# intercorr_cont_nb2: Calculate Intermediate MVN Correlation for Continuous -... In SimCorrMix: Simulation of Correlated Data with Multiple Variable Types Including Continuous and Count Mixture Distributions

## 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:

ρ_{Y2,Z2} = \frac{1}{σ_{Y2}} ∑_{j = 1}^{r-1} φ(τ_{j})(y2_{j+1} - y2_{j})

where

φ(τ) = (2π)^{-1/2} * exp(-0.5 τ^2)

Here, y_{j} is the j-th support value and τ_{j} is Φ^{-1}(∑_{i=1}^{j} Pr(Y = y_{i})). The power method correlation is given by:

ρ_{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

 1 2 intercorr_cont_nb2(method = c("Fleishman", "Polynomial"), constants = NULL, rho_cont_nb = NULL, nb_marg = list(), nb_support = list()) 

## Arguments

 method the method used to generate the k_cont continuous variables. "Fleishman" uses Fleishman's third-order polynomial transformation and "Polynomial" uses Headrick's fifth-order transformation. constants a matrix with k_cont rows, each a vector of constants c0, c1, c2, c3 (if method = "Fleishman") or c0, c1, c2, c3, c4, c5 (if method = "Polynomial"), like that returned by find_constants rho_cont_nb a k_cont x k_nb matrix of target correlations among continuous and Negative Binomial variables; the NB variables should be ordered 1st regular, 2nd zero-inflated nb_marg a list of length equal to k_nb ordered 1st regular and 2nd zero-inflated; 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); this is created within intercorr2 and corrvar2 nb_support a list of length equal to k_nb ordered 1st regular and 2nd zero-inflated; the i-th element is a vector of containing the r ordered support values, with a minimum of 0 and maximum determined via maxcount_support

## 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.

find_constants, power_norm_corr, intercorr2, corrvar2