# Fixed Item Calibration Estimates unknown item parameters using Marginal Maximum Likelihood via Expectation-Maximization Algorithm. Uses a custom Bounded Newton-Raphson solver. Supports mixed-format data containing dichotomous and polytomous responses ```r fixed_item(response_df, item_params_df, control = list()) ``` ## Arguments - `response_df`: A data.frame of responses. Rows=Students, Cols=Items. Data MUST be from 0-indexed (0, 1, 2...). - `item_params_df`: A data.frame of known parameters. Required: "item", "model". - `control`: A `list` of control parameters for the estimation algorithm: * `max_iter`: Maximum number of EM iterations (default = 50). * `conv_crit`: Convergence criterion for parameter change (default = 0.005). * `verbose`: Logical; if `TRUE`, prints progress to console. ## Returns A list containing: * `item_params`: Estimated parameters for unknown items. * `person_params`: Estimated person parameters. * `model_fit`: A data frame containing number off estimated parameters and fit statistics such as Akaike’s Information Criterion (AIC), the Bayesian Information Criterion (BIC), and Log-Likelihood. ## Examples ```r # 1. TOY EXAMPLE # =========================================================================== set.seed(123) # Create a very small dataset (N=50, J=4) N_toy <- 50 df_toy <- data.frame( I1 = rbinom(N_toy, 1, 0.5), I2 = rbinom(N_toy, 1, 0.6), # Known items U1 = rbinom(N_toy, 1, 0.5), U2 = rbinom(N_toy, 1, 0.4) # Unknown items ) # Define the "Known" parameters for I1 and I2 known_params <- data.frame( item = c("I1", "I2"), model = c("2PL", "2PL"), a = c(1.0, 1.2), b = c(-0.5, 0.5) ) # Run Fixed Item Calibration with very low iterations fit_toy <- fixed_item(df_toy, known_params, control=list(max_iter=2, verbose=FALSE)) print(head(fit_toy$item_params)) # --- Example 2: Simulation --- set.seed(123) N <- 500 true_theta <- rnorm(N, 0, 1) # 1. Simulation Helpers sim_2pl <- function(theta, a, b) { p <- 1 / (1 + exp(-1.7 * a * (theta - b))) rbinom(N, 1, p) } sim_poly <- function(theta, a, steps) { n_cat <- length(steps) + 1 probs <- matrix(0, length(theta), n_cat) for(k in 1:n_cat) { score <- k - 1 if(score == 0) num <- 0 else num <- a * (score * theta - sum(steps[1:score])) probs[, k] <- exp(num) } probs <- probs / rowSums(probs) apply(probs, 1, function(x) sample(0:(n_cat-1), 1, prob=x)) } # 2. Generate Data (Mixed Known/Unknown Items) # Items 1-5: Known Binary (2PL) # Items 6-10: Unknown Binary (2PL) # Items 11-12: Known Poly (GPCM) # Items 13-15: Unknown Poly (GPCM) resp_mat <- matrix(NA, N, 15) colnames(resp_mat) <- paste0("Item_", 1:15) # Known Binary Parameters a_bin <- c(1.0, 1.2, 0.9, 1.1, 0.8) b_bin <- c(-1, -0.5, 0, 0.5, 1) for(i in 1:5) resp_mat[,i] <- sim_2pl(true_theta, a_bin[i], b_bin[i]) for(i in 6:10) resp_mat[,i] <- sim_2pl(true_theta, runif(1,0.8,1.2), rnorm(1)) # Known Poly Parameters a_poly <- c(1.0, 0.8) d_poly <- list(c(-1, 1), c(-0.5, 0.5)) resp_mat[,11] <- sim_poly(true_theta, a_poly[1], d_poly[[1]]) resp_mat[,12] <- sim_poly(true_theta, a_poly[2], d_poly[[2]]) for(i in 13:15) resp_mat[,i] <- sim_poly(true_theta, 1.0, c(-0.5, 0.5)) df_resp <- as.data.frame(resp_mat) # 3. Create 'Known Parameters' Dataframe # This tells the function: "Fix these, Estimate the rest" known_df <- data.frame( item = c(paste0("Item_", 1:5), "Item_11", "Item_12"), model = c(rep("2PL", 5), rep("GPCM", 2)), a = c(a_bin, a_poly), b = c(b_bin, NA, NA), # Binary difficulty step_1 = c(rep(NA, 5), -1, -0.5), # Poly steps step_2 = c(rep(NA, 5), 1, 0.5), stringsAsFactors = FALSE ) # 4. Run Estimation res <- fixed_item(df_resp, known_df, control=list(max_iter=20)) # View Results # Notice Items 1-5 and 11-12 have Status "Fixed" head(res$item_params, 12) # --- Example 2: With Package Data --- data("ela1", package = "tirt") # Let's treat the first 5 items as "Known" with arbitrary parameters # just to demonstrate syntax. df_real <- ela1[, 1:20] known_real <- data.frame( item = paste0("Q", 1:5), model = "2PL", a = 1.0, b = seq(-1, 1, length.out=5) ) # Ideally, column names in df_real should match 'item' column in known_real colnames(df_real)[1:5] <- paste0("Q", 1:5) real_res <- fixed_item(df_real, known_real, control=list(max_iter=10)) head(real_res$item_params) ```

fixed_item function