# Unidimensional Polytomous Testlet Response Theory Estimation Estimates item and person parameters for Polytomous Testlet models using Robust Newton-Raphson optimization. ```r trt_poly( data, group, model = c("GRT", "PCMT", "BiFT"), method = c("MLE", "EM"), control = list() ) ``` ## Arguments - `data`: A data.frame of polytomous responses. Rows=persons, Cols=items in testlets. - `group`: A list defining testlet structures. Example: `list(c(1,2,3), c(4,5,6))`. - `model`: Character. "GRT" (Graded Response Model), "PCMT" (Partial Credit Model for Testlet), or "BiFT" (Biffactor). - `method`: Character. "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`: A data frame of estimated item parameters. * `person_params`: A data frame of estimated person abilities and testlet effects . * `model_fit`: A data frame containing iterations and fit statistics such as Akaike’s Information Criterion (AIC), the Bayesian Information Criterion (BIC), and Log-Likelihood. ## Examples ```r # --- Example: Simulation (Mixed Categories GRT) --- set.seed(42) N <- 500; J <- 16 # Define Groups (4 Testlets) groups <- list(c(1:4), c(5:8), c(9:12), c(13:16)) # Define Categories (Binary, 3-cat, 4-cat, Mixed) # Items 1-4: 2 cats; 5-8: 3 cats; 9-12: 4 cats; 13-16: mixed cats <- c(rep(2, 4), rep(3, 4), rep(4, 4), 3, 5, 3, 5) # 1. Generate Parameters theta <- rnorm(N) # Gamma for 4 testlets (SD = 0.8) gamma <- matrix(rnorm(N * 4, 0, 0.8), N, 4) a <- rlnorm(J, 0, 0.2) b_list <- vector("list", J) # Generate Thresholds based on category count for(j in 1:J) { n_thresh <- cats[j] - 1 if(n_thresh == 1) { b_list[[j]] <- rnorm(1) } else { # Spread thresholds b_list[[j]] <- sort(rnorm(1) + seq(-1, 1, length.out=n_thresh)) } } # 2. Generate Responses (GRT Logic) resp <- matrix(NA, N, J) colnames(resp) <- paste0("Item_", 1:J) for(i in 1:N) { for(j in 1:J) { # Identify Testlet ID tid <- which(sapply(groups, function(x) j %in% x)) eff <- theta[i] + gamma[i, tid] # Calculate Probabilities (Graded Response) K <- cats[j] probs <- numeric(K) P_prev <- 1 for(k in 1:(K-1)) { term <- a[j] * (eff - b_list[[j]][k]) P_star <- 1 / (1 + exp(-term)) probs[k] <- P_prev - P_star P_prev <- P_star } probs[K] <- P_prev # Sample Response resp[i, j] <- sample(0:(K-1), 1, prob = probs) } } df_sim <- as.data.frame(resp) # 3. Run Estimation fit <- trt_poly( data = df_sim, group = groups, model = "GRT", method = "EM", control = list(max_iter = 20, verbose = FALSE) ) head(fit$item_params) head(fit$person_params) ```

trt_poly function