calc_corr_y: Calculate Expected Correlation Matrix of Outcomes (Y) for Correlated Systems of Continuous Variables

Description

This function calculates the expected correlation matrix for outcomes (Y) in a correlated system of continuous variables. This system is generated with nonnormsys using the techniques of Headrick and Beasley (10.1081/SAC-120028431). These correlations are determined based on the beta (slope) coefficients calculated with calc_betas, the correlations between independent variables \(X_{(pj)}\) for a given outcome \(Y_p\), for p = 1, ..., M, the correlations between error terms, and the variances. The result can be used to compare the simulated correlation matrix to the theoretical correlation matrix. If there are continuous mixture variables and the betas are specified in terms of non-mixture and mixture variables and/or error_type = "mix", then the correlations in corr.x and/or corr.e will be calculated in terms of non-mixture and mixture variables using rho_M1M2 and rho_M1Y. In this case, the dimensions of the matrices in corr.x should not match the number of columns of betas. The vignette Theory and Equations for Correlated Systems of Continuous Variables gives the equations, and the vignette Correlated Systems of Statistical Equations with Non-Mixture and Mixture Continuous Variables gives examples. There are also vignettes in SimCorrMix which provide more details on continuous non-mixture and mixture variables.

Usage

calc_corr_y(betas = NULL, corr.x = list(), corr.e = NULL, vars = list(),
  mix_pis = list(), mix_mus = list(), mix_sigmas = list(),
  error_type = c("non_mix", "mix"))

Arguments

betas

a matrix of the slope coefficients calculated with calc_betas, rows represent the outcomes

corr.x

list of length M, each component a list of length M; corr.x[[p]][[q]] is matrix of correlations for independent variables in equations p (\(X_{(pj)}\) for outcome \(Y_p\)) and q (\(X_{(qj)}\) for outcome \(Y_q\)); if p = q, corr.x[[p]][[q]] is a correlation matrix with nrow(corr.x[[p]][[q]]) = # \(X_{(pj)}\) for outcome \(Y_p\); if p != q, corr.x[[p]][[q]] is a non-symmetric matrix of correlations where rows correspond to covariates for \(Y_p\) so that nrow(corr.x[[p]][[q]]) = # \(X_{(pj)}\) for outcome \(Y_p\) and columns correspond to covariates for \(Y_q\) so that ncol(corr.x[[p]][[q]]) = # \(X_{(qj)}\) for outcome \(Y_q\); order is 1st continuous non-mixture and 2nd components of continuous mixture variables

corr.e

correlation matrix for continuous non-mixture or components of mixture error terms

vars

a list of same length as corr.x of vectors of variances for \(X_{(pj)}, E\); E term should be last; order should be the same as in corr.x

mix_pis

a list of same length as corr.x, where mix_pis[[p]][[j]] is a vector of mixing probabilities for \(X_{mix(pj)}\) that sum to 1, the j-th mixture covariate for outcome \(Y_p\); the last element of mix_pis[[p]] is for \(E_p\) (if error_type = "mix"); if \(Y_p\) has no mixture variables, use mix_pis[[p]] = NULL

mix_mus

a list of same length as corr.x, where mix_mus[[p]][[j]] is a vector of means for \(X_{mix(pj)}\), the j-th mixture covariate for outcome \(Y_p\); the last element of mix_mus[[p]] is for \(E_p\) (if error_type = "mix"); if \(Y_p\) has no mixture variables, use mix_mus[[p]] = NULL

mix_sigmas

a list of same length as corr.x, where mix_sigmas[[p]][[j]] is a vector of standard deviations for \(X_{mix(pj)}\), the j-th mixture covariate for outcome \(Y_p\); the last element of mix_sigmas[[p]] is for \(E_p\) (if error_type = "mix"); if \(Y_p\) has no mixture variables, use mix_sigmas[[p]] = NULL

error_type

"non_mix" if all error terms have continuous non-mixture distributions, "mix" if all error terms have continuous mixture distributions, defaults to "non_mix"

Value

corr.y the correlation matrix for the outcomes \(Y\)

References

Headrick TC, Beasley TM (2004). A Method for Simulating Correlated Non-Normal Systems of Linear Statistical Equations. Communications in Statistics - Simulation and Computation, 33(1). 10.1081/SAC-120028431

Examples

Run this code

# NOT RUN {
# Example: system of three equations for 2 independent variables, where each
# error term has unit variance, from Headrick & Beasley (2002)
corr.yx <- list(matrix(c(0.4, 0.4), 1), matrix(c(0.5, 0.5), 1),
  matrix(c(0.6, 0.6), 1))
corr.x <- list()
corr.x[[1]] <- corr.x[[2]] <- corr.x[[3]] <- list()
corr.x[[1]][[1]] <- matrix(c(1, 0.1, 0.1, 1), 2, 2)
corr.x[[1]][[2]] <- matrix(c(0.1974318, 0.1859656, 0.1879483, 0.1858601),
  2, 2, byrow = TRUE)
corr.x[[1]][[3]] <- matrix(c(0.2873190, 0.2589830, 0.2682057, 0.2589542),
  2, 2, byrow = TRUE)
corr.x[[2]][[1]] <- t(corr.x[[1]][[2]])
corr.x[[2]][[2]] <- matrix(c(1, 0.35, 0.35, 1), 2, 2)
corr.x[[2]][[3]] <- matrix(c(0.5723303, 0.4883054, 0.5004441, 0.4841808),
  2, 2, byrow = TRUE)
corr.x[[3]][[1]] <- t(corr.x[[1]][[3]])
corr.x[[3]][[2]] <- t(corr.x[[2]][[3]])
corr.x[[3]][[3]] <- matrix(c(1, 0.7, 0.7, 1), 2, 2)
corr.e <- matrix(0.4, nrow = 3, ncol = 3)
diag(corr.e) <- 1
vars <- list(rep(1, 3), rep(1, 3), rep(1, 3))
betas <- calc_betas(corr.yx, corr.x, vars)
calc_corr_y(betas, corr.x, corr.e, vars)

# }

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