# irtQ: Unidimensional Item Response Theory Modeling The `irtQ` package provides tools for fitting unidimensional item response theory (IRT) models to test data that include both dichotomous and polytomous items. It enables the calibration of pretest item parameters, estimation of examinees' abilities, and offers a comprehensive suite of utilities for unidimensional IRT analysis, including model-data fit evaluation, differential item functioning (DIF) detection, and more. For item parameter estimation, the package employs marginal maximum likelihood estimation via the expectation-maximization (MMLE-EM) algorithm (Bock & Aitkin, 1981). To calibrate pretest (newly developed) items, it supports both fixed item parameter calibration (FIPC; Kim, 2006) and fixed ability parameter calibration (FAPC; Ban et al., 2001; Stocking, 1988), also known as Stocking’s Method A. In addition, the package offers multiple-group item calibration via the MMLE-EM algorithm. For ability estimation, several widely used scoring methods are available, including: * Maximum likelihood estimation (ML) * Maximum likelihood estimation with fences (MLF; Han, 2016) * Weighted likelihood estimation (WL; Warm, 1989) * Maximum a posteriori estimation (MAP; Hambleton et al., 1991) * Expected a posteriori estimation (EAP; Bock & Mislevy, 1982) * EAP summed scoring (Thissen et al., 1995; Thissen & Orlando, 2001) * Inverse test characteristic curve (TCC) scoring (e.g., Kolen & Brennan, 2004; Kolen & Tong, 2010; Stocking, 1996) In addition, the package offers a variety of utilities for IRT analysis, including: * Evaluating model-data fit * Detecting DIF * Computing classification accuracy and consistency indices * Importing item or ability parameters from popular IRT software * Running flexMIRT (Cai, 2017) directly from R * Simulating response data * Computing the conditional distribution of observed scores using the Lord-Wingersky recursion * Calculating item and test information and characteristic functions * Visualizing item and test characteristic and information curves * Supporting additional tools for flexible and practical IRT analyses ||| |:--|:--| |Package:|irtQ| |Version:|1.0.0| |Date:|2025-07-16| |Depends:|R (>= 4.3)| |License:|GPL (>= 2)| package ## Details The following five sections provide an overview of: 1. The IRT models implemented in the `irtQ` package 2. How to estimate item parameters for a linear test form 3. How to perform pretest item calibration using the fixed item parameter calibration (FIPC) method 4. How to perform pretest item calibration using the fixed ability parameter calibration (FAPC) method 5. Illustrative examples of item calibration for a linear test form and for pretest items ## IRT Models The `irtQ` package supports both dichotomous and polytomous item response theory (IRT) models. For dichotomous items, the one-, two-, and three-parameter logistic models (1PLM, 2PLM, and 3PLM) are available. For polytomous items, the graded response model (GRM) and the (generalized) partial credit model (GPCM) are implemented. When fitting the partial credit model (PCM), note that item discrimination (or slope) parameters should be fixed to 1. In the following, let $Y$ be the response of an examinee with latent ability $\theta$ on an item, and suppose that each polytomous item has $K$ unique score categories. - **IRT 1–3PL models**: For the IRT 1PLM, 2PLM, and 3PLM models, the probability that an examinee with ability $\theta$ responds correctly to an item is given by: $$ P(Y = 1 \mid \theta) = g + \frac{(1 - g)}{1 + \exp(-Da(\theta - b))}, $$ where $a$ is the item discrimination (slope) parameter, $b$ is the item difficulty parameter, and $g$ is the guessing parameter. The constant $D$ is a scaling factor used to make the logistic function approximate the normal ogive function, typically set to $D = 1.702$. When the 1PLM is used, the discrimination parameter $a$ is either fixed to a constant value (e.g., $a = 1$) or constrained to be equal across all items. For 1PLM and 2PLM models, the guessing parameter is fixed at $g = 0$. - **GRM**: For the graded response model (GRM), the probability that an examinee with latent ability $\theta$ responds in score category $k$ ($k = 0, 1, \ldots, K - 1$) is given by: $$ P(Y = k \mid \theta) = P^{*}(Y \ge k \mid \theta) - P^{*}(Y \ge k + 1 \mid \theta), $$ $$ P^{*}(Y \ge k \mid \theta) = \frac{1}{1 + \exp(-Da(\theta - b_{k}))}, $$ $$ P^{*}(Y \ge k + 1 \mid \theta) = \frac{1}{1 + \exp(-Da(\theta - b_{k+1}))}, $$ where $P^{*}(Y \ge k \mid \theta)$ is the cumulative probability (or category boundary function) for score category $k$, with a functional form similar to the 2PL model. The parameter $b_{k}$ represents the difficulty (threshold) parameter associated with the boundary between categories $k - 1$ and $k$. Note that the probability of responding in the lowest or highest score category can be written as: $$ P(Y = 0 \mid \theta) = 1 - P^{*}(Y \ge 1 \mid \theta), $$ $$ P(Y = K - 1 \mid \theta) = P^{*}(Y \ge K - 1 \mid \theta). $$ - **GPCM**: For the generalized partial credit model (GPCM), the probability that an examinee with latent ability $\theta$ responds in score category $k$ ($k = 0, 1, \ldots, K - 1$) is given by: $$ P(Y = k \mid \theta) =\frac{\exp\left(\sum_{v = 0}^{k} Da(\theta - b_v)\right)}{\sum_{h = 0}^{K - 1} \exp\left(\sum_{v = 0}^{h} Da(\theta - b_v)\right)}, $$ where $b_v$ is the threshold (or step) parameter associated with category boundary $v$ of the item. In alternative parameterizations, $b_v$ can be expressed as $b_v = \beta - \tau_v$, where $\beta$ is the overall location (difficulty) parameter of the item, and $\tau_v$ is the threshold for score category $v$. In the `irtQ` package, an item with $K$ unique score categories requires $K - 1$ threshold parameters, as $b_0 = 0$ by convention. When fitting the partial credit model, the item discrimination parameter $a$ is fixed to 1. ## Item Calibration for a Linear Test Form Item parameter estimation for a linear test form can be performed using the `est_irt()` function, which implements marginal maximum likelihood estimation via the expectation-maximization (MMLE-EM) algorithm (Bock & Aitkin, 1981). The function returns item parameter estimates along with their standard errors, computed using the cross-product approximation method (Meilijson, 1989). The `irtQ` package supports calibration for mixed-format tests containing both dichotomous and polytomous items. It also provides a flexible set of options to address various practical calibration needs. For example, users can: * Specify prior distributions for item parameters * Fix specific parameters (e.g., the guessing parameter in the 3PL model) * Estimate the latent ability distribution using a nonparametric histogram method (Woods, 2007) In the `irtQ` package, item calibration for a linear test form typically involves two main steps: 1. Prepare the examinees' response data set for the linear test form To estimate item parameters using the `est_irt()` function, a response data set for the linear test form must first be prepared. The data should be provided in either a matrix or data frame format, where rows represent examinees and columns represent items. If there are missing responses, they should be properly coded (e.g., `NA`). 2. Estimate item parameters using the `est_irt()` function To estimate item parameters, several key input arguments must be specified in the `est_irt()` function: * `data`: A matrix or data frame containing examinees' item responses. * `model`: A character vector specifying the IRT model for each item (e.g., `"1PLM"`, `"2PLM"`, `"3PLM"`, `"GRM"`, `"GPCM"`). * `cats`: A numeric vector indicating the number of score categories for each item. For dichotomous items, use 2. * `D`: A scaling constant (typically 1.702) to align the logistic function with the normal ogive model. Optionally, you may incorporate prior distributions for item parameters: * `use.aprior`, `use.bprior`, `use.gprior`: Logical indicators specifying whether to apply prior distributions to the discrimination (`a`), difficulty (`b`), and guessing (`g`) parameters, respectively. * `aprior`, `bprior`, `gprior`: Lists specifying the distributional form and corresponding parameters for each prior. Supported distributions include Beta, Log-normal, and Normal. If the response data contain missing values, you must specify the missing value code via the `missing` argument. By default, the latent ability distribution is assumed to follow a standard normal distribution (i.e., N(0, 1)). However, users can estimate the empirical histogram of the latent distribution by setting `EmpHist = TRUE`, based on the nonparametric method proposed by Woods (2007). ## Pretest Item Calibration with the Fixed Item Parameter Calibration (FIPC) Method (e.g., Kim, 2006) The fixed item parameter calibration (FIPC) method is a widely used approach for calibrating pretest items in computerized adaptive testing (CAT). It enables the placement of parameter estimates for newly developed items onto the same scale as the operational item parameters (i.e., the scale of the item bank), without the need for post hoc linking or rescaling procedures (Ban et al., 2001; Chen & Wang, 2016). In FIPC, the parameters of the operational items are fixed, and the prior distribution of the latent ability variable is estimated during the calibration process. This estimated prior is used to place the pretest item parameters on the same scale as the fixed operational items (Kim, 2006). In the `irtQ` package, FIPC is implemented through the following three steps: 1. Prepare the item metadata, including both the operational items (to be fixed) and the pretest items. To perform FIPC using the `est_irt()` function, the item metadata must first be prepared. The item metadata is a structured data frame that includes essential information for each item, such as the number of score categories and the IRT model type. For more details, refer to the Details section of the `est_irt()` documentation. In the FIPC procedure, the metadata must contain both: * Operational items (whose parameters will be fixed), and * Pretest items (whose parameters will be freely estimated). For the pretest items, the `cats` (number of score categories) and `model` (IRT model type) must be accurately specified. However, the item parameter values (e.g., `par.1`, `par.2`, `par.3`) in the metadata serve only as placeholders and can be arbitrary, since the actual parameter estimates will be obtained during calibration. To facilitate creation of the metadata for FIPC, the helper function `shape_df_fipc()` can be used. 2. Prepare the response data set from examinees who answered both the operational and pretest items. To implement FIPC using the `est_irt()` function, examinees' response data for the test form must be provided, including both operational and pretest items. The response data should be in a matrix or data frame format, where rows represent examinees and columns represent items. Note that the column order of the response data must exactly match the row order of the item metadata. 3. Perform FIPC using the `est_irt()` function to calibrate the pretest items. When FIPC is performed using the `est_irt()` function, the parameters of pretest items are estimated while the parameters of operational items are fixed. To implement FIPC, you must provide the following arguments to `est_irt()`: * `x`: The item metadata, including both operational and pretest items. * `data`: The examinee response data corresponding to the item metadata. * `fipc = TRUE`: Enables fixed item parameter calibration. * `fipc.method`: Specifies the FIPC method to be used (e.g., `"MEM"`). * `fix.loc`: A vector indicating the positions of the operational items to be fixed. Optionally, you may estimate the empirical histogram and scale of the latent ability distribution by setting `EmpHist = TRUE`. If `EmpHist = FALSE`, a normal prior is assumed and its scale is updated iteratively during the EM algorithm. For additional details on implementing FIPC, refer to the documentation for `est_irt()`. ## Pretest Item Calibration with the Fixed Ability Parameter Calibration (FAPC) Method (e.g., Stocking, 1988) In computerized adaptive testing (CAT), the fixed ability parameter calibration (FAPC) method—also known as Stocking’s Method A (Stocking, 1988)—is one of the simplest and most straightforward approaches for calibrating pretest items. It involves estimating item parameters using maximum likelihood estimation, conditional on known or estimated proficiency values. FAPC is primarily used to place the parameter estimates of pretest items onto the same scale as the operational item parameters. It can also be used to recalibrate operational items when evaluating potential item parameter drift (Chen & Wang, 2016; Stocking, 1988). This method is known to produce accurate and unbiased item parameter estimates when items are randomly administered to examinees, rather than adaptively, which is often the case for pretest items (Ban et al., 2001; Chen & Wang, 2016). In the `irtQ` package, FAPC can be conducted in two main steps: 1. Prepare a data set containing both the item response data and the corresponding ability (proficiency) estimates. To use the `est_item()` function, two input data sets are required: * Ability estimates: A numeric vector containing examinees' ability (or proficiency) estimates. * Item response data: A matrix or data frame containing item responses, where rows represent examinees and columns represent items. The order of examinees in the response data must exactly match the order of the ability estimates. 2. Estimate the item parameters using the `est_item()` function. The `est_item()` function estimates pretest item parameters based on provided ability estimates. To use this function, you must specify the following arguments: * `data`: A matrix or data frame containing examinees' item responses. * `score`: A numeric vector of examinees' ability (proficiency) estimates. * `model`: A character vector specifying the IRT model for each item (e.g., `"1PLM"`, `"2PLM"`, `"3PLM"`, `"GRM"`, `"GPCM"`). * `cats`: A numeric vector indicating the number of score categories for each item. For dichotomous items, use 2. * `D`: A scaling constant (typically 1.702) to align the logistic function with the normal ogive model. For additional details on implementing FAPC, refer to the documentation for `est_item()`. ## Three examples of R scripts The examples below demonstrate how to estimate item parameters for a linear test form, and how to calibrate pretest items using the FIPC and FAPC methods. All examples use simulated data sets, where examinees' response data are generated based on true item parameters and true ability values using the `simdat()` function. The examples utilize item parameter estimates imported from a flexMIRT output file sample, which includes a mixed-format test of 55 items: 50 dichotomous items modeled with the 3PLM and 5 polytomous items modeled with the GRM. These item parameters are treated as the "true" values for data generation. Specifically: * Items 1–38: 3PLM items * Items 39–40: GRM items * Items 41–52: 3PLM items * Items 53–55: GRM items (all with 5 score categories) Example 1 – Estimating item parameters for a linear test form This example uses the first 40 items (38 3PLM and 2 GRM) to form a linear test. The test is assumed to be administered to a reference group of 2,000 examinees whose latent ability follows a standard normal distribution, N(0, 1). Examples 2 and 3 – Calibrating pretest items using FIPC and FAPC These examples assume a new test form consisting of all 55 items described above. The first 40 items are treated as the fixed operational items (i.e., the same as in Example 1), while the remaining 15 items are newly developed pretest items to be calibrated. The new test form is administered to a separate group of 2,000 examinees, whose ability distribution is assumed to follow N(0.5, 1.3²). The calibration of the pretest items is then conducted on the scale of the old form by using FIPC and FAPC procedures, respectively. ``` # Attach the packages library(irtQ) ##--------------------------------------------------------------------------- ## 1. Item parameter estimation for a linear test form ##--------------------------------------------------------------------------- ## Step 1: Prepare response data for the reference group ## Import the "-prm.txt" output file from flexMIRT meta_true <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ") # Extract item metadata using `irtQ::bring.flexmirt()` # This will serve as the base test form for later pretest item examples x_new <- irtQ::bring.flexmirt(file = meta_true, "par")$Group1$full_df # Extract items 1 to 40 to define the linear test form used in this illustration x_ref <- x_new[1:40, ] # Generate true ability values (N = 2,000) from N(0, 1) for the reference group set.seed(20) theta_ref <- rnorm(2000, mean = 0, sd = 1) # Simulate response data for the linear test form # Scaling factor D = 1 assumes a logistic IRT model data_ref <- irtQ::simdat(x = x_ref, theta = theta_ref, D = 1) ## Step 2: Estimate item parameters for the linear test form mod_ref <- irtQ::est_irt( data = data_ref, # Response data D = 1, # Scaling factor model = c(rep("3PLM", 38), rep("GRM", 2)), # Item models cats = c(rep(2, 38), rep(5, 2)), # Score categories per item item.id = paste0("Ref_I", 1:40), # Item IDs use.gprior = TRUE, # Use prior for guessing parameter gprior = list(dist = "beta", params = c(5, 16)),# Prior: Beta(5,16) for g Quadrature = c(49, 6), # 49 quadrature points from -6 to 6 group.mean = 0, group.var = 1, # Fixed latent ability: N(0,1) EmpHist = TRUE, # Estimate empirical ability distribution Etol = 1e-3, # E-step convergence tolerance MaxE = 500) # Max EM iterations # Summarize estimation results irtQ::summary(mod_ref) # Extract item parameter estimates est_ref <- mod_ref$par.est print(est_ref) ##------------------------------------------------------------------------------ ## 2. Pretest item calibration using Fixed Item Parameter Calibration (FIPC) ##------------------------------------------------------------------------------ ## Step 1: Prepare item metadata for both fixed operational items and pretest items # Define anchor item positions (items to be fixed) fixed_pos <- c(1:40) # Specify IDs, models, and categories for 15 pretest items # Includes 12 3PLM and 3 GRM items (each GRM has 5 categories) new_ids <- paste0("New_I", 1:15) new_models <- c(rep("3PLM", 12), rep("GRM", 3)) new_cats <- c(rep(2, 12), rep(5, 3)) # Construct item metadata using `shape_df_fipc()`. See Details of `shape_df_fipc()` # for more information # First 40 items are anchor items (fixed); last 15 are pretest (freely estimated) meta_fipc <- irtQ::shape_df_fipc(x = est_ref, fix.loc = fixed_pos, item.id = new_ids, cats = new_cats, model = new_models) ## Step 2: Prepare response data for the new test form # Generate latent abilities for 2,000 new examinees from N(0.5, 1.3²) set.seed(21) theta_new <- rnorm(2000, mean = 0.5, sd = 1.3) # Simulate response data using true item parameters and true abilities data_new <- irtQ::simdat(x = x_new, theta = theta_new, D = 1) ## Step 3: Calibrate pretest items using FIPC # Fit 3PLM to dichotomous and GRM to polytomous items # Fix first 40 items and freely estimate the remaining 15 pretest items mod_fipc <- irtQ::est_irt( x = meta_fipc, # Combined item metadata data = data_new, # Response data D = 1, # Scaling constant use.gprior = TRUE, # Use prior for guessing parameter gprior = list(dist = "beta", params = c(5, 16)), # Prior: Beta(5,16) for g Quadrature = c(49, 6), # 49 quadrature points from -6 to 6 EmpHist = TRUE, # Estimate empirical ability distribution Etol = 1e-3, # E-step convergence tolerance MaxE = 500, # Max EM iterations fipc = TRUE, # Enable FIPC fipc.method = "MEM", # Use Multiple EM cycles fix.loc = c(1:40)) # Anchor item positions to fix # Summarize estimation results irtQ::summary(mod_fipc) # Extract item parameter estimates est_new_fipc <- mod_fipc$par.est print(est_new_fipc) # Plot estimated empirical distribution of ability emphist <- irtQ::getirt(mod_fipc, what="weights") plot(emphist$weight ~ emphist$theta, xlab="Theta", ylab="Density", type = "h") ##------------------------------------------------------------------------------ ## 3. Pretest item calibration using Fixed Ability Parameter Calibration (FAPC) ##------------------------------------------------------------------------------ ## Step 1: Prepare response data and ability estimates # In FAPC, ability estimates are assumed known and fixed. # Estimate abilities for new examinees using the first 40 fixed operational (anchor) items only. # Pretest items are not used for scoring, as their parameters are not yet calibrated. # Estimate abilities using ML method via `irtQ::est_score()` # Based on fixed anchor item parameters and corresponding responses score_ml <- irtQ::est_score( x = est_ref, # Metadata with operational item parameters data = data_new[, 1:40], # Responses to anchor items D = 1, # Scaling constant method = "ML", # Scoring method: Maximum Likelihood range = c(-5, 5)) # Scoring bounds # Extract estimated abilities theta_est <- score_ml$est.theta ## Step 2: Calibrate pretest items using FAPC # Only the 15 pretest items are included in the calibration mod_fapc <- irtQ::est_item( data = data_new[, 41:55], # Responses to pretest items score = theta_est, # Fixed ability estimates D = 1, # Scaling constant model = c(rep("3PLM", 12), rep("GRM", 3)), # Item models cats = c(rep(2, 12), rep(5, 3)), # Score categories item.id = paste0("New_I", 1:15), # Item IDs use.gprior = TRUE, # Use prior for guessing parameter gprior = list(dist = "beta", params = c(5, 16)) # Prior: Beta(5,16) for g ) # Summarize estimation results irtQ::summary(mod_fapc) # Extract item parameter estimates est_new_fapc <- mod_fapc$par.est print(est_new_fapc) ``` ## References Ames, A. 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Chicago, IL: Scientific Software International. URL http://www.ssicentral.com ## Author(s) Hwanggyu Lim hglim83@gmail.com

irtQ-package