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EM Algorithm Estimation of the Binary Mediator Misclassification Model

COMMA_EM.Rd

Jointly estimate \(\beta\), \(\gamma\), and \(\theta\) parameters from the true mediator, observed mediator, and outcome mechanisms, respectively, in a binary mediator misclassification model.

Usage

COMMA_EM(
  Mstar,
  outcome,
  outcome_distribution,
  interaction_indicator,
  x_matrix,
  z_matrix,
  c_matrix,
  beta_start,
  gamma_start,
  theta_start,
  sigma_start = NULL,
  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.

outcome_distribution

A character string specifying the distribution of the outcome variable. Options are "Bernoulli", "Normal", or "Poisson".

interaction_indicator

A logical value indicating if an interaction between x and m should be used to generate the outcome variable, y.

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 (if interaction_indicator is FALSE) or 3 (if interaction_indicator is TRUE). 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, and, optionally, slope coefficient for xm).

sigma_start

A numeric value specifying the starting value for the standard deviation. This value is only required if outcome_distribution is "Normal". Otherwise, this value is set to NULL.

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_EM 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, followed by the standard error estimates, SE. The final column, Convergence, reports whether or not the algorithm converged for a given parameter estimate.

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 = "Bernoulli",
                           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"]]
                           
EM_results <- COMMA_EM(Mstar, outcome, "Bernoulli", FALSE,
                       x_matrix, z_matrix, c_matrix,
                       beta_start, gamma_start, theta_start)
#> Loading required package: doParallel
#> Loading required package: foreach
#> Loading required package: iterators
#> Loading required package: parallel
#> Loading required package: numDeriv
#> Loading required package: quantreg
#> Loading required package: SparseM
#> 
#> Attaching package: 'turboEM'
#> The following objects are masked from 'package:numDeriv':
#> 
#>     grad, hessian
#> Warning: non-integer #successes in a binomial glm!
#> Warning: non-integer #successes in a binomial glm!
#> Warning: non-integer #successes in a binomial glm!
#> Warning: non-integer #successes in a binomial glm!
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EM_results
#>    Parameter  Estimates Convergence
#> 1     beta_0  0.8355459        TRUE
#> 2     beta_1 -1.6157072        TRUE
#> 3     beta_2  0.3864456        TRUE
#> 4    gamma11  1.0924272        TRUE
#> 5    gamma21  1.4723205        TRUE
#> 6    gamma12 -0.2875945        TRUE
#> 7    gamma22 -1.7222641        TRUE
#> 8    theta_0  0.8914868        TRUE
#> 9   theta_x1  1.7143072        TRUE
#> 10   theta_m -1.9727559        TRUE
#> 11  theta_c1 -0.1759259        TRUE