# Estimate Bootstrap Standard Errors using EM ```r COMMA_EM_bootstrap_SE( parameter_estimates, sigma_estimate = 1, n_bootstrap, n_parallel, outcome_distribution, interaction_indicator, x_matrix, z_matrix, c_matrix, tolerance = 1e-07, max_em_iterations = 1500, em_method = "squarem", random_seed = NULL ) ``` ## Arguments - `parameter_estimates`: A column matrix of $\beta$, $\gamma$, and $\theta$ parameter values obtained from a COMMA analysis function. Parameter estimates should be supplied in the following order: 1) $\beta$ (intercept, slope), 2) $\gamma$ (intercept and slope from the M = 1 mechanism, intercept and slope from the M = 2 mechanism), and 3) $\theta$ (intercept, slope, coefficient for `x`, slope coefficient for `m`, slope coefficient for `c`, and, optionally, slope coefficient for `xm` if using). - `sigma_estimate`: A numeric value specifying the estimated standard deviation. This value is only required if `outcome_distribution` is `"Normal"`. Default is 1. For non-Normal outcome distributions, the value should be `NULL`. - `n_bootstrap`: A numeric value specifying the number of bootstrap samples to draw. - `n_parallel`: A numeric value specifying the number of parallel cores to run the computation on. - `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`. - `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"`. - `random_seed`: A numeric value specifying the random seed to set for bootstrap sampling. Default is `NULL`. ## Returns `COMMA_EM_bootstrap_SE` returns a list with two elements: 1) `bootstrap_df` and 2) `bootstrap_SE`. `bootstrap_df` is a data frame containing `COMMA_EM` output for each bootstrap sample. `bootstrap_SE` is a data frame containing bootstrap standard error estimates for each parameter. ## 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 = "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) EM_results EM_SEs <- COMMA_EM_bootstrap_SE(EM_results$Estimates, sigma_estimate = NULL, n_bootstrap = 3, n_parallel = 1, outcome_distribution = "Bernoulli", interaction_indicator = FALSE, x_matrix, z_matrix, c_matrix, random_seed = 1) EM_SEs$bootstrap_SE ```

COMMA_EM_bootstrap_SE function