Concurrent generation of binary, ordinal and continuous data

Description

This package implements a procedure for generating samples from a mix of binary, ordinal and continuous random variables with a pre-specified correlation matrix and marginal distributions based on the methodology proposed by Demirtas et al. (2012) and its extensions.

This package consists of nine functions. The function Fleishman.coef.NN computes the Fleishman coefficients for each continuous variable with pre-specified skewness and kurtosis values. The functions LimitforNN and LimitforONN return the lower and upper correlation bounds of a pairwise correlation between two continuous variables, and between a binary/ordinal variable and a continuous variable, respectively. The function valid.limits.BinOrdNN computes the lower and upper bounds for the correlation entries based on the marginal distributions of the variables. The function validate.target.cormat.BinOrdNN checks the validity of the values of pairwise correlations. The function IntermediateNonNor and IntermediateONN compute the intermediate correlations for continuous pairs, and binary/ordinal-continuous pairs, respectively. The function cmat.star.BinOrdNN assembles the intermediate correlation matrix. The engine function genBinOrdNN generates mixed data in accordance with a given correlation matrix and marginal distributions.

The key packages and functions that we call in this package include GenOrd, OrdNor, BBsolve, rmvnorm, and nearPD.

Details

Author(s)

Hakan Demirtas, Yue Wang, Rawan Allozi, Ran Gao

Maintainer: Ran Gao <rgao8@uic.edu>

References

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

Demirtas, H. and Hedeker, D. (2016). Computing the point-biserial correlation under any underlying continuous distribution. Communications in Statistics - Simulation and Computation, 45(8), 2744-2751.

Demirtas, H., Hedeker, D., and Mermelstein, R.J. (2012). Simulation of massive public health data by power polynomials. Statistics in Medicine, 31(27), 3337-3346.

Demirtas, H. and Yavuz Y. (2015). Concurrent generation of ordinal and normal data. Journal of Biopharmaceutical Statistics, 25(4), 635-650.

Fleishman, A.I. (1978). A method for simulating non-normal distributions. Psychometrika, 43(4), 521-532.

Vale, C.D., and Maurelli, V.A. (1983). Simulating multivariate nonnormal distributions. Psychometrika, 48(3), 465-471.

Computes the Fleishman coefficients for each continuous variable

Description

The function checks whether the skewness and kurtosis parameters violates the universal equality given in Demirtas, Hedeker, Mermelstein (2012) and computes the Fleishman coefficients for each continuous variable with pre-specified skewness and kurtosis values by solving the Fleishman's polynomial equations using BBsolve function in BB package.

Usage

Fleishman.coef.NN(skew.vec, kurto.vec)

Arguments

Value

An matrix with four columns corresponding to the four Fleishman coefficients, and number of rows corresponding to number of continuous variables. The i-th row contains the estimates of the four Fleishman coefficients a, b, c and d for the i-th continuous variable with i-th pre-specified skewness and kurtosis values.

References

Demirtas, H., Hedeker, D., and Mermelstein, R.J. (2012). Simulation of massive public health data by power polynomials. Statistics in Medicine, 31(27), 3337-3346.

Fleishman, A.I. (1978). A method for simulating non-normal distributions. Psychometrika, 43(4), 521-532.

Examples

# Consider four continuous variables, which come from
# Exp(1),Beta(4,4),Beta(4,2) and Gamma(10,10), respectively.
# Skewness and kurtosis values of these variables are as follows:

skew.vec <- c(2,0,-0.4677,0.6325)
kurto.vec <- c(6,-0.5455,-0.3750,0.6)
coef.est <- Fleishman.coef.NN(skew.vec, kurto.vec)

Computes the intermediate correlations for all continuous pairs

Description

The function computes the intermediate correlation values of pairwise correlations between continuous variables.

Usage

IntermediateNonNor(skew.vec, kurto.vec, cormat)

Arguments

Value

A pairwise correlation matrix of intermediate correlation for continuous variables.

References

Demirtas, H., Hedeker, D., and Mermelstein, R.J. (2012). Simulation of massive public health data by power polynomials. Statistics in Medicine, 31(27), 3337-3346.

Vale, C.D., and Maurelli, V.A.(1983). Simulating multivariate nonnormal distributions. Psychometrika, 48(3), 465-471.

See Also

IntermediateONN, cmat.star.BinOrdNN

Examples

IntermediateNonNor(skew.vec=c(1,2), kurto.vec=c(2, 7), 
                   cormat=matrix(c(1,-0.47,-0.47,1),2,2))

Computes the intermediate (biserial/polyserial) correlations given the point-biserial/polyserial correlations for binary/ordinal-continuous pairs prior to dichotomization/ordinalization

Description

This function computes the intermediate correlation values of pairwise correlations between binary/ordinal and continuous variables.

Usage

IntermediateONN(plist, skew.vec, kurto.vec, ONNCorrMat)

Arguments

Value

A pairwise correlation matrix of intermediate correlations, where rows and columns represent continuous and binary/ordinal variables, respectively.

References

Demirtas, H., Hedeker, D., and Mermelstein, R.J. (2012). Simulation of massive public health data by power polynomials. Statistics in Medicine, 31(27), 3337-3346.

Demirtas, H. and Hedeker, D. (2016). Computing the point-biserial correlation under any underlying continuous distribution. Communications in Statistics - Simulation and Computation, 45(8), 2744-2751.

See Also

IntermediateNonNor, cmat.star.BinOrdNN

Examples

no.bin <- 1
no.ord <- 2
no.NN <- 4
q <- no.bin + no.ord + no.NN
set.seed(54321)

Sigma <- diag(q)
Sigma[lower.tri(Sigma)] <- runif((q*(q-1)/2),-0.4,0.4)
Sigma <- Sigma + t(Sigma)
diag(Sigma) <- 1

marginal <- list(0.3, cumsum( c(0.30, 0.40) ), cumsum(c(0.4, 0.2, 0.3) ) )
ONNCorrMat <- Sigma[4:7, 1:3]
IntermediateONN(marginal, skew.vec=c(1,2,2,3), kurto.vec=c(2,7,25,25), ONNCorrMat)

Finds the feasible correlation range for a pair of continuous variables

Description

The function computes the lower and upper correlation bounds of a pairwise correlation between two continuous variables using generate, sort, and correlate (GSC) algorithm in Demirtas and Hedeker (2011).

Usage

LimitforNN(skew.vec, kurto.vec)
Limit_forNN(skew.vec, kurto.vec) #Deprecated

Arguments

Value

A vector of two elements. The first element is the lower bound and the second element is the upper bound.

References

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

See Also

Fleishman.coef.NN

Examples

LimitforNN(skew.vec=c(1,2),kurto.vec=c(2,7))

Finds the feasible correlation range for a pair of binary/ordinal and continuous variables

Description

The function computes the lower and upper correlation bounds of a pairwise correlation between a binary/ordinal variable and a continuous variable using GSC algorithm in Demirtas and Hedeker (2011).

Usage

LimitforONN(pvec1, skew1, kurto1)
Limit_forONN(pvec1, skew1, kurto1) #Deprecated

Arguments

Value

A vector of two elements. The first element is the lower correlation bound and the second element is the upper correlation bound.

References

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

See Also

Fleishman.coef.NN

Examples

LimitforONN(pvec1=c(0.2, 0.5), skew1=1, kurto1=2)

Computes the intermediate correlation matrix

Description

The function computes the correlations of intermediate multivariate normal data prior to subsequent dichotomization (for binary variables), ordinalization (for ordinal variables), and transformation (for continuous variables)

Usage

cmat.star.BinOrdNN(plist, skew.vec, kurto.vec, no.bin, no.ord, no.NN, CorrMat)

Arguments

Value

An intermediate correlation of size (no.bin + no.ord + no.NN)*(no.bin + no.ord + no.NN)

See Also

validate.target.cormat.BinOrdNN, IntermediateNonNor, IntermediateONN

Examples

## Not run:
no.bin <- 1
no.ord <- 2
no.NN <- 4
q <- no.bin + no.ord + no.NN
set.seed(54321)

Sigma <- diag(q)
Sigma[lower.tri(Sigma)] <- runif((q*(q-1)/2),-0.4,0.4)
Sigma <- Sigma + t(Sigma)
diag(Sigma) <- 1

marginal <- list(0.3, cumsum(c(0.30, 0.40) ), cumsum(c(0.4, 0.2, 0.3) ) )
cmat.star <- cmat.star.BinOrdNN(plist=marginal, skew.vec=c(1,2,2,3), 
kurto.vec=c(2,7,25,25),no.bin=1, no.ord=2, no.NN=4, CorrMat=Sigma) 
## End(Not run)

Generates a data set with binary, ordinal and continuous variables

Description

The function simulates a sample of size n from a multivariate binary, ordinal and continuous variables with intermediate correlation matrix cmat.star, and pre-specified marginal distributions.

Usage

genBinOrdNN(n, plist, mean.vec, var.vec, skew.vec, kurto.vec, no.bin, no.ord, 
no.NN, cmat.star)

Arguments

Value

A matrix of size n*(no.bin + no.ord + no.NN), of which the first no.bin columns are binary variables, the next no.ord columns are ordinal variables, and the last no.NN columns are continuous variables.

References

Demirtas, H., Hedeker, D., and Mermelstein, R.J. (2012). Simulation of massive public health data by power polynomials. Statistics in Medicine, 31(27), 3337-3346.

Demirtas, H. and Yavuz Y. (2015). Concurrent generation of ordinal and normal data. Journal of Biopharmaceutical Statistics, 25(4), 635-650.

Vale, C.D., and Maurelli, V.A. (1983). Simulating multivariate nonnormal distributions. Psychometrika, 48(3), 465-471.

See Also

cmat.star.BinOrdNN, Fleishman.coef.NN

Examples

## Not run:
set.seed(54321)
no.bin <- 1
no.ord <- 1
no.NN <- 4
q <- no.bin + no.ord + no.NN

marginal <- list(0.4, cumsum(c(0.4, 0.2, 0.3)))

skewness.vec <- c(2,0,-0.4677,0.6325)
kurtosis.vec <- c(6,-0.5455,-0.3750,0.6)

corr.mat <- matrix(c(1.0,-0.3,-0.3,-0.3,-0.3,-0.3,
                    -0.3, 1.0,-0.3,-0.3,-0.3,-0.3,
                    -0.3,-0.3, 1.0, 0.4, 0.5, 0.6,
                    -0.3,-0.3, 0.4, 1.0, 0.7, 0.8,
                    -0.3,-0.3, 0.5, 0.7, 1.0, 0.9,
                    -0.3,-0.3, 0.6, 0.8, 0.9, 1.0),
                    q,byrow=TRUE)

corr.mat.star <- cmat.star.BinOrdNN(plist=marginal, skew.vec=skewness.vec, 
kurto.vec=kurtosis.vec, no.bin=1, no.ord=1, no.NN=4, CorrMat=corr.mat)

sim.data <- genBinOrdNN(n=100000, plist=marginal, mean.vec=c(2,3,4,5), 
var.vec=c(3,5,10,20), skew.vec=skewness.vec, kurto.vec=kurtosis.vec,
no.bin=1, no.ord=1, no.NN=4, cmat.star=corr.mat.star) 

## End(Not run)

Computes the lower and upper bounds of correlation in the form of two matrices

Description

The function computes the lower and upper bounds for the correlation entries based on the marginal distributions of the variables.

Usage

valid.limits.BinOrdNN(plist, skew.vec, kurto.vec, no.bin, no.ord, no.NN)

Arguments

Value

A list of two matrices. The one named lower contains the lower bounds and the other named upper contains the upper bounds of the feasible correlations.

See Also

LimitforNN, LimitforONN

Examples

marginal <- list(0.2, c(0.4, 0.7, 0.9))
valid.limits.BinOrdNN(plist=marginal, skew.vec=c(1,2), kurto.vec=c(2,7), 
                      no.bin=1, no.ord=1, no.NN=2)

Checks the validity of the target correlation matrix

Description

The function checks the validity of pairwise correlations. In addition, it checks positive definiteness, symmetry, and correct dimensions.

Usage

validate.target.cormat.BinOrdNN(plist, skew.vec, kurto.vec, no.bin, no.ord, 
no.NN, CorrMat)

Arguments

Value

In addition to being positive definite and symmetric, the values of pairwise correlations in the target correlation matrix must also fall within the limits imposed by the marginal distributions of the variables. The function ensures that the supplied correlation matrix is valid for simulation. If a violation occurs, an error message is displayed that identifies the violation. The function returns a logical value TRUE when no such violation occurs.

See Also

valid.limits.BinOrdNN

Examples

Sigma <- diag(4)
Sigma[lower.tri(Sigma)] <- c(0.42, 0.55, 0.29, 0.37, 0.14, 0.26)
Sigma <- Sigma + t(Sigma)
diag(Sigma) <- 1

marginal <- list(0.2, c(0.4, 0.7, 0.9))

validate.target.cormat.BinOrdNN(plist=marginal, skew.vec=c(1,2), kurto.vec=c(2,7), 
                                no.bin=1, no.ord=1, no.NN=2, CorrMat=Sigma)