# Mixed Item Response Model Estimation (Dichotomous & Polytomous) Provides a estimation framework for a broad class of different item response theory models. This function can model different combinations of item categories. ```r mixed_irt(data, model = "2PL", method = "EM", control = list()) ``` ## Arguments - `data`: A N x J data.frame. Binary items must be 0/1. Polytomous items should be continuous integers (0, 1, 2...). - `model`: A character vector of length J (one model per item). Supported: "Rasch", "2PL" (2-Parameter Logistic), "3PL" (3-Parameter Logistic), "GRM" (Graded Response Model), "GPCM" (Generalized Partial Credit Model), "PCM" (Partial Credit Model). If a single string is provided, it is applied to all items. - `method`: String. "EM" (Marginal Maximum Likelihood via Expectation-Maximization) or "MLE" (Joint Maximum Likelihood). - `control`: A `list` of control parameters for the estimation algorithm: * `max_iter`: Maximum number of EM iterations (default = 100). * `converge_tol`: Convergence criterion for parameter change (default = 1e-4). * `theta_range`: Numeric vector of length 2 specifying the integration grid bounds (default = c(-4, 4)). * `quad_points`: Number of quadrature points (default = 21). * `verbose`: Logical; if `TRUE`, prints progress to console. ## Returns A list containing: * `item_params`: Data frame of item parameters (discrimination, difficulty/thresholds, guessing). * `person_params`: A data frame of estimated person abilities (theta) and standard errors. * `model_fit`: A data frame containing fit statistics such as Akaike’s Information Criterion (AIC) and the Bayesian Information Criterion (BIC). * `settings`: A list of control parameters used in the estimation. ## Examples ```r # --- Example 1: Simulation (Mixed 2PL + GPCM) --- set.seed(2025) N <- 100 n_bin <- 5 n_poly <- 2 J <- n_bin + n_poly # 1. Generate Theta (Wide range to match user request) true_theta <- rnorm(N, mean = 0, sd = 3) # 2. Simulation Helper: GPCM sim_gpcm <- function(theta, a, steps) { n_cat <- length(steps) + 1 probs <- matrix(0, length(theta), n_cat) for(k in 1:n_cat) { score <- k - 1 if(score == 0) numer <- rep(0, length(theta)) else numer <- a * (score * theta - sum(steps[1:score])) probs[, k] <- exp(numer) } probs <- probs / rowSums(probs) apply(probs, 1, function(p) sample(0:(n_cat-1), 1, prob=p)) } # 3. Create Data data_sim <- data.frame(matrix(NA, nrow = N, ncol = J)) colnames(data_sim) <- paste0("Item_", 1:J) # Binary Items (2PL) a_bin <- runif(n_bin, 0.8, 1.5) b_bin <- seq(-3, 3, length.out = n_bin) for(j in 1:n_bin) { prob <- 1 / (1 + exp(-(a_bin[j] * (true_theta - b_bin[j])))) data_sim[, j] <- rbinom(N, 1, prob) } # Polytomous Items (GPCM) # Item 6: 2 steps (-2, 2) data_sim[, 6] <- sim_gpcm(true_theta, a=1.0, steps=c(-2, 2)) # Item 7: 5 steps data_sim[, 7] <- sim_gpcm(true_theta, a=1.2, steps=c(-5, -2.5, 0, 2.5, 5)) # 4. Run Estimation # Note: Wide theta_range needed due to SD=3 in simulation my_models <- c(rep("2PL", n_bin), rep("GPCM", n_poly)) res <- mixed_irt(data = data_sim, model = my_models, method = "EM", control = list(max_iter = 20, theta_range = c(-6, 6))) head(res$item_params) print(res$model_fit) # --- Example 2: With Package Data --- data("ela2", package = "tirt") # Define Models (7 Binary, 3 Poly) real_models <- c(rep("2PL", 7), rep("GRM", 3)) # Run Estimation real_res <- mixed_irt(ela2, model = real_models, method = "EM", control = list(max_iter = 10)) head(real_res$item_params) print(real_res$model_fit) ```

mixed_irt function