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

findintercorr_cont_cat {SimMultiCorrData}

R Documentation

Calculate Intermediate MVN Correlation for Continuous - Ordinal Variables

Description

This function calculates a k_cont x k_cat intermediate matrix of correlations for the k_cont continuous and k_cat ordinal (r >= 2 categories) variables. It extends the method of Demirtas et al. (2012, doi: 10.1198/tast.2011.10090) in simulating binary and non-normal data using the Fleishman transformation by:

1) allowing the continuous variables to be generated via Fleishman's third-order or Headrick's fifth-order transformation, and

2) allowing for ordinal variables with more than 2 categories.

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 discretized to produce an ordinal variable Y2) 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. 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 findintercorr and findintercorr2. This function would not ordinarily be called by the user.

Usage

findintercorr_cont_cat(method = c("Fleishman", "Polynomial"), constants,
  rho_cont_cat, marginal, support)

Arguments

`method`	the method used to generate the k_cont continuous variables. "Fleishman" uses a 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_cat`	a `k_cont x k_cat` matrix of target correlations among continuous and ordinal variables
`marginal`	a list of length equal to `k_cat`; 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)
`support`	a list of length equal to `k_cat`; the i-th element is a vector of containing the r ordered support values

Value

a k_cont x k_cat matrix whose rows represent the k_cont continuous variables and columns represent the k_cat ordinal variables

References

Demirtas H, Hedeker D, & Mermelstein RJ (2012). Simulation of massive public health data by power polynomials. Statistics in Medicine, 31(27): 3337-3346. doi: 10.1002/sim.5362.

Fleishman AI (1978). A Method for Simulating Non-normal Distributions. Psychometrika, 43, 521-532. doi: 10.1007/BF02293811.

Headrick TC (2002). Fast Fifth-order Polynomial Transforms for Generating Univariate and Multivariate Non-normal Distributions. Computational Statistics & Data Analysis, 40(4):685-711. doi: 10.1016/S0167-9473(02)00072-5. (ScienceDirect)

Headrick TC (2004). On Polynomial Transformations for Simulating Multivariate Nonnormal Distributions. Journal of Modern Applied Statistical Methods, 3(1), 65-71. doi: 10.22237/jmasm/1083370080.

Headrick TC, Kowalchuk RK (2007). The Power Method Transformation: Its Probability Density Function, Distribution Function, and Its Further Use for Fitting Data. Journal of Statistical Computation and Simulation, 77, 229-249. doi: 10.1080/10629360600605065.

Headrick TC, Sawilowsky SS (1999). Simulating Correlated Non-normal Distributions: Extending the Fleishman Power Method. Psychometrika, 64, 25-35. doi: 10.1007/BF02294317.

Headrick TC, Sheng Y, & Hodis FA (2007). Numerical Computing and Graphics for the Power Method Transformation Using Mathematica. Journal of Statistical Software, 19(3), 1 - 17. doi: 10.18637/jss.v019.i03.