# Item Response Theory Equating / Linking Conducts item response theory scale linking using Mean-Mean, Mean-Sigma, and Stocking-Lord methods. Supports mixed formats of both dichotomous and polytomous models. Automatically detects anchor items and validates model consistency. ```r equate_irt(base_params, new_params, person_params = NULL, methods = NULL) ``` ## Arguments - `base_params`: Data frame of reference item parameters (Form X). - `new_params`: Data frame of new item parameters to be transformed (Form Y). - `person_params`: (Optional) Data frame of person parameters from Form Y. - `methods`: Character vector. Options: "Mean-Mean", "Mean-Sigma", "Stocking-Lord". If NULL, defaults to all three. ## Returns A list containing three data frames: - **transformed_item_params**: New items transformed to Base scale (with SEs). - **transformed_person_params**: New persons transformed to Base scale (if provided). - **linking_constants**: The A (slope) and B (intercept) constants for each method. ## Examples ```r # =========================================================================== # Example: Equating Form Y (New) to Form X (Base) # =========================================================================== set.seed(123) # 1. Generate "True" Base Parameters (Form X) # --------------------------------------------------------------------------- # 10 Common Items (Anchors) + 10 Unique Items # 2PL and GRM mixed gen_item_params <- function(n, type="2PL") { if(type=="2PL") { data.frame( item = paste0("Item_", 1:n), model = "2PL", a = round(runif(n, 0.8, 1.5), 2), b = round(rnorm(n, 0, 1), 2), stringsAsFactors = FALSE ) } else { # GRM with 3 thresholds d <- t(apply(matrix(rnorm(n*3, 0, 0.5), n, 3), 1, sort)) df <- data.frame( item = paste0("Poly_", 1:n), model = "GRM", a = round(runif(n, 0.8, 1.5), 2), stringsAsFactors = FALSE ) df <- cbind(df, setNames(as.data.frame(d), paste0("step_", 1:3))) df } } # Anchors anchor_2pl <- gen_item_params(5, "2PL") anchor_grm <- gen_item_params(3, "GRM") # Unique Form X unique_x <- gen_item_params(5, "2PL") unique_x$item <- paste0("X_", unique_x$item) base_params <- dplyr::bind_rows(anchor_2pl, anchor_grm, unique_x) # 2. Generate "New" Form Y Parameters (with Scale Shift) # --------------------------------------------------------------------------- # Scale Transformation: Theta_base = 1.2 * Theta_new + 0.5 # True Constants: A = 1.2, B = 0.5 TRUE_A <- 1.2 TRUE_B <- 0.5 # Transform Anchor Parameters to "New" scale (Inverse Logic) # a_new = a_base * A # b_new = (b_base - B) / A anchor_2pl_new <- anchor_2pl anchor_2pl_new$a <- anchor_2pl$a * TRUE_A anchor_2pl_new$b <- (anchor_2pl$b - TRUE_B) / TRUE_A anchor_grm_new <- anchor_grm anchor_grm_new$a <- anchor_grm$a * TRUE_A step_cols <- grep("step_", names(anchor_grm_new)) anchor_grm_new[, step_cols] <- (anchor_grm[, step_cols] - TRUE_B) / TRUE_A # Unique Form Y unique_y <- gen_item_params(5, "2PL") unique_y$item <- paste0("Y_", unique_y$item) new_params <- dplyr::bind_rows(anchor_2pl_new, anchor_grm_new, unique_y) # 3. Create Dummy Person Parameters for Form Y # --------------------------------------------------------------------------- person_params <- data.frame( id = paste0("P", 1:50), theta = rnorm(50, 0, 1), theta_se = runif(50, 0.2, 0.5) ) # 4. Perform Equating # --------------------------------------------------------------------------- # We expect to recover A approx 1.2 and B approx 0.5 results <- equate_irt( base_params = base_params, new_params = new_params, person_params = person_params, methods = c("Mean-Mean", "Stocking-Lord") ) # 5. Inspect Results # --------------------------------------------------------------------------- # Linking Constants print(results$linking_constants) # Transformed Items (Form Y items on Form X scale) head(results$transformed_item_params) # Transformed Persons head(results$transformed_person_params) ```

equate_irt function