This is a stepbystep guideline for correlated data simulation. More information about the different variable types can be found in the Variable Types vignette. More information about the differences between correlation methods 1 and 2 can be found in the Comparison of Correlation Methods 1 and 2 vignette. Some functions have been modified from the SimMultiCorrData package (Fialkowski 2018).
Obtain the distributional parameters for the desired variables.
SimMultiCorrData::calc_theory
. If the goal is to mimic an empirical data set, these values can be found using SimMultiCorrData::calc_moments
(using the method of moments) or SimMultiCorrData::calc_fisherk
(using Fisher’s kstatistics). 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. Due to the nature of the integration involved in calc_theory
, the results are approximations. Greater accuracy can be achieved by increasing the number of subdivisions (sub
) used in the integration process.For mixture variables, the parameters are specified at the component level by the inputs mix_skews
, mix_skurts
, mix_fifths
, mix_sixths
, and mix_Six
. The mixing probabilities, means, and standard deviations of the component variables are given by mix_pis
, mix_mus
and mix_sigmas
.
The means and variances of nonmixture and mixture variables are specified by means
and vars
. These are at the variable level, i.e., they refer to the continuous nonmixture and mixture variables themselves. The function calc_mixmoments
calculates the expected mean, standard deviation, and standardized cumulants for mixture variables based on the component distributions.
For some sets of cumulants, it is either not possible to find power method constants or the calculated constants do not generate valid power method PDF’s. In these situations, adding a value to the sixth cumulant may provide solutions (see find_constants
). If simulation results indicate that a continuous variable does not generate a valid PDF, the user can try find_constants
with various sixth cumulant correction vectors to determine if a valid PDF can be found. These sixth cumulant corrections are specified in the simulation functions using Six
or mix_Six
.
Choice of Fleishman (1978)’s or Headrick (2002)’s Method: Using the fifthorder PMT (method
= “Polynomial”) 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 the thirdorder method (method
= “Fleishman”). For example, Fleishman’s method can not be used to generate a nonnormal distribution with a ratio of \(\gamma_{1}^2/\gamma_{2} > 9/14\) (Headrick and Kowalchuk 2007). This eliminates the family of distributions, which has a constant ratio of \(\gamma_{1}^2/\gamma_{2} = 2/3\). The fifthorder method also generates more distributions with valid PDF’s, However, if the fifth and sixth cumulants do not exist, the thirdorder PMT should be used.
Ordinal variables (\(r \ge 2\) categories): these are the cumulative marginal probabilities and support values (if desired). The probabilities should be combined into a list of length equal to the number of ordinal variables. The \(i^{th}\) element is a vector of the cumulative probabilities defining the marginal distribution of the \(i^{th}\) variable. If the variable can take \(r\) values, the vector will contain \(r  1\) probabilities (the \(r^{th}\) is assumed to be \(1\)). For binary variables, the usersupplied probability should be the probability of the \(1^{st}\) (lower) support value. This would ordinarily be considered the probability of failure (\(q\)), while the probability of the \(2^{nd}\) (upper) support value would be considered the probability of success (\(p = 1  q\)). The support values should be combined into a separate list. The \(i^{th}\) element is a vector containing the \(r\) ordered support values. If not provided, the default is for the \(i^{th}\) element to be the vector \(1, ..., r\).
Poisson variables: the lambda (mean > 0) values should be given as a vector (see stats::dpois
). For zeroinflated Poisson variables, the probability of a structural zero is specified in p_zip
(see VGAM::dzipois
). The default is p_zip = 0
for all variables. For correlation method 2, the total cumulative probability truncation values are specified in pois_eps
, with the default of \(0.0001\) for all variables. The order for parameters should be regular then zeroinflated for all inputs. The distribution functions are taken from the VGAM package (Yee 2018).
Negative Binomial variables: the sizes (target number of successes) and either the success probabilities or the means should be given as vectors (see stats::dnbinom
). The variable represents the number of failures which occur in a sequence of Bernoulli trials before the target number of successes is achieved. For zeroinflated NB variables, the probability of a structural zero is specified in p_zinb
(see VGAM::dzinegbin
). The default is p_zinb = 0
for all variables. For correlation method 2, the total cumulative probability truncation values are specified in nb_eps
, with the default of \(0.0001\) for all variables. The order for parameters should be regular then zeroinflated for all inputs. The distribution functions are taken from the VGAM package (Yee 2018).
Check that all parameter inputs have the correct format using validpar
. There are no checks within the correlation validation or simulation functions in order to decrease simulation time.
If continuous variables are desired, verify that the standardized kurtoses are greater than the lower skurtosis bounds. These bounds can be calculated using SimMultiCorrData::calc_lower_skurt
, given the skewness (for method
= “Fleishman”) and standardized fifth and sixth cumulants (for method
= “Polynomial”) for each variable. Different seeds should be examined to see if a lower boundary can be found. If a lower bound produces power method constants that yield an invalid PDF, the user may wish to provide a Skurt
vector of kurtosis corrections. In this case, calc_lower_skurt
will attempt to find the smallest value that produces a kurtosis which yields a valid power method PDF. In addition, if method
= “Polynomial”, a sixth cumulant correction vector (Six
) may be used to facilitate convergence of the rootsolving algorithm. Since this step can take considerable computation time, the user may instead wish to perform this check after simulation if any of the variables have invalid power method PDF’s.
Check if the target correlation matrix rho
falls within the feasible correlation bounds, given the parameters for the desired distributions. The ordering of the variables in rho
must be 1st ordinal, 2nd continuous nonmixture, 3rd components of continuous mixture variables, 4th regular Poisson, 5th zeroinflated Poisson, 6th regular NB, and 7th zeroinflated NB. These bounds can be calculated using either validcorr
(correlation method 1) or validcorr2
(correlation method 2). Note that falling within these bounds does not guarantee that the target correlation can be achieved. However, the check can alert the user to pairwise correlations that obviously fall outside the bounds.
Generate the variables using either correlation method 1 and corrvar
or correlation method 2 and corrvar2
. The user may want to try both to see which gives a better approximation to the variables and correlation matrix. The accuracy and simulation time will vary by situation. In addition, the error loop can minimize the correlation errors in most situations. See the Error Loop Algorithm vignette for details about the error loop.
Summarize the results numerically. The functions corrvar
and corrvar2
do not provide variable or correlation summaries in order to decrease simulation time. These can be obtained using summary_var
, which gives summaries by variable type, the final correlation matrix, and the maximum error between the final and target correlation matrices. Additional summary functions include: SimMultiCorrData::sim_cdf_prob
(to calculate a cumulative probability up to a given continuous y value), SimMultiCorrData::power_norm_corr
(to calculate the correlation between a continuous variable and the generating standard normal variable), and SimMultiCorrData::stats_pdf
(to calculate the \(100 \alpha \%\) symmetric trimmedmean, median, mode, and maximum height of a valid power method PDF).
Summarize the results graphically. Comparing the simulated data to the target distribution demonstrates simulation accuracy. The graphing functions provided in this package and the SimMultiCorrData package can be used to display simulated data values, PDF’s, or CDF’s. The target distributions (either by theoretical distribution name or given an empirical data set) can be added to the data value or PDF plots. Cumulative probabilities can be added to the CDF plots (for continuous variables).
The following examples demonstrate the use of the corrvar and corrvar2 functions to simulate the following correlated variables (n = 10,000):
Ordinal variable: \(O1\) is binary with \(p = 0.3\).
Continuous nonmixture variables: \(C1\) and \(C2\) have a Logistic(0, 1) distribution.
Poisson variable: \(P1\) with \(\lambda = 0.5\) is a zeroinflated Poisson variable with the probability of a structural zero set at \(0.1\).
Negative Binomial variable: \(NB1\) with \(size = 2\) and \(\mu = 2/3\) is a zeroinflated NB variable with the probability of a structural zero set at \(0.2\).
Headrick (2002)’s fifthorder transformation (method
= “Polynomial”) is used for the continuous variables.
The target pairwise correlation is set at \(0.35\) between \(O1\), \(C1\), \(C2\), \(M1_1\), \(M1_2\), \(M2_1\), \(M2_2\), \(M2_3\), \(P1\), and \(NB1\). The correlation between the components of the same mixture variable (i.e., \(M1_1\) and \(M1_2\)) is set at \(0\). Therefore, the correlation is controlled at the component level for the mixture variables. However, the expected correlations for the mixture variables can be approximated (see the Expected Cumulants and Correlations for Continuous Mixture Variables vignette).
library("SimCorrMix")
library("printr")
options(scipen = 999)
seed < 276
n < 10000
# Continuous variables
L < calc_theory("Logistic", c(0, 1))
C < calc_theory("Chisq", 4)
B < calc_theory("Beta", c(4, 1.5))
# Nonmixture variables
skews < rep(L[3], 2)
skurts < rep(L[4], 2)
fifths < rep(L[5], 2)
sixths < rep(L[6], 2)
Six < list(1.75, 1.75)
# Mixture variables
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(L[1], L[1], Nstcum[1], Mstcum[1])
vars < c(L[2]^2, L[2]^2, 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
mu < size * (1  prob)/prob
p_zinb < 0.2
k_cat < length(marginal)
k_cont < length(Six)
k_mix < length(mix_pis)
k_comp < sum(unlist(lapply(mix_pis, length)))
k_pois < length(lam)
k_nb < length(size)
k_total < k_cat + k_cont + k_comp + k_pois + k_nb
Rey < matrix(0.35, k_total, k_total)
diag(Rey) < 1
rownames(Rey) < colnames(Rey) < c("O1", "C1", "C2", "M1_1", "M1_2", "M2_1",
"M2_2", "M2_3", "P1", "NB1")
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"] <
Rey["M2_3", "M2_1"] < Rey["M2_2", "M2_3"] < Rey["M2_3", "M2_2"] < 0
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)
## [1] TRUE
Since this step takes considerable computation time, the user may wish to calculate these after simulation if any of the simulated continuous variables have invalid PDF’s. The calculation will be demonstrated for the Chisq(4) distribution using both the third and fifthorder PMT’s for comparison.
Using Fleishman (1978)’s thirdorder method:
Lower_third < calc_lower_skurt(method = "Fleishman", skews = C[3],
Skurt = seq(1.161, 1.17, 0.001), seed = 104)
knitr::kable(Lower_third$Min[, c("skew", "valid.pdf", "skurtosis")],
row.names = FALSE, caption = "ThirdOrder Lower Skurtosis Bound")
skew  valid.pdf  skurtosis 

1.414214  TRUE  3.141272 
The original lower skurtosis boundary (see Lower_third$Invalid.C
) of 1.979272 has been increased to 3.141272, so that the skurtosis correction is 1.162. The skurtosis for the distribution (\(3\)) is lower than this boundary and the thirdorder PMT should not be used to simulate this variable.
Using Headrick (2002)’s fifthorder method:
Lower_fifth < calc_lower_skurt(method = "Polynomial", skews = C[3],
fifths = C[5], sixths = C[6], Skurt = seq(0.022, 0.03, 0.001), seed = 104)
knitr::kable(Lower_fifth$Min[, c("skew", "fifth", "sixth", "valid.pdf",
"skurtosis")], row.names = FALSE,
caption = "FifthOrder Lower Skurtosis Bound")
skew  fifth  sixth  valid.pdf  skurtosis 

1.414214  8.485281  30  TRUE  2.998959 
The original lower skurtosis boundary (see Lower_fifth$Invalid.C
) of 2.975959 has been increased to 2.998959, so that the skurtosis correction is 0.023. The skurtosis for the distribution (\(3\)) is approximately equal to this boundary. This does not cause a problem since the simulated variable has a valid power method PDF.
The remaining steps vary by simulation method:
valid1 < 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)
## All correlations are in feasible range!
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)
## Total Simulation time: 0.037 minutes
Sum1 < summary_var(Sim1$Y_cat, Sim1$Y_cont, 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  Sum1$rho_calc)
Summary of correlation errors:
summary(as.numeric(Sim1_error))
Min.  1st Qu.  Median  Mean  3rd Qu.  Max. 

0  0.0003912  0.0017661  0.0062285  0.0057543  0.0480139 
Simulated correlation matrix for \(O1\), \(C1\), \(C2\), \(M1\), \(M2\), \(P1\), and \(NB1\):
rho_mix < Sum1$rho_mix
rownames(rho_mix) < c("01", "C1", "C2", "M1", "M2", "P1", "NB1")
colnames(rho_mix) < rownames(rho_mix)
rho_mix
01  C1  C2  M1  M2  P1  NB1  

01  1.0000000  0.3449509  0.3515655  0.1642942  0.1987437  0.3090497  0.3019861 
C1  0.3449509  1.0000000  0.3499182  0.1566208  0.1926035  0.3532803  0.3521983 
C2  0.3515655  0.3499182  1.0000000  0.1656771  0.2035505  0.3391623  0.3607957 
M1  0.1642942  0.1566208  0.1656771  1.0000000  0.0859799  0.1717365  0.1569002 
M2  0.1987437  0.1926035  0.2035505  0.0859799  1.0000000  0.2051287  0.2173914 
P1  0.3090497  0.3532803  0.3391623  0.1717365  0.2051287  1.0000000  0.3626053 
NB1  0.3019861  0.3521983  0.3607957  0.1569002  0.2173914  0.3626053  1.0000000 
We can approximate the expected correlations using the formulas in the Expected Cumulants and Correlations for Continuous Mixture Variables vignette and the rho_M1M2
and rho_M1Y
functions:
p_M11M21 < p_M11M22 < p_M11M23 < 0.35
p_M12M21 < p_M12M22 < p_M12M23 < 0.35
p_M1M2 < matrix(c(p_M11M21, p_M11M22, p_M11M23, p_M12M21, p_M12M22, p_M12M23),
2, 3, byrow = TRUE)
rhoM1M2 < rho_M1M2(mix_pis, mix_mus, mix_sigmas, p_M1M2)
p_M11C1 < p_M12C1 < 0.35
p_M1C1 < c(p_M11C1, p_M12C1)
rho_M1C1 < rho_M1Y(mix_pis[[1]], mix_mus[[1]], mix_sigmas[[1]], p_M1C1)
p_M21C1 < p_M22C1 < p_M23C1 < 0.35
p_M2C1 < c(p_M21C1, p_M22C1, p_M23C1)
rho_M2C1 < rho_M1Y(mix_pis[[2]], mix_mus[[2]], mix_sigmas[[2]], p_M2C1)
Do all continuous variables have valid PDF’s?
Sim1$valid.pdf
## [1] "TRUE" "TRUE" "TRUE" "TRUE" "TRUE" "TRUE" "TRUE"
Sim1$sixth_correction
## [1] 1.75 1.75 NA NA 1.75 NA 0.03
Nonmixture continuous variables and components of mixture variables:
target_sum < Sum1$target_sum
cont_sum < Sum1$cont_sum
rownames(target_sum) < rownames(cont_sum) < c("C1", "C2", "M1_1", "M1_2",
"M2_1", "M2_2", "M2_3")
knitr::kable(target_sum, digits = 5, row.names = TRUE,
caption = "Summary of Target Distributions")
Distribution  Mean  SD  Skew  Skurtosis  Fifth  Sixth  

C1  1  0.00000  3.28987  0.00000  1.2000  0.00000  6.85714 
C2  2  0.00000  3.28987  0.00000  1.2000  0.00000  6.85714 
M1_1  3  2.00000  1.00000  0.00000  0.0000  0.00000  0.00000 
M1_2  4  2.00000  1.00000  0.00000  0.0000  0.00000  0.00000 
M2_1  5  0.00000  3.28987  0.00000  1.2000  0.00000  6.85714 
M2_2  6  4.00000  8.00000  1.41421  3.0000  8.48528  30.00000 
M2_3  7  0.72727  0.03051  0.69388  0.0686  1.82825  3.37911 
knitr::kable(cont_sum[, c(2, 5:7)], digits = 5, row.names = TRUE,
caption = "Summary of Simulated Distributions")
Distribution  Mean  SD  Skew  Skurtosis  Fifth  Sixth  

C1  1  0.00287  1.81664  0.12011  1.40786  2.24058  11.41755 
C2  2  0.00235  1.81326  0.07377  1.14818  0.75267  5.28291 
M1_1  3  2.00000  0.99995  0.00682  0.03108  0.06622  0.06271 
M1_2  4  2.00000  0.99995  0.00946  0.00582  0.08557  0.05229 
M2_1  5  0.00136  1.81194  0.08924  1.17706  1.85416  7.93594 
M2_2  6  4.00055  2.84733  1.48311  3.42233  10.47408  36.91613 
M2_3  7  0.72752  0.17529  0.71763  0.01946  1.81549  3.72666 
Mixture continuous variables:
target_mix < Sum1$target_mix
mix_sum < Sum1$mix_sum
rownames(target_mix) < rownames(mix_sum) < c("M1", "M2")
knitr::kable(target_mix, digits = 5, row.names = TRUE,
caption = "Summary of Target Distributions")
Distribution  Mean  SD  Skew  Skurtosis  Fifth  Sixth  

M1  1  0.40000  2.20000  0.28850  1.15402  1.79302  6.17327 
M2  2  1.16364  2.17086  2.01328  8.78954  36.48103  192.72198 
knitr::kable(mix_sum[, c(2, 5:7)], digits = 5, row.names = TRUE,
caption = "Summary of Simulated Distributions")
Distribution  Mean  SD  Skew  Skurtosis  Fifth  Sixth  

M1  1  0.40000  2.19989  0.30573  1.12584  1.86882  5.83223 
M2  2  1.16364  2.17075  2.02355  9.04578  39.87345  223.50952 
Nplot < plot_simpdf_theory(sim_y = Sim1$Y_mix[, 1], ylower = 10,
yupper = 10, title = "PDF of Mixture of N(2, 1) and N(2, 1) Distributions",
fx = function(x) mix_pis[[1]][1] * dnorm(x, mix_mus[[1]][1],
mix_sigmas[[1]][1]) + mix_pis[[1]][2] * dnorm(x, mix_mus[[1]][2],
mix_sigmas[[1]][2]), lower = Inf, upper = Inf, sim_size = 0.5,
target_size = 0.5)
Nplot
Mplot < plot_simpdf_theory(sim_y = Sim1$Y_mix[, 2],
title = paste("PDF of Mixture of Logistic(0, 1), Chisq(4),",
"\nand Beta(4, 1.5) Distributions", sep = ""),
fx = function(x) mix_pis[[2]][1] * dlogis(x, 0, 1) + mix_pis[[2]][2] *
dchisq(x, 4) + mix_pis[[2]][3] * dbeta(x, 4, 1.5),
lower = Inf, upper = Inf, sim_size = 0.5, target_size = 0.5)
Mplot
knitr::kable(Sum1$ord_sum, caption = "Summary of Ordinal Variables")

knitr::kable(Sum1$pois_sum[, c(2, 9:11)],
caption = "Summary of Poisson Variables")
Distribution  P0  Exp_P0  Mean  Exp_Mean  Var  Exp_Var  Skew  Skurtosis  

mean  1  0.6486  0.6458776  0.4498  0.45  0.48008  0.5277778  1.58424  2.460644 
Pplot < plot_simpdf_theory(sim_y = Sim1$Y_pois[, 1],
title = "PMF of ZeroInflated Poisson Distribution", Dist = "Poisson",
params = c(lam, p_zip), cont_var = FALSE, col_width = 0.25)
Pplot
knitr::kable(Sum1$nb_sum[, c(2, 10:12)],
caption = "Summary of Negative Binomial Variables")
Distribution  P0  Exp_P0  Prob  Mean  Exp_Mean  Var  Exp_Var  Skew  Skurtosis  

mean  1  0.6464  0.65  0.75  0.5353  0.5333333  0.7613539  0.7822222  1.954248  4.490933 
NBplot < plot_simtheory(sim_y = Sim1$Y_nb[, 1],
title = "Simulated ZeroInflated NB Values", binwidth = 0.5,
Dist = "Negative_Binomial", params = c(size, mu, p_zinb),
cont_var = FALSE)
NBplot
For this example, mu
is used to describe \(NB1\) instead of prob
for demonstration purposes.
pois_eps < 0.0001
nb_eps < 0.0001
valid2 < 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 = NULL, mu, p_zinb, pois_eps, nb_eps, Rey, seed)
## All correlations are in feasible range!
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 = NULL, mu,
p_zinb, pois_eps, nb_eps, Rey, seed, epsilon = 0.01)
## Total Simulation time: 0.006 minutes
Sum2 < summary_var(Sim2$Y_cat, Sim2$Y_cont, Sim2$Y_comp, Sim2$Y_mix,
Sim2$Y_pois, Sim2$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 = NULL, mu, p_zinb, Rey)
Sim2_error < abs(Rey  Sum2$rho_calc)
Summary of correlation errors:
summary(as.numeric(Sim2_error))
Min.  1st Qu.  Median  Mean  3rd Qu.  Max. 

0  0.0008558  0.0018499  0.0053624  0.0061373  0.0370054 
Simulated correlation matrix for \(O1\), \(C1\), \(C2\), \(M1\), \(M2\), \(P1\), and \(NB1\):
rho_mix < Sum2$rho_mix
rownames(rho_mix) < c("01", "C1", "C2", "M1", "M2", "P1", "NB1")
colnames(rho_mix) < rownames(rho_mix)
rho_mix
01  C1  C2  M1  M2  P1  NB1  

01  1.0000000  0.3526467  0.3494859  0.1522525  0.1967726  0.3481883  0.3412737 
C1  0.3526467  1.0000000  0.3477938  0.1564539  0.1878836  0.3481095  0.3561373 
C2  0.3494859  0.3477938  1.0000000  0.1632542  0.1985422  0.3505951  0.3607467 
M1  0.1522525  0.1564539  0.1632542  1.0000000  0.0907563  0.1539732  0.1692886 
M2  0.1967726  0.1878836  0.1985422  0.0907563  1.0000000  0.2018016  0.1964330 
P1  0.3481883  0.3481095  0.3505951  0.1539732  0.2018016  1.0000000  0.3495915 
NB1  0.3412737  0.3561373  0.3607467  0.1692886  0.1964330  0.3495915  1.0000000 
Do all continuous variables have valid PDF’s?
Sim2$valid.pdf
## [1] "TRUE" "TRUE" "TRUE" "TRUE" "TRUE" "TRUE" "TRUE"
Sim2$sixth_correction
## [1] 1.75 1.75 NA NA 1.75 NA 0.03
Nonmixture continuous variables and components of mixture variables:
target_sum < Sum2$target_sum
cont_sum < Sum2$cont_sum
rownames(target_sum) < rownames(cont_sum) < c("C1", "C2", "M1_1", "M1_2",
"M2_1", "M2_2", "M2_3")
knitr::kable(target_sum, digits = 5, row.names = TRUE,
caption = "Summary of Target Distributions")
Distribution  Mean  SD  Skew  Skurtosis  Fifth  Sixth  

C1  1  0.00000  3.28987  0.00000  1.2000  0.00000  6.85714 
C2  2  0.00000  3.28987  0.00000  1.2000  0.00000  6.85714 
M1_1  3  2.00000  1.00000  0.00000  0.0000  0.00000  0.00000 
M1_2  4  2.00000  1.00000  0.00000  0.0000  0.00000  0.00000 
M2_1  5  0.00000  3.28987  0.00000  1.2000  0.00000  6.85714 
M2_2  6  4.00000  8.00000  1.41421  3.0000  8.48528  30.00000 
M2_3  7  0.72727  0.03051  0.69388  0.0686  1.82825  3.37911 
knitr::kable(cont_sum[, c(2, 5:7)], digits = 5, row.names = TRUE,
caption = "Summary of Simulated Distributions")
Distribution  Mean  SD  Skew  Skurtosis  Fifth  Sixth  

C1  1  0.00137  1.81558  0.06184  1.32049  1.15791  11.32372 
C2  2  0.00316  1.81408  0.05852  1.20886  0.33438  6.66564 
M1_1  3  2.00000  0.99995  0.01645  0.03766  0.14312  0.11787 
M1_2  4  2.00000  0.99995  0.00513  0.00950  0.04604  0.03334 
M2_1  5  0.00114  1.81206  0.08295  1.16386  1.84537  7.37171 
M2_2  6  4.00044  2.84502  1.46747  3.27444  9.44722  30.91024 
M2_3  7  0.72749  0.17520  0.71409  0.02244  1.79531  3.66029 
Mixture continuous variables:
target_mix < Sum2$target_mix
mix_sum < Sum2$mix_sum
rownames(target_mix) < rownames(mix_sum) < c("M1", "M2")
knitr::kable(target_mix, digits = 5, row.names = TRUE,
caption = "Summary of Target Distributions")
Distribution  Mean  SD  Skew  Skurtosis  Fifth  Sixth  

M1  1  0.40000  2.20000  0.28850  1.15402  1.79302  6.17327 
M2  2  1.16364  2.17086  2.01328  8.78954  36.48103  192.72198 
knitr::kable(mix_sum[, c(2, 5:7)], digits = 5, row.names = TRUE,
caption = "Summary of Simulated Distributions")
Distribution  Mean  SD  Skew  Skurtosis  Fifth  Sixth  

M1  1  0.40000  2.19989  0.30025  1.13041  1.85623  5.88668 
M2  2  1.16364  2.17075  2.00831  8.85771  37.06375  191.28657 
knitr::kable(Sum2$ord_sum, caption = "Summary of Ordinal Variables")

knitr::kable(Sum2$pois_sum[, c(2, 9:11)],
caption = "Summary of Poisson Variables")
Distribution  P0  Exp_P0  Mean  Exp_Mean  Var  Exp_Var  Skew  Skurtosis  

mean  1  0.6468  0.6458776  0.4504  0.45  0.4749398  0.5277778  1.554168  2.331026 
knitr::kable(Sum2$nb_sum[, c(2, 10:12)],
caption = "Summary of Negative Binomial Variables")
Distribution  P0  Exp_P0  Prob  Mean  Exp_Mean  Var  Exp_Var  Skew  Skurtosis  

mean  1  0.6509  0.65  0.75  0.5342  0.5333333  0.7904304  0.7822222  2.084995  5.33719 
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Headrick, T C. 2002. “Fast FifthOrder Polynomial Transforms for Generating Univariate and Multivariate NonNormal Distributions.” Computational Statistics and Data Analysis 40 (4): 685–711. https://doi.org/10.1016/S01679473(02)000725.
Headrick, T C, and R K Kowalchuk. 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: 229–49. https://doi.org/10.1080/10629360600605065.
Yee, Thomas W. 2018. VGAM: Vector Generalized Linear and Additive Models. https://CRAN.Rproject.org/package=VGAM.