Simultaneous Generation of Multivariate Data with Poisson and Normal Marginals

Description

The package implements a procedure for simultaneous generation of multivariate data with count and continuous variables with a pre-specified correlation matrix. The count and continuous variables are assumed to have Poisson and normal marginals, respectively. The data generation mechanism is a combination of the normal to anything principle and a connection between Poisson and normal correlations in the mixture. Data generation is accomplished by first calculating an intermediate correlation matrix (cmat.star) which is used to generate a sample from multivariate normal distribution. Then, the first few components (corresponding to number of Poisson variables) are transformed to Poisson variables via the inverse CDF method. The resulting data are composed of a mixture of Poisson and normal variables that conform with pre-specified marginal distributions and correlation structure.

The function Valid.correlation returns the lower and upper bounds of the correlation coefficients of Poisson-Poisson and Poisson-normal pairs given their marginal distributions, i.e. returns the range of feasible pairwise correlations. The function Validate.correlation checks the validity of the values of pairwise correlations. Additionally, it checks positive definiteness, symmetry and correctness of the dimensions. The engine function genPoisNor generates mixed data in accordance with the specified marginal and correlational quantities.

Details

Author(s)

Anup Amatya, Hakan Demirtas, Ran Gao

Maintainer: Ran Gao <rgao8@uic.edu>

Finds the feasible correlation range for a pair of Poisson and normal variables

Description

The function computes the lower and upper bounds of a pairwise correlation between a Poisson and a normal variable via the method of Demirtas and Hedeker (2011).

Usage

Cor.PN.Limit(lam)

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.

Examples

Cor.PN.Limit(0.05)

Finds the feasible correlation range for a pair of Poisson variables

Description

The function computes the lower and upper bounds of a pairwise correlation between a pair of Poisson variables.

Usage

Cor.PP.Limit(lamvec)

Arguments

Value

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

Examples

Cor.PP.Limit(c(0.05, 0.07) )

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

Description

The function computes the lower and upper bounds for the target correlations based on the marginal rates.

Usage

Valid.correlation(no.pois, no.norm, lamvec)

Arguments

Details

The function returns a list of two matrices. The min contains the lower bounds and the max contains the upper bounds of the feasible correlations.

Examples

lamvec= c(0.05,0.07,0.09)
Valid.correlation(no.pois=3, no.norm=3,lamvec)

Checks the target correlation matrix

Description

The function checks the validity of the values of pairwise correlations. Additionally, it checks positive definiteness, symmetry and correctness of the dimensions.

Usage

Validate.correlation(no.pois, no.norm, corMat, lamvec)

Arguments

Details

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 in the system. 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.

Examples

## Not run: 
# An example with a valid target correlation matrix.

lamvec= c(0.05,0.07,0.09)
M=c(0.352, 0.265, 0.342, 0.09, 0.141, 0.121, 0.297, 
	-0.022, 0.177, 0.294, -0.044, 0.129, 0.1, 0.354, 0.386)

N=diag(6)
N[lower.tri(N)]=M
TV=N+t(N)
diag(TV)<-1

Validate.correlation(no.pois=3, no.norm=3, corMat=TV, lamvec)


# An example with an invalid target correlation matrix (bound violation).

lamvec= c(0.05,0.07,0.09)
M=c(-0.151, -0.085, -0.11, 0.29, 0.6, 0.132, 0.161, 0.139, 
	-0.088, 0.075, -0.025, -0.293, -0.67, -0.03, 0.61)
N=diag(6)
N[lower.tri(N)]=M
TV1=N+t(N)
diag(TV1)<-1
Validate.correlation(no.pois=3, no.norm=3, corMat=TV1, lamvec)

# Examples with an incorrect dimension specification.

lamvec= c(0.05,0.07,0.09)
Validate.correlation(no.pois=3, no.norm=2, corMat=TV, lamvec)
Validate.correlation(no.pois=2, no.norm=3, corMat=TV, lamvec)

# An example with a non-positive definite correlation matrix.

TV1=TV
TV1[5,1]=TV1[1,5] = 1.5
Validate.correlation(no.pois=3, no.norm=3, corMat=TV1, lamvec)

# An example with a non-symmetric correlation matrix.

TV1=TV
TV1[5,1] = 0.1
Validate.correlation(no.pois=3, no.norm=3, corMat=TV1, lamvec)

# An example with an invalid diagonal element in the correlation matrix.

TV1=TV
TV1[5,5] = 2
Validate.correlation(no.pois=3, no.norm=3, corMat=TV1, lamvec)

## End(Not run)

Computes the correlation of intermediate multivariate normal data before subsequent inverse CDF transformation

Description

The function computes an intermediate correlation matrix which leads to the target correlation matrix after inverse CDF transformation of the samples generated from a multivariate normal distribution with the intermediate correlation matrix.

Usage

cmat.star(no.pois, no.norm, corMat, lamvec)

Arguments

Value

An intermediate correlation matrix of size (no.pois + no.norm) \times (no.pois+ no.norm)

References

I. Yahav and G. Shmueli (2012). On generating multivariate poisson data in management science applications. Applied Stochastic Models in Business and Industry; 28(1):91-102.

See Also

Validate.correlation

Examples

## Not run: 
lamvec= c(0.5,0.7,0.9)
M=c(0.352, 0.265, 0.342, 0.09, 0.141, 0.121, 0.297, 
  -0.022, 0.177, 0.294, -0.044, 0.129, 0.1, 0.354, 0.386)

N=diag(6)
N[lower.tri(N)]=M
TV=N+t(N)
diag(TV)<-1

cstar = cmat.star(no.pois=3, no.norm=3, TV, lamvec)

## End(Not run)

Generates a data set with Poisson and normal variables

Description

The function simulates multivariate data with Poisson and normal components with a pre-specified correlation matrix and marginal distributions.

Usage

genPoisNor(n, no.norm, no.pois, cmat.star, lamvec, sd.vec, mean.vec)

Arguments

Value

A matrix of size n \times (no.pois + no.norm), of which first no.pois are Poisson variables.

Examples

## Not run: 
  lamvec= c(0.05,0.07,0.09)
  M=c(0.352, 0.265, 0.342, 0.09, 0.141, 0.121, 0.297, 
		-0.022, 0.177, 0.294, -0.044, 0.129, 0.1, 0.354, 0.386)
  N=diag(6)
  N[lower.tri(N)]=M
  TV=N+t(N)
  diag(TV)<-1
  
  cstar = cmat.star(no.pois=3, no.norm=3, TV, lamvec)
  mydata=genPoisNor(n=200, no.norm=3,  no.pois=3, cmat.star=cstar,
		 lamvec, sd.vec=c(1,1,1), mean.vec=c(0,0,0))

## End(Not run)