Package 'Qval'

Parameter estimation for cognitive diagnosis models (CDMs) by MMLE/EM or MMLE/BM algorithm.

Description

A function to estimate parameters for cognitive diagnosis models by MMLE/EM (de la Torre, 2009; de la Torre, 2011) or MMLE/BM (Ma & Jiang, 2020) algorithm.The function imports various functions from the GDINA package, parameter estimation for Cognitive Diagnostic Models (CDMs) was performed and extended. The CDM function not only accomplishes parameter estimation for most commonly used models (e.g., GDINA, DINA, DINO, ACDM, LLM, or rRUM). Furthermore, it incorporates Bayes modal estimation (BM; Ma & Jiang, 2020) to obtain more reliable estimation results, especially in small sample sizes. The monotonic constraints are able to be satisfied.

Usage

CDM(
  Y,
  Q,
  model = "GDINA",
  method = "EM",
  mono.constraint = TRUE,
  maxitr = 2000,
  verbose = 1
)

Arguments

Y	A required $N$ × $I$ matrix or data.frame consisting of the responses of N individuals to $N$ × $I$ items. Missing values need to be coded as NA.
Q	A required binary $I$ × $K$ matrix containing the attributes not required or required master the items. The ith row of the matrix is a binary indicator vector indicating which attributes are not required (coded by 0) and which attributes are required (coded by 1) to master item $i$.
model	Type of model to be fitted; can be "GDINA", "LCDM", "DINA", "DINO", "ACDM", "LLM", or "rRUM". Default = "GDINA".
method	Type of method to estimate CDMs' parameters; one out of "EM" and "BM". Default = "EM". However, "BM" is only avaible when method = "GDINA".
mono.constraint	Logical indicating whether monotonicity constraints should be fulfilled in estimation. Default = TRUE.
maxitr	A vector for each item or nonzero category, or a scalar which will be used for all items to specify the maximum number of EM or BM cycles allowed. Default = 2000.
verbose	Can be 0, 1 or 2, indicating to print no information, information for current iteration, or information for all iterations. Default = 1.

Details

CDMs are statistical models that fully integrates cognitive structure variables, which define the response probability of examinees on items by assuming the mechanism between attributes. In the dichotomous test, this probability is the probability of answering correctly. According to the specificity or generality of CDM assumptions, it can be divided into reduced CDM and saturated CDM.

Reduced CDMs possess specific assumptions about the mechanisms of attribute interactions, leading to clear interactions between attributes. Representative reduced models include the Deterministic Input, Noisy and Gate (DINA) model (Haertel, 1989; Junker & Sijtsma, 2001; de la Torre & Douglas, 2004), the Deterministic Input, Noisy or Gate (DINO) model (Templin & Henson, 2006), and the Additive Cognitive Diagnosis Model (A-CDM; de la Torre, 2011), the reduced Reparametrized Unified Model (rRUM; Hartz, 2002), among others. Compared to reduced models, saturated models, such as the Log-Linear Cognitive Diagnosis Model (LCDM; Henson et al., 2009) and the general Deterministic Input, Noisy and Gate model (G-DINA; de la Torre, 2011), do not have strict assumptions about the mechanisms of attribute interactions. When appropriate constraints are applied, saturated models can be transformed into various reduced models (Henson et al., 2008; de la Torre, 2011).

The LCDM is a saturated CDM fully proposed within the framework of cognitive diagnosis. Unlike reduced models that only discuss the main effects of attributes, it also considers the interaction between attributes, thus having more generalized assumptions about attributes. Its definition of the probability of correct response is as follows:

$P(X_{pi}=1|\boldsymbol{\alpha}_{l}) = \frac{\exp \left[\lambda_{i0} + \boldsymbol{\lambda}_{i}^{T} \boldsymbol{h} (\boldsymbol{q}_{i}, \boldsymbol{\alpha}_{l}) \right]} {1 + \exp \left[\lambda_{i0} + \boldsymbol{\lambda}_{i}^{T} \boldsymbol{h} (\boldsymbol{q}_{i}, \boldsymbol{\alpha}_{l}) \right]}$

$\boldsymbol{\lambda}_{i}^{T} \boldsymbol{h}(\boldsymbol{q}_{i}, \boldsymbol{\alpha}_{l}) = \sum_{k=1}^{K^\ast}\lambda_{ik}\alpha_{lk} +\sum_{k=1}^{K^\ast-1}\sum_{k'=k+1}^{K^\ast} \lambda_{ikk'}\alpha_{lk}\alpha_{lk'} + \cdots + \lambda_{12 \cdots K^\ast}\prod_{k=1}^{K^\ast}\alpha_{lk}$

Where, $P(X_{pi}=1|\boldsymbol{\alpha}_{l})$ represents the probability of an examinee with attribute mastery pattern $\boldsymbol{\alpha}_{l}$ ($l=1,2,\cdots,L$ and $L=2^{K^\ast}$) correctly answering item i. Here, $K^\ast = \sum_{k=1}^{K} q_{ik}$ denotes the number of attributes in the collapsed q-vector, $\lambda_{i0}$ is the intercept parameter, and $\boldsymbol{\lambda}_{i}=(\lambda_{i1}, \lambda_{i2}, \cdots, \lambda_{i12}, \cdots, \lambda_{i12{\cdots}K^\ast})$ represents the effect vector of the attributes. Specifically, $\lambda_{ik}$ is the main effect of attribute $k$, $\lambda_{ikk'}$ is the interaction effect between attributes $k$ and $k'$, and $\lambda_{j12{\cdots}K^\ast}$ represents the interaction effect of all required attributes.

The G-DINA, proposed by de la Torre (2011), is another saturated model that offers three types of link functions: identity link, log link, and logit link, which are defined as follows:

$P(X_{pi}=1|\boldsymbol{\alpha}_{l}) = \delta_{i0} + \sum_{k=1}^{K^\ast}\delta_{ik}\alpha_{lk} +\sum_{k=1}^{K^\ast-1}\sum_{k'=k+1}^{K^\ast}\delta_{ikk'}\alpha_{lk}\alpha_{lk'} + \cdots + \delta_{12{\cdots}K^\ast}\prod_{k=1}^{K^\ast}\alpha_{lk}$

$log \left[P(X_{pi}=1|\boldsymbol{\alpha}_{l}) \right] = v_{i0} + \sum_{k=1}^{K^\ast}v_{ik}\alpha_{lk} +\sum_{k=1}^{K^\ast-1}\sum_{k'=k+1}^{K^\ast}v_{ikk'}\alpha_{lk}\alpha_{lk'} + \cdots + v_{12{\cdots}K^\ast}\prod_{k=1}^{K^\ast}\alpha_{lk}$

$logit \left[P(X_{pi}=1|\boldsymbol{\alpha}_{l}) \right] = \lambda_{i0} + \sum_{k=1}^{K^\ast}\lambda_{ik}\alpha_{lk} +\sum_{k=1}^{K^\ast-1}\sum_{k'=k+1}^{K^\ast}\lambda_{ikk'}\alpha_{lk}\alpha_{lk'} + \cdots + \lambda_{12{\cdots}K^\ast}\prod_{k=1}^{K^\ast}\alpha_{lk}$

Where $\delta_{i0}$, $v_{i0}$, and $\lambda_{i0}$ are the intercept parameters for the three link functions, respectively; $\delta_{ik}$, $v_{ik}$, and $\lambda_{ik}$ are the main effect parameters of $\alpha_{lk}$ for the three link functions, respectively; $\delta_{ikk'}$, $v_{ikk'}$, and $\lambda_{ikk'}$ are the interaction effect parameters between $\alpha_{lk}$ and $\alpha_{lk'}$ for the three link functions, respectively; and $\delta_{i12{\cdots }K^\ast}$, $v_{i12{\cdots}K^\ast}$, and $\lambda_{i12{\cdots}K^\ast}$ are the interaction effect parameters of $\alpha_{l1}{\cdots}\alpha_{lK^\ast}$ for the three link functions, respectively. It can be observed that when the logit link is adopted, the G-DINA model is equivalent to the LCDM model.

Specifically, the A-CDM can be formulated as:

$P(X_{pi}=1|\boldsymbol{\alpha}_{l}) = \delta_{i0} + \sum_{k=1}^{K^\ast}\delta_{ik}\alpha_{lk}$

The rRUM, can be written as:

$log \left[P(X_{pi}=1|\boldsymbol{\alpha}_{l}) \right] = \lambda_{i0} + \sum_{k=1}^{K^\ast}\lambda_{ik}\alpha_{lk}$

The item response function for the linear logistic model (LLM) can be given by:

$logit\left[P(X_{pi}=1|\boldsymbol{\alpha}_{l}) \right] = \lambda_{i0} + \sum_{k=1}^{K^\ast}\lambda_{ik}\alpha_{lk}$

In the DINA model, every item is characterized by two key parameters: guessing (g) and slip (s). Within the traditional framework of DINA model parameterization, a latent variable $\eta$, specific to examinee $p$ who has the attribute mastery pattern $\boldsymbol{\alpha}_{l}$ and responses to $i$, is defined as follows:

$\eta_{li}=\prod_{k=1}^{K}\alpha_{lk}^{q_{ik}}$

If examinee $p$ whose attribute mastery pattern is $\boldsymbol{\alpha}_{l}$ has acquired every attribute required by item i, $\eta_{pi}$ is given a value of 1. If not, $\eta_{pi}$ is set to 0. The DINA model's item response function can be concisely formulated as such:

$P(X_{pi}=1|\boldsymbol{\alpha}_{l}) = (1-s_j)^{\eta_{li}}g_j^{(1-\eta_{li})} = \delta_{i0}+\delta_{i12{\cdots}K}\prod_{k=1}^{K^\ast}\alpha_{lk}$

$(1-s_j)^{\eta_{li}}g_j^{(1-\eta_{li})}$ is the original expression of the DINA model, while $\delta_{i0}+\delta_{i12{\cdots}K}\prod_{k=1}^{K^\ast}\alpha_{lk}$ is an equivalent form of the DINA model after adding constraints in the G-DINA model. Here, $g_j = \delta_{i0}$ and $1-s_j = \delta_{i0}+\delta_{i12{\cdots}K}\prod_{k=1}^{K^\ast}\alpha_{lk}$.

In contrast to the DINA model, the DINO model suggests that an examinee can correctly respond to an item if he/she have mastered at least one of the item's measured attributes. Additionally, like the DINA model, the DINO model also accounts for parameters related to guessing and slipping. Therefore, the main difference between DINO and DINA lies in their respective $\eta_{li}$ formulations. The DINO model can be given by:

$\eta_{li} = 1-\prod_{k=1}^{K}(1 - \alpha_{lk})^{q_{lk}}$

Value

An object of class CDM containing the following components:

analysis.obj

An GDINA object gained from GDINA package or an list after BM algorithm, depending on which estimation is used.

alpha

Individuals' attribute parameters calculated by EAP method

P.alpha.Xi

Individual's posterior probability

alpha.P

Individuals' marginal mastery probabilities matrix

P.alpha

Attribute prior weights for calculating marginalized likelihood in the last iteration

model.fit

Some basic model-fit indeces, including Deviance, npar, AIC, BIC. @seealso fit

Author(s)

Haijiang Qin <[email protected]>

References

de la Torre, J. (2009). DINA Model and Parameter Estimation: A Didactic. Journal of Educational and Behavioral Statistics, 34(1), 115-130. DOI: 10.3102/1076998607309474.

de la Torre, J., & Douglas, J. A. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69(3), 333-353. DOI: 10.1007/BF02295640.

de la Torre, J. (2011). The Generalized DINA Model Framework. Psychometrika, 76(2), 179-199. DOI: 10.1007/s11336-011-9207-7.

Haertel, E. H. (1989). Using restricted latent class models to map the skill structure of achievement items. Journal of Educational Measurement, 26(4), 301-323. DOI: 10.1111/j.1745-3984.1989.tb00336.x.

Hartz, S. M. (2002). A Bayesian framework for the unified model for assessing cognitive abilities: Blending theory with practicality (Unpublished doctoral dissertation). University of Illinois at Urbana-Champaign.

Henson, R. A., Templin, J. L., & Willse, J. T. (2008). Defining a Family of Cognitive Diagnosis Models Using Log-Linear Models with Latent Variables. Psychometrika, 74(2), 191-210. DOI: 10.1007/s11336-008-9089-5.

Huebner, A., & Wang, C. (2011). A note on comparing examinee classification methods for cognitive diagnosis models. Educational and Psychological Measurement, 71, 407-419. DOI: 10.1177/0013164410388832.

Junker, B. W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25(3), 258-272. DOI: 10.1177/01466210122032064.

Ma, W., & Jiang, Z. (2020). Estimating Cognitive Diagnosis Models in Small Samples: Bayes Modal Estimation and Monotonic Constraints. Applied Psychological Measurement, 45(2), 95-111. DOI: 10.1177/0146621620977681.

Templin, J. L., & Henson, R. A. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological methods, 11(3), 287-305. DOI: 10.1037/1082-989X.11.3.287.

Tu, D., Chiu, J., Ma, W., Wang, D., Cai, Y., & Ouyang, X. (2022). A multiple logistic regression-based (MLR-B) Q-matrix validation method for cognitive diagnosis models: A confirmatory approach. Behavior Research Methods. DOI: 10.3758/s13428-022-01880-x.

See Also

validation.

Examples

################################################################
#                           Example 1                          #
#            fit using MMLE/EM to fit the GDINA models         #
################################################################
set.seed(123)

library(Qval)

## generate Q-matrix and data to fit
K <- 3
I <- 30
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ,
                         model = "GDINA", distribute = "horder")


## using MMLE/EM to fit GDINA model
example.CDM.obj <- CDM(example.data$dat, example.Q, model = "GDINA",
                       method = "EM", maxitr = 2000, verbose = 1)



################################################################
#                           Example 2                          #
#               fit using MMLE/BM to fit the DINA              #
################################################################
set.seed(123)

library(Qval)

## generate Q-matrix and data to fit
K <- 5
I <- 30
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ,
                         model = "DINA", distribute = "horder")


## using MMLE/BM to fit GDINA model
example.CDM.obj <- CDM(example.data$dat, example.Q, model = "GDINA",
                       method = "BM", maxitr = 1000, verbose = 2)


################################################################
#                           Example 3                          #
#              fit using MMLE/EM to fit the ACDM               #
################################################################
set.seed(123)

library(Qval)

## generate Q-matrix and data to fit
K <- 5
I <- 30
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ,
                         model = "ACDM", distribute = "horder")


## using MMLE/EM to fit GDINA model
example.CDM.obj <- CDM(example.data$dat, example.Q, model = "ACDM",
                       method = "EM", maxitr = 2000, verbose = 1)

---

Extract elements from objects of various classes

Description

A generic function to extract elements from objects of class CDM, validation or sim.data.

Objects which can be extracted from CDM object include:

analysis.obj

An GDINA object (@seealso GDINA) gained from GDINA package or an list after BM algorithm, depending on which estimation is used.

alpha

Individuals' attribute parameters calculated by EAP method

P.alpha.Xi

Individual's posterior probability

alpha.P

Individuals' marginal mastery probabilities matrix

P.alpha

Attribute prior weights for calculating marginalized likelihood in the last iteration

Deviance

deviance, or negative two times observed marginal log likelihood

npar

The number of parameters

AIC

AIC

BIC

BIC

Objects which can be extracted from validation object include:

Q.orig

The original Q-matrix that maybe contain some mis-specifications and need to be validated.

Q.sug

The Q-matrix that suggested by certain validation method.

time.cost

The time that CPU cost to finish the validation.

process

A matrix that contains the modification process of each item during each iteration. Each row represents an iteration, and each column corresponds to the q-vector index of respective item. The order of the indices is consistent with the row number in the matrix generated by the attributepattern function in the GDINA package. Only when maxitr > 1, the value is available.

iter

The number of iteration. Only when maxitr > 1, the value is available.

priority

An I × K matrix that contains the priority of every attribute for each item. Only when the search.method is "PAA", the value is available.

Hull.fit

A list containing all the information needed to plot the Hull plot, which is available only when method = "Hull".

Objects which can be extracted from sim.data object include:

dat

An N × I simulated item response matrix.

Q

The Q-matrix.

attribute

An N × K matrix for inviduals' attribute patterns.

catprob.parm

A list of non-zero category success probabilities for each attribute mastery pattern.

delta.parm

A list of delta parameters.

higher.order.parm

Higher-order parameters.

mvnorm.parm

Multivariate normal distribution parameters.

LCprob.parm

A matrix of item/category success probabilities for each attribute mastery pattern.

Usage

extract(object, what, ...)

## S3 method for class 'CDM'
extract(object, what, ...)

## S3 method for class 'validation'
extract(object, what, ...)

## S3 method for class 'sim.data'
extract(object, what, ...)

Arguments

object	objects from class CDM, validation, sim.data
what	what to extract
...	Additional arguments.

Methods (by class)

• extract(CDM): various elements of CDM object

• extract(validation): various elements of validation object

• extract(sim.data): various elements of sim.data object

Examples

set.seed(123)

library(Qval)

## generate Q-matrix and data to fit
K <- 3
I <- 30
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)
example.data <- sim.data(Q = example.Q, N = 1000, IQ = IQ,
                         model = "GDINA", distribute = "horder")
extract(example.data,"dat")


## using MMLE/EM to fit GDINA model
example.CDM.obj <- CDM(example.data$dat, example.Q, model = "GDINA",
                       method = "EM", maxitr = 2000, verbose = 1)
extract(example.CDM.obj,"alpha")
extract(example.CDM.obj,"npar")

example.MQ <- sim.MQ(example.Q, 0.1)
example.CDM.obj <- CDM(example.data$dat, example.MQ, model = "GDINA",
                       method = "EM", maxitr = 2000, verbose = 1)
                       
validation.obj <- validation(example.data$dat, example.MQ, 
                             example.CDM.obj, method = "MLR-B", eps = 0.90)
extract(validation.obj,"Q.sug")

---

Calculate fit indeces

Description

Calculate relative fit indices (-2LL, AIC, BIC, CAIC, SABIC) and absolute fit indices ($M_2$ test, $RMSEA_2$, SRMSR) using the modelfit function in the GDINA package.

Usage

fit(Y, Q, model = "GDINA")

Arguments

Y	A required $N$ × $I$ matrix or data.frame consisting of the responses of N individuals to $N$ × $I$ items. Missing values need to be coded as NA.
Q	A required binary $I$ × $K$ matrix containing the attributes not required or required master the items. The ith row of the matrix is a binary indicator vector indicating which attributes are not required (coded by 0) and which attributes are required (coded by 1) to master item $i$.
model	Type of model to be fitted; can be "GDINA", "LCDM", "DINA", "DINO", "ACDM", "LLM", or "rRUM". Default = "GDINA".

Value

An object of class list. The list contains various fit indices:

npar

The number of parameters.

-2LL

The Deviance.

AIC

The Akaike information criterion.

BIC

The Bayesian information criterion.

CAIC

The consistent Akaike information criterion.

SABIC

The Sample-size Adjusted BIC.

M2

A vector consisting of $M_2$ statistic, degrees of freedom, significance level, and $RMSEA_2$ (Liu, Tian, & Xin, 2016).

SRMSR

The standardized root mean squared residual (SRMSR; Ravand & Robitzsch, 2018).

Author(s)

Haijiang Qin <[email protected]>

References

Khaldi, R., Chiheb, R., & Afa, A.E. (2018). Feed-forward and Recurrent Neural Networks for Time Series Forecasting: Comparative Study. In: Proceedings of the International Conference on Learning and Optimization Algorithms: Theory and Applications (LOPAL 18). Association for Computing Machinery, New York, NY, USA, Article 18, 1–6. DOI: 10.1145/3230905.3230946.

Liu, Y., Tian, W., & Xin, T. (2016). An application of M2 statistic to evaluate the fit of cognitive diagnostic models. Journal of Educational and Behavioral Statistics, 41, 3–26. DOI: 10.3102/1076998615621293.

Ravand, H., & Robitzsch, A. (2018). Cognitive diagnostic model of best choice: a study of reading comprehension. Educational Psychology, 38, 1255–1277. DOI: 10.1080/01443410.2018.1489524.

Examples

set.seed(123)

library(Qval)

## generate Q-matrix and data to fit
K <- 5
I <- 30
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ, model = "GDINA", distribute = "horder")

## calculate fit indices
fit.indices <- fit(Y = example.data$dat, Q = example.Q, model = "GDINA")
print(fit.indices)

---

Calculate $\beta$

Description

The function is able to calculate the $\beta$ index for all items after fitting CDM or directly.

Usage

get.beta(Y = NULL, Q = NULL, CDM.obj = NULL, model = "GDINA")

Arguments

Y	A required $N$ × $I$ matrix or data.frame consisting of the responses of N individuals to $N$ × $I$ items. Missing values need to be coded as NA.
Q	A required binary $I$ × $K$ matrix containing the attributes not required or required master the items. The ith row of the matrix is a binary indicator vector indicating which attributes are not required (coded by 0) and which attributes are required (coded by 1) to master item $i$.
CDM.obj	An object of class CDM.obj. Can be NULL, but when it is not NULL, it enables rapid validation of the Q-matrix without the need for parameter estimation. @seealso CDM.
model	Type of model to be fitted; can be "GDINA", "LCDM", "DINA", "DINO", "ACDM", "LLM", or "rRUM". Default = "GDINA".

Details

For item $i$ with the q-vector of the $c$-th ($c = 1, 2, ..., 2^{K}$) type, the $\beta$ index is computed as follows:

$\beta_{ic} = \sum_{l=1}^{2^K} \left| \frac{r_{li}}{n_l} P_{ic}(\boldsymbol{\alpha}_{l}) - \left(1 - \frac{r_{li}}{n_l}\right) \left[1 - P_{ic}(\boldsymbol{\alpha}_{l})\right] \right| = \sum_{l=1}^{2^K} \left| \frac{r_{li}}{n_l} - \left[1 - P_{ic}(\boldsymbol{\alpha}_{l}) \right] \right|$

In the formula, $r_{li}$ represents the number of examinees in attribute mastery pattern $\boldsymbol{\alpha}_{l}$ who correctly answered item $i$, while $n_l$ is the total number of examinees in attribute mastery pattern $\boldsymbol{\alpha}_{l}$. $P_{ic}(\boldsymbol{\alpha}_{l})$ denotes the probability that an examinee in attribute mastery pattern $\boldsymbol{\alpha}_{l}$ answers item $i$ correctly when the q-vector for item $i$ is of the $c$-th type. In fact, $\frac{r_{li}}{n_l}$ is the observed probability that an examinee in attribute mastery pattern $\boldsymbol{\alpha}_{l}$ answers item $i$ correctly, and $\beta_{jc}$ represents the difference between the actual proportion of correct answers for item $i$ in each attribute mastery pattern and the expected probability of answering the item incorrectly in that state. Therefore, to some extent, $\beta_{jc}$ can be considered as a measure of discriminability.

Value

An object of class matrix, which consisted of $\beta$ index for each item and each possible attribute mastery pattern.

Author(s)

Haijiang Qin <[email protected]>

References

Li, J., & Chen, P. (2024). A new Q-matrix validation method based on signal detection theory. British Journal of Mathematical and Statistical Psychology, 00, 1–33. DOI: 10.1111/bmsp.12371

See Also

validation

Examples

library(Qval)

set.seed(123)

## generate Q-matrix and data
K <- 3
I <- 20
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)

model <- "DINA"
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ, model = model, distribute = "horder")

## calculate beta directly
beta <-get.beta(Y = example.data$dat, Q = example.Q, model = model)
print(beta)

## calculate beta after fitting CDM
example.CDM.obj <- CDM(example.data$dat, example.Q, model=model)
beta <-get.beta(CDM.obj = example.CDM.obj)
print(beta)

---

Calculate $\mathbf{M}$ matrix

Description

Calculate $\mathbf{M}$ matrix for stauted CDMs (de la Torre, 2011). The $\mathbf{M}$ matrix is a matrix used to represent the interaction mechanisms between attributes.

Usage

get.Mmatrix(K = NULL, pattern = NULL)

Arguments

K	The number of attributes. Can be NULL if the argument pattern is not NULL.
pattern	The attribute mastery pattern matrix containing all possible attribute mastery pattern. Can be gained from attributepattern. Also can be NULL if K is not NULL.

Value

An object of class matrix.

Author(s)

Haijiang Qin <[email protected]>

References

de la Torre, J. (2011). The Generalized DINA Model Framework. Psychometrika, 76(2), 179-199. DOI: 10.1007/s11336-011-9207-7.

Examples

library(Qval)

example.Mmatrix <-  get.Mmatrix(K = 5)

---

Priority of Attribute

Description

This function will provide the priorities of attributes for all items.

Usage

get.priority(Y = NULL, Q = NULL, CDM.obj = NULL, model = "GDINA")

Arguments

Y	A required $N$ × $I$ matrix or data.frame consisting of the responses of N individuals to $N$ × $I$ items. Missing values need to be coded as NA.
Q	A required binary $I$ × $K$ matrix containing the attributes not required or required master the items. The ith row of the matrix is a binary indicator vector indicating which attributes are not required (coded by 0) and which attributes are required (coded by 1) to master item $i$.
CDM.obj	An object of class CDM.obj. When it is not NULL, it enables rapid validation of the Q-matrix without the need for parameter estimation. @seealso CDM.
model	Type of model to fit; can be "GDINA", "LCDM", "DINA", "DINO" , "ACDM", "LLM", or "rRUM". Default = "GDINA". @seealso CDM.

Details

The calculation of priorities is straightforward (Qin & Guo, 2025): the priority of an attribute is the regression coefficient obtained from a LASSO multinomial logistic regression, with the attribute as the independent variable and the response data from the examinees as the dependent variable. The formula (Tu et al., 2022) is as follows:

$\log[\frac{P(X_{pi} = 1 | \boldsymbol{\Lambda}_{p})}{P(X_{pi} = 0 | \boldsymbol{\Lambda}_{p})}] = logit[P(X_{pi} = 1 | \boldsymbol{\Lambda}_{p})] = \beta_{i0} + \beta_{i1} \Lambda_{p1} + \ldots + \beta_{ik} \Lambda_{pk} + \ldots + \beta_{iK} \Lambda_{pK}$

Where $X_{pi}$ represents the response of examinee $p$ on item $i$, $\boldsymbol{\Lambda}_{p}$ denotes the marginal mastery probabilities of examinee $p$ (which can be obtained from the return value alpha.P of the CDM function), $\beta_{i0}$ is the intercept term, and $\beta_{ik}$ represents the regression coefficient.

The LASSO loss function can be expressed as:

$l_{lasso}(\boldsymbol{X}_i | \boldsymbol{\Lambda}) = l(\boldsymbol{X}_i | \boldsymbol{\Lambda}) - \lambda |\boldsymbol{\beta}_i|$

Where $l_{lasso}(\boldsymbol{X}_i | \boldsymbol{\Lambda})$ is the penalized likelihood, $l(\boldsymbol{X}_i | \boldsymbol{\Lambda})$ is the original likelihood, and $\lambda$ is the tuning parameter for penalization (a larger value imposes a stronger penalty on $\boldsymbol{\beta}_i = [\beta_{i1}, \ldots, \beta_{ik}, \ldots, \beta_{iK}]$). The priority for attribute $i$ is defined as: $\boldsymbol{priority}_i = \boldsymbol{\beta}_i = [\beta_{i1}, \ldots, \beta_{ik}, \ldots, \beta_{iK}]$

Value

A matrix containing all attribute priorities.

References

Qin, H., & Guo, L. (2025). Priority attribute algorithm for Q-matrix validation: A didactic. Behavior Research Methods, 57(1), 31. DOI: 10.3758/s13428-024-02547-5.

Tu, D., Chiu, J., Ma, W., Wang, D., Cai, Y., & Ouyang, X. (2022). A multiple logistic regression-based (MLR-B) Q-matrix validation method for cognitive diagnosis models: A confirmatory approach. Behavior Research Methods. DOI: 10.3758/s13428-022-01880-x.

Examples

set.seed(123)
library(Qval)

## generate Q-matrix and data
K <- 5
I <- 20
IQ <- list(
  P0 = runif(I, 0.1, 0.3),
  P1 = runif(I, 0.7, 0.9)
)


Q <- sim.Q(K, I)
data <- sim.data(Q = Q, N = 500, IQ = IQ, model = "GDINA", distribute = "horder")
MQ <- sim.MQ(Q, 0.1)

CDM.obj <- CDM(data$dat, MQ)

priority <- get.priority(data$dat, Q, CDM.obj)
head(priority)

---

Calculate $PVAF$

Description

The function is able to calculate the proportion of variance accounted for ($PVAF$) for all items after fitting CDM or directly.

Usage

get.PVAF(Y = NULL, Q = NULL, CDM.obj = NULL, model = "GDINA")

Arguments

Y	A required $N$ × $I$ matrix or data.frame consisting of the responses of N individuals to $N$ × $I$ items. Missing values need to be coded as NA.
Q	A required binary $I$ × $K$ matrix containing the attributes not required or required master the items. The ith row of the matrix is a binary indicator vector indicating which attributes are not required (coded by 0) and which attributes are required (coded by 1) to master item $i$.
CDM.obj	An object of class CDM.obj. Can can be NULL, but when it is not NULL, it enables rapid verification of the Q-matrix without the need for parameter estimation. @seealso CDM.
model	Type of model to be fitted; can be "GDINA", "LCDM", "DINA", "DINO", "ACDM", "LLM", or "rRUM". Default = "GDINA".

Details

The intrinsic essence of the GDI index (as denoted by $\zeta_{2}$) is the weighted variance of all $2^{K\ast}$ attribute mastery patterns' probabilities of correctly responding to item $i$, which can be computed as:

$\zeta^2 = \sum_{l=1}^{2^K} \pi_{l}{(P(X_{pi}=1|\boldsymbol{\alpha}_{l}) - \bar{P}_{i})}^2$

where $\pi_{l}$ represents the prior probability of mastery pattern $l$; $\bar{P}_{i}=\sum_{k=1}^{2^K}\pi_{l}P(X_{pi}=1|\boldsymbol{\alpha}_{l})$ is the weighted average of the correct response probabilities across all attribute mastery patterns. When the q-vector is correctly specified, the calculated $\zeta^2$ should be maximized, indicating the maximum discrimination of the item.

Theoretically, $\zeta^{2}$ is larger when $\boldsymbol{q}_{i}$ is either specified correctly or over-specified, unlike when $\boldsymbol{q}_{i}$ is under-specified, and that when $\boldsymbol{q}_{i}$ is over-specified, $\zeta^{2}$ is larger than but close to the value of $\boldsymbol{q}_{i}$ when specified correctly. The value of $\zeta^{2}$ continues to increase slightly as the number of over-specified attributes increases, until $\boldsymbol{q}_{i}$ becomes $\boldsymbol{q}_{i1:K}$ ($\boldsymbol{q}_{i1:K}$ = [11...1]). Thus, $\zeta^{2} / \zeta_{max}^{2}$ is computed to indicate the proportion of variance accounted for by $\boldsymbol{q}_{i}$, called the $PVAF$.

Value

An object of class matrix, which consisted of $PVAF$ for each item and each possible q-vector.

Author(s)

Haijiang Qin <[email protected]>

References

de la Torre, J., & Chiu, C. Y. (2016). A General Method of Empirical Q-matrix Validation. Psychometrika, 81(2), 253-273. DOI: 10.1007/s11336-015-9467-8.

See Also

validation

Examples

library(Qval)

set.seed(123)

## generate Q-matrix and data
K <- 3
I <- 20
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ, model = "GDINA", distribute = "horder")

## calculate PVAF directly
PVAF <-get.PVAF(Y = example.data$dat, Q = example.Q)
print(PVAF)

## calculate PVAF after fitting CDM
example.CDM.obj <- CDM(example.data$dat, example.Q, model="GDINA")
PVAF <-get.PVAF(CDM.obj = example.CDM.obj)
print(PVAF)

---

Calculate McFadden pseudo-R2

Description

The function is able to calculate the McFadden pseudo-$R^{2}$ ($R^{2}$) for all items after fitting CDM or directly.

Usage

get.R2(Y = NULL, Q = NULL, CDM.obj = NULL, model = "GDINA")

Arguments

Y	A required $N$ × $I$ matrix or data.frame consisting of the responses of N individuals to $N$ × $I$ items. Missing values need to be coded as NA.
Q	A required binary $I$ × $K$ matrix containing the attributes not required or required master the items. The ith row of the matrix is a binary indicator vector indicating which attributes are not required (coded by 0) and which attributes are required (coded by 1) to master item $i$.
CDM.obj	An object of class CDM.obj. Can can be NULL, but when it is not NULL, it enables rapid verification of the Q-matrix without the need for parameter estimation. @seealso CDM.
model	Type of model to fit; can be "GDINA", "LCDM", "DINA", "DINO", "ACDM", "LLM", or "rRUM". Default = "GDINA".

Details

The McFadden pseudo-$R^{2}$ (McFadden, 1974) serves as a definitive model-fit index, quantifying the proportion of variance explained by the observed responses. Comparable to the squared multiple-correlation coefficient in linear statistical models, this coefficient of determination finds its application in logistic regression models. Specifically, in the context of the CDM, where probabilities of accurate item responses are predicted for each examinee, the McFadden pseudo-$R^{2}$ provides a metric to assess the alignment between these predictions and the actual responses observed. Its computation is straightforward, following the formula:

$R_{i}^{2} = 1 - \frac{\log(L_{im})}{\log(L_{i0})}$

where $\log(L_{im})$ is the log-likelihood of the model, and $\log(L_{i0})$ is the log-likelihood of the null model. If there were $N$ examinees taking a test comprising $I$ items, then $\log(L_{im})$ would be computed as:

$\log(L_{im}) = \sum_{p}^{N} \log \sum_{l=1}^{2^{K^\ast}} \pi(\boldsymbol{\alpha}_{l}^{\ast} | \boldsymbol{X}_{p}) P_{i}(\boldsymbol{\alpha}_{l}^{\ast})^{X_{pi}} \left[ 1-P_{i}(\boldsymbol{\alpha}_{l}^{\ast}) \right] ^{(1-X_{pi})}$

where $\pi(\boldsymbol{\alpha}_{l}^{\ast} | \boldsymbol{X}_{p})$ is the posterior probability of examinee $p$ with attribute mastery pattern $\boldsymbol{\alpha}_{l}^{\ast}$ when their response vector is $\boldsymbol{X}_{p}$, and $X_{pi}$ is examinee $p$'s response to item $i$. Let $\bar{X}_{i}$ be the average probability of correctly responding to item $i$ across all $N$ examinees; then $\log(L_{i0})$ could be computed as:

$\log(L_{i0}) = \sum_{p}^{N} \log {\bar{X}_{i}}^{X_{pi}} {(1-\bar{X}_{i})}^{(1-X_{pi})}$

Value

An object of class matrix, which consisted of $R^{2}$ for each item and each possible attribute mastery pattern.

Author(s)

Haijiang Qin <[email protected]>

References

McFadden, D. (1974). Conditional logit analysis of qualitative choice behavior. In P. Zarembka (Ed.), Frontiers in economics (pp.105–142). Academic Press.

Najera, P., Sorrel, M. A., de la Torre, J., & Abad, F. J. (2021). Balancing ft and parsimony to improve Q-matrix validation. British Journal of Mathematical and Statistical Psychology, 74, 110–130. DOI: 10.1111/bmsp.12228.

Qin, H., & Guo, L. (2023). Using machine learning to improve Q-matrix validation. Behavior Research Methods. DOI: 10.3758/s13428-023-02126-0.

See Also

validation

Examples

library(Qval)

set.seed(123)

## generate Q-matrix and data
K <- 3
I <- 20
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ, model = "GDINA", distribute = "horder")

## calculate R2 directly
R2 <-get.R2(Y = example.data$dat, Q = example.Q)
print(R2)

## calculate R2 after fitting CDM
example.CDM.obj <- CDM(example.data$dat, example.Q, model="GDINA")
R2 <-get.R2(CDM.obj = example.CDM.obj)
print(R2)

---

Restriction matrix

Description

This function returns the restriction matrix (de la Torre, 2011; Ma & de la Torre, 2020) based on two q-vectors, where the two q-vectors can only differ by one attribute.

Usage

get.Rmatrix(q1, q2)

Arguments

q1	A q-vector
q2	Another q-vector

Value

A restriction matrix

References

de la Torre, J. (2011). The Generalized DINA Model Framework. Psychometrika, 76(2), 179-199. DOI: 10.1007/s11336-011-9207-7.

Ma, W., & de la Torre, J. (2020). An empirical Q-matrix validation method for the sequential generalized DINA model. British Journal of Mathematical and Statistical Psychology, 73(1), 142-163. DOI: 10.1111/bmsp.12156.

Examples

q1 <- c(1, 1, 0)
q2 <- c(1, 1, 1)

Rmatrix <- get.Rmatrix(q1, q2)

print(Rmatrix)

---

A tool for the $\beta$ Method

Description

This function performs a single iteration of the $\beta$ method for one item's validation. It is designed to be used in parallel computing environments to speed up the validation process of the $\beta$ method. The function is a utility function for validation. When the user calls the validation function with method = "beta", parallel_iter runs automatically, so there is no need for the user to call parallel_iter. It may seem that parallel_iter, as an internal function, could better serve users. However, we found that the Qval package must export it to resolve variable environment conflicts in R and enable parallel computation. Perhaps a better solution will be found in the future.

Usage

parallel_iter(
  i,
  Y,
  criter.index,
  P.alpha.Xi,
  P.alpha,
  pattern,
  ri,
  Ni,
  Q.pattern.ini,
  model,
  criter,
  search.method,
  P_GDINA,
  Q.beta,
  L,
  K,
  alpha.P,
  get.MLRlasso
)

Arguments

i	An integer indicating the item number that needs to be validated.
Y	A matrix of observed data used for validation.
criter.index	An integer representing the index of the criterion.
P.alpha.Xi	A matrix representing individual posterior probability.
P.alpha	A vector of attribute prior weights.
pattern	A matrix representing the attribute mastery patterns.
ri	A vector containing the number of examinees in each knowledge state who correctly answered item $i$.
Ni	A vector containing the total number of examinees in each knowledge state.
Q.pattern.ini	An integer representing the initial pattern order for the model.
model	A model object used for fitting, such as the GDINA model.
criter	A character string specifying the fit criterion. Possible values are "AIC", "BIC", "CAIC", or "SABIC".
search.method	A character string specifying the search method for model selection. Options include "beta", "ESA", "SSA", or "PAA".
P_GDINA	A function that calculates probabilities for the GDINA model.
Q.beta	A Q-matrix used for validation.
L	An integer representing the number of all attribute mastery patterns.
K	An integer representing the number of attributes.
alpha.P	A matrix of individuals' marginal mastery probabilities (Tu et al., 2022).
get.MLRlasso	A function for Lasso regression with multiple linear regression.

Value

A list containing the following components:

fit.index.pre

The previous fit index value after applying the selected search method.

fit.index.cur

The current fit index value after applying the selected search method.

Q.pattern.cur

The pattern that corresponds to the optimal model configuration for the current iteration.

priority

The priority vector used in the PAA method, if applicable.

---

Hull Plot

Description

This function can provide the Hull plot. The point suggested by the Hull method is marked in red.

Usage

## S3 method for class 'validation'
plot(x, i, ...)

Arguments

x	A validation in which method = "Hull".
i	A numeric, which represents the item you want to plot Hull curve.
...	Additional arguments.

Value

None. This function is used for side effects (plotting).

Examples

set.seed(123)
library(Qval)

## generate Q-matrix and data
K <- 4
I <- 20
IQ <- list(
  P0 = runif(I, 0.2, 0.4),
  P1 = runif(I, 0.6, 0.8)
)


Q <- sim.Q(K, I)
data <- sim.data(Q = Q, N = 500, IQ = IQ, model = "GDINA", distribute = "horder")
MQ <- sim.MQ(Q, 0.1)

CDM.obj <- CDM(data$dat, MQ)

############### ESA ###############
Hull.obj <- validation(data$dat, MQ, CDM.obj, method = "Hull", search.method = "ESA") 

## plot Hull curve for item 20
plot(Hull.obj, 20)

############### PAA ###############
Hull.obj <- validation(data$dat, MQ, CDM.obj, method = "Hull", search.method = "PAA") 

## plot Hull curve for item 20
plot(Hull.obj, 20)

---

Print Method for CDM Objects

Description

This function prints the details of a CDM object. It outputs the call used to create the object, the version and the date of the Qval package.

Usage

## S3 method for class 'CDM'
print(x, ...)

Arguments

x	A CDM object to be printed.
...	Additional arguments.

---

Print Method for sim.data Objects

Description

This function prints the details of a sim.data object. It outputs the call used to create the object, the version and the date of the Qval package.

Usage

## S3 method for class 'sim.data'
print(x, ...)

Arguments

x	A sim.data object to be printed.
...	Additional arguments.

---

Print Method for Validation Objects

Description

This function prints the details of a validation object. It outputs the call used to create the object, the version and the date of the Qval package.

Usage

## S3 method for class 'validation'
print(x, ...)

Arguments

x	A validation object to be printed.
...	Additional arguments.

---

generate response

Description

randomly generate response matrix according to certain conditions, including attributes distribution, item quality, sample size, Q-matrix and cognitive diagnosis models (CDMs).

Usage

sim.data(
  Q = NULL,
  N = NULL,
  IQ = list(P0 = NULL, P1 = NULL),
  model = "GDINA",
  distribute = "uniform",
  control = NULL,
  verbose = TRUE
)

Arguments

Q	The Q-matrix. A random 30 × 5 Q-matrix (sim.Q) will be used if Q = NULL.
N	Sample size. Default = 500.
IQ	A list containing two $I$-length vectors: P0 and P1. P0 represents the probability of examinees who have not mastered any attributes ($[00...0]$) correctly answering the item, while P1 represents the probability of examinees who have mastered all attributes ($[11...1]$) correctly answering the item.
model	Type of model to be fitted; can be "GDINA", "LCDM", "DINA", "DINO", "ACDM", "LLM", or "rRUM".
distribute	Attribute distributions; can be "uniform" for the uniform distribution, "mvnorm" for the multivariate normal distribution (Chiu, Douglas, & Li, 2009) and "horder" for the higher-order distribution (Tu et al., 2022).
control	A list of control parameters with elements: sigma A positive-definite symmetric matrix specifying the variance-covariance matrix when distribute = "mvnorm". Default = 0.5 (Chiu, Douglas, & Li, 2009). cutoffs A vector giving the cutoff for each attribute when distribute = "mvnorm". Default = $k/(1+K)$ (Chiu, Douglas, & Li, 2009). theta A vector of length N representing the higher-order ability for each examinee. By default, generate randomly from the standard normal distribution (Tu et al, 2022). a The slopes for the higher-order model when distribute = "horder". Default = 1.5 (Tu et al, 2022). b The intercepts when distribute = "horder". By default, select equally spaced values between -1.5 and 1.5 according to the number of attributes (Tu et al, 2022).
verbose	Logical indicating to print information or not. Default is TRUE

Value

Object of class sim.data. An sim.data object initially gained by simGDINA function form GDINA package. Elements that can be extracted using method extract include:

dat

An N × I simulated item response matrix.

Q

The Q-matrix.

attribute

An N × K matrix for inviduals' attribute patterns.

catprob.parm

A list of non-zero success probabilities for each attribute mastery pattern.

delta.parm

A list of delta parameters.

higher.order.parm

Higher-order parameters.

mvnorm.parm

Multivariate normal distribution parameters.

LCprob.parm

A matrix of success probabilities for each attribute mastery pattern.

Author(s)

Haijiang Qin <[email protected]>

References

Chiu, C.-Y., Douglas, J. A., & Li, X. (2009). Cluster Analysis for Cognitive Diagnosis: Theory and Applications. Psychometrika, 74(4), 633-665. DOI: 10.1007/s11336-009-9125-0.

Tu, D., Chiu, J., Ma, W., Wang, D., Cai, Y., & Ouyang, X. (2022). A multiple logistic regression-based (MLR-B) Q-matrix validation method for cognitive diagnosis models:A confirmatory approach. Behavior Research Methods. DOI: 10.3758/s13428-022-01880-x.

Examples

################################################################
#                           Example 1                          #
#          generate data follow the uniform distrbution        #
################################################################
library(Qval)

set.seed(123)

K <- 5
I <- 10
Q <- sim.Q(K, I)

IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)

data <- sim.data(Q = Q, N = 10, IQ=IQ, model = "GDINA", distribute = "uniform")

print(data$dat)

################################################################
#                           Example 2                          #
#          generate data follow the mvnorm distrbution         #
################################################################
set.seed(123)
K <- 5
I <- 10
Q <- sim.Q(K, I)

IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)

example_cutoffs <- sample(qnorm(c(1:K)/(K+1)), ncol(Q))
data <- sim.data(Q = Q, N = 10, IQ=IQ, model = "GDINA", distribute = "mvnorm",
                 control = list(sigma = 0.5, cutoffs = example_cutoffs))

print(data$dat)

#################################################################
#                            Example 3                          #
#           generate data follow the horder distrbution         #
#################################################################
set.seed(123)
K <- 5
I <- 10
Q <- sim.Q(K, I)

IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)

example_theta <- rnorm(10, 0, 1)
example_b <- seq(-1.5,1.5,length.out=K)
data <- sim.data(Q = Q, N = 10, IQ=IQ, model = "GDINA", distribute = "horder",
                 control = list(theta = example_theta, a = 1.5, b = example_b))

print(data$dat)

---

Simulate mis-specifications in the Q-matrix

Description

simulate certen rate mis-specifications in the Q-matrix.

Usage

sim.MQ(Q, rate, verbose = TRUE)

Arguments

Q	The Q-matrix (sim.Q) that need to simulate mis-specifications.
rate	The ratio of mis-specifications in the Q-matrix.
verbose	Logical indicating to print information or not. Default is TRUE

Value

An object of class matrix.

Author(s)

Haijiang Qin <[email protected]>

Examples

library(Qval)

set.seed(123)

Q <- sim.Q(5, 10)
print(Q)

MQ <- sim.MQ(Q, 0.1)
print(MQ)

---

generate a random Q-matrix

Description

generate a $I$ × $K$ Q-matrix randomly, which consisted of one-attribute q-vectors (50 This function ensures that the generated Q-matrix contains at least two identity matrices as a priority. Therefore, this function must also satisfy the condition that the number of items ($I$) must be at least twice the number of attributes ($K$).

Usage

sim.Q(K, I)

Arguments

K	The number of attributes of the Q-matrix.
I	The number of items.

Value

An object of class matrix.

Author(s)

Haijiang Qin <[email protected]>

References

Najera, P., Sorrel, M. A., de la Torre, J., & Abad, F. J. (2021). Balancing fit and parsimony to improve Q-matrix validation. Br J Math Stat Psychol, 74 Suppl 1, 110-130. DOI: 10.1111/bmsp.12228.

Examples

library(Qval)

set.seed(123)

Q <- sim.Q(5, 10)
print(Q)

---

Perform Q-matrix validation methods

Description

This function uses generalized Q-matrix validation methods to validate the Q-matrix, including commonly used methods such as GDI (de la Torre, & Chiu, 2016; Najera, Sorrel, & Abad, 2019; Najera et al., 2020), Wald (Ma, & de la Torre, 2020), Hull (Najera et al., 2021), and MLR-B (Tu et al., 2022). It supports different iteration methods (test level or item level; Najera et al., 2020; Najera et al., 2021; Tu et al., 2022) and can apply various attribute search methods (ESA, SSA, PAA; de la Torre, 2008; Terzi, & de la Torre, 2018).

Usage

validation(
  Y,
  Q,
  CDM.obj = NULL,
  par.method = "EM",
  mono.constraint = TRUE,
  model = "GDINA",
  method = "GDI",
  search.method = "PAA",
  iter.level = "no",
  maxitr = 1,
  eps = 0.95,
  alpha.level = 0.05,
  criter = NULL,
  verbose = TRUE
)

Arguments

Y	A required $N$ × $I$ matrix or data.frame consisting of the responses of N individuals to $N$ × $I$ items. Missing values need to be coded as NA.
Q	A required binary $I$ × $K$ matrix containing the attributes not required or required master the items. The ith row of the matrix is a binary indicator vector indicating which attributes are not required (coded by 0) and which attributes are required (coded by 1) to master item $i$.
CDM.obj	An object of class CDM.obj. When it is not NULL, it enables rapid validation of the Q-matrix without the need for parameter estimation. @seealso CDM.
par.method	Type of mtehod to estimate CDMs' parameters; one out of "EM", "BM". Default = "EM". However, "BM" is only available when method = "GDINA".
mono.constraint	Logical indicating whether monotonicity constraints should be fulfilled in estimation. Default = TRUE.
model	Type of model to fit; can be "GDINA", "LCDM", "DINA", "DINO" , "ACDM", "LLM", or "rRUM". Default = "GDINA". @seealso CDM.
method	The methods to validata Q-matrix, can be "GDI", "Wald", "Hull", "MLR-B" and "beta". The "model" must be "GDINA" when method = "Wald". Please note that the $\beta$ method has different meanings when applying different search algorithms. For more details, see section 'Search algorithm' below. Default = "GDI". See details.
search.method	Character string specifying the search method to use during validation. "ESA" for exhaustive search algorithm. Can not for the "Wald" method. "SSA" for sequential search algorithm (see de la Torre, 2008; Terzi & de la Torre, 2018). It will be equal to "forward" when method = "Wald". "PAA" for priority attribute algorithm. "stepwise" only for the "Wald" method "beta" only for the "beta" method
iter.level	Can be "no", "item" level, "test.att" or "test" level. Default = "no" and "test.att" can not for "Wald" and "MLR-B". See details.
maxitr	Number of max iterations. Default = 1.
eps	Cut-off points of $PVAF$, will work when the method is "GDI" or "Wald". Default = 0.95. When eps = 'logit', the predicted eps by Najera et al. (2019) will be used. See details.
alpha.level	alpha level for the wald test. Default = 0.05
criter	The kind of fit-index value. When method = "Hull", it can be R2 for $R_{McFadden}^2$ @seealso get.R2 or 'PVAF' for the proportion of variance accounted for ($PVAF$) @seealso get.PVAF. When method = "beta", it can be 'AIC', 'BIC', 'CAIC' or 'SABIC'. Default = "PVAF" for 'Hull' and default = "AIC" for 'beta'. See details.
verbose	Logical indicating to print iterative information or not. Default is TRUE

Value

An object of class validation containing the following components:

Q.orig

The original Q-matrix that maybe contain some mis-specifications and need to be validated.

Q.sug

The Q-matrix that suggested by certain validation method.

time.cost

The time that CPU cost to finish the function.

process

A matrix that contains the modification process of each question during each iteration. Each row represents an iteration, and each column corresponds to the q-vector index of the respective question. The order of the indices is consistent with the row numbering in the matrix generated by the attributepattern function in the GDINA package. Only when maxitr > 1, the, the value is available.

iter

The number of iteration. Only when maxitr > 1, the value is available.

priority

An I × K matrix that contains the priority of every attribute for each item. Only when the search.method is "PAA", the value is available. See details.

Hull.fit

A list containing all the information needed to plot the Hull plot, which is available only when method = "Hull".

The GDI method

The GDI method (de la Torre & Chiu, 2016), as the first Q-matrix validation method applicable to saturated models, serves as an important foundation for various mainstream Q-matrix validation methods.

The method calculates the proportion of variance accounted for ($PVAF$; @seealso get.PVAF) for all possible q-vectors for each item, selects the q-vector with a $PVAF$ just greater than the cut-off point (or Epsilon, EPS) as the correction result, and the variance $\zeta^2$ is the generalized discriminating index (GDI; de la Torre & Chiu, 2016). Therefore, the GDI method is also considered as a generalized extension of the $delta$ method (de la Torre, 2008), which also takes maximizing discrimination as its basic idea. In the GDI method, $\zeta^2$ is defined as the weighted variance of the correct response probabilities across all mastery patterns, that is:

$\zeta^2 = \sum_{l=1}^{2^K} \pi_{l} \left[ P(X_{pi}=1|\boldsymbol{\alpha}_{l}) - \bar{P}_{i} \right]^2$

where $\pi_{l}$ represents the prior probability of mastery pattern $l$; $\bar{P}_{i}=\sum_{k=1}^{K}\pi_{l}P(X_{pi}=1|\boldsymbol{\alpha}_{l})$ is the weighted average of the correct response probabilities across all attribute mastery patterns. When the q-vector is correctly specified, the calculated $\zeta^2$ should be maximized, indicating the maximum discrimination of the item. However, in reality, $\zeta^2$ continues to increase when the q-vector is over-specified, and the more attributes that are over-specified, the larger $\zeta^2$ becomes. The q-vector with all attributes set to 1 (i.e., $\boldsymbol{q}_{1:K}$) has the largest $\zeta^2$ (de la Torre, 2016). This is because an increase in attributes in the q-vector leads to an increase in item parameters, resulting in greater differences in correct response probabilities across attribute patterns and, consequently, increased variance. However, this increase in variance is spurious. Therefore, de la Torre et al. calculated $PVAF = \frac{\zeta^2}{\zeta_{1:K}^2}$ to describe the degree to which the discrimination of the current q-vector explains the maximum discrimination. They selected an appropriate $PVAF$ cut-off point to achieve a balance between q-vector fit and parsimony. According to previous studies, the $PVAF$ cut-off point is typically set at 0.95 (Ma & de la Torre, 2020; Najera et al., 2021). Najera et al. (2019) proposed using multinomial logistic regression to predict a more appropriate cut-off point for $PVAF$. The cut-off point is denoted as $eps$, and the predicted regression equation is as follows:

$\log \left( \frac{eps}{1-eps} \right) = \text{logit}(eps) = -0.405 + 2.867 \cdot IQ + 4.840 \times 10^4 \cdot N - 3.316 \times 10^3 \cdot I$

Where $IQ$ represents the question quality, calculated as the negative difference between the probability of an examinee with all attributes answering the question correctly and the probability of an examinee with no attributes answering the question correctly ($IQ = - \left\{ P\left( \boldsymbol{1} \right) - \left[ 1 - P\left( \boldsymbol{0} \right) \right] \right\}$), and $N$ and $I$ represent the number of examinees and the number of questions, respectively.

The Wald method

The Wald method (Ma & de la Torre, 2020) combines the Wald test with $PVAF$ to correct the Q-matrix at the item level. Its basic logic is as follows: when correcting item $i$, the single attribute that maximizes the $PVAF$ value is added to a vector with all attributes set to $\boldsymbol{0}$ (i.e., $\boldsymbol{q} = (0, 0, \ldots, 0)$) as a starting point. In subsequent iterations, attributes in this vector are continuously added or removed through the Wald test. The correction process ends when the $PVAF$ exceeds the cut-off point or when no further attribute changes occur. The Wald statistic follows an asymptotic $\chi^{2}$ distribution with a degree of freedom of $2^{K^\ast} - 1$.

The calculation method is as follows:

$Wald = \left[\boldsymbol{R} \times \boldsymbol{P}_{i}(\boldsymbol{\alpha})\right]^{'} (\boldsymbol{R} \times \boldsymbol{V}_{i} \times \boldsymbol{R})^{-1} \left[\boldsymbol{R} \times P_{i}(\boldsymbol{\alpha})\right]$

$\boldsymbol{R}$ represents the restriction matrix (@seealso get.Rmatrix); $\boldsymbol{P}_{i}(\boldsymbol{\alpha})$ denotes the vector of correct response probabilities for item $i$; $\boldsymbol{V}_i$ is the variance-covariance matrix of the correct response probabilities for item $i$, which can be obtained by multiplying the $\boldsymbol{M}_i$ matrix (de la Torre, 2011) with the variance-covariance matrix of item parameters $\boldsymbol{\Sigma}_i$, i.e., $\boldsymbol{V}_i = \boldsymbol{M}_i \times \boldsymbol{\Sigma}_i$. The $\boldsymbol{\Sigma}_i$ can be derived by inverting the information matrix. Using the the empirical cross-product information matrix (de la Torre, 2011) to calculate $\boldsymbol{\Sigma}_i$.

$\boldsymbol{M}_i$ is a $2^{K^\ast} \times 2^{K^\ast}$ matrix (@seealso get.Mmatrix) that represents the relationship between the parameters of item $i$ and the attribute mastery patterns. The rows represent different mastery patterns, while the columns represent different item parameters.

The Hull method

The Hull method (Najera et al., 2021) addresses the issue of the cut-off point in the GDI method and demonstrates good performance in simulation studies. Najera et al. applied the Hull method for determining the number of factors to retain in exploratory factor analysis (Lorenzo-Seva et al., 2011) to the retention of attribute quantities in the q-vector, specifically for Q-matrix validation. The Hull method aligns with the GDI approach in its philosophy of seeking a balance between fit and parsimony. While GDI relies on a preset, arbitrary cut-off point to determine this balance, the Hull method utilizes the most pronounced elbow in the Hull plot to make this judgment. The the most pronounced elbow is determined using the following formula:

$st = \frac{(f_k - f_{k-1}) / (np_k - np_{k-1})}{(f_{k+1} - f_k) / (np_{k+1} - np_k)}$

where $f_k$ represents the fit-index value (can be $PVAF$ @seealso get.PVAF or $R2$ @seealso get.R2) when the q-vector contains $k$ attributes, similarly, $f_{k-1}$ and $f_{k+1}$ represent the fit-index value when the q-vector contains $k-1$ and $k+1$ attributes, respectively. ${np}_k$ denotes the number of parameters when the q-vector has $k$ attributes, which is $2^k$ for a saturated model. Likewise, ${np}_{k-1}$ and ${np}_{k+1}$ represent the number of parameters when the q-vector has $k-1$ and $k+1$ attributes, respectively. The Hull method calculates the $st$ index for all possible q-vectors and retains the q-vector with the maximum $st$ index as the corrected result. Najera et al. (2021) removed any concave points from the Hull plot, and when only the first and last points remained in the plot, the saturated q-vector was selected.

The MLR-B method

The MLR-B method proposed by Tu et al. (2022) differs from the GDI, Wald and Hull method in that it does not employ $PVAF$. Instead, it directly uses the marginal probabilities of attribute mastery for examinees to perform multivariate logistic regression on their observed scores. This approach assumes all possible q-vectors and conducts $2^K-1$ regression modelings. After proposing regression equations that exclude any insignificant regression coefficients, it selects the q-vector corresponding to the equation with the minimum $AIC$ value as the validation result. The performance of this method in both the LCDM and GDM models even surpasses that of the Hull method (Tu et al., 2022), making it an efficient and reliable approach for Q-matrix validation.

The $\beta$ method

The $\beta$ method (Li & Chen, 2024) addresses the Q-matrix validation problem from the perspective of signal detection theory. Signal detection theory posits that any stimulus is a signal embedded in noise, where the signal always overlaps with noise. The $\beta$ method treats the correct q-vector as the signal and other possible q-vectors as noise. The goal is to identify the signal from the noise, i.e., to correctly identify the q-vector. For item $i$ with the q-vector of the $c$-th type, the $\beta$ index is computed as follows:

$\beta_{ic} = \sum_{l=1}^{2^K} \left| \frac{r_{li}}{n_l} P_{ic}(\boldsymbol{\alpha_l}) - \left(1 - \frac{r_{li}}{n_l}\right) \left[1 - P_{ic}(\boldsymbol{\alpha_l})\right] \right| = \sum_{l=1}^{2^K} \left| \frac{r_{li}}{n_l} - \left[1 - P_{ic}(\boldsymbol{\alpha_l}) \right] \right|$

In the formula, $r_{li}$ represents the number of examinees in knowledge state $l$ who correctly answered item $i$, while $n_l$ is the total number of examinees in knowledge state $l$. $P_{ic}(\boldsymbol{\alpha_l})$ denotes the probability that an examinee in knowledge state $l$ answers item $i$ correctly when the q-vector for item $i$ is of the $c$-th type. In fact, $\frac{r_{li}}{n_l}$ is the observed probability that an examinee in knowledge state $l$ answers item $i$ correctly, and $\beta_{jc}$ represents the difference between the actual proportion of correct answers for item $i$ in each knowledge state and the expected probability of answering the item incorrectly in that state. Therefore, to some extent, $\beta_{jc}$ can be considered as a measure of discriminability, and the $\beta$ method posits that the correct q-vector maximizes $\beta_{jc}$, i.e.:

$\boldsymbol{q}_i = \arg\max_{\boldsymbol{q}} \left( \beta_{jc} : \boldsymbol{q} \in \left\{ \boldsymbol{q}_{ic}, \, c = 1, 2, \dots, 2^{K} - 1 \right\} \right)$

Therefore, essentially, $\beta_{jc}$ is an index similar to GDI. Both increase as the number of attributes in the q-vector increases. Unlike the GDI method, the $\beta$ method does not continue to compute $\beta_{jc} / \beta_{j[11...1]}$ but instead uses the minimum $AIC$ value to determine whether the attributes in the q-vector are sufficient. In Package Qval, parLapply will be used to accelerate the $\beta$ method.

Please note that the $\beta$ method has different meanings when applying different search algorithms. For more details, see section 'Search algorithm' below.

Iterative procedure

The iterative procedure that one item modification at a time is item level iteration ( iter.level = "item") in (Najera et al., 2020, 2021). The steps of the item level iterative procedure algorithm are as follows:

Step1

Fit the CDM according to the item responses and the provisional Q-matrix ($\boldsymbol{Q}^0$).

Step2

Validate the provisional Q-matrix and gain a suggested Q-matrix ($\boldsymbol{Q}^1$).

Step3

for each item, $PVAF_{0i}$ as the $PVAF$ of the provisional q-vector specified in $\boldsymbol{Q}^0$, and $PVAF_{1i}$ as the $PVAF$ of the suggested q-vector in $\boldsymbol{Q}^1$.

Step4

Calculate all items' $\Delta PVAF_{i}$, defined as $\Delta PVAF_{i} = |PVAF_{1i} - PVAF_{0i}|$

Step5

Define the hit item as the item with the highest $\Delta PVAF_{i}$.

Step6

Update $\boldsymbol{Q}^0$ by changing the provisional q-vector by the suggested q-vector of the hit item.

Step7

Iterate over Steps 1 to 6 until $\sum_{i=1}^{I} \Delta PVAF_{i} = 0$

When the Q-matrix validation method is "MLR-B" or "Hull" when criter = "AIC" or criter = "R2", $PVAF$ is not used. In this case, the criterion for determining which item's index will be replaced is $AIC$ or $R^2$, respectively.

The iterative procedure that the entire Q-matrix is modified at each iteration is test level iteration ( iter.level = "test") (Najera et al., 2020; Tu et al., 2022). The steps of the test level iterative procedure algorithm are as follows:

Step1

Fit the CDM according to the item responses and the provisional Q-matrix ($\boldsymbol{Q}^0$).

Step2

Validate the provisional Q-matrix and gain a suggested Q-matrix ($\boldsymbol{Q}^1$).

Step3

Check whether $\boldsymbol{Q}^1 = \boldsymbol{Q}^0$. If TRUE, terminate the iterative algorithm. If FALSE, Update $\boldsymbol{Q}^0$ as $\boldsymbol{Q}^1$.

Step4

Iterate over Steps 1 and 3 until one of conditions as follows is satisfied: 1. $\boldsymbol{Q}^1 = \boldsymbol{Q}^0$; 2. Reach the maximum number of iterations (maxitr); 3. $\boldsymbol{Q}^1$ does not satisfy the condition that an attribute is measured by one item at least.

iter.level = 'test.att' will use a method called the test-attribute iterative procedure (Najera et al., 2021), which modifies all items in each iteration while following the principle of minimizing changes in the number of attributes. Therefore, the test-attribute iterative procedure and the test-level iterative procedure follow the same process for large items. The key difference is that the test-attribute iterative procedure only allows minimal adjustments to the $q$-vector in each iteration. For example, if the original $q$-vector is $[0010]$ and the validation methods suggest $[1110]$, the test-level iterative procedure can directly update the $q$-vector to $[1110]$. In contrast, the test-attribute iterative procedure can only make a gradual adjustment, first modifying the $q$-vector to either $[1010]$ or $[0110]$. As a result, the test-attribute iterative procedure is more cautious than the test-level iterative procedure and may require more iterations.

Search algorithm

Three search algorithms are available: Exhaustive Search Algorithm (ESA), Sequential Search Algorithm (SSA), and Priority Attribute Algorithm (PAA). ESA is a brute-force algorithm. When validating the q-vector of a particular item, it traverses all possible q-vectors and selects the most appropriate one based on the chosen Q-matrix validation method. Since there are $2^{K-1}$ possible q-vectors with $K$ attributes, ESA requires $2^{K-1}$ searches for each item.

SSA reduces the number of searches by adding one attribute at a time to the q-vector in a stepwise manner. Therefore, in the worst-case scenario, SSA requires $K(K-1)/2$ searches. The detailed steps are as follows:

Step 1

Define an empty q-vector $\boldsymbol{q}^0=[00...0]$ of length $K$, where all elements are 0.

Step 2

Examine all single-attribute q-vectors, which are those formed by changing one of the 0s in $\boldsymbol{q}^0$ to 1. According to the criteria of the chosen Q-matrix validation method, select the optimal single-attribute q-vector, denoted as $\boldsymbol{q}^1$.

Step 3

Examine all two-attribute q-vectors, which are those formed by changing one of the 0s in $\boldsymbol{q}^1$ to 1. According to the criteria of the chosen Q-matrix validation method, select the optimal two-attribute q-vector, denoted as $\boldsymbol{q}^2$.

Step 4

Repeat this process until $\boldsymbol{q}^K$ is found, or the stopping criterion of the chosen Q-matrix validation method is met.

PAA is a highly efficient and concise algorithm that evaluates whether each attribute needs to be included in the q-vector based on the priority of the attributes. @seealso get.priority. Therefore, even in the worst-case scenario, PAA only requires $K$ searches. The detailed process is as follows:

Step 1

Using the applicable CDM (e.g. the G-DINA model) to estimate the model parameters and obtain the marginal attribute mastery probabilities matrix $\boldsymbol{\Lambda}$

Step 2

Use LASSO regression to calculate the priority of each attribute in the q-vector for item $i$

Step 3

Check whether each attribute is included in the optimal q-vector based on the attribute priorities from high to low seriatim and output the final suggested q-vector according to the criteria of the chosen Q-matrix validation method.

The calculation of priorities is straightforward (Qin & Guo, 2025): the priority of an attribute is the regression coefficient obtained from a LASSO multinomial logistic regression, with the attribute as the independent variable and the response data from the examinees as the dependent variable. The formula (Tu et al., 2022) is as follows:

$\log[\frac{P(X_{pi} = 1 | \boldsymbol{\Lambda}_{p})}{P(X_{pi} = 0 | \boldsymbol{\Lambda}_{p})}] = logit[P(X_{pi} = 1 | \boldsymbol{\Lambda}_{p})] = \beta_{i0} + \beta_{i1} \Lambda_{p1} + \ldots + \beta_{ik} \Lambda_{pk} + \ldots + \beta_{iK} \Lambda_{pK}$

Where $X_{pi}$ represents the response of examinee $p$ on item $i$, $\boldsymbol{\Lambda}_{p}$ denotes the marginal mastery probabilities of examinee $p$ (which can be obtained from the return value alpha.P of the CDM function), $\beta_{i0}$ is the intercept term, and $\beta_{ik}$ represents the regression coefficient.

The LASSO loss function can be expressed as:

$l_{lasso}(\boldsymbol{X}_i | \boldsymbol{\Lambda}) = l(\boldsymbol{X}_i | \boldsymbol{\Lambda}) - \lambda |\boldsymbol{\beta}_i|$

Where $l_{lasso}(\boldsymbol{X}_i | \boldsymbol{\Lambda})$ is the penalized likelihood, $l(\boldsymbol{X}_i | \boldsymbol{\Lambda})$ is the original likelihood, and $\lambda$ is the tuning parameter for penalization (a larger value imposes a stronger penalty on $\boldsymbol{\beta}_i = [\beta_{i1}, \ldots, \beta_{ik}, \ldots, \beta_{iK}]$). The priority for attribute $i$ is defined as: $\boldsymbol{priority}_i = \boldsymbol{\beta}_i = [\beta_{i1}, \ldots, \beta_{ik}, \ldots, \beta_{iK}]$

It should be noted that the Wald method proposed by Ma and de la Torre (2020) uses a "stepwise" search approach. This approach involves incrementally adding or removing 1 from the q-vector and evaluating the significance of the change using the Wald test: 1. If removing a 1 results in non-significance (indicating that the 1 is unnecessary), the 1 is removed from the q-vector; otherwise, the q-vector remains unchanged. 2. If adding a 1 results in significance (indicating that the 1 is necessary), the 1 is added to the q-vector; otherwise, the q-vector remains unchanged. The process stops when the q-vector no longer changes or when the PVAF reaches the preset cut-off point (i.e., 0.95). Stepwise are unique search approach of the Wald method, and users should be aware of this. Since stepwise is inefficient and differs significantly from the extremely high efficiency of PAA, Qval package also provides PAA for q-vector search in the Wald method. When applying the PAA version of the Wald method, the search still examines whether each attribute is necessary (by checking if the Wald test reaches significance after adding the attribute) according to attribute priority. The search stops when no further necessary attributes are found or when the PVAF reaches the preset cut-off point (i.e., 0.95). The "forward" search approach is another search method available for the Wald method, which is equivalent to "SSA". When "Wald" uses search.method = "SSA", it means that the Wald method is employing the forward search approach. Its basic process is the same as 'stepwise', except that it does not remove elements from the q-vector. Therefore, the "forward" search approach is essentially equivalent to SSA.

Please note that, since the $\beta$ method essentially selects q-vectors based on $AIC$, even without using the iterative process, the $\beta$ method requires multiple parameter estimations to obtain the AIC values for different q-vectors. Therefore, the $\beta$ method is more time-consuming and computationally intensive compared to the other methods. Li and Chen (2024) introduced a specialized search approach for the $\beta$ method, which is referred to as the $\beta$ search (search.method = 'beta'). The number of searches required is $2^{K-2} + K + 1$, and the specific steps are as follows:

Step 1

For item $i$, sequentially examine the $\beta$ values for each single-attribute q-vector, select the largest $\beta_{most}$ and the smallest $\beta_{least}$, along with the corresponding attributes $k_{most}$ and $k_{least}$. (K searches)

Step 2

Then, add all possible q-vectors (a total of $2^K - 1$) containing attribute $k_{most}$ and not containing $k_{least}$ to the search space $\boldsymbol{S}_i$ (a total of $2^{K-2}$)), and unconditionally add the saturated q-vector $[11\ldots1]$ to $\boldsymbol{S}_i$ to ensure that it is tested.

Step 3

Select the q-vector with the minimum AIC from $\boldsymbol{S}_i$ as the final output of the $\beta$ method. (The remaining $2^{K-2} + 1$ searches)

The Qval package also provides three search methods, ESA, SSA, and PAA, for the $\beta$ method. When the $\beta$ method applies these three search methods, Q-matrix validation can be completed without calculating any $\beta$ values, as the $\beta$ method essentially uses AIC for selecting q-vectors. For example, when applying ESA, the $\beta$ method does not need to perform Step 1 of the $\beta$ search and only needs to include all possible q-vectors (a total of $2^K - 1$) in $\boldsymbol{S}_i$, then outputs the corresponding q-vector based on the minimum $AIC$. When applying SSA or PAA, the $\beta$ method also does not require any calculation of $\beta$ values. In this case, the $\beta$ method is consistent with the Q-matrix validation process described by Chen et al. (2013) using relative fit indices. Therefore, when the $\beta$ method does not use $\beta$ search, it is equivalent to the method of Chen et al. (2013). To better implement Chen et al. (2013)'s Q-matrix validation method using relative fit indices, the Qval package also provides $BIC$, $CAIC$, and $SABIC$ as alternatives to validate q-vectors, in addition to $AIC$.

Author(s)

Haijiang Qin <[email protected]>

References

Chen, J., de la Torre, J., & Zhang, Z. (2013). Relative and Absolute Fit Evaluation in Cognitive Diagnosis Modeling. Journal of Educational Measurement, 50(2), 123-140. DOI: 10.1111/j.1745-3984.2012.00185.x

de la Torre, J., & Chiu, C. Y. (2016). A General Method of Empirical Q-matrix Validation. Psychometrika, 81(2), 253-273. DOI: 10.1007/s11336-015-9467-8.

de la Torre, J. (2008). An Empirically Based Method of Q-Matrix Validation for the DINA Model: Development and Applications. Journal of Education Measurement, 45(4), 343-362. DOI: 10.1111/j.1745-3984.2008.00069.x.

Li, J., & Chen, P. (2024). A new Q-matrix validation method based on signal detection theory. British Journal of Mathematical and Statistical Psychology, 00, 1–33. DOI: 10.1111/bmsp.12371

Lorenzo-Seva, U., Timmerman, M. E., & Kiers, H. A. (2011). The Hull method for selecting the number of common factors. Multivariate Behavioral Research, 46, 340–364. DOI: 10.1080/00273171.2011.564527.

Ma, W., & de la Torre, J. (2020). An empirical Q-matrix validation method for the sequential generalized DINA model. British Journal of Mathematical and Statistical Psychology, 73(1), 142-163. DOI: 10.1111/bmsp.12156.

McFadden, D. (1974). Conditional logit analysis of qualitative choice behavior. In P. Zarembka (Ed.), Frontiers in economics (pp. 105–142). New York, NY: Academic Press.

Najera, P., Sorrel, M. A., & Abad, F. J. (2019). Reconsidering Cutoff Points in the General Method of Empirical Q-Matrix Validation. Educational and Psychological Measurement, 79(4), 727-753. DOI: 10.1177/0013164418822700.

Najera, P., Sorrel, M. A., de la Torre, J., & Abad, F. J. (2020). Improving Robustness in Q-Matrix Validation Using an Iterative and Dynamic Procedure. Applied Psychological Measurement, 44(6), 431-446. DOI: 10.1177/0146621620909904.

Najera, P., Sorrel, M. A., de la Torre, J., & Abad, F. J. (2021). Balancing fit and parsimony to improve Q-matrix validation. British Journal of Mathematical and Statistical Psychology, 74 Suppl 1, 110-130. DOI: 10.1111/bmsp.12228.

Qin, H., & Guo, L. (2025). Priority attribute algorithm for Q-matrix validation: A didactic. Behavior Research Methods, 57(1), 31. DOI: 10.3758/s13428-024-02547-5.

Terzi, R., & de la Torre, J. (2018). An Iterative Method for Empirically-Based Q-Matrix Validation. International Journal of Assessment Tools in Education, 248-262. DOI: 10.21449/ijate.40719.

Tu, D., Chiu, J., Ma, W., Wang, D., Cai, Y., & Ouyang, X. (2022). A multiple logistic regression-based (MLR-B) Q-matrix validation method for cognitive diagnosis models: A confirmatory approach. Behavior Research Methods. DOI: 10.3758/s13428-022-01880-x.

Examples

################################################################
#                           Example 1                          #
#             The GDI method to validate Q-matrix              #
################################################################
set.seed(123)

library(Qval)

## generate Q-matrix and data
K <- 4
I <- 20
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ,
                         model = "GDINA", distribute = "horder")

## simulate random mis-specifications
example.MQ <- sim.MQ(example.Q, 0.1)


## using MMLE/EM to fit CDM model first
example.CDM.obj <- CDM(example.data$dat, example.MQ)

## using the fitted CDM.obj to avoid extra parameter estimation.
Q.GDI.obj <- validation(example.data$dat, example.MQ, example.CDM.obj, method = "GDI")


## also can validate the Q-matrix directly
Q.GDI.obj <- validation(example.data$dat, example.MQ)

## item level iteration
Q.GDI.obj <- validation(example.data$dat, example.MQ, method = "GDI",
                        iter.level = "item", maxitr = 150)

## search method
Q.GDI.obj <- validation(example.data$dat, example.MQ, method = "GDI",
                        search.method = "ESA")

## cut-off point
Q.GDI.obj <- validation(example.data$dat, example.MQ, method = "GDI",
                        eps = 0.90)

## check QRR
print(zQRR(example.Q, Q.GDI.obj$Q.sug))




################################################################
#                           Example 2                          #
#             The Wald method to validate Q-matrix             #
################################################################
set.seed(123)

library(Qval)

## generate Q-matrix and data
K <- 4
I <- 20
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ, model = "GDINA",
                         distribute = "horder")

## simulate random mis-specifications
example.MQ <- sim.MQ(example.Q, 0.1)


## using MMLE/EM to fit CDM first
example.CDM.obj <- CDM(example.data$dat, example.MQ)

## using the fitted CDM.obj to avoid extra parameter estimation.
Q.Wald.obj <- validation(example.data$dat, example.MQ, example.CDM.obj, method = "Wald")


## also can validate the Q-matrix directly
Q.Wald.obj <- validation(example.data$dat, example.MQ, method = "Wald")

## check QRR
print(zQRR(example.Q, Q.Wald.obj$Q.sug))




################################################################
#                           Example 3                          #
#             The Hull method to validate Q-matrix             #
################################################################
set.seed(123)

library(Qval)

## generate Q-matrix and data
K <- 4
I <- 20
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ, model = "GDINA",
                         distribute = "horder")

## simulate random mis-specifications
example.MQ <- sim.MQ(example.Q, 0.1)


## using MMLE/EM to fit CDM first
example.CDM.obj <- CDM(example.data$dat, example.MQ)

## using the fitted CDM.obj to avoid extra parameter estimation.
Q.Hull.obj <- validation(example.data$dat, example.MQ, example.CDM.obj, method = "Hull")


## also can validate the Q-matrix directly
Q.Hull.obj <- validation(example.data$dat, example.MQ, method = "Hull")

## change PVAF to R2 as fit-index
Q.Hull.obj <- validation(example.data$dat, example.MQ, method = "Hull", criter = "R2")

## check QRR
print(zQRR(example.Q, Q.Hull.obj$Q.sug))




################################################################
#                           Example 4                          #
#             The MLR-B method to validate Q-matrix            #
################################################################
set.seed(123)

library(Qval)

## generate Q-matrix and data
K <- 4
I <- 20
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ, model = "GDINA",
                         distribute = "horder")

## simulate random mis-specifications
example.MQ <- sim.MQ(example.Q, 0.1)


## using MMLE/EM to fit CDM first
example.CDM.obj <- CDM(example.data$dat, example.MQ)

## using the fitted CDM.obj to avoid extra parameter estimation.
Q.MLR.obj <- validation(example.data$dat, example.MQ, example.CDM.obj, method = "MLR-B")


## also can validate the Q-matrix directly
Q.MLR.obj <- validation(example.data$dat, example.MQ, method  = "MLR-B")

## check QRR
print(zQRR(example.Q, Q.MLR.obj$Q.sug))

---