# 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. ```r 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"`. ## Returns `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 ```r 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 ```

COMMA_OLS function