Ordinary Least Squares Estimation of the Binary Mediator Misclassification Model

Estimate \(\beta\), \(\gamma\), and \(\theta\) parameters from the true mediator, observed mediator, and outcome mechanisms, respectively, in a binary mediator misclassification model using an ordinary least squares correction.

Usage

COMMA_OLS(
  Mstar,
  outcome,
  x_matrix,
  z_matrix,
  c_matrix,
  beta_start,
  gamma_start,
  theta_start,
  tolerance = 1e-07,
  max_em_iterations = 1500,
  em_method = "squarem"
)

Arguments

Mstar: A numeric vector of indicator variables (1, 2) for the observed mediator M*. There should be no NA terms. The reference category is 2.
outcome: A vector containing the outcome variables of interest. There should be no NA terms.
x_matrix: A numeric matrix of predictors in the true mediator and outcome mechanisms. x_matrix should not contain an intercept and no values should be NA.
z_matrix: A numeric matrix of covariates in the observation mechanism. z_matrix should not contain an intercept and no values should be NA.
c_matrix: A numeric matrix of covariates in the true mediator and outcome mechanisms. c_matrix should not contain an intercept and no values should be NA.
beta_start: A numeric vector or column matrix of starting values for the \(\beta\) parameters in the true mediator mechanism. The number of elements in beta_start should be equal to the number of columns of x_matrix and c_matrix plus 1. Starting values should be provided in the following order: intercept, slope coefficient for the x_matrix term, slope coefficient for first column of the c_matrix, ..., slope coefficient for the final column of the c_matrix.
gamma_start: A numeric vector or matrix of starting values for the \(\gamma\) parameters in the observation mechanism. In matrix form, the gamma_start matrix rows correspond to parameters for the M* = 1 observed mediator, with the dimensions of z_matrix plus 1, and the gamma parameter matrix columns correspond to the true mediator categories \(M \in \{1, 2\}\). A numeric vector for gamma_start is obtained by concatenating the gamma matrix, i.e. gamma_start <- c(gamma_matrix). Starting values should be provided in the following order within each column: intercept, slope coefficient for first column of the z_matrix, ..., slope coefficient for the final column of the z_matrix.
theta_start: A numeric vector or column matrix of starting values for the \(\theta\) parameters in the outcome mechanism. The number of elements in theta_start should be equal to the number of columns of x_matrix and c_matrix plus 2. Starting values should be provided in the following order: intercept, slope coefficient for the x_matrix term, slope coefficient for the mediator m term, slope coefficient for first column of the c_matrix, ..., slope coefficient for the final column of the c_matrix.
tolerance: A numeric value specifying when to stop estimation, based on the difference of subsequent log-likelihood estimates. The default is 1e-7.
max_em_iterations: A numeric value specifying when to stop estimation, based on the difference of subsequent log-likelihood estimates. The default is 1e-7.
em_method: A character string specifying which EM algorithm will be applied. Options are "em", "squarem", or "pem". The default and recommended option is "squarem".

Value

COMMA_PVW returns a data frame containing four columns. The first column, Parameter, represents a unique parameter value for each row. The next column contains the parameter Estimates. The third column, Convergence, reports whether or not the algorithm converged for a given parameter estimate. The final column, Method, reports that the estimates are obtained from the "PVW" procedure.

Details

Note that this method can only be used for Normal outcome models, and interaction terms (between x and m) are not supported.

Examples

set.seed(20240709)
sample_size <- 2000

n_cat <- 2 # Number of categories in the binary mediator

# Data generation settings
x_mu <- 0
x_sigma <- 1
z_shape <- 1
c_shape <- 1

# True parameter values (gamma terms set the misclassification rate)
true_beta <- matrix(c(1, -2, .5), ncol = 1)
true_gamma <- matrix(c(1, 1, -.5, -1.5), nrow = 2, byrow = FALSE)
true_theta <- matrix(c(1, 1.5, -2, 2), ncol = 1)

example_data <- COMMA_data(sample_size, x_mu, x_sigma, z_shape, c_shape,
                           interaction_indicator = FALSE,
                           outcome_distribution = "Normal",
                           true_beta, true_gamma, true_theta)
                           
beta_start <- matrix(rep(1, 3), ncol = 1)
gamma_start <- matrix(rep(1, 4), nrow = 2, ncol = 2)
theta_start <- matrix(rep(1, 4), ncol = 1)

Mstar = example_data[["obs_mediator"]]
outcome = example_data[["outcome"]]
x_matrix = example_data[["x"]]
z_matrix = example_data[["z"]]
c_matrix = example_data[["c"]]
                           
OLS_results <- COMMA_OLS(Mstar, outcome,
                         x_matrix, z_matrix, c_matrix,
                         beta_start, gamma_start, theta_start)
                         
OLS_results
#>    Parameter  Estimates Convergence Method
#> 1      beta1  0.8272721        TRUE    OLS
#> 2      beta2 -1.6154039        TRUE    OLS
#> 3      beta3  0.3586729        TRUE    OLS
#> 4    gamma11  1.2279060        TRUE    OLS
#> 5    gamma21  1.3535571        TRUE    OLS
#> 6    gamma12 -0.4846708        TRUE    OLS
#> 7    gamma22 -1.4126826        TRUE    OLS
#> 8     theta0  0.8900722        TRUE    OLS
#> 9    theta_m -1.9041094        TRUE    OLS
#> 10   theta_x  1.5529579        TRUE    OLS
#> 11  theta_c1  2.0215448        TRUE    OLS