k_cat <- k_pois <- k_nb <- 1
k_cont <- 0
k_mix <- 2
Rey <- matrix(0.39, 8, 8)
diag(Rey) <- 1
rownames(Rey) <- colnames(Rey) <- c("O1", "M1_1", "M1_2", "M2_1", "M2_2",
  "M2_3", "P1", "NB1")

# set correlation between components of the same mixture variable to 0
Rey["M1_1", "M1_2"] <- Rey["M1_2", "M1_1"] <- 0
Rey["M2_1", "M2_2"] <- Rey["M2_2", "M2_1"] <- Rey["M2_1", "M2_3"] <- 0
Rey["M2_3", "M2_1"] <- Rey["M2_2", "M2_3"] <- Rey["M2_3", "M2_2"] <- 0

# check parameter inputs
validpar(k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal, support,
  lam, p_zip, size, prob, mu = NULL, p_zinb, rho = Rey)

# check to make sure Rey is within the feasible correlation boundaries
validcorr(n, k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal,
  lam, p_zip, size, prob, mu = NULL, p_zinb, Rey, seed)

# simulate without the error loop
Sim1 <- corrvar(n, k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal, support,
  lam, p_zip, size, prob, mu = NULL, p_zinb, Rey, seed, epsilon = 0.01)

Summ1 <- summary_var(Sim1$Y_cat, Y_cont = NULL, Sim1$Y_comp, Sim1$Y_mix,
  Sim1$Y_pois, Sim1$Y_nb, means, vars, skews, skurts, fifths, sixths,
  mix_pis, mix_mus, mix_sigmas, mix_skews, mix_skurts, mix_fifths,
  mix_sixths, marginal, lam, p_zip, size, prob, mu = NULL, p_zinb, Rey)

Sim1_error <- abs(Rey - Summ1$rho_calc)
summary(as.numeric(Sim1_error))


## End(Not run)

Determine Correlation Bounds for Ordinal, Continuous, Poisson, and/or Negative Binomial Variables: Correlation Method 1

Description

This function calculates the lower and upper correlation bounds for the given distributions and checks if a given target correlation matrix rho is within the bounds. It should be used before simulation with corrvar. However, even if all pairwise correlations fall within the bounds, it is still possible that the desired correlation matrix is not feasible. This is particularly true when ordinal variables (r \ge 2 categories) are generated or negative correlations are desired. Therefore, this function should be used as a general check to eliminate pairwise correlations that are obviously not reproducible. It will help prevent errors when executing the simulation. The ordering of the variables in rho must be 1st ordinal, 2nd continuous non-mixture, 3rd components of continuous mixture, 4th regular Poisson, 5th zero-inflated Poisson, 6th regular NB, and 7th zero-inflated NB. Note that it is possible for k_cat, k_cont, k_mix, k_pois, and/or k_nb to be 0. The target correlations are specified with respect to the components of the continuous mixture variables. There are no parameter input checks in order to decrease simulation time. All inputs should be checked prior to simulation with validpar.

Please see the Comparison of Correlation Methods 1 and 2 vignette for the differences between the two correlation methods, and the Variable Types vignette for a detailed explanation of how the correlation boundaries are calculated.

Usage

validcorr(n = 10000, k_cat = 0, k_cont = 0, k_mix = 0, k_pois = 0,
  k_nb = 0, method = c("Fleishman", "Polynomial"), means = NULL,
  vars = NULL, skews = NULL, skurts = NULL, fifths = NULL,
  sixths = NULL, Six = list(), mix_pis = list(), mix_mus = list(),
  mix_sigmas = list(), mix_skews = list(), mix_skurts = list(),
  mix_fifths = list(), mix_sixths = list(), mix_Six = list(),
  marginal = list(), lam = NULL, p_zip = 0, size = NULL, prob = NULL,
  mu = NULL, p_zinb = 0, rho = NULL, seed = 1234, use.nearPD = TRUE,
  quiet = FALSE)

Arguments

Value

A list with components:

rho the target correlation matrix, which will differ from the supplied matrix (if provided) if it was converted to the nearest positive-definite matrix

L_rho the lower correlation bound

U_rho the upper correlation bound

If continuous variables are desired, additional components are:

constants the calculated constants

sixth_correction a vector of the sixth cumulant correction values

valid.pdf a vector with i-th component equal to "TRUE" if variable Y_i has a valid power method PDF, else "FALSE"

If a target correlation matrix rho is provided, each pairwise correlation is checked to see if it is within the lower and upper bounds. If the correlation is outside the bounds, the indices of the variable pair are given.

valid.rho TRUE if all entries of rho are within the bounds, else FALSE

Reasons for Function Errors

1) The most likely cause for function errors is that no solutions to fleish or poly converged when using find_constants. If this happens, the function will stop. It may help to first use find_constants for each continuous variable to determine if a sixth cumulant correction value is needed. If the standardized cumulants are obtained from calc_theory, the user may need to use rounded values as inputs (i.e. skews = round(skews, 8)). For example, in order to ensure that skew is exactly 0 for symmetric distributions.

2) The kurtosis may be outside the region of possible values. There is an associated lower boundary for kurtosis associated with a given skew (for Fleishman's method) or skew and fifth and sixth cumulants (for Headrick's method). Use calc_lower_skurt to determine the boundary for a given set of cumulants.

References

Please see references for SimCorrMix.

See Also

find_constants, corrvar, validpar

Examples

validcorr(n = 1000, k_cat = 1, k_cont = 1, method = "Polynomial",
  means = 0, vars = 1, skews = 0, skurts = 0, fifths = 0, sixths = 0,
  marginal = list(c(1/3, 2/3)), rho = matrix(c(1, 0.4, 0.4, 1), 2, 2),
  quiet = TRUE)
## Not run: 

# 2 continuous mixture, 1 binary, 1 zero-inflated Poisson, and
# 1 zero-inflated NB variable
n <- 10000
seed <- 1234

# Mixture variables: Normal mixture with 2 components;
# mixture of Logistic(0, 1), Chisq(4), Beta(4, 1.5)
# Find cumulants of components of 2nd mixture variable
L <- calc_theory("Logistic", c(0, 1))
C <- calc_theory("Chisq", 4)
B <- calc_theory("Beta", c(4, 1.5))

skews <- skurts <- fifths <- sixths <- NULL
Six <- list()
mix_pis <- list(c(0.4, 0.6), c(0.3, 0.2, 0.5))
mix_mus <- list(c(-2, 2), c(L[1], C[1], B[1]))
mix_sigmas <- list(c(1, 1), c(L[2], C[2], B[2]))
mix_skews <- list(rep(0, 2), c(L[3], C[3], B[3]))
mix_skurts <- list(rep(0, 2), c(L[4], C[4], B[4]))
mix_fifths <- list(rep(0, 2), c(L[5], C[5], B[5]))
mix_sixths <- list(rep(0, 2), c(L[6], C[6], B[6]))
mix_Six <- list(list(NULL, NULL), list(1.75, NULL, 0.03))
Nstcum <- calc_mixmoments(mix_pis[[1]], mix_mus[[1]], mix_sigmas[[1]],
  mix_skews[[1]], mix_skurts[[1]], mix_fifths[[1]], mix_sixths[[1]])
Mstcum <- calc_mixmoments(mix_pis[[2]], mix_mus[[2]], mix_sigmas[[2]],
  mix_skews[[2]], mix_skurts[[2]], mix_fifths[[2]], mix_sixths[[2]])
means <- c(Nstcum[1], Mstcum[1])
vars <- c(Nstcum[2]^2, Mstcum[2]^2)

marginal <- list(0.3)
support <- list(c(0, 1))
lam <- 0.5
p_zip <- 0.1
size <- 2
prob <- 0.75
p_zinb <- 0.2

k_cat <- k_pois <- k_nb <- 1
k_cont <- 0
k_mix <- 2
Rey <- matrix(0.39, 8, 8)
diag(Rey) <- 1
rownames(Rey) <- colnames(Rey) <- c("O1", "M1_1", "M1_2", "M2_1", "M2_2",
  "M2_3", "P1", "NB1")

# set correlation between components of the same mixture variable to 0
Rey["M1_1", "M1_2"] <- Rey["M1_2", "M1_1"] <- 0
Rey["M2_1", "M2_2"] <- Rey["M2_2", "M2_1"] <- Rey["M2_1", "M2_3"] <- 0
Rey["M2_3", "M2_1"] <- Rey["M2_2", "M2_3"] <- Rey["M2_3", "M2_2"] <- 0

# check parameter inputs
validpar(k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal, support,
  lam, p_zip, size, prob, mu = NULL, rho = Rey)

# check to make sure Rey is within the feasible correlation boundaries
validcorr(n, k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal,
  lam, p_zip, size, prob, mu = NULL, p_zinb, Rey, seed)

## End(Not run)

Determine Correlation Bounds for Ordinal, Continuous, Poisson, and/or Negative Binomial Variables: Correlation Method 2

Description

This function calculates the lower and upper correlation bounds for the given distributions and checks if a given target correlation matrix rho is within the bounds. It should be used before simulation with corrvar2. However, even if all pairwise correlations fall within the bounds, it is still possible that the desired correlation matrix is not feasible. This is particularly true when ordinal variables (r \ge 2 categories) are generated or negative correlations are desired. Therefore, this function should be used as a general check to eliminate pairwise correlations that are obviously not reproducible. It will help prevent errors when executing the simulation. The ordering of the variables in rho must be 1st ordinal, 2nd continuous non-mixture, 3rd components of continuous mixture, 4th regular Poisson, 5th zero-inflated Poisson, 6th regular NB, and 7th zero-inflated NB. Note that it is possible for k_cat, k_cont, k_mix, k_pois, and/or k_nb to be 0. The target correlations are specified with respect to the components of the continuous mixture variables. There are no parameter input checks in order to decrease simulation time. All inputs should be checked prior to simulation with validpar.

Please see the Comparison of Correlation Methods 1 and 2 vignette for the differences between the two correlation methods, and the Variable Types vignette for a detailed explanation of how the correlation boundaries are calculated.

Usage

validcorr2(n = 10000, k_cat = 0, k_cont = 0, k_mix = 0, k_pois = 0,
  k_nb = 0, method = c("Fleishman", "Polynomial"), means = NULL,
  vars = NULL, skews = NULL, skurts = NULL, fifths = NULL,
  sixths = NULL, Six = list(), mix_pis = list(), mix_mus = list(),
  mix_sigmas = list(), mix_skews = list(), mix_skurts = list(),
  mix_fifths = list(), mix_sixths = list(), mix_Six = list(),
  marginal = list(), lam = NULL, p_zip = 0, size = NULL, prob = NULL,
  mu = NULL, p_zinb = 0, pois_eps = 0.0001, nb_eps = 0.0001,
  rho = NULL, seed = 1234, use.nearPD = TRUE, quiet = FALSE)

Arguments

Value

A list with components:

rho the target correlation matrix, which will differ from the supplied matrix (if provided) if it was converted to the nearest positive-definite matrix

L_rho the lower correlation bound

U_rho the upper correlation bound

If continuous variables are desired, additional components are:

constants the calculated constants

sixth_correction a vector of the sixth cumulant correction values

valid.pdf a vector with i-th component equal to "TRUE" if variable Y_i has a valid power method PDF, else "FALSE"

If a target correlation matrix rho is provided, each pairwise correlation is checked to see if it is within the lower and upper bounds. If the correlation is outside the bounds, the indices of the variable pair are given.

valid.rho TRUE if all entries of rho are within the bounds, else FALSE

Reasons for Function Errors

1) The most likely cause for function errors is that no solutions to fleish or poly converged when using find_constants. If this happens, the function will stop. It may help to first use find_constants for each continuous variable to determine if a sixth cumulant correction value is needed. If the standardized cumulants are obtained from calc_theory, the user may need to use rounded values as inputs (i.e. skews = round(skews, 8)). For example, in order to ensure that skew is exactly 0 for symmetric distributions.

2) The kurtosis may be outside the region of possible values. There is an associated lower boundary for kurtosis associated with a given skew (for Fleishman's method) or skew and fifth and sixth cumulants (for Headrick's method). Use calc_lower_skurt to determine the boundary for a given set of cumulants.

References

Please see references for SimCorrMix.

See Also

find_constants, corrvar2, validpar

Examples

validcorr2(n = 1000, k_cat = 1, k_cont = 1, method = "Polynomial",
  means = 0, vars = 1, skews = 0, skurts = 0, fifths = 0, sixths = 0,
  marginal = list(c(1/3, 2/3)), rho = matrix(c(1, 0.4, 0.4, 1), 2, 2),
  quiet = TRUE)
## Not run: 

# 2 continuous mixture, 1 binary, 1 zero-inflated Poisson, and
# 1 zero-inflated NB variable
n <- 10000
seed <- 1234

# Mixture variables: Normal mixture with 2 components;
# mixture of Logistic(0, 1), Chisq(4), Beta(4, 1.5)
# Find cumulants of components of 2nd mixture variable
L <- calc_theory("Logistic", c(0, 1))
C <- calc_theory("Chisq", 4)
B <- calc_theory("Beta", c(4, 1.5))

skews <- skurts <- fifths <- sixths <- NULL
Six <- list()
mix_pis <- list(c(0.4, 0.6), c(0.3, 0.2, 0.5))
mix_mus <- list(c(-2, 2), c(L[1], C[1], B[1]))
mix_sigmas <- list(c(1, 1), c(L[2], C[2], B[2]))
mix_skews <- list(rep(0, 2), c(L[3], C[3], B[3]))
mix_skurts <- list(rep(0, 2), c(L[4], C[4], B[4]))
mix_fifths <- list(rep(0, 2), c(L[5], C[5], B[5]))
mix_sixths <- list(rep(0, 2), c(L[6], C[6], B[6]))
mix_Six <- list(list(NULL, NULL), list(1.75, NULL, 0.03))
Nstcum <- calc_mixmoments(mix_pis[[1]], mix_mus[[1]], mix_sigmas[[1]],
  mix_skews[[1]], mix_skurts[[1]], mix_fifths[[1]], mix_sixths[[1]])
Mstcum <- calc_mixmoments(mix_pis[[2]], mix_mus[[2]], mix_sigmas[[2]],
  mix_skews[[2]], mix_skurts[[2]], mix_fifths[[2]], mix_sixths[[2]])
means <- c(Nstcum[1], Mstcum[1])
vars <- c(Nstcum[2]^2, Mstcum[2]^2)

marginal <- list(0.3)
support <- list(c(0, 1))
lam <- 0.5
p_zip <- 0.1
pois_eps <- 0.0001
size <- 2
prob <- 0.75
p_zinb <- 0.2
nb_eps <- 0.0001

k_cat <- k_pois <- k_nb <- 1
k_cont <- 0
k_mix <- 2
Rey <- matrix(0.39, 8, 8)
diag(Rey) <- 1
rownames(Rey) <- colnames(Rey) <- c("O1", "M1_1", "M1_2", "M2_1", "M2_2",
  "M2_3", "P1", "NB1")

# set correlation between components of the same mixture variable to 0
Rey["M1_1", "M1_2"] <- Rey["M1_2", "M1_1"] <- 0
Rey["M2_1", "M2_2"] <- Rey["M2_2", "M2_1"] <- Rey["M2_1", "M2_3"] <- 0
Rey["M2_3", "M2_1"] <- Rey["M2_2", "M2_3"] <- Rey["M2_3", "M2_2"] <- 0

# check parameter inputs
validpar(k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal, support,
  lam, p_zip, size, prob, mu = NULL, p_zinb, pois_eps, nb_eps, Rey)

# check to make sure Rey is within the feasible correlation boundaries
validcorr2(n, k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal,
  lam, p_zip, size, prob, mu = NULL, p_zinb, pois_eps, nb_eps, Rey, seed)

## End(Not run)

Parameter Check for Simulation or Correlation Validation Functions

Description

This function checks the parameter inputs to the simulation functions contmixvar1, corrvar, and corrvar2 and to the correlation validation functions validcorr and validcorr2. It should be used prior to execution of these functions to ensure all inputs are of the correct format. Those functions do not contain parameter checks in order to decrease simulation time. This would be important if the user is running several simulation repetitions so that the inputs only have to be checked once. Note that the inputs do not include all of the inputs to the simulation functions. See the appropriate function documentation for more details about parameter inputs.

Usage

validpar(k_cat = 0, k_cont = 0, k_mix = 0, k_pois = 0, k_nb = 0,
  method = c("Fleishman", "Polynomial"), means = NULL, vars = NULL,
  skews = NULL, skurts = NULL, fifths = NULL, sixths = NULL,
  Six = list(), mix_pis = list(), mix_mus = list(), mix_sigmas = list(),
  mix_skews = list(), mix_skurts = list(), mix_fifths = list(),
  mix_sixths = list(), mix_Six = list(), marginal = list(),
  support = list(), lam = NULL, p_zip = 0, size = NULL, prob = NULL,
  mu = NULL, p_zinb = 0, pois_eps = 0.0001, nb_eps = 0.0001,
  rho = NULL, Sigma = NULL, cstart = list(), quiet = FALSE)

Arguments

Value

TRUE if all inputs are correct, else it will stop with a correction message

See Also

contmixvar1, corrvar, corrvar2, validcorr, validcorr2

Examples

validpar(k_cat = 1, k_cont = 1, method = "Polynomial", means = 0,
  vars = 1, skews = 0, skurts = 0, fifths = 0, sixths = 0,
  marginal = list(c(1/3, 2/3)), rho = matrix(c(1, 0.4, 0.4, 1), 2, 2),
  quiet = TRUE)
## Not run: 
# 2 continuous mixture, 1 binary, 1 zero-inflated Poisson, and
# 1 zero-inflated NB variable

# Mixture variables: Normal mixture with 2 components;
# mixture of Logistic(0, 1), Chisq(4), Beta(4, 1.5)
# Find cumulants of components of 2nd mixture variable
L <- calc_theory("Logistic", c(0, 1))
C <- calc_theory("Chisq", 4)
B <- calc_theory("Beta", c(4, 1.5))

skews <- skurts <- fifths <- sixths <- NULL
Six <- list()
mix_pis <- list(c(0.4, 0.6), c(0.3, 0.2, 0.5))
mix_mus <- list(c(-2, 2), c(L[1], C[1], B[1]))
mix_sigmas <- list(c(1, 1), c(L[2], C[2], B[2]))
mix_skews <- list(rep(0, 2), c(L[3], C[3], B[3]))
mix_skurts <- list(rep(0, 2), c(L[4], C[4], B[4]))
mix_fifths <- list(rep(0, 2), c(L[5], C[5], B[5]))
mix_sixths <- list(rep(0, 2), c(L[6], C[6], B[6]))
mix_Six <- list(list(NULL, NULL), list(1.75, NULL, 0.03))
Nstcum <- calc_mixmoments(mix_pis[[1]], mix_mus[[1]], mix_sigmas[[1]],
  mix_skews[[1]], mix_skurts[[1]], mix_fifths[[1]], mix_sixths[[1]])
Mstcum <- calc_mixmoments(mix_pis[[2]], mix_mus[[2]], mix_sigmas[[2]],
  mix_skews[[2]], mix_skurts[[2]], mix_fifths[[2]], mix_sixths[[2]])
means <- c(Nstcum[1], Mstcum[1])
vars <- c(Nstcum[2]^2, Mstcum[2]^2)

marginal <- list(0.3)
support <- list(c(0, 1))
lam <- 0.5
p_zip <- 0.1
size <- 2
prob <- 0.75
p_zinb <- 0.2

k_cat <- k_pois <- k_nb <- 1
k_cont <- 0
k_mix <- 2
Rey <- matrix(0.39, 8, 8)
diag(Rey) <- 1
rownames(Rey) <- colnames(Rey) <- c("O1", "M1_1", "M1_2", "M2_1", "M2_2",
  "M2_3", "P1", "NB1")

# set correlation between components of the same mixture variable to 0
Rey["M1_1", "M1_2"] <- Rey["M1_2", "M1_1"] <- 0
Rey["M2_1", "M2_2"] <- Rey["M2_2", "M2_1"] <- Rey["M2_1", "M2_3"] <- 0
Rey["M2_3", "M2_1"] <- Rey["M2_2", "M2_3"] <- Rey["M2_3", "M2_2"] <- 0

# use before contmixvar1 with 1st mixture variable:
# change mix_pis to not sum to 1

check1 <- validpar(k_mix = 1, method = "Polynomial", means = Nstcum[1],
  vars = Nstcum[2]^2, mix_pis = C(0.4, 0.5), mix_mus = mix_mus[[1]],
  mix_sigmas = mix_sigmas[[1]], mix_skews = mix_skews[[1]],
  mix_skurts = mix_skurts[[1]], mix_fifths = mix_fifths[[1]],
  mix_sixths = mix_sixths[[1]])

# use before validcorr: should return TRUE

check2 <- validpar(k_cat, k_cont, k_mix, k_pois, k_nb, "Polynomial", means,
  vars, skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas,
  mix_skews, mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal, support,
  lam, p_zip, size, prob, mu = NULL, p_zinb, rho = Rey)


## End(Not run)