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, 10.1002/sim.5362) and
Barbiero & Ferrari (2015, 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,
10.1007/BF02294164) and the power method correlation (described in Headrick & Kowalchuk, 2007, 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:
$$\rho_{Y2,Z2} = \frac{1}{\sigma_{Y2}} \sum_{j = 1}^{r-1} \phi(\tau_{j})(y2_{j+1} - y2_{j})$$ where
$$\phi(\tau) = (2\pi)^{-1/2} * exp(-0.5 \tau^2)$$ Here, \(y_{j}\) is the j-th support
value and \(\tau_{j}\) is \(\Phi^{-1}(\sum_{i=1}^{j} Pr(Y = y_{i}))\). The power method correlation is given by:
$$\rho_{Y1, Z1} = c_1 + 3c_3 + 15c_5,$$ 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.
intercorr_cont_nb2(method = c("Fleishman", "Polynomial"), constants = NULL,
rho_cont_nb = NULL, nb_marg = list(), nb_support = list())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.
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
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
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
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
a k_cont x k_nb matrix whose rows represent the k_cont continuous variables and columns represent the
k_nb Negative Binomial variables
Please see references in intercorr_cont_pois2.