validcorr2: Determine Correlation Bounds for Ordinal, Continuous,...

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

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 ≥ 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

n	the sample size (i.e. the length of each simulated variable; default = 10000)
k_cat	the number of ordinal (r >= 2 categories) variables (default = 0)
k_cont	the number of continuous non-mixture variables (default = 0)
k_mix	the number of continuous mixture variables (default = 0)
k_pois	the number of regular Poisson and zero-inflated Poisson variables (default = 0)
k_nb	the number of regular Negative Binomial and zero-inflated Negative Binomial variables (default = 0)
method	the method used to generate the k_cont non-mixture and k_mix mixture continuous variables. "Fleishman" uses Fleishman's third-order polynomial transformation and "Polynomial" uses Headrick's fifth-order transformation.
means	a vector of means for the k_cont non-mixture and k_mix mixture continuous variables (i.e. rep(0, (k_cont + k_mix)))
vars	a vector of variances for the k_cont non-mixture and k_mix mixture continuous variables (i.e. rep(1, (k_cont + k_mix)))
skews	a vector of skewness values for the k_cont non-mixture continuous variables
skurts	a vector of standardized kurtoses (kurtosis - 3, so that normal variables have a value of 0) for the k_cont non-mixture continuous variables
fifths	a vector of standardized fifth cumulants for the k_cont non-mixture continuous variables (not necessary for method = "Fleishman")
sixths	a vector of standardized sixth cumulants for the k_cont non-mixture continuous variables (not necessary for method = "Fleishman")
Six	a list of vectors of sixth cumulant correction values for the k_cont non-mixture continuous variables if no valid PDF constants are found, ex: Six = list(seq(0.01, 2, 0.01), seq(1, 10, 0.5)); if no correction is desired for variable Y_{cont_i}, set set the i-th list component equal to NULL; if no correction is desired for any of the Y_{cont} keep as Six = list() (not necessary for method = "Fleishman")
mix_pis	a list of length k_mix with i-th component a vector of mixing probabilities that sum to 1 for component distributions of Y_{mix_i}
mix_mus	a list of length k_mix with i-th component a vector of means for component distributions of Y_{mix_i}
mix_sigmas	a list of length k_mix with i-th component a vector of standard deviations for component distributions of Y_{mix_i}
mix_skews	a list of length k_mix with i-th component a vector of skew values for component distributions of Y_{mix_i}
mix_skurts	a list of length k_mix with i-th component a vector of standardized kurtoses for component distributions of Y_{mix_i}
mix_fifths	a list of length k_mix with i-th component a vector of standardized fifth cumulants for component distributions of Y_{mix_i} (not necessary for method = "Fleishman")
mix_sixths	a list of length k_mix with i-th component a vector of standardized sixth cumulants for component distributions of Y_{mix_i} (not necessary for method = "Fleishman")
mix_Six	a list of length k_mix with i-th component a list of vectors of sixth cumulant correction values for component distributions of Y_{mix_i}; use NULL if no correction is desired for a given component or mixture variable; if no correction is desired for any of the Y_{mix} keep as mix_Six = list() (not necessary for method = "Fleishman")
marginal	a list of length equal to k_cat; the i-th element is a vector of the cumulative probabilities defining the marginal distribution of the i-th variable; if the variable can take r values, the vector will contain r - 1 probabilities (the r-th is assumed to be 1); for binary variables, these should be input the same as for ordinal variables with more than 2 categories (i.e. the user-specified probability is the probability of the 1st category, which has the smaller support value)
lam	a vector of lambda (> 0) constants for the Poisson variables (see stats::dpois); the order should be 1st regular Poisson variables, 2nd zero-inflated Poisson variables
p_zip	a vector of probabilities of structural zeros (not including zeros from the Poisson distribution) for the zero-inflated Poisson variables (see VGAM::dzipois); if p_zip = 0, Y_{pois} has a regular Poisson distribution; if p_zip is in (0, 1), Y_{pois} has a zero-inflated Poisson distribution; if p_zip is in (-(exp(lam) - 1)^(-1), 0), Y_{pois} has a zero-deflated Poisson distribution and p_zip is not a probability; if p_zip = -(exp(lam) - 1)^(-1), Y_{pois} has a positive-Poisson distribution (see VGAM::dpospois); if length(p_zip) < length(lam), the missing values are set to 0 (and ordered 1st)
size	a vector of size parameters for the Negative Binomial variables (see stats::dnbinom); the order should be 1st regular NB variables, 2nd zero-inflated NB variables
prob	a vector of success probability parameters for the NB variables; order the same as in size
mu	a vector of mean parameters for the NB variables (*Note: either prob or mu should be supplied for all Negative Binomial variables, not a mixture; default = NULL); order the same as in size; for zero-inflated NB this refers to the mean of the NB distribution (see VGAM::dzinegbin)
p_zinb	a vector of probabilities of structural zeros (not including zeros from the NB distribution) for the zero-inflated NB variables (see VGAM::dzinegbin); if p_zinb = 0, Y_{nb} has a regular NB distribution; if p_zinb is in (-prob^size/(1 - prob^size), 0), Y_{nb} has a zero-deflated NB distribution and p_zinb is not a probability; if p_zinb = -prob^size/(1 - prob^size), Y_{nb} has a positive-NB distribution (see VGAM::dposnegbin); if length(p_zinb) < length(size), the missing values are set to 0 (and ordered 1st)
pois_eps	a vector of length k_pois containing total cumulative probability truncation values; if none are provided, the default is 0.0001 for each variable
nb_eps	a vector of length k_nb containing total cumulative probability truncation values; if none are provided, the default is 0.0001 for each variable
rho	the target correlation matrix which must be ordered 1st ordinal, 2nd continuous non-mixture, 3rd components of continuous mixtures, 4th regular Poisson, 5th zero-inflated Poisson, 6th regular NB, 7th zero-inflated NB; note that rho is specified in terms of the components of Y_{mix}
seed	the seed value for random number generation (default = 1234)
use.nearPD	TRUE to convert rho to the nearest positive definite matrix with Matrix::nearPD if necessary
quiet	if FALSE prints messages, if TRUE suppresses message printing

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)

SimCorrMix documentation built on May 2, 2019, 1:24 p.m.