Generate the Sampling Distribution of Regression Parameters Using the Monte Carlo Method

Usage

MC(
  object,
  R = 20000L,
  type = "hc3",
  g1 = 1,
  g2 = 1.5,
  k = 0.7,
  decomposition = "eigen",
  pd = TRUE,
  tol = 1e-06,
  fixed_x = FALSE,
  seed = NULL
)

Arguments

object: Object of class lm.
R: Positive integer. Number of Monte Carlo replications.
type: Character string. Sampling covariance matrix type. Possible values are "mvn", "adf", "hc0", "hc1", "hc2", "hc3", "hc4", "hc4m", and "hc5". type = "mvn" uses the normal-theory sampling covariance matrix. type = "adf" uses the asymptotic distribution-free sampling covariance matrix. type = "hc0" through "hc5" uses different versions of heteroskedasticity-consistent sampling covariance matrix.
g1: Numeric. g1 value for type = "hc4m".
g2: Numeric. g2 value for type = "hc4m".
k: Numeric. Constant for type = "hc5"
decomposition: Character string. Matrix decomposition of the sampling variance-covariance matrix for the data generation. If decomposition = "chol", use Cholesky decomposition. If decomposition = "eigen", use eigenvalue decomposition. If decomposition = "svd", use singular value decomposition.
pd: Logical. If pd = TRUE, check if the sampling variance-covariance matrix is positive definite using tol.
tol: Numeric. Tolerance used for pd.
fixed_x: Logical. If fixed_x = TRUE, treat the regressors as fixed. If fixed_x = FALSE, treat the regressors as random.
seed: Integer. Seed number for reproducibility.

Value

Returns an object of class mc which is a list with the following elements:

call: Function call.
args: Function arguments.
lm_process: Processed lm object.
scale: Sampling variance-covariance matrix of parameter estimates.
location: Parameter estimates.
thetahatstar: Sampling distribution of parameter estimates.
fun: Function used ("MC").

Details

Let the parameter vector of the unstandardized regression model be given by $$ \boldsymbol{\theta} = \left\{ \mathbf{b}, \sigma^{2}, \mathrm{vech} \left( \boldsymbol{\Sigma}_{\mathbf{X}\mathbf{X}} \right) \right\} $$ where $\mathbf{b}$ is the vector of regression slopes, $\sigma^{2}$ is the error variance, and $ \mathrm{vech} \left( \boldsymbol{\Sigma}_{\mathbf{X}\mathbf{X}} \right) $ is the vector of unique elements of the covariance matrix of the regressor variables. The empirical sampling distribution of $\boldsymbol{\theta}$ is generated using the Monte Carlo method, that is, random values of parameter estimates are sampled from the multivariate normal distribution using the estimated parameter vector as the mean vector and the specified sampling covariance matrix using the type argument as the covariance matrix. A replacement sampling approach is implemented to ensure that the model-implied covariance matrix is positive definite.

References

Dudgeon, P. (2017). Some improvements in confidence intervals for standardized regression coefficients. Psychometrika, 82(4), 928–951. doi:10.1007/s11336-017-9563-z

MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39(1), 99-128. doi:10.1207/s15327906mbr3901_4

Pesigan, I. J. A., & Cheung, S. F. (2023). Monte Carlo confidence intervals for the indirect effect with missing data. Behavior Research Methods. doi:10.3758/s13428-023-02114-4

Preacher, K. J., & Selig, J. P. (2012). Advantages of Monte Carlo confidence intervals for indirect effects. Communication Methods and Measures, 6(2), 77–98. doi:10.1080/19312458.2012.679848

Author

Ivan Jacob Agaloos Pesigan

Examples

# Data ---------------------------------------------------------------------
data("nas1982", package = "betaMC")

# Fit Model in lm ----------------------------------------------------------
object <- lm(QUALITY ~ NARTIC + PCTGRT + PCTSUPP, data = nas1982)

# MC -----------------------------------------------------------------------
mc <- MC(
  object,
  R = 100, # use a large value e.g., 20000L for actual research
  seed = 0508
)
mc
#> Call:
#> MC(object = object, R = 100, seed = 508)
#> The first set of simulated parameter estimates
#> and model-implied covariance matrix.
#> 
#> $coef
#> [1] 0.08131373 0.18654775 0.15899393
#> 
#> $sigmasq
#> [1] 19.0325
#> 
#> $vechsigmacapx
#> [1] 3796.5610  605.8221  607.1206  306.2053  191.5167  529.1519
#> 
#> $sigmacapx
#>           [,1]     [,2]     [,3]
#> [1,] 3796.5610 605.8221 607.1206
#> [2,]  605.8221 306.2053 191.5167
#> [3,]  607.1206 191.5167 529.1519
#> 
#> $sigmaysq
#> [1] 113.6057
#> 
#> $sigmayx
#> [1] 518.2558 136.8336 169.2262
#> 
#> $sigmacap
#>          [,1]      [,2]     [,3]     [,4]
#> [1,] 113.6057  518.2558 136.8336 169.2262
#> [2,] 518.2558 3796.5610 605.8221 607.1206
#> [3,] 136.8336  605.8221 306.2053 191.5167
#> [4,] 169.2262  607.1206 191.5167 529.1519
#> 
#> $pd
#> [1] TRUE
#> 
# The `mc` object can be passed as the first argument
# to the following functions
#   - BetaMC
#   - DeltaRSqMC
#   - DiffBetaMC
#   - PCorMC
#   - RSqMC
#   - SCorMC