calc_betas: Calculate Beta Coefficients for Correlated Systems of...

In SimRepeat: Simulation of Correlated Systems of Equations with Multiple Variable Types

Description

This function calculates the beta (slope) coefficients used in nonnormsys by the techniques of Headrick and Beasley (doi: 10.1081/SAC-120028431). These coefficients are determined based on the correlations between independent variables X_{(pj)} for a given outcome Y_p, for p = 1, ..., M, the correlations between that outcome Y_p and the X_{(pj)} terms, and the variances. If there are continuous mixture variables and the matrices in corr.yx are specified in terms of correlations between outcomes and non-mixture and mixture variables, then the solutions are the slope coefficients for the non-mixture and mixture variables. In this case, the number of columns of the matrices of corr.yx should not match the dimensions of the matrices in corr.x. The correlations in corr.x will be calculated in terms of non-mixture and mixture variables using rho_M1M2 and rho_M1Y. If there are continuous mixture variables and the matrices in corr.yx are specified in terms of correlations between outcomes and non-mixture and components of mixture variables, then the solutions are the slope coefficients for the non-mixture and components of mixture variables. In this case, the number of columns of the matrices of corr.yx should match the dimensions of the matrices in corr.x. 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_betas(corr.yx = list(), corr.x = list(), vars = list(),
  mix_pis = list(), mix_mus = list(), mix_sigmas = list(),
  error_type = c("non_mix", "mix"), n = 25, seed = 1234)

Arguments

corr.yx	a list of length M = # of equations, where the p-th component is a 1 row matrix of correlations between Y_p and X_{(pj)}; if there are mixture variables and the betas are desired in terms of these (and not the components), then corr.yx should be specified in terms of correlations between outcomes and non-mixture or mixture variables, and the number of columns of the matrices of corr.yx should not match the dimensions of the matrices in corr.x; if the betas are desired in terms of the components, then corr.yx should be specified in terms of correlations between outcomes and non-mixture or components of mixture variables, and the number of columns of the matrices of corr.yx should match the dimensions of the matrices in corr.x
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
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"
n	the number of sets of random uniform(0, 1) numbers used as starting values in nleqslv to find the betas
seed	the seed for random number generation

Value

betas a matrix of slope coefficients where rows represent the outcomes; extra zeros are appended at the end of a row if that outcome has fewer X_{(pj)} terms

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). doi: 10.1081/SAC-120028431

See Also

nonnormsys, rho_M1M2, rho_M1Y

Examples

# 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)
vars <- list(rep(1, 3), rep(1, 3), rep(1, 3))
calc_betas(corr.yx, corr.x, vars)

SimRepeat documentation built on May 2, 2019, 9:32 a.m.