R: Calculate Intermediate MVN Correlation for Continuous -...

findintercorr_cont_nb2 {SimMultiCorrData}

R Documentation

Calculate Intermediate MVN Correlation for Continuous - Negative Binomial Variables: Correlation Method 2

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 count (Poisson and 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).

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 max_count_support, the point-polyserial correlation is given by:

\rho_{y2,z2} = (1/\sigma_{y2})\sum_{j = 1}^{r-1} \phi(\tau_{j})(y2_{j+1} - y2_{j})

where

\phi(\tau) = (2\pi)^{-1/2}*exp(-\tau^2/2)

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} = c1 + 3c3 + 15c5

, where c5 = 0 if method = "Fleishman". The function is used in findintercorr2 and rcorrvar2. This function would not ordinarily be called by the user.

Usage

findintercorr_cont_nb2(method, constants, rho_cont_nb, nb_marg, nb_support)

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
`nb_marg`	a list of length equal to `k_nb`; 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 `findintercorr2` and `rcorrvar2`
`nb_support`	a list of length equal to `k_nb`; the i-th element is a vector of containing the r ordered support values, with a minimum of 0 and maximum determined via `max_count_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 additional references in findintercorr_cont_cat.