Package 'variationalDCM'

Artificial data generating function for the DINA model based on the given Q-matrix

Description

dina_data_gen() returns the artificially generated item response data for the DINA model

Usage

dina_data_gen(Q, N, attr_cor = 0.1, s = 0.2, g = 0.2, seed = 17)

Arguments

Q	the $J \times K$ binary matrix
N	the number of assumed respondents
attr_cor	the true value of the correlation among attributes (default: 0.1)
s	the true value of the slip parameter (default: 0.2)
g	the true value of the guessing parameter (default: 0.2)
seed	the seed value used for random number generation (default: 17)

Value

A list including:

X

the generated artificial item response data

attr_pat

the generated true vale of the attribute mastery pattern

References

Oka, M., & Okada, K. (2023). Scalable Bayesian Approach for the Dina Q-Matrix Estimation Combining Stochastic Optimization and Variational Inference. Psychometrika, 88, 302–331. doi:10.1007/s11336-022-09884-4

Examples

# load Q-matrix
Q = sim_Q_J80K5
sim_data = dina_data_gen(Q=Q,N=200)

---

Artificial data generating function for the hidden-Markov DCM based on the given Q-matrix

Description

hm_dcm_data_gen() returns the artificially generated item response data for the HM-DCM

Usage

hm_dcm_data_gen(
  N,
  Q,
  min_theta = 0.2,
  max_theta = 0.8,
  attr_cor = 0.1,
  seed = 17
)

Arguments

N	the number of assumed respondents
Q	the $J \times K$ binary matrix
min_theta	the minimum value of the item parameter $\theta_{jht}$
max_theta	the maximum value of the item parameter $\theta_{jht}$
attr_cor	the true value of the correlation among attributes (default: 0.1)
seed	the seed value used for random number generation (default: 17)

Value

A list including:

X

the generated artificial item response data

attr_pat

the generated true vale of the attribute mastery pattern, matrix form

attr_pat_string

the generated true vale of the attribute mastery pattern, string form

References

Yamaguchi, K., & Martinez, A. J. (2024). Variational Bayes inference for hidden Markov diagnostic classification models. British Journal of Mathematical and Statistical Psychology, 77(1), 55–79. doi:10.1111/bmsp.12308

Examples

indT = 3
Q = sim_Q_J30K3
hm_sim_Q = lapply(1:indT,function(time_point) Q)
hm_sim_data = hm_dcm_data_gen(Q=hm_sim_Q,N=200)

---

Artificial data generating function for the multiple-choice DINA model based on the given Q-matrix

Description

mc_dina_data_gen() returns the artificially generated item response data for the MC-DINA model

Usage

mc_dina_data_gen(N, Q, attr_cor = 0.1, seed = 17)

Arguments

N	the number of assumed respondents
Q	the $J \times K$ binary matrix
attr_cor	the true value of the correlation among attributes (default: 0.1)
seed	the seed value used for random number generation (default: 17)

Value

A list including:

X

the generated artificial item response data

attr_pat

the generated true vale of the attribute mastery pattern

References

Yamaguchi, K. (2020). Variational Bayesian inference for the multiple-choice DINA model. Behaviormetrika, 47(1), 159-187. doi:10.1007/s41237-020-00104-w

Examples

# load a simulated Q-matrix
mc_Q = mc_sim_Q
mc_sim_data = mc_dina_data_gen(Q=mc_Q,N=200)

---

Artificial Q-matrix for MC-DINA model

Description

Artificial Q-matrix for a 30-item test measuring 5 attributes.

Usage

mc_sim_Q

Format

A matrix with components

column 1

Item number

column 2

Stem

column 3 to end

attributes

References

Yamaguchi, K. (2020). Variational Bayesian inference for the multiple-choice DINA model. Behaviormetrika, 47(1), 159-187. doi:10.1007/s41237-020-00104-w

---

Artificial Q-matrix for 30 items 3 attributes

Description

this matrix represents an artificial Q-matrix for 30 items and 3 attributes

Usage

sim_Q_J30K3

Format

An object of class matrix (inherits from array) with 30 rows and 3 columns.

Source

artificially simulated

---

Artificial Q-matrix for 80 items 5 attributes

Description

Artificial Q-matrix for a 80-item test measuring 5 attributes

Usage

sim_Q_J80K5

Format

An object of class matrix (inherits from array) with 80 rows and 5 columns.

Source

artificially simulated

---

Variational Bayesian estimation for DCMs

Description

variationalDCM() fits DCMs by VB algorithms.

Usage

variationalDCM(X, Q, model, max_it = 500, epsilon = 1e-04, verbose = TRUE, ...)

## S3 method for class 'variationalDCM'
summary(object, ...)

Arguments

X	$N \times J$ item response data for the DINA, DINO, MC-DINA, and saturated DCM models. Alternatively, $T$-length list or 3-dim array whose elements are $N \times J/T$ binary item response data matrices for the HM-DCM
Q	$J \times K$ binary Q-matrix for the DINA, DINO, and saturated DCM models. For the MC-DINA model, its size should be $J \times (K+2)$. Alternatively, $T$-length list or 3-dim array whose elements are $J/T \times K$ Q-matrices for the HM-DCM
model	specify one of "dina", "dino", "mc_dina", "satu_dcm", and "hm_dcm"
max_it	Maximum number of iterations (default: 500)
epsilon	convergence tolerance for iterations (default: 1e-4)
verbose	logical, controls whether to print progress (default: TRUE)
...	additional arguments such as hyperparameter values
object	the return of the variationalDCM function and the argument of our summary function

Value

variationalDCM returns an object of class variationalDCM. We provide the summary function to summarize a result and users can check the following information:

(model parameters)

estimates of posteror means and posterior standard deviations of model parameters that vary up to the model

attr_mastery_pat

MAP etimates of attribute mastery patterns

ELBO

resulting value of evidence lower bound

time

time spent in computation

Methods (by generic)

• summary(variationalDCM): print summary information

variationalDCM

The variationalDCM() function performs recently-developed variational Bayesian inference for various DCMs. The current version can support the DINA, DINO, MC-DINA, saturated DCM, HM-DCM models. We briefly introduce additional arguments that are specific to each model.

DINA model

The DINA model has two types of model parameters: slip $s_j$ and guessing $g_j$ for $j=1,\cdots,J$. We name the hyperparameters for the DINA model: delta_0 is a L-dimensional vector, which is a hyperparameter $\boldsymbol{\delta}^0$ for the Dirichlet distribution for the class mixing parameter $\boldsymbol{\pi}$ (default: NULL). When delta_0 is specified as NULL, we set $\boldsymbol{\delta}^0=\boldsymbol{1}_L$. alpha_s, beta_s, alpha_g, and beta_g are positive values. They are hyperparameters {$\alpha_s$, $\beta_s$, $\alpha_g$, $\beta_g$} that determines the shape of prior beta distribution for the slip and guessing parameters (default: NULL). When they are specified as NULL, they are set $1$.

DINO model

The DINO model has the same model parameters and hyperparameters as the DINA model. We thus refer the readers to the DINA model.

MC-DINA model

The MC-DINA model has additional arguments delta_0 and a_0. a_0 corresponds to positive hyperparamters $\mathbf{a}_{jc^\prime}^0$ for all $j$ and $c^\prime$. a_0 is by default set to NULL, and then it is specified as $1$ for all elements.

Saturated DCM

The saturated DCM is a generalized model such as the G-DINA and GDM. In the saturated DCM, we have hyperparameters $\mathbf{A}^0$ and $\mathbf{B}^0$ in addition to $\boldsymbol{\delta}^0$, which can be specified as arguments A_0 and B_0. They are specified by default as NULL, and then we set weakly informative priors.

HM-DCM

When model is specified as "hm_dcm", users have additional arguments nondecreasing_attribute, measurement_model, random_block_design, Test_versions, Test_order, random_start, A_0, B_0, delta_0, and omega_0. Users can accommodate the nondecreasing attribute constraint, which represents the assumption that mastered attributes are not forgotten, by setting the logical valued argument nondecreasing_attribute as TRUE (default: FALSE). Users can also control the measurement model by specifying measurement_model (default: "general"), and the current version can deal with the HM-general DCM ("general") and HM-DINA ("dina") models. This function can also handle the datasets collected by a random block design by specifying the logical valued argument random_block_design (default: FALSE). When it is specified as TRUE, users must enter Test_versions and Test_order. Test_versions is an argument indicating which version of the test each respondent has been assigned to based on a random block design, while Test_order indicates the sequence in which items are rearranged based on the random block design. A_0, B_0, delta_0, and omega_0 correspond to hyperparameters $\mathbf{A}^0$, $\mathbf{B}^0$, $\boldsymbol{\delta}^0$, and $\boldsymbol{\Omega}^0$. $\boldsymbol{\Omega}^0$ is nonnegative hyperparameters of Dirichlet distributions for attribute transition probabilities. omega_0 is by default set to NULL, and then we set $\boldsymbol{\Omega}^0=\mathbf{1}_L\mathbf{1}_L^\top$.

References

Yamaguchi, K., & Okada, K. (2020). Variational Bayes inference for the DINA model. Journal of Educational and Behavioral Statistics, 45(5), 569-597. doi:10.3102/1076998620911934

Yamaguchi, K. (2020). Variational Bayesian inference for the multiple-choice DINA model. Behaviormetrika, 47(1), 159-187. doi:10.1007/s41237-020-00104-w

Yamaguchi, K., Okada, K. (2020). Variational Bayes Inference Algorithm for the Saturated Diagnostic Classification Model. Psychometrika, 85(4), 973–995. doi:10.1007/s11336-020-09739-w

Yamaguchi, K., & Martinez, A. J. (2024). Variational Bayes inference for hidden Markov diagnostic classification models. British Journal of Mathematical and Statistical Psychology, 77(1), 55–79. doi:10.1111/bmsp.12308

Examples

# fit the DINA model
Q = sim_Q_J80K5
sim_data = dina_data_gen(Q=Q,N=200)
res = variationalDCM(X=sim_data$X, Q=Q, model="dina")
summary(res)

---