The goal of irtQ is to fit unidimensional item response
theory (IRT) models to data that may include both dichotomous and
polytomous items. The package enables:
Item parameter estimation is conducted using marginal maximum
likelihood estimation via the expectation-maximization (MMLE-EM)
algorithm (Bock & Aitkin, 1981).
For pretest item calibration, irtQ supports:
For ability estimation, several widely used scoring methods are available, including:
Also, model fit assessment includes item fit statistics such as:
In addition, the package offers a variety of utilities for IRT analysis, including:
mirt R
package; the latter requires the suggested mirt
package)Beyond these IRT-based analyses, the package also provides a small
set of classical test theory (CTT) functions (ctt(),
freq_score(), ctt_distr(), and
score_resp()) for computing traditional item- and
test-level statistics and for scoring selected-response item data.
For full documentation, including function references and tutorial
articles, visit the package website: https://hwangQ.github.io/irtQ/. A short introduction is
available with vignette("irtQ", package = "irtQ").
You can install the released version of irtQ from CRAN with:
install.packages("irtQ")You can also install the latest development version of
irtQ from GitHub using the devtools
package:
install.packages("devtools")
devtools::install_github("hwangQ/irtQ")Item parameter estimation for a linear test form can be performed
using the irtQ::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:
In the irtQ package, item calibration for a linear test
form typically involves two main steps:
Prepare the examinees’ response data set for the linear test form
To estimate item parameters using the irtQ::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).
Estimate item parameters using the irtQ::est_irt()
function
To estimate item parameters, several key input arguments must be
specified in the irtQ::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; the default, 1, gives the
logistic metric, and 1.702 approximates the normal ogive.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).
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:
Prepare the item metadata, including both the operational items (to be fixed) and the pretest items.
To perform FIPC using the irtQ::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
irtQ::est_irt() documentation.
In the FIPC procedure, the metadata must contain both:
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
irtQ::shape_df_fipc() can be used.
Prepare the response data set from examinees who answered both the operational and pretest items.
To implement FIPC using the irtQ::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.
Perform FIPC using the irtQ::est_irt() function to
calibrate the pretest items.
When FIPC is performed using the irtQ::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
irtQ::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 irtQ::est_irt().
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:
Prepare a data set containing both the item response data and the corresponding ability (proficiency) estimates.
To use the irtQ::est_item() function, two input data
sets are required:
Estimate the item parameters using the
irtQ::est_item() function.
The irtQ::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; the default, 1, gives the
logistic metric, and 1.702 approximates the normal ogive.For additional details on implementing FAPC, refer to the
documentation for irtQ::est_item().
Evaluating how well an item response theory (IRT) model fits observed
response data is a critical step in psychometric analysis. The
irtQ package provides both statistical and graphical tools
for evaluating item-level model fit. These include traditional fit
statistics (e.g., X^2, G^2, infit, outfit, and S-X^2) and diagnostic
residual plots.
Model fit evaluation using irtQ typically involves the
following three steps:
Before conducting the IRT model fit analysis, three key data sets must be prepared:
Item metadata: A data frame containing item-level information, including:
You can either construct this data frame manually or generate it
using the irtQ::shape_df() function. Additionally, if item
parameters were estimated using other IRT software (e.g., BILOG-MG 3,
PARSCALE 4, flexMIRT, or the mirt R package), you can
import them using the corresponding irtQ::bring.*()
functions (e.g., irtQ::bring.flexmirt(),
irtQ::bring.bilog()).
Ability estimates: A numeric vector of examinees’ estimated proficiency values.
Response data: A matrix or data frame in which rows represent examinees and columns represent items. The order of examinees in this matrix must exactly match that of the ability estimates, and the column order must match the item metadata.
irtQ::irtfit()The irtQ::irtfit() function calculates widely used item
fit statistics, including:
To compute X^2 and G^2 statistics, the latent ability scale must be divided into several groups. Two grouping methods are available:
"equal.width": Divides the scale into intervals of
equal length"equal.freq": Divides the scale into groups with equal
numbers of examineesYou must also specify where the expected probabilities of item responses are calculated within each group. Two options are available:
"average": Uses the average ability estimate within
each group"middle": Uses the midpoint of each intervalTo implement this step, specify the item metadata (x),
ability estimates (score), and response data
(data) as arguments in the irtQ::irtfit()
function. If you want to use more or fewer than ten ability groups,
adjust the n.width argument accordingly. If the response
data contain missing values, specify the missing value code using the
missing argument.
Upon execution, the function returns item fit statistics and contingency tables used to compute the X^2 and G^2 statistics.
Note that the model-fit evaluation using the S-X^2 statistic can be
implemented using the irtQ::sx2_fit() function.
plot() methodAfter obtaining fit statistics using the irtQ::irtfit()
function, you can use the plot() method to visualize
residuals for individual items. Two types of plots are available: raw
residual plot and standardized residual plot.
To generate a plot, specify the item to be examined using the
item.loc argument. Only one item can be plotted at a
time.
The ci.method argument controls how confidence intervals
are computed in the raw residual plots. Supported methods include:
"wald": Wald interval based on the normal approximation
(Laplace,
1812)"wilson": Wilson score interval (Wilson, 1927)"wilson.cr": Wilson score interval with continuity
correction (Newcombe, 1998)These graphical diagnostics complement the statistical fit measures, allowing for deeper investigation into the adequacy of model fit for individual items.
# 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
## using the following arguments:
# 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 # Fix the latent scale: N(0, 1)
# EmpHist = TRUE # Estimate empirical ability distribution
# Etol = 1e-3 # E-step convergence tolerance
# MaxE = 500 # Max EM iterations
mod_ref <- irtQ::est_irt(
data = data_ref,
D = 1,
model = c(rep("3PLM", 38), rep("GRM", 2)),
cats = c(rep(2, 38), rep(5, 2)),
item.id = paste0("Ref_I", 1:40),
use.gprior = TRUE,
gprior = list(dist = "beta", params = c(5, 16)),
Quadrature = c(49, 6),
group.mean = 0,
group.var = 1,
EmpHist = TRUE,
Etol = 1e-3,
MaxE = 500)
#> Parsing input...
#> Estimating item parameters...
#> EM iteration: 1, Loglike: -53907.8298, Max-Change: 1.917401 EM iteration: 2, Loglike: -47810.7610, Max-Change: 0.333348 EM iteration: 3, Loglike: -47780.1401, Max-Change: 0.130911 EM iteration: 4, Loglike: -47777.7493, Max-Change: 0.064179 EM iteration: 5, Loglike: -47776.9296, Max-Change: 0.038227 EM iteration: 6, Loglike: -47776.4542, Max-Change: 0.026209 EM iteration: 7, Loglike: -47776.1402, Max-Change: 0.019566 EM iteration: 8, Loglike: -47775.9185, Max-Change: 0.015306 EM iteration: 9, Loglike: -47775.7539, Max-Change: 0.012285 EM iteration: 10, Loglike: -47775.6263, Max-Change: 0.01001 EM iteration: 11, Loglike: -47775.5239, Max-Change: 0.008238 EM iteration: 12, Loglike: -47775.4394, Max-Change: 0.006834 EM iteration: 13, Loglike: -47775.3679, Max-Change: 0.005706 EM iteration: 14, Loglike: -47775.3064, Max-Change: 0.004795 EM iteration: 15, Loglike: -47775.2525, Max-Change: 0.004052 EM iteration: 16, Loglike: -47775.2048, Max-Change: 0.003444 EM iteration: 17, Loglike: -47775.1621, Max-Change: 0.002944 EM iteration: 18, Loglike: -47775.1234, Max-Change: 0.002529 EM iteration: 19, Loglike: -47775.0882, Max-Change: 0.002184 EM iteration: 20, Loglike: -47775.0558, Max-Change: 0.001895 EM iteration: 21, Loglike: -47775.0259, Max-Change: 0.001652 EM iteration: 22, Loglike: -47774.9980, Max-Change: 0.001446 EM iteration: 23, Loglike: -47774.9719, Max-Change: 0.001271 EM iteration: 24, Loglike: -47774.9473, Max-Change: 0.001121 EM iteration: 25, Loglike: -47774.9241, Max-Change: 0.000993
#> Computing item parameter var-covariance matrix...
#> Estimation is finished in 1.32 seconds.
# Summarize estimation results
irtQ::summary(mod_ref)
#>
#> Call:
#> irtQ::est_irt(data = data_ref, D = 1, model = c(rep("3PLM", 38),
#> rep("GRM", 2)), cats = c(rep(2, 38), rep(5, 2)), item.id = paste0("Ref_I",
#> 1:40), use.gprior = TRUE, gprior = list(dist = "beta", params = c(5,
#> 16)), Quadrature = c(49, 6), group.mean = 0, group.var = 1,
#> EmpHist = TRUE, Etol = 0.001, MaxE = 500)
#>
#> Summary of the Data
#> Number of Items: 40
#> Number of Cases: 2000
#>
#> Summary of Estimation Process
#> Maximum number of EM cycles: 500
#> Convergence criterion of E-step: 0.001
#> Number of rectangular quadrature points: 49
#> Minimum & Maximum quadrature points: -6, 6
#> Number of free parameters: 124
#> Number of fixed items: 0
#> Number of E-step cycles completed: 25
#> Maximum parameter change: 0.0009933655
#>
#> Processing time (in seconds)
#> EM algorithm: 1.2
#> Standard error computation: 0.06
#> Total computation: 1.32
#>
#> Convergence and Stability of Solution
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix:
#> Variance-covariance matrix of item parameter estimates is obtainable.
#>
#> Summary of Estimation Results
#> -2loglikelihood: 95549.84
#> Akaike Information Criterion (AIC): 95797.84
#> Bayesian Information Criterion (BIC): 96492.35
#> Item Parameters:
#> id cats model par.1 se.1 par.2 se.2 par.3 se.3 par.4 se.4
#> 1 Ref_I1 2 3PLM 0.69 0.15 1.14 0.27 0.19 0.07 NA NA
#> 2 Ref_I2 2 3PLM 1.77 0.17 -1.02 0.14 0.20 0.07 NA NA
#> 3 Ref_I3 2 3PLM 1.52 0.21 0.71 0.10 0.24 0.03 NA NA
#> 4 Ref_I4 2 3PLM 1.05 0.12 -0.44 0.20 0.18 0.07 NA NA
#> 5 Ref_I5 2 3PLM 0.92 0.16 0.27 0.26 0.26 0.07 NA NA
#> 6 Ref_I6 2 3PLM 1.87 0.20 0.73 0.06 0.09 0.02 NA NA
#> 7 Ref_I7 2 3PLM 1.06 0.19 1.20 0.13 0.18 0.04 NA NA
#> 8 Ref_I8 2 3PLM 0.97 0.16 0.95 0.15 0.16 0.05 NA NA
#> 9 Ref_I9 2 3PLM 1.30 0.22 0.73 0.13 0.28 0.04 NA NA
#> 10 Ref_I10 2 3PLM 1.86 0.20 0.19 0.08 0.16 0.03 NA NA
#> 11 Ref_I11 2 3PLM 0.99 0.12 -0.27 0.19 0.16 0.06 NA NA
#> 12 Ref_I12 2 3PLM 1.04 0.16 1.16 0.12 0.12 0.04 NA NA
#> 13 Ref_I13 2 3PLM 1.22 0.21 1.33 0.11 0.15 0.03 NA NA
#> 14 Ref_I14 2 3PLM 1.54 0.19 0.24 0.11 0.24 0.04 NA NA
#> 15 Ref_I15 2 3PLM 1.48 0.16 0.01 0.11 0.17 0.05 NA NA
#> 16 Ref_I16 2 3PLM 2.35 0.19 0.01 0.05 0.07 0.02 NA NA
#> 17 Ref_I17 2 3PLM 1.48 0.17 -0.03 0.12 0.21 0.05 NA NA
#> 18 Ref_I18 2 3PLM 1.70 0.30 1.24 0.09 0.27 0.02 NA NA
#> 19 Ref_I19 2 3PLM 2.15 0.19 -1.01 0.10 0.15 0.06 NA NA
#> 20 Ref_I20 2 3PLM 1.76 0.20 -1.28 0.19 0.30 0.09 NA NA
#> 21 Ref_I21 2 3PLM 1.73 0.20 -0.94 0.16 0.26 0.08 NA NA
#> 22 Ref_I22 2 3PLM 0.90 0.14 -0.23 0.29 0.25 0.08 NA NA
#> 23 Ref_I23 2 3PLM 0.96 0.10 -0.27 0.17 0.13 0.06 NA NA
#> 24 Ref_I24 2 3PLM 1.05 0.25 1.54 0.15 0.23 0.04 NA NA
#> 25 Ref_I25 2 3PLM 0.71 0.09 -1.60 0.37 0.22 0.09 NA NA
#> 26 Ref_I26 2 3PLM 0.88 0.10 -1.98 0.31 0.22 0.10 NA NA
#> 27 Ref_I27 2 3PLM 1.41 0.16 0.13 0.11 0.16 0.04 NA NA
#> 28 Ref_I28 2 3PLM 2.73 0.30 0.12 0.06 0.25 0.03 NA NA
#> 29 Ref_I29 2 3PLM 1.27 0.13 -1.38 0.21 0.22 0.09 NA NA
#> 30 Ref_I30 2 3PLM 1.69 0.27 0.89 0.10 0.35 0.03 NA NA
#> 31 Ref_I31 2 3PLM 1.04 0.16 0.84 0.13 0.14 0.04 NA NA
#> 32 Ref_I32 2 3PLM 1.69 0.19 -0.70 0.16 0.30 0.07 NA NA
#> 33 Ref_I33 2 3PLM 1.24 0.11 -1.38 0.18 0.17 0.07 NA NA
#> 34 Ref_I34 2 3PLM 1.38 0.18 0.37 0.12 0.21 0.04 NA NA
#> 35 Ref_I35 2 3PLM 1.68 0.21 0.01 0.12 0.28 0.05 NA NA
#> 36 Ref_I36 2 3PLM 1.02 0.19 1.35 0.13 0.16 0.04 NA NA
#> 37 Ref_I37 2 3PLM 1.92 0.19 -0.21 0.09 0.17 0.04 NA NA
#> 38 Ref_I38 2 3PLM 0.72 0.10 -0.43 0.34 0.21 0.09 NA NA
#> 39 Ref_I39 5 GRM 1.96 0.09 -1.83 0.08 -1.17 0.05 -0.62 0.04
#> 40 Ref_I40 5 GRM 1.33 0.06 -0.73 0.06 -0.07 0.05 0.58 0.05
#> par.5 se.5
#> 1 NA NA
#> 2 NA NA
#> 3 NA NA
#> 4 NA NA
#> 5 NA NA
#> 6 NA NA
#> 7 NA NA
#> 8 NA NA
#> 9 NA NA
#> 10 NA NA
#> 11 NA NA
#> 12 NA NA
#> 13 NA NA
#> 14 NA NA
#> 15 NA NA
#> 16 NA NA
#> 17 NA NA
#> 18 NA NA
#> 19 NA NA
#> 20 NA NA
#> 21 NA NA
#> 22 NA NA
#> 23 NA NA
#> 24 NA NA
#> 25 NA NA
#> 26 NA NA
#> 27 NA NA
#> 28 NA NA
#> 29 NA NA
#> 30 NA NA
#> 31 NA NA
#> 32 NA NA
#> 33 NA NA
#> 34 NA NA
#> 35 NA NA
#> 36 NA NA
#> 37 NA NA
#> 38 NA NA
#> 39 -0.17 0.04
#> 40 1.10 0.06
#> Group Parameters:
#> mu sigma2 sigma
#> estimates 0 1 1
#> se NA NA NA
# Extract item parameter estimates
est_ref <- mod_ref$par.est
print(est_ref)
#> id cats model par.1 par.2 par.3 par.4 par.5
#> 1 Ref_I1 2 3PLM 0.6919957 1.14484402 0.18976499 NA NA
#> 2 Ref_I2 2 3PLM 1.7724464 -1.02099189 0.19616040 NA NA
#> 3 Ref_I3 2 3PLM 1.5150979 0.70966856 0.23867404 NA NA
#> 4 Ref_I4 2 3PLM 1.0524673 -0.43526130 0.17728798 NA NA
#> 5 Ref_I5 2 3PLM 0.9167739 0.26820573 0.26351619 NA NA
#> 6 Ref_I6 2 3PLM 1.8666135 0.72927989 0.08652424 NA NA
#> 7 Ref_I7 2 3PLM 1.0562015 1.19815132 0.18056280 NA NA
#> 8 Ref_I8 2 3PLM 0.9748064 0.94777748 0.15848887 NA NA
#> 9 Ref_I9 2 3PLM 1.3025729 0.73175285 0.28029919 NA NA
#> 10 Ref_I10 2 3PLM 1.8639741 0.18640838 0.16366457 NA NA
#> 11 Ref_I11 2 3PLM 0.9939469 -0.26792404 0.15892792 NA NA
#> 12 Ref_I12 2 3PLM 1.0372451 1.16353752 0.11695580 NA NA
#> 13 Ref_I13 2 3PLM 1.2190362 1.33299399 0.15362908 NA NA
#> 14 Ref_I14 2 3PLM 1.5357356 0.23646403 0.23961503 NA NA
#> 15 Ref_I15 2 3PLM 1.4824881 0.01278565 0.16718700 NA NA
#> 16 Ref_I16 2 3PLM 2.3536532 0.01021881 0.06936722 NA NA
#> 17 Ref_I17 2 3PLM 1.4812118 -0.02787402 0.21368590 NA NA
#> 18 Ref_I18 2 3PLM 1.7011526 1.23768149 0.26906305 NA NA
#> 19 Ref_I19 2 3PLM 2.1548402 -1.01143124 0.14829321 NA NA
#> 20 Ref_I20 2 3PLM 1.7590705 -1.27947219 0.30164290 NA NA
#> 21 Ref_I21 2 3PLM 1.7311826 -0.93888503 0.26145478 NA NA
#> 22 Ref_I22 2 3PLM 0.8950656 -0.23309090 0.24895260 NA NA
#> 23 Ref_I23 2 3PLM 0.9604247 -0.27453531 0.13157552 NA NA
#> 24 Ref_I24 2 3PLM 1.0475077 1.53775301 0.22975399 NA NA
#> 25 Ref_I25 2 3PLM 0.7093919 -1.60381822 0.21525701 NA NA
#> 26 Ref_I26 2 3PLM 0.8782582 -1.97974491 0.22367121 NA NA
#> 27 Ref_I27 2 3PLM 1.4142228 0.12900789 0.16073407 NA NA
#> 28 Ref_I28 2 3PLM 2.7271259 0.11722864 0.24875028 NA NA
#> 29 Ref_I29 2 3PLM 1.2736643 -1.38227009 0.22190752 NA NA
#> 30 Ref_I30 2 3PLM 1.6924004 0.89383796 0.35473129 NA NA
#> 31 Ref_I31 2 3PLM 1.0382178 0.83916998 0.14081448 NA NA
#> 32 Ref_I32 2 3PLM 1.6949111 -0.69941722 0.30436382 NA NA
#> 33 Ref_I33 2 3PLM 1.2377865 -1.38353599 0.16635813 NA NA
#> 34 Ref_I34 2 3PLM 1.3780871 0.37216894 0.20560614 NA NA
#> 35 Ref_I35 2 3PLM 1.6786172 0.01343367 0.28045264 NA NA
#> 36 Ref_I36 2 3PLM 1.0208415 1.35077167 0.16012085 NA NA
#> 37 Ref_I37 2 3PLM 1.9186509 -0.21446010 0.17043569 NA NA
#> 38 Ref_I38 2 3PLM 0.7239630 -0.42947088 0.21019599 NA NA
#> 39 Ref_I39 5 GRM 1.9602130 -1.83262071 -1.16744768 -0.6208679 -0.1692025
#> 40 Ref_I40 5 GRM 1.3329010 -0.72583244 -0.06982294 0.5783162 1.1047434
##------------------------------------------------------------------------------
## 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, 1.3^2)
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
## using the following arguments:
# 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
mod_fipc <- irtQ::est_irt(
x = meta_fipc,
data = data_new,
D = 1,
use.gprior = TRUE,
gprior = list(dist = "beta", params = c(5, 16)),
Quadrature = c(49, 6),
EmpHist = TRUE,
Etol = 1e-3,
MaxE = 500,
fipc = TRUE,
fipc.method = "MEM",
fix.loc = c(1:40))
#> Parsing input...
#> Estimating item parameters...
#> EM iteration: 1, Loglike: -41799.5018, Max-Change: 2.177366 EM iteration: 2, Loglike: -60177.8990, Max-Change: 0.660102 EM iteration: 3, Loglike: -60143.0624, Max-Change: 0.22625 EM iteration: 4, Loglike: -60141.1866, Max-Change: 0.082367 EM iteration: 5, Loglike: -60140.7281, Max-Change: 0.031397 EM iteration: 6, Loglike: -60140.4860, Max-Change: 0.012555 EM iteration: 7, Loglike: -60140.3168, Max-Change: 0.005361 EM iteration: 8, Loglike: -60140.1888, Max-Change: 0.002513 EM iteration: 9, Loglike: -60140.0888, Max-Change: 0.001386 EM iteration: 10, Loglike: -60140.0086, Max-Change: 0.001119 EM iteration: 11, Loglike: -60139.9427, Max-Change: 0.000908
#> Computing item parameter var-covariance matrix...
#> Estimation is finished in 0.51 seconds.
# Summarize estimation results
irtQ::summary(mod_fipc)
#>
#> Call:
#> irtQ::est_irt(x = meta_fipc, data = data_new, D = 1, use.gprior = TRUE,
#> gprior = list(dist = "beta", params = c(5, 16)), Quadrature = c(49,
#> 6), EmpHist = TRUE, Etol = 0.001, MaxE = 500, fipc = TRUE,
#> fipc.method = "MEM", fix.loc = c(1:40))
#>
#> Summary of the Data
#> Number of Items: 55
#> Number of Cases: 2000
#>
#> Summary of Estimation Process
#> Maximum number of EM cycles: 500
#> Convergence criterion of E-step: 0.001
#> Number of rectangular quadrature points: 49
#> Minimum & Maximum quadrature points: -6, 6
#> Number of free parameters: 53
#> Number of fixed items: 40
#> Number of E-step cycles completed: 11
#> Maximum parameter change: 0.0009076875
#>
#> Processing time (in seconds)
#> EM algorithm: 0.44
#> Standard error computation: 0.02
#> Total computation: 0.51
#>
#> Convergence and Stability of Solution
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix:
#> Variance-covariance matrix of item parameter estimates is obtainable.
#>
#> Summary of Estimation Results
#> -2loglikelihood: 120279.9
#> Akaike Information Criterion (AIC): 120385.9
#> Bayesian Information Criterion (BIC): 120682.7
#> Item Parameters:
#> id cats model par.1 se.1 par.2 se.2 par.3 se.3 par.4 se.4
#> 1 Ref_I1 2 3PLM 0.69 NA 1.14 NA 0.19 NA NA NA
#> 2 Ref_I2 2 3PLM 1.77 NA -1.02 NA 0.20 NA NA NA
#> 3 Ref_I3 2 3PLM 1.52 NA 0.71 NA 0.24 NA NA NA
#> 4 Ref_I4 2 3PLM 1.05 NA -0.44 NA 0.18 NA NA NA
#> 5 Ref_I5 2 3PLM 0.92 NA 0.27 NA 0.26 NA NA NA
#> 6 Ref_I6 2 3PLM 1.87 NA 0.73 NA 0.09 NA NA NA
#> 7 Ref_I7 2 3PLM 1.06 NA 1.20 NA 0.18 NA NA NA
#> 8 Ref_I8 2 3PLM 0.97 NA 0.95 NA 0.16 NA NA NA
#> 9 Ref_I9 2 3PLM 1.30 NA 0.73 NA 0.28 NA NA NA
#> 10 Ref_I10 2 3PLM 1.86 NA 0.19 NA 0.16 NA NA NA
#> 11 Ref_I11 2 3PLM 0.99 NA -0.27 NA 0.16 NA NA NA
#> 12 Ref_I12 2 3PLM 1.04 NA 1.16 NA 0.12 NA NA NA
#> 13 Ref_I13 2 3PLM 1.22 NA 1.33 NA 0.15 NA NA NA
#> 14 Ref_I14 2 3PLM 1.54 NA 0.24 NA 0.24 NA NA NA
#> 15 Ref_I15 2 3PLM 1.48 NA 0.01 NA 0.17 NA NA NA
#> 16 Ref_I16 2 3PLM 2.35 NA 0.01 NA 0.07 NA NA NA
#> 17 Ref_I17 2 3PLM 1.48 NA -0.03 NA 0.21 NA NA NA
#> 18 Ref_I18 2 3PLM 1.70 NA 1.24 NA 0.27 NA NA NA
#> 19 Ref_I19 2 3PLM 2.15 NA -1.01 NA 0.15 NA NA NA
#> 20 Ref_I20 2 3PLM 1.76 NA -1.28 NA 0.30 NA NA NA
#> 21 Ref_I21 2 3PLM 1.73 NA -0.94 NA 0.26 NA NA NA
#> 22 Ref_I22 2 3PLM 0.90 NA -0.23 NA 0.25 NA NA NA
#> 23 Ref_I23 2 3PLM 0.96 NA -0.27 NA 0.13 NA NA NA
#> 24 Ref_I24 2 3PLM 1.05 NA 1.54 NA 0.23 NA NA NA
#> 25 Ref_I25 2 3PLM 0.71 NA -1.60 NA 0.22 NA NA NA
#> 26 Ref_I26 2 3PLM 0.88 NA -1.98 NA 0.22 NA NA NA
#> 27 Ref_I27 2 3PLM 1.41 NA 0.13 NA 0.16 NA NA NA
#> 28 Ref_I28 2 3PLM 2.73 NA 0.12 NA 0.25 NA NA NA
#> 29 Ref_I29 2 3PLM 1.27 NA -1.38 NA 0.22 NA NA NA
#> 30 Ref_I30 2 3PLM 1.69 NA 0.89 NA 0.35 NA NA NA
#> 31 Ref_I31 2 3PLM 1.04 NA 0.84 NA 0.14 NA NA NA
#> 32 Ref_I32 2 3PLM 1.69 NA -0.70 NA 0.30 NA NA NA
#> 33 Ref_I33 2 3PLM 1.24 NA -1.38 NA 0.17 NA NA NA
#> 34 Ref_I34 2 3PLM 1.38 NA 0.37 NA 0.21 NA NA NA
#> 35 Ref_I35 2 3PLM 1.68 NA 0.01 NA 0.28 NA NA NA
#> 36 Ref_I36 2 3PLM 1.02 NA 1.35 NA 0.16 NA NA NA
#> 37 Ref_I37 2 3PLM 1.92 NA -0.21 NA 0.17 NA NA NA
#> 38 Ref_I38 2 3PLM 0.72 NA -0.43 NA 0.21 NA NA NA
#> 39 Ref_I39 5 GRM 1.96 NA -1.83 NA -1.17 NA -0.62 NA
#> 40 Ref_I40 5 GRM 1.33 NA -0.73 NA -0.07 NA 0.58 NA
#> 41 New_I1 2 3PLM 1.75 0.17 0.61 0.08 0.25 0.03 NA NA
#> 42 New_I2 2 3PLM 1.85 0.17 -1.21 0.14 0.20 0.07 NA NA
#> 43 New_I3 2 3PLM 1.61 0.13 0.49 0.08 0.14 0.03 NA NA
#> 44 New_I4 2 3PLM 1.06 0.10 -0.24 0.17 0.15 0.06 NA NA
#> 45 New_I5 2 3PLM 1.09 0.17 2.21 0.12 0.15 0.03 NA NA
#> 46 New_I6 2 3PLM 2.85 0.35 1.54 0.05 0.20 0.02 NA NA
#> 47 New_I7 2 3PLM 1.38 0.11 0.10 0.11 0.17 0.04 NA NA
#> 48 New_I8 2 3PLM 1.72 0.15 0.15 0.09 0.18 0.04 NA NA
#> 49 New_I9 2 3PLM 1.34 0.10 0.32 0.08 0.09 0.03 NA NA
#> 50 New_I10 2 3PLM 1.53 0.14 1.24 0.06 0.09 0.02 NA NA
#> 51 New_I11 2 3PLM 1.90 0.18 -0.99 0.13 0.21 0.06 NA NA
#> 52 New_I12 2 3PLM 1.35 0.16 -0.16 0.19 0.38 0.06 NA NA
#> 53 New_I13 5 GRM 1.25 0.05 -0.39 0.05 0.19 0.05 0.77 0.04
#> 54 New_I14 5 GRM 1.28 0.06 -2.17 0.11 -1.46 0.08 -0.74 0.06
#> 55 New_I15 5 GRM 0.91 0.05 -0.76 0.08 -0.04 0.06 0.61 0.05
#> par.5 se.5
#> 1 NA NA
#> 2 NA NA
#> 3 NA NA
#> 4 NA NA
#> 5 NA NA
#> 6 NA NA
#> 7 NA NA
#> 8 NA NA
#> 9 NA NA
#> 10 NA NA
#> 11 NA NA
#> 12 NA NA
#> 13 NA NA
#> 14 NA NA
#> 15 NA NA
#> 16 NA NA
#> 17 NA NA
#> 18 NA NA
#> 19 NA NA
#> 20 NA NA
#> 21 NA NA
#> 22 NA NA
#> 23 NA NA
#> 24 NA NA
#> 25 NA NA
#> 26 NA NA
#> 27 NA NA
#> 28 NA NA
#> 29 NA NA
#> 30 NA NA
#> 31 NA NA
#> 32 NA NA
#> 33 NA NA
#> 34 NA NA
#> 35 NA NA
#> 36 NA NA
#> 37 NA NA
#> 38 NA NA
#> 39 -0.17 NA
#> 40 1.10 NA
#> 41 NA NA
#> 42 NA NA
#> 43 NA NA
#> 44 NA NA
#> 45 NA NA
#> 46 NA NA
#> 47 NA NA
#> 48 NA NA
#> 49 NA NA
#> 50 NA NA
#> 51 NA NA
#> 52 NA NA
#> 53 1.22 0.05
#> 54 -0.13 0.05
#> 55 1.13 0.06
#> Group Parameters:
#> mu sigma2 sigma
#> estimates 0.55 1.50 1.22
#> se 0.03 0.05 0.02
# Extract item parameter estimates
est_new_fipc <- mod_fipc$par.est
print(est_new_fipc)
#> id cats model par.1 par.2 par.3 par.4 par.5
#> 1 Ref_I1 2 3PLM 0.6919957 1.14484402 0.18976499 NA NA
#> 2 Ref_I2 2 3PLM 1.7724464 -1.02099189 0.19616040 NA NA
#> 3 Ref_I3 2 3PLM 1.5150979 0.70966856 0.23867404 NA NA
#> 4 Ref_I4 2 3PLM 1.0524673 -0.43526130 0.17728798 NA NA
#> 5 Ref_I5 2 3PLM 0.9167739 0.26820573 0.26351619 NA NA
#> 6 Ref_I6 2 3PLM 1.8666135 0.72927989 0.08652424 NA NA
#> 7 Ref_I7 2 3PLM 1.0562015 1.19815132 0.18056280 NA NA
#> 8 Ref_I8 2 3PLM 0.9748064 0.94777748 0.15848887 NA NA
#> 9 Ref_I9 2 3PLM 1.3025729 0.73175285 0.28029919 NA NA
#> 10 Ref_I10 2 3PLM 1.8639741 0.18640838 0.16366457 NA NA
#> 11 Ref_I11 2 3PLM 0.9939469 -0.26792404 0.15892792 NA NA
#> 12 Ref_I12 2 3PLM 1.0372451 1.16353752 0.11695580 NA NA
#> 13 Ref_I13 2 3PLM 1.2190362 1.33299399 0.15362908 NA NA
#> 14 Ref_I14 2 3PLM 1.5357356 0.23646403 0.23961503 NA NA
#> 15 Ref_I15 2 3PLM 1.4824881 0.01278565 0.16718700 NA NA
#> 16 Ref_I16 2 3PLM 2.3536532 0.01021881 0.06936722 NA NA
#> 17 Ref_I17 2 3PLM 1.4812118 -0.02787402 0.21368590 NA NA
#> 18 Ref_I18 2 3PLM 1.7011526 1.23768149 0.26906305 NA NA
#> 19 Ref_I19 2 3PLM 2.1548402 -1.01143124 0.14829321 NA NA
#> 20 Ref_I20 2 3PLM 1.7590705 -1.27947219 0.30164290 NA NA
#> 21 Ref_I21 2 3PLM 1.7311826 -0.93888503 0.26145478 NA NA
#> 22 Ref_I22 2 3PLM 0.8950656 -0.23309090 0.24895260 NA NA
#> 23 Ref_I23 2 3PLM 0.9604247 -0.27453531 0.13157552 NA NA
#> 24 Ref_I24 2 3PLM 1.0475077 1.53775301 0.22975399 NA NA
#> 25 Ref_I25 2 3PLM 0.7093919 -1.60381822 0.21525701 NA NA
#> 26 Ref_I26 2 3PLM 0.8782582 -1.97974491 0.22367121 NA NA
#> 27 Ref_I27 2 3PLM 1.4142228 0.12900789 0.16073407 NA NA
#> 28 Ref_I28 2 3PLM 2.7271259 0.11722864 0.24875028 NA NA
#> 29 Ref_I29 2 3PLM 1.2736643 -1.38227009 0.22190752 NA NA
#> 30 Ref_I30 2 3PLM 1.6924004 0.89383796 0.35473129 NA NA
#> 31 Ref_I31 2 3PLM 1.0382178 0.83916998 0.14081448 NA NA
#> 32 Ref_I32 2 3PLM 1.6949111 -0.69941722 0.30436382 NA NA
#> 33 Ref_I33 2 3PLM 1.2377865 -1.38353599 0.16635813 NA NA
#> 34 Ref_I34 2 3PLM 1.3780871 0.37216894 0.20560614 NA NA
#> 35 Ref_I35 2 3PLM 1.6786172 0.01343367 0.28045264 NA NA
#> 36 Ref_I36 2 3PLM 1.0208415 1.35077167 0.16012085 NA NA
#> 37 Ref_I37 2 3PLM 1.9186509 -0.21446010 0.17043569 NA NA
#> 38 Ref_I38 2 3PLM 0.7239630 -0.42947088 0.21019599 NA NA
#> 39 Ref_I39 5 GRM 1.9602130 -1.83262071 -1.16744768 -0.6208679 -0.1692025
#> 40 Ref_I40 5 GRM 1.3329010 -0.72583244 -0.06982294 0.5783162 1.1047434
#> 41 New_I1 2 3PLM 1.7543402 0.60768748 0.24512492 NA NA
#> 42 New_I2 2 3PLM 1.8507316 -1.20642098 0.20158480 NA NA
#> 43 New_I3 2 3PLM 1.6143166 0.49324431 0.13562223 NA NA
#> 44 New_I4 2 3PLM 1.0553544 -0.23865064 0.15287748 NA NA
#> 45 New_I5 2 3PLM 1.0900633 2.21190002 0.15223745 NA NA
#> 46 New_I6 2 3PLM 2.8521623 1.54040816 0.19668704 NA NA
#> 47 New_I7 2 3PLM 1.3791948 0.09594720 0.17338939 NA NA
#> 48 New_I8 2 3PLM 1.7222468 0.15443907 0.17622375 NA NA
#> 49 New_I9 2 3PLM 1.3362979 0.32068908 0.08720578 NA NA
#> 50 New_I10 2 3PLM 1.5308617 1.24212064 0.08671002 NA NA
#> 51 New_I11 2 3PLM 1.9021076 -0.98959007 0.21130200 NA NA
#> 52 New_I12 2 3PLM 1.3469003 -0.16129939 0.37842809 NA NA
#> 53 New_I13 5 GRM 1.2493662 -0.38617153 0.18897747 0.7711592 1.2172181
#> 54 New_I14 5 GRM 1.2823294 -2.16859443 -1.45652159 -0.7445505 -0.1294246
#> 55 New_I15 5 GRM 0.9145699 -0.76032211 -0.03746317 0.6086947 1.1283965
# 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
# using the following arguments:
# 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
score_ml <- irtQ::est_score(
x = est_ref,
data = data_new[, 1:40],
D = 1,
method = "ML",
range = c(-5, 5))
# 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
# using the following arguments:
# 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
mod_fapc <- irtQ::est_item(
data = data_new[, 41:55],
score = theta_est,
D = 1,
model = c(rep("3PLM", 12), rep("GRM", 3)),
cats = c(rep(2, 12), rep(5, 3)),
item.id = paste0("New_I", 1:15),
use.gprior = TRUE,
gprior = list(dist = "beta", params = c(5, 16))
)
#> Starting...
#> Parsing input...
#> Estimating item parameters...
#> Estimation is finished.
# Summarize estimation results
irtQ::summary(mod_fapc)
#>
#> Call:
#> irtQ::est_item(data = data_new[, 41:55], score = theta_est, D = 1,
#> model = c(rep("3PLM", 12), rep("GRM", 3)), cats = c(rep(2,
#> 12), rep(5, 3)), item.id = paste0("New_I", 1:15), use.gprior = TRUE,
#> gprior = list(dist = "beta", params = c(5, 16)))
#>
#> Summary of the Data
#> Number of Items in Response Data: 15
#> Number of Excluded Items: 0
#> Number of free parameters: 51
#> Number of Responses for Each Item:
#> id n
#> 1 New_I1 2000
#> 2 New_I2 2000
#> 3 New_I3 2000
#> 4 New_I4 2000
#> 5 New_I5 2000
#> 6 New_I6 2000
#> 7 New_I7 2000
#> 8 New_I8 2000
#> 9 New_I9 2000
#> 10 New_I10 2000
#> 11 New_I11 2000
#> 12 New_I12 2000
#> 13 New_I13 2000
#> 14 New_I14 2000
#> 15 New_I15 2000
#>
#> Processing time (in seconds)
#> Total computation: 0.54
#>
#> Convergence of Solution
#> All item parameters were successfully converged.
#>
#> Summary of Estimation Results
#> -2loglikelihood: 36901.01
#> Item Parameters:
#> id cats model par.1 se.1 par.2 se.2 par.3 se.3 par.4 se.4
#> 1 New_I1 2 3PLM 1.41 0.12 0.58 0.09 0.23 0.03 NA NA
#> 2 New_I2 2 3PLM 1.73 0.16 -1.17 0.15 0.28 0.06 NA NA
#> 3 New_I3 2 3PLM 1.34 0.10 0.46 0.08 0.12 0.03 NA NA
#> 4 New_I4 2 3PLM 0.94 0.08 -0.27 0.16 0.16 0.05 NA NA
#> 5 New_I5 2 3PLM 0.72 0.09 2.58 0.17 0.12 0.03 NA NA
#> 6 New_I6 2 3PLM 1.73 0.15 1.66 0.06 0.18 0.02 NA NA
#> 7 New_I7 2 3PLM 1.15 0.09 0.04 0.12 0.17 0.04 NA NA
#> 8 New_I8 2 3PLM 1.46 0.11 0.10 0.09 0.17 0.03 NA NA
#> 9 New_I9 2 3PLM 1.14 0.07 0.29 0.08 0.08 0.03 NA NA
#> 10 New_I10 2 3PLM 1.17 0.09 1.32 0.07 0.07 0.02 NA NA
#> 11 New_I11 2 3PLM 1.75 0.18 -0.97 0.16 0.28 0.06 NA NA
#> 12 New_I12 2 3PLM 1.15 0.12 -0.22 0.19 0.38 0.05 NA NA
#> 13 New_I13 5 GRM 1.04 0.04 -0.50 0.06 0.16 0.05 0.83 0.05
#> 14 New_I14 5 GRM 1.06 0.05 -2.59 0.13 -1.75 0.10 -0.92 0.07
#> 15 New_I15 5 GRM 0.77 0.04 -0.94 0.09 -0.10 0.07 0.65 0.06
#> par.5 se.5
#> 1 NA NA
#> 2 NA NA
#> 3 NA NA
#> 4 NA NA
#> 5 NA NA
#> 6 NA NA
#> 7 NA NA
#> 8 NA NA
#> 9 NA NA
#> 10 NA NA
#> 11 NA NA
#> 12 NA NA
#> 13 1.34 0.06
#> 14 -0.21 0.05
#> 15 1.25 0.07
#>
#> Group Parameters:
#> mu sigma
#> 0.58 1.42
# Extract item parameter estimates
est_new_fapc <- mod_fapc$par.est
print(est_new_fapc)
#> id cats model par.1 par.2 par.3 par.4 par.5
#> 1 New_I1 2 3PLM 1.4108912 0.5752356 0.22780986 NA NA
#> 2 New_I2 2 3PLM 1.7292652 -1.1664492 0.28099633 NA NA
#> 3 New_I3 2 3PLM 1.3445775 0.4589801 0.11966233 NA NA
#> 4 New_I4 2 3PLM 0.9392278 -0.2741224 0.16246450 NA NA
#> 5 New_I5 2 3PLM 0.7187061 2.5813126 0.12067576 NA NA
#> 6 New_I6 2 3PLM 1.7274273 1.6586861 0.17543426 NA NA
#> 7 New_I7 2 3PLM 1.1541765 0.0407375 0.16859783 NA NA
#> 8 New_I8 2 3PLM 1.4568544 0.1037810 0.16684661 NA NA
#> 9 New_I9 2 3PLM 1.1397906 0.2946300 0.08121185 NA NA
#> 10 New_I10 2 3PLM 1.1743444 1.3207411 0.07014496 NA NA
#> 11 New_I11 2 3PLM 1.7464945 -0.9654624 0.27579057 NA NA
#> 12 New_I12 2 3PLM 1.1510321 -0.2226698 0.38023720 NA NA
#> 13 New_I13 5 GRM 1.0422956 -0.5008251 0.15722021 0.8253882 1.3390717
#> 14 New_I14 5 GRM 1.0561218 -2.5854106 -1.75056858 -0.9230562 -0.2103185
#> 15 New_I15 5 GRM 0.7683113 -0.9434727 -0.10444162 0.6454301 1.2493058
## ----------------------------------------------------------------------------
## 4. IRT model-data fit evaluation using `irtQ::irtfit()`
## ----------------------------------------------------------------------------
## Step 1: Prepare the data set for IRT model fit analysis
## In this example, we use a simulated mixed-format CAT data set.
## Only items with more than 1,000 non-missing responses are evaluated.
# Identify items with more than 1,000 valid responses
over1000 <- which(colSums(!is.na(simCAT_MX$res.dat)) > 1000)
# (1) Item metadata
x <- simCAT_MX$item.prm[over1000, ]
dim(x)
#> [1] 122 7
print(x[1:10, ])
#> id cats model par.1 par.2 par.3 par.4
#> 1 V1 2 2PLM 0.9664715 -0.8408555 NA NA
#> 2 V2 2 2PLM 0.9152754 1.3843593 NA NA
#> 3 V3 2 2PLM 1.3454796 -1.2554919 NA NA
#> 5 V5 2 2PLM 1.0862914 1.7114409 NA NA
#> 6 V6 2 2PLM 1.1311496 -0.6029080 NA NA
#> 7 V7 2 2PLM 1.2012407 -0.4721664 NA NA
#> 8 V8 2 2PLM 1.3244155 -0.6353713 NA NA
#> 10 V10 2 2PLM 1.2487125 0.1381082 NA NA
#> 11 V11 2 2PLM 1.4413208 1.2276303 NA NA
#> 12 V12 2 2PLM 1.2077273 -0.8017795 NA NA
# (2) Examinees' ability estimates
score <- simCAT_MX$score
length(score)
#> [1] 30000
print(score[1:100])
#> [1] -0.30311440 -0.67224807 -0.73474583 1.76935738 -0.91017203 -0.28448278
#> [7] 0.81656431 -1.66434615 0.59312008 -0.35182937 0.23129679 -0.93107524
#> [13] -0.29971993 -0.32700449 -0.22271651 1.48912121 -0.92927809 0.43453041
#> [19] -0.01795450 -0.28365286 0.01115173 -0.76101441 0.12144273 0.83096135
#> [25] 1.96600585 -0.83510402 -0.40268865 -0.05605526 0.72398446 -0.16026059
#> [31] -1.09011778 1.22126764 -0.13340360 -1.28230720 -1.05581980 0.83484173
#> [37] -0.52136360 -0.66913590 -1.08580804 1.73214834 0.56950387 0.48016332
#> [43] -0.03472720 -2.17577824 0.44127032 0.98913071 1.43861714 -1.08133809
#> [49] -0.69016072 0.19325797 0.89998383 1.25383167 -1.09600809 0.50519143
#> [55] -0.51707395 -0.39474484 -0.45031102 1.85675021 1.50768131 1.06011811
#> [61] -0.41064797 1.10960278 -0.68853387 -0.59397660 -0.65326436 0.29147751
#> [67] -1.86787473 1.04838050 -1.14582092 1.07395234 -0.03828693 0.08445559
#> [73] 0.34582524 0.72300905 0.84448992 -1.86488055 0.77121937 1.66573208
#> [79] 0.10311673 -0.50768866 -1.60992457 -0.23074682 0.16162326 0.26091160
#> [85] 0.60682182 0.65415304 -0.69923141 1.07545766 0.24060267 -0.93542383
#> [91] 1.24988766 -0.01826940 1.27403936 0.10985621 -1.19092047 0.79614598
#> [97] 0.62302338 -0.89455596 -0.03472720 0.20250837
# (3) Response data
data <- simCAT_MX$res.dat[, over1000]
dim(data)
#> [1] 30000 122
print(data[1:20, 1:6])
#> Item.dc.1 Item.dc.2 Item.dc.3 Item.dc.5 Item.dc.6 Item.dc.7
#> [1,] NA NA NA NA NA 0
#> [2,] NA NA NA NA NA 0
#> [3,] NA NA NA NA NA 1
#> [4,] NA NA NA 0 NA NA
#> [5,] NA NA 1 NA 0 1
#> [6,] NA NA 0 NA 0 1
#> [7,] NA NA NA NA NA NA
#> [8,] 1 NA 1 NA 1 1
#> [9,] NA NA NA 0 NA NA
#> [10,] NA NA 0 NA 1 1
#> [11,] NA NA NA 0 NA NA
#> [12,] NA NA 1 NA 0 1
#> [13,] NA NA 0 NA 1 1
#> [14,] NA NA NA NA NA 1
#> [15,] NA NA NA NA 1 1
#> [16,] NA 1 NA 0 NA NA
#> [17,] NA NA 0 NA 0 1
#> [18,] NA NA NA NA NA NA
#> [19,] NA NA 0 NA 0 1
#> [20,] NA NA NA NA NA NA
## Step 2: Compute IRT model-data fit statistics
# (1) Using the "equal.width" method to form ability groups
fit1 <- irtfit(
x = x, score = score, data = data, group.method = "equal.width",
n.width = 11, loc.theta = "average", range.score = c(-4, 4),
D = 1, alpha = 0.05, missing = NA, overSR = 2.5
)
# Inspect the structure of the returned object
names(fit1)
#> [1] "fit_stat" "contingency.fitstat" "contingency.plot"
#> [4] "item_df" "individual.info" "ancillary"
#> [7] "call"
# View the first 10 rows of fit statistics
fit1$fit_stat[1:10, ]
#> id X2 G2 df.X2 df.G2 crit.val.X2 crit.val.G2 p.X2 p.G2 outfit
#> 1 V1 82.646 84.105 8 10 15.51 18.31 0 0 0.940
#> 2 V2 75.070 75.209 8 10 15.51 18.31 0 0 1.018
#> 3 V3 186.880 168.082 8 10 15.51 18.31 0 0 1.124
#> 4 V5 151.329 139.213 8 10 15.51 18.31 0 0 1.133
#> 5 V6 178.409 157.911 8 10 15.51 18.31 0 0 1.056
#> 6 V7 185.438 170.360 9 11 16.92 19.68 0 0 1.078
#> 7 V8 209.653 193.001 8 10 15.51 18.31 0 0 1.098
#> 8 V10 267.444 239.563 9 11 16.92 19.68 0 0 1.097
#> 9 V11 148.896 133.209 7 9 14.07 16.92 0 0 1.129
#> 10 V12 139.295 125.647 9 11 16.92 19.68 0 0 1.065
#> infit N overSR.prop
#> 1 0.945 2558 0.545
#> 2 1.016 2018 0.364
#> 3 1.090 11041 0.636
#> 4 1.111 5181 0.727
#> 5 1.045 13599 0.545
#> 6 1.059 18293 0.455
#> 7 1.075 16163 0.636
#> 8 1.073 19702 0.727
#> 9 1.083 13885 0.455
#> 10 1.051 12118 0.636
# View the contingency table for the first item (dichotomous)
fit1$contingency.fitstat[[1]]
#> total obs.freq.0 obs.freq.1 exp.freq.0 exp.freq.1 obs.prop.0 obs.prop.1
#> 1 238 196 42 186.633670 51.366330 0.8235294 0.1764706
#> 2 445 352 93 324.652889 120.347111 0.7910112 0.2089888
#> 3 455 317 138 308.205375 146.794625 0.6967033 0.3032967
#> 4 458 313 145 286.290084 171.709916 0.6834061 0.3165939
#> 5 299 205 94 174.952852 124.047148 0.6856187 0.3143813
#> 6 297 196 101 159.760191 137.239809 0.6599327 0.3400673
#> 7 203 127 76 99.726943 103.273057 0.6256158 0.3743842
#> 8 125 77 48 55.552477 69.447523 0.6160000 0.3840000
#> 9 25 15 10 9.782949 15.217051 0.6000000 0.4000000
#> 10 13 5 8 4.319149 8.680851 0.3846154 0.6153846
#> exp.prob.0 exp.prob.1 raw.rsd.0 raw.rsd.1
#> 1 0.7841751 0.2158249 0.03935433 -0.03935433
#> 2 0.7295571 0.2704429 0.06145418 -0.06145418
#> 3 0.6773745 0.3226255 0.01932885 -0.01932885
#> 4 0.6250875 0.3749125 0.05831859 -0.05831859
#> 5 0.5851266 0.4148734 0.10049213 -0.10049213
#> 6 0.5379131 0.4620869 0.12201956 -0.12201956
#> 7 0.4912657 0.5087343 0.13435004 -0.13435004
#> 8 0.4444198 0.5555802 0.17158019 -0.17158019
#> 9 0.3913179 0.6086821 0.20868206 -0.20868206
#> 10 0.3322423 0.6677577 0.05237312 -0.05237312
# (2) Using the "equal.freq" method to form ability groups
fit2 <- irtfit(
x = x, score = score, data = data, group.method = "equal.freq",
n.width = 11, loc.theta = "average", range.score = c(-4, 4),
D = 1, alpha = 0.05, missing = NA
)
# View the first 10 rows of fit statistics
fit2$fit_stat[1:10, ]
#> id X2 G2 df.X2 df.G2 crit.val.X2 crit.val.G2 p.X2 p.G2 outfit
#> 1 V1 83.193 85.029 8 10 15.51 18.31 0 0 0.940
#> 2 V2 79.629 79.941 9 11 16.92 19.68 0 0 1.018
#> 3 V3 200.266 180.620 9 11 16.92 19.68 0 0 1.124
#> 4 V5 148.742 138.244 9 11 16.92 19.68 0 0 1.133
#> 5 V6 141.905 135.027 9 11 16.92 19.68 0 0 1.056
#> 6 V7 189.680 178.200 9 11 16.92 19.68 0 0 1.078
#> 7 V8 214.014 198.621 9 11 16.92 19.68 0 0 1.098
#> 8 V10 258.335 237.874 9 11 16.92 19.68 0 0 1.097
#> 9 V11 162.225 146.413 9 11 16.92 19.68 0 0 1.129
#> 10 V12 147.600 136.192 9 11 16.92 19.68 0 0 1.065
#> infit N overSR.prop
#> 1 0.945 2558 0.600
#> 2 1.016 2018 0.636
#> 3 1.090 11041 0.636
#> 4 1.111 5181 0.727
#> 5 1.045 13599 0.636
#> 6 1.059 18293 0.455
#> 7 1.075 16163 0.545
#> 8 1.073 19702 0.636
#> 9 1.083 13885 0.636
#> 10 1.051 12118 0.455
# View the contingency table for the 113th item (polytomous)
fit2$contingency.fitstat[[113]]
#> total obs.freq.0 obs.freq.1 obs.freq.2 obs.freq.3 exp.freq.0 exp.freq.1
#> 1 398 226 172 0 0 111.87605 135.75420
#> 2 404 161 177 66 0 84.44214 125.75188
#> 3 421 122 96 91 112 70.71786 120.13631
#> 4 406 63 202 74 67 58.60545 108.32861
#> 5 378 35 125 70 148 47.42153 94.39910
#> 6 438 1 168 122 147 46.19867 100.34772
#> 7 391 0 115 96 180 34.59119 81.74650
#> 8 422 0 0 111 311 31.61964 80.66715
#> 9 369 0 0 79 290 24.10207 65.37509
#> 10 443 0 0 61 382 22.68985 68.35441
#> 11 421 0 0 3 418 13.81829 49.97358
#> exp.freq.2 exp.freq.3 obs.prop.0 obs.prop.1 obs.prop.2 obs.prop.3
#> 1 61.89381 88.47593 0.567839196 0.4321608 0.000000000 0.0000000
#> 2 70.36352 123.44246 0.398514851 0.4381188 0.163366337 0.0000000
#> 3 76.68267 153.46316 0.289786223 0.2280285 0.216152019 0.2660333
#> 4 75.23608 163.82986 0.155172414 0.4975369 0.182266010 0.1650246
#> 5 70.60541 165.57396 0.092592593 0.3306878 0.185185185 0.3915344
#> 6 81.89613 209.55747 0.002283105 0.3835616 0.278538813 0.3356164
#> 7 72.58564 202.07667 0.000000000 0.2941176 0.245524297 0.4603581
#> 8 77.32401 232.38920 0.000000000 0.0000000 0.263033175 0.7369668
#> 9 66.62667 212.89617 0.000000000 0.0000000 0.214092141 0.7859079
#> 10 77.37120 274.58454 0.000000000 0.0000000 0.137697517 0.8623025
#> 11 67.90543 289.30270 0.000000000 0.0000000 0.007125891 0.9928741
#> exp.prob.0 exp.prob.1 exp.prob.2 exp.prob.3 raw.rsd.0 raw.rsd.1
#> 1 0.28109561 0.3410910 0.1555121 0.2223013 0.28674359 0.09106984
#> 2 0.20901519 0.3112670 0.1741671 0.3055507 0.18949966 0.12685179
#> 3 0.16797592 0.2853594 0.1821441 0.3645206 0.12181030 -0.05733089
#> 4 0.14434840 0.2668192 0.1853105 0.4035218 0.01082401 0.23071771
#> 5 0.12545378 0.2497331 0.1867868 0.4380263 -0.03286119 0.08095476
#> 6 0.10547642 0.2291044 0.1869775 0.4784417 -0.10319332 0.15445725
#> 7 0.08846851 0.2090703 0.1856410 0.5168201 -0.08846851 0.08504731
#> 8 0.07492806 0.1911544 0.1832322 0.5506853 -0.07492806 -0.19115439
#> 9 0.06531726 0.1771683 0.1805601 0.5769544 -0.06531726 -0.17716827
#> 10 0.05121861 0.1542989 0.1746528 0.6198297 -0.05121861 -0.15429889
#> 11 0.03282254 0.1187021 0.1612956 0.6871798 -0.03282254 -0.11870210
#> raw.rsd.2 raw.rsd.3
#> 1 -0.155512095 -0.22230134
#> 2 -0.010800794 -0.30555065
#> 3 0.034007906 -0.09848731
#> 4 -0.003044527 -0.23849719
#> 5 -0.001601613 -0.04649195
#> 6 0.091561350 -0.14282528
#> 7 0.059883275 -0.05646208
#> 8 0.079800926 0.18628152
#> 9 0.033532060 0.20895346
#> 10 -0.036955308 0.24247281
#> 11 -0.154169672 0.30569431
## Step 3: Draw residual plots for IRT model-data fit diagnostics
# 1. Dichotomous item
# (1) Both raw and standardized residual plots
plot(x = fit1, item.loc = 1, type = "both", ci.method = "wald",
ylim.sr.adjust = TRUE)
#> interval point total obs.freq.0 obs.freq.1 obs.prop.0
#> 1 [-2.175778,-1.961629) -2.17577824 238 196 42 0.8235294
#> 2 [-1.961629,-1.747479) -1.86765906 445 352 93 0.7910112
#> 3 [-1.747479,-1.533329) -1.60831924 455 317 138 0.6967033
#> 4 [-1.533329,-1.31918) -1.36978887 458 313 145 0.6834061
#> 5 [-1.31918,-1.10503) -1.19663917 299 205 94 0.6856187
#> 6 [-1.10503,-0.8908802) -0.99807073 297 196 101 0.6599327
#> 7 [-0.8908802,-0.6767305) -0.80470269 203 127 76 0.6256158
#> 8 [-0.6767305,-0.4625808) -0.60986749 125 77 48 0.6160000
#> 9 [-0.4625808,-0.2484312) -0.38375387 25 15 10 0.6000000
#> 10 [-0.2484312,-0.03428149) -0.15648039 11 4 7 0.3636364
#> 11 [-0.03428149,0.1798682] 0.09933816 2 1 1 0.5000000
#> obs.prop.1 exp.prob.0 exp.prob.1 raw.rsd.0 raw.rsd.1 se.0
#> 1 0.1764706 0.7841751 0.2158249 0.03935433 -0.03935433 0.02666667
#> 2 0.2089888 0.7295571 0.2704429 0.06145418 -0.06145418 0.02105656
#> 3 0.3032967 0.6773745 0.3226255 0.01932885 -0.01932885 0.02191584
#> 4 0.3165939 0.6250875 0.3749125 0.05831859 -0.05831859 0.02262052
#> 5 0.3143813 0.5851266 0.4148734 0.10049213 -0.10049213 0.02849359
#> 6 0.3400673 0.5379131 0.4620869 0.12201956 -0.12201956 0.02892942
#> 7 0.3743842 0.4912657 0.5087343 0.13435004 -0.13435004 0.03508777
#> 8 0.3840000 0.4444198 0.5555802 0.17158019 -0.17158019 0.04444420
#> 9 0.4000000 0.3913179 0.6086821 0.20868206 -0.20868206 0.09760906
#> 10 0.6363636 0.3404187 0.6595813 0.02321769 -0.02321769 0.14287114
#> 11 0.5000000 0.2872720 0.7127280 0.21272800 -0.21272800 0.31995843
#> se.1 std.rsd.0 std.rsd.1
#> 1 0.02666667 1.4757869 -1.4757869
#> 2 0.02105656 2.9185289 -2.9185289
#> 3 0.02191584 0.8819579 -0.8819579
#> 4 0.02262052 2.5781277 -2.5781277
#> 5 0.02849359 3.5268334 -3.5268334
#> 6 0.02892942 4.2178369 -4.2178369
#> 7 0.03508777 3.8289710 -3.8289710
#> 8 0.04444420 3.8605756 -3.8605756
#> 9 0.09760906 2.1379374 -2.1379374
#> 10 0.14287114 0.1625079 -0.1625079
#> 11 0.31995843 0.6648614 -0.6648614
# (2) Raw residual plot only
plot(x = fit1, item.loc = 1, type = "icc", ci.method = "wald",
ylim.sr.adjust = TRUE)
#> interval point total obs.freq.0 obs.freq.1 obs.prop.0
#> 1 [-2.175778,-1.961629) -2.17577824 238 196 42 0.8235294
#> 2 [-1.961629,-1.747479) -1.86765906 445 352 93 0.7910112
#> 3 [-1.747479,-1.533329) -1.60831924 455 317 138 0.6967033
#> 4 [-1.533329,-1.31918) -1.36978887 458 313 145 0.6834061
#> 5 [-1.31918,-1.10503) -1.19663917 299 205 94 0.6856187
#> 6 [-1.10503,-0.8908802) -0.99807073 297 196 101 0.6599327
#> 7 [-0.8908802,-0.6767305) -0.80470269 203 127 76 0.6256158
#> 8 [-0.6767305,-0.4625808) -0.60986749 125 77 48 0.6160000
#> 9 [-0.4625808,-0.2484312) -0.38375387 25 15 10 0.6000000
#> 10 [-0.2484312,-0.03428149) -0.15648039 11 4 7 0.3636364
#> 11 [-0.03428149,0.1798682] 0.09933816 2 1 1 0.5000000
#> obs.prop.1 exp.prob.0 exp.prob.1 raw.rsd.0 raw.rsd.1 se.0
#> 1 0.1764706 0.7841751 0.2158249 0.03935433 -0.03935433 0.02666667
#> 2 0.2089888 0.7295571 0.2704429 0.06145418 -0.06145418 0.02105656
#> 3 0.3032967 0.6773745 0.3226255 0.01932885 -0.01932885 0.02191584
#> 4 0.3165939 0.6250875 0.3749125 0.05831859 -0.05831859 0.02262052
#> 5 0.3143813 0.5851266 0.4148734 0.10049213 -0.10049213 0.02849359
#> 6 0.3400673 0.5379131 0.4620869 0.12201956 -0.12201956 0.02892942
#> 7 0.3743842 0.4912657 0.5087343 0.13435004 -0.13435004 0.03508777
#> 8 0.3840000 0.4444198 0.5555802 0.17158019 -0.17158019 0.04444420
#> 9 0.4000000 0.3913179 0.6086821 0.20868206 -0.20868206 0.09760906
#> 10 0.6363636 0.3404187 0.6595813 0.02321769 -0.02321769 0.14287114
#> 11 0.5000000 0.2872720 0.7127280 0.21272800 -0.21272800 0.31995843
#> se.1 std.rsd.0 std.rsd.1
#> 1 0.02666667 1.4757869 -1.4757869
#> 2 0.02105656 2.9185289 -2.9185289
#> 3 0.02191584 0.8819579 -0.8819579
#> 4 0.02262052 2.5781277 -2.5781277
#> 5 0.02849359 3.5268334 -3.5268334
#> 6 0.02892942 4.2178369 -4.2178369
#> 7 0.03508777 3.8289710 -3.8289710
#> 8 0.04444420 3.8605756 -3.8605756
#> 9 0.09760906 2.1379374 -2.1379374
#> 10 0.14287114 0.1625079 -0.1625079
#> 11 0.31995843 0.6648614 -0.6648614
# (3) Standardized residual plot only
plot(x = fit1, item.loc = 1, type = "sr", ci.method = "wald",
ylim.sr.adjust = TRUE)
#> interval point total obs.freq.0 obs.freq.1 obs.prop.0
#> 1 [-2.175778,-1.961629) -2.17577824 238 196 42 0.8235294
#> 2 [-1.961629,-1.747479) -1.86765906 445 352 93 0.7910112
#> 3 [-1.747479,-1.533329) -1.60831924 455 317 138 0.6967033
#> 4 [-1.533329,-1.31918) -1.36978887 458 313 145 0.6834061
#> 5 [-1.31918,-1.10503) -1.19663917 299 205 94 0.6856187
#> 6 [-1.10503,-0.8908802) -0.99807073 297 196 101 0.6599327
#> 7 [-0.8908802,-0.6767305) -0.80470269 203 127 76 0.6256158
#> 8 [-0.6767305,-0.4625808) -0.60986749 125 77 48 0.6160000
#> 9 [-0.4625808,-0.2484312) -0.38375387 25 15 10 0.6000000
#> 10 [-0.2484312,-0.03428149) -0.15648039 11 4 7 0.3636364
#> 11 [-0.03428149,0.1798682] 0.09933816 2 1 1 0.5000000
#> obs.prop.1 exp.prob.0 exp.prob.1 raw.rsd.0 raw.rsd.1 se.0
#> 1 0.1764706 0.7841751 0.2158249 0.03935433 -0.03935433 0.02666667
#> 2 0.2089888 0.7295571 0.2704429 0.06145418 -0.06145418 0.02105656
#> 3 0.3032967 0.6773745 0.3226255 0.01932885 -0.01932885 0.02191584
#> 4 0.3165939 0.6250875 0.3749125 0.05831859 -0.05831859 0.02262052
#> 5 0.3143813 0.5851266 0.4148734 0.10049213 -0.10049213 0.02849359
#> 6 0.3400673 0.5379131 0.4620869 0.12201956 -0.12201956 0.02892942
#> 7 0.3743842 0.4912657 0.5087343 0.13435004 -0.13435004 0.03508777
#> 8 0.3840000 0.4444198 0.5555802 0.17158019 -0.17158019 0.04444420
#> 9 0.4000000 0.3913179 0.6086821 0.20868206 -0.20868206 0.09760906
#> 10 0.6363636 0.3404187 0.6595813 0.02321769 -0.02321769 0.14287114
#> 11 0.5000000 0.2872720 0.7127280 0.21272800 -0.21272800 0.31995843
#> se.1 std.rsd.0 std.rsd.1
#> 1 0.02666667 1.4757869 -1.4757869
#> 2 0.02105656 2.9185289 -2.9185289
#> 3 0.02191584 0.8819579 -0.8819579
#> 4 0.02262052 2.5781277 -2.5781277
#> 5 0.02849359 3.5268334 -3.5268334
#> 6 0.02892942 4.2178369 -4.2178369
#> 7 0.03508777 3.8289710 -3.8289710
#> 8 0.04444420 3.8605756 -3.8605756
#> 9 0.09760906 2.1379374 -2.1379374
#> 10 0.14287114 0.1625079 -0.1625079
#> 11 0.31995843 0.6648614 -0.6648614
# 2. Polytomous item
# (1) Both raw and standardized residual plots
plot(x = fit1, item.loc = 113, type = "both", ci.method = "wald",
ylim.sr.adjust = TRUE)
#> interval point total obs.freq.0 obs.freq.1 obs.freq.2
#> 1 [-0.1202983,0.01795651) -0.03804602 29 29 0 0
#> 2 [0.01795651,0.1562113) 0.09936557 133 112 21 0
#> 3 [0.1562113,0.2944662) 0.22102526 257 93 151 13
#> 4 [0.2944662,0.432721) 0.36224032 420 187 180 53
#> 5 [0.432721,0.5709758) 0.50694958 671 141 214 137
#> 6 [0.5709758,0.7092307) 0.63423837 709 46 308 157
#> 7 [0.7092307,0.8474855) 0.77733968 755 0 181 219
#> 8 [0.8474855,0.9857403) 0.89421166 653 0 0 130
#> 9 [0.9857403,1.123995) 1.02377099 516 0 0 63
#> 10 [1.123995,1.26225) 1.19789074 326 0 0 1
#> 11 [1.26225,1.400505] 1.32465982 22 0 0 0
#> obs.freq.3 obs.prop.0 obs.prop.1 obs.prop.2 obs.prop.3 exp.prob.0
#> 1 0 1.00000000 0.0000000 0.000000000 0.0000000 0.36102600
#> 2 0 0.84210526 0.1578947 0.000000000 0.0000000 0.30481458
#> 3 0 0.36186770 0.5875486 0.050583658 0.0000000 0.25690066
#> 4 0 0.44523810 0.4285714 0.126190476 0.0000000 0.20519378
#> 5 179 0.21013413 0.3189270 0.204172876 0.2667660 0.15821250
#> 6 198 0.06488011 0.4344147 0.221438646 0.2792666 0.12283649
#> 7 355 0.00000000 0.2397351 0.290066225 0.4701987 0.09007484
#> 8 523 0.00000000 0.0000000 0.199081164 0.8009188 0.06863423
#> 9 453 0.00000000 0.0000000 0.122093023 0.8779070 0.04990471
#> 10 325 0.00000000 0.0000000 0.003067485 0.9969325 0.03172799
#> 11 22 0.00000000 0.0000000 0.000000000 1.0000000 0.02247920
#> exp.prob.1 exp.prob.2 exp.prob.3 raw.rsd.0 raw.rsd.1 raw.rsd.2
#> 1 0.35529236 0.1313745 0.1523071 0.63897400 -0.35529236 -0.13137450
#> 2 0.34719988 0.1485940 0.1993916 0.53729068 -0.18930514 -0.14859398
#> 3 0.33306323 0.1622430 0.2477931 0.10496704 0.25448540 -0.11165935
#> 4 0.30915739 0.1750140 0.3106348 0.24004431 0.11941404 -0.04882356
#> 5 0.27805125 0.1836059 0.3801303 0.05192163 0.04087573 0.02056694
#> 6 0.24718962 0.1869007 0.4430732 -0.05795638 0.18722504 0.03453796
#> 7 0.21107263 0.1858396 0.5130130 -0.09007484 0.02866247 0.10422667
#> 8 0.18212707 0.1815875 0.5676512 -0.06863423 -0.18212707 0.01749362
#> 9 0.15199988 0.1739493 0.6241461 -0.04990471 -0.15199988 -0.05185631
#> 10 0.11630628 0.1601923 0.6917735 -0.03172799 -0.11630628 -0.15712479
#> 11 0.09430184 0.1486405 0.7345784 -0.02247920 -0.09430184 -0.14864051
#> raw.rsd.3 se.0 se.1 se.2 se.3 std.rsd.0
#> 1 -0.15230713 0.089189111 0.08887413 0.06272965 0.06672374 7.1642602
#> 2 -0.19939157 0.039915574 0.04128137 0.03084204 0.03464477 13.4606780
#> 3 -0.24779310 0.027254580 0.02939944 0.02299723 0.02693064 3.8513542
#> 4 -0.31063479 0.019705528 0.02255042 0.01854108 0.02258006 12.1815722
#> 5 -0.11336430 0.014088358 0.01729635 0.01494624 0.01873938 3.6854278
#> 6 -0.16380663 0.012327666 0.01620074 0.01464044 0.01865579 -4.7013261
#> 7 -0.04281429 0.010419122 0.01485118 0.01415633 0.01819070 -8.6451471
#> 8 0.23326768 0.009894046 0.01510336 0.01508595 0.01938659 -6.9369226
#> 9 0.25376091 0.009585825 0.01580501 0.01668745 0.02132199 -5.2060944
#> 10 0.30515905 0.009707584 0.01775594 0.02031430 0.02557456 -3.2683714
#> 11 0.26542156 0.031604006 0.06230752 0.07584269 0.09414036 -0.7112771
#> std.rsd.1 std.rsd.2 std.rsd.3
#> 1 -3.997703 -2.094297 -2.282653
#> 2 -4.585728 -4.817903 -5.755315
#> 3 8.656130 -4.855340 -9.201158
#> 4 5.295423 -2.633264 -13.757040
#> 5 2.363258 1.376061 -6.049523
#> 6 11.556575 2.359080 -8.780471
#> 7 1.929979 7.362550 -2.353637
#> 8 -12.058712 1.159597 12.032427
#> 9 -9.617197 -3.107504 11.901368
#> 10 -6.550274 -7.734688 11.932133
#> 11 -1.513490 -1.959853 2.819424
# (2) Raw residual plot only, with two columns in layout
plot(x = fit1, item.loc = 113, type = "icc", ci.method = "wald",
layout.col = 2, ylim.sr.adjust = TRUE)
#> interval point total obs.freq.0 obs.freq.1 obs.freq.2
#> 1 [-0.1202983,0.01795651) -0.03804602 29 29 0 0
#> 2 [0.01795651,0.1562113) 0.09936557 133 112 21 0
#> 3 [0.1562113,0.2944662) 0.22102526 257 93 151 13
#> 4 [0.2944662,0.432721) 0.36224032 420 187 180 53
#> 5 [0.432721,0.5709758) 0.50694958 671 141 214 137
#> 6 [0.5709758,0.7092307) 0.63423837 709 46 308 157
#> 7 [0.7092307,0.8474855) 0.77733968 755 0 181 219
#> 8 [0.8474855,0.9857403) 0.89421166 653 0 0 130
#> 9 [0.9857403,1.123995) 1.02377099 516 0 0 63
#> 10 [1.123995,1.26225) 1.19789074 326 0 0 1
#> 11 [1.26225,1.400505] 1.32465982 22 0 0 0
#> obs.freq.3 obs.prop.0 obs.prop.1 obs.prop.2 obs.prop.3 exp.prob.0
#> 1 0 1.00000000 0.0000000 0.000000000 0.0000000 0.36102600
#> 2 0 0.84210526 0.1578947 0.000000000 0.0000000 0.30481458
#> 3 0 0.36186770 0.5875486 0.050583658 0.0000000 0.25690066
#> 4 0 0.44523810 0.4285714 0.126190476 0.0000000 0.20519378
#> 5 179 0.21013413 0.3189270 0.204172876 0.2667660 0.15821250
#> 6 198 0.06488011 0.4344147 0.221438646 0.2792666 0.12283649
#> 7 355 0.00000000 0.2397351 0.290066225 0.4701987 0.09007484
#> 8 523 0.00000000 0.0000000 0.199081164 0.8009188 0.06863423
#> 9 453 0.00000000 0.0000000 0.122093023 0.8779070 0.04990471
#> 10 325 0.00000000 0.0000000 0.003067485 0.9969325 0.03172799
#> 11 22 0.00000000 0.0000000 0.000000000 1.0000000 0.02247920
#> exp.prob.1 exp.prob.2 exp.prob.3 raw.rsd.0 raw.rsd.1 raw.rsd.2
#> 1 0.35529236 0.1313745 0.1523071 0.63897400 -0.35529236 -0.13137450
#> 2 0.34719988 0.1485940 0.1993916 0.53729068 -0.18930514 -0.14859398
#> 3 0.33306323 0.1622430 0.2477931 0.10496704 0.25448540 -0.11165935
#> 4 0.30915739 0.1750140 0.3106348 0.24004431 0.11941404 -0.04882356
#> 5 0.27805125 0.1836059 0.3801303 0.05192163 0.04087573 0.02056694
#> 6 0.24718962 0.1869007 0.4430732 -0.05795638 0.18722504 0.03453796
#> 7 0.21107263 0.1858396 0.5130130 -0.09007484 0.02866247 0.10422667
#> 8 0.18212707 0.1815875 0.5676512 -0.06863423 -0.18212707 0.01749362
#> 9 0.15199988 0.1739493 0.6241461 -0.04990471 -0.15199988 -0.05185631
#> 10 0.11630628 0.1601923 0.6917735 -0.03172799 -0.11630628 -0.15712479
#> 11 0.09430184 0.1486405 0.7345784 -0.02247920 -0.09430184 -0.14864051
#> raw.rsd.3 se.0 se.1 se.2 se.3 std.rsd.0
#> 1 -0.15230713 0.089189111 0.08887413 0.06272965 0.06672374 7.1642602
#> 2 -0.19939157 0.039915574 0.04128137 0.03084204 0.03464477 13.4606780
#> 3 -0.24779310 0.027254580 0.02939944 0.02299723 0.02693064 3.8513542
#> 4 -0.31063479 0.019705528 0.02255042 0.01854108 0.02258006 12.1815722
#> 5 -0.11336430 0.014088358 0.01729635 0.01494624 0.01873938 3.6854278
#> 6 -0.16380663 0.012327666 0.01620074 0.01464044 0.01865579 -4.7013261
#> 7 -0.04281429 0.010419122 0.01485118 0.01415633 0.01819070 -8.6451471
#> 8 0.23326768 0.009894046 0.01510336 0.01508595 0.01938659 -6.9369226
#> 9 0.25376091 0.009585825 0.01580501 0.01668745 0.02132199 -5.2060944
#> 10 0.30515905 0.009707584 0.01775594 0.02031430 0.02557456 -3.2683714
#> 11 0.26542156 0.031604006 0.06230752 0.07584269 0.09414036 -0.7112771
#> std.rsd.1 std.rsd.2 std.rsd.3
#> 1 -3.997703 -2.094297 -2.282653
#> 2 -4.585728 -4.817903 -5.755315
#> 3 8.656130 -4.855340 -9.201158
#> 4 5.295423 -2.633264 -13.757040
#> 5 2.363258 1.376061 -6.049523
#> 6 11.556575 2.359080 -8.780471
#> 7 1.929979 7.362550 -2.353637
#> 8 -12.058712 1.159597 12.032427
#> 9 -9.617197 -3.107504 11.901368
#> 10 -6.550274 -7.734688 11.932133
#> 11 -1.513490 -1.959853 2.819424
# (3) Standardized residual plot only, with four columns in layout
plot(x = fit1, item.loc = 113, type = "sr", ci.method = "wald",
layout.col = 4, ylim.sr.adjust = TRUE)
#> interval point total obs.freq.0 obs.freq.1 obs.freq.2
#> 1 [-0.1202983,0.01795651) -0.03804602 29 29 0 0
#> 2 [0.01795651,0.1562113) 0.09936557 133 112 21 0
#> 3 [0.1562113,0.2944662) 0.22102526 257 93 151 13
#> 4 [0.2944662,0.432721) 0.36224032 420 187 180 53
#> 5 [0.432721,0.5709758) 0.50694958 671 141 214 137
#> 6 [0.5709758,0.7092307) 0.63423837 709 46 308 157
#> 7 [0.7092307,0.8474855) 0.77733968 755 0 181 219
#> 8 [0.8474855,0.9857403) 0.89421166 653 0 0 130
#> 9 [0.9857403,1.123995) 1.02377099 516 0 0 63
#> 10 [1.123995,1.26225) 1.19789074 326 0 0 1
#> 11 [1.26225,1.400505] 1.32465982 22 0 0 0
#> obs.freq.3 obs.prop.0 obs.prop.1 obs.prop.2 obs.prop.3 exp.prob.0
#> 1 0 1.00000000 0.0000000 0.000000000 0.0000000 0.36102600
#> 2 0 0.84210526 0.1578947 0.000000000 0.0000000 0.30481458
#> 3 0 0.36186770 0.5875486 0.050583658 0.0000000 0.25690066
#> 4 0 0.44523810 0.4285714 0.126190476 0.0000000 0.20519378
#> 5 179 0.21013413 0.3189270 0.204172876 0.2667660 0.15821250
#> 6 198 0.06488011 0.4344147 0.221438646 0.2792666 0.12283649
#> 7 355 0.00000000 0.2397351 0.290066225 0.4701987 0.09007484
#> 8 523 0.00000000 0.0000000 0.199081164 0.8009188 0.06863423
#> 9 453 0.00000000 0.0000000 0.122093023 0.8779070 0.04990471
#> 10 325 0.00000000 0.0000000 0.003067485 0.9969325 0.03172799
#> 11 22 0.00000000 0.0000000 0.000000000 1.0000000 0.02247920
#> exp.prob.1 exp.prob.2 exp.prob.3 raw.rsd.0 raw.rsd.1 raw.rsd.2
#> 1 0.35529236 0.1313745 0.1523071 0.63897400 -0.35529236 -0.13137450
#> 2 0.34719988 0.1485940 0.1993916 0.53729068 -0.18930514 -0.14859398
#> 3 0.33306323 0.1622430 0.2477931 0.10496704 0.25448540 -0.11165935
#> 4 0.30915739 0.1750140 0.3106348 0.24004431 0.11941404 -0.04882356
#> 5 0.27805125 0.1836059 0.3801303 0.05192163 0.04087573 0.02056694
#> 6 0.24718962 0.1869007 0.4430732 -0.05795638 0.18722504 0.03453796
#> 7 0.21107263 0.1858396 0.5130130 -0.09007484 0.02866247 0.10422667
#> 8 0.18212707 0.1815875 0.5676512 -0.06863423 -0.18212707 0.01749362
#> 9 0.15199988 0.1739493 0.6241461 -0.04990471 -0.15199988 -0.05185631
#> 10 0.11630628 0.1601923 0.6917735 -0.03172799 -0.11630628 -0.15712479
#> 11 0.09430184 0.1486405 0.7345784 -0.02247920 -0.09430184 -0.14864051
#> raw.rsd.3 se.0 se.1 se.2 se.3 std.rsd.0
#> 1 -0.15230713 0.089189111 0.08887413 0.06272965 0.06672374 7.1642602
#> 2 -0.19939157 0.039915574 0.04128137 0.03084204 0.03464477 13.4606780
#> 3 -0.24779310 0.027254580 0.02939944 0.02299723 0.02693064 3.8513542
#> 4 -0.31063479 0.019705528 0.02255042 0.01854108 0.02258006 12.1815722
#> 5 -0.11336430 0.014088358 0.01729635 0.01494624 0.01873938 3.6854278
#> 6 -0.16380663 0.012327666 0.01620074 0.01464044 0.01865579 -4.7013261
#> 7 -0.04281429 0.010419122 0.01485118 0.01415633 0.01819070 -8.6451471
#> 8 0.23326768 0.009894046 0.01510336 0.01508595 0.01938659 -6.9369226
#> 9 0.25376091 0.009585825 0.01580501 0.01668745 0.02132199 -5.2060944
#> 10 0.30515905 0.009707584 0.01775594 0.02031430 0.02557456 -3.2683714
#> 11 0.26542156 0.031604006 0.06230752 0.07584269 0.09414036 -0.7112771
#> std.rsd.1 std.rsd.2 std.rsd.3
#> 1 -3.997703 -2.094297 -2.282653
#> 2 -4.585728 -4.817903 -5.755315
#> 3 8.656130 -4.855340 -9.201158
#> 4 5.295423 -2.633264 -13.757040
#> 5 2.363258 1.376061 -6.049523
#> 6 11.556575 2.359080 -8.780471
#> 7 1.929979 7.362550 -2.353637
#> 8 -12.058712 1.159597 12.032427
#> 9 -9.617197 -3.107504 11.901368
#> 10 -6.550274 -7.734688 11.932133
#> 11 -1.513490 -1.959853 2.819424