Package 'GDINA'

The Generalized DINA Model Framework

Description

For conducting CDM analysis within the G-DINA model framework

Details

This package (Ma & de la Torre, 2020a) provides a framework for a series of cognitively diagnostic analyses for dichotomous and polytomous responses.

Various cognitive diagnosis models (CDMs) can be calibrated using the GDINA function, including the G-DINA model (de la Torre, 2011), the deterministic inputs, noisy and gate (DINA; de la Torre, 2009; Junker & Sijtsma, 2001) model, the deterministic inputs, noisy or gate (DINO; Templin & Henson, 2006) model, the reduced reparametrized unified model (R-RUM; Hartz, 2002), the additive CDM (A-CDM; de la Torre, 2011), and the linear logistic model (LLM; Maris, 1999), the multiple-strategy DINA model (de la Torre, & Douglas, 2008) and models defined by users under the G-DINA framework using different link functions and design matrices (de la Torre, 2011). Note that the LLM is also called compensatory RUM and the RRUM is equivalent to the generalized NIDA model.

For ordinal and nominal responses, the sequential G-DINA model (Ma, & de la Torre, 2016) can be fitted and most of the aforementioned CDMs can be used as the processing functions (Ma, & de la Torre, 2016) at the category level. Different CDMs can be assigned to different items within a single assessment. Item parameters are estimated using the MMLE/EM algorithm. Details about the estimation algorithm can be found in Ma and de la Torre (2020). The joint attribute distribution can be modeled using an independent model, a higher-order IRT model (de la Torre, & Douglas, 2004), a loglinear model (Xu & von Davier, 2008), a saturated model or a hierarchical structures (e.g., linear, divergent). Monotonicity constraints for item/category success probabilities can also be specified.

In addition, to handle multiple strategies, generalized multiple-strategy CDMs for dichotomous response (Ma & Guo, 2019) can be fitted using GMSCDM function and diagnostic tree model (Ma, 2019) can also be estimated using DTM function for polytomous responses. Note that these functions are experimental, and are expected to be further extended in the future. Other diagnostic approaches include the multiple-choice model (de la Torre, 2009) and an iterative latent class analysis (ILCA; Jiang, 2019).

Various Q-matrix validation methods (de la Torre, & Chiu, 2016; de la Torre & Ma, 2016; Ma & de la Torre, 2020b; Najera, Sorrel, & Abad, 2019; see Qval), model-data fit statistics (Chen, de la Torre, & Zhang, 2013; Hansen, Cai, Monroe, & Li, 2016; Liu, Tian, & Xin, 2016; Ma, 2020; see modelfit and itemfit), model comparison at test and item level (de la Torre, 2011; de la Torre, & Lee, 2013; Ma, Iaconangelo, & de la Torre, 2016; Ma & de la Torre, 2019; Sorrel, Abad, Olea, de la Torre, & Barrada, 2017; Sorrel, de la Torre, Abad, & Olea, 2017; see modelcomp), and differential item functioning (Hou, de la Torre, & Nandakumar, 2014; Ma, Terzi, Lee, & de la Torre, 2017; see dif) can also be conducted.

To use the graphical user interface, check startGDINA.

Author(s)

Wenchao Ma, The University of Alabama, [email protected]
Jimmy de la Torre, The University of Hong Kong

References

Chen, J., & de la Torre, J. (2013). A General Cognitive Diagnosis Model for Expert-Defined Polytomous Attributes. Applied Psychological Measurement, 37, 419-437.

Chen, J., de la Torre, J., & Zhang, Z. (2013). Relative and Absolute Fit Evaluation in Cognitive Diagnosis Modeling. Journal of Educational Measurement, 50, 123-140.

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

de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179-199.

de la Torre, J. & Chiu, C-Y. (2016). A General Method of Empirical Q-matrix Validation. Psychometrika, 81, 253-273.

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

de La Torre, J., & Douglas, J. A. (2008). Model evaluation and multiple strategies in cognitive diagnosis: An analysis of fraction subtraction data. Psychometrika, 73, 595.

de la Torre, J., & Lee, Y. S. (2013). Evaluating the wald test for item-level comparison of saturated and reduced models in cognitive diagnosis. Journal of Educational Measurement, 50, 355-373.

de la Torre, J., & Ma, W. (2016, August). Cognitive diagnosis modeling: A general framework approach and its implementation in R. A Short Course at the Fourth Conference on Statistical Methods in Psychometrics, Columbia University, New York.

Haertel, E. H. (1989). Using restricted latent class models to map the skill structure of achievement items. Journal of Educational Measurement, 26, 301-321.

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.

Hou, L., de la Torre, J., & Nandakumar, R. (2014). Differential item functioning assessment in cognitive diagnostic modeling: Application of the Wald test to investigate DIF in the DINA model. Journal of Educational Measurement, 51, 98-125.

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

Ma, W. (2019). A diagnostic tree model for polytomous responses with multiple strategies. British Journal of Mathematical and Statistical Psychology, 72, 61-82.

Ma, W. (2020). Evaluating the fit of sequential G-DINA model using limited-information measures. Applied Psychological Measurement, 44, 167-181.

Ma, W., & de la Torre, J. (2016). A sequential cognitive diagnosis model for polytomous responses. British Journal of Mathematical and Statistical Psychology. 69, 253-275.

Ma, W., & de la Torre, J. (2019). Category-Level Model Selection for the Sequential G-DINA Model. Journal of Educational and Behavioral Statistics. 44, 61-82.

Ma, W., & de la Torre, J. (2019). Digital Module 05: Diagnostic measurement-The G-DINA framework. Educational Measurement: Issues and Practice, 39, 114-115.

Ma, W., & de la Torre, J. (2020a). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Ma, W., & de la Torre, J. (2020b). An empirical Q-matrix validation method for the sequential G-DINA model. British Journal of Mathematical and Statistical Psychology, 73, 142-163.

Ma, W., & Guo, W. (2019). Cognitive diagnosis models for multiple strategies. British Journal of Mathematical and Statistical Psychology, 72, 370-392.

Ma, W., Iaconangelo, C., & de la Torre, J. (2016). Model similarity, model selection and attribute classification. Applied Psychological Measurement, 40, 200-217.

Ma, W., Terzi, R., Lee, S., & de la Torre, J. (2017, April). Multiple group cognitive diagnosis models and their applications in detecting differential item functioning. Paper presented at the Annual Meeting ofthe American Educational Research Association, San Antonio, TX.

Maris, E. (1999). Estimating multiple classification latent class models. Psychometrika, 64, 187-212.

Najera, P., Sorrel, M., & Abad, P. (2019). Reconsidering Cutoff Points in the General Method of Empirical Q-Matrix Validation. Educational and Psychological Measurement.

Sorrel, M. A., Abad, F. J., Olea, J., de la Torre, J., & Barrada, J. R. (2017). Inferential Item-Fit Evaluation in Cognitive Diagnosis Modeling. Applied Psychological Measurement, 41, 614-631.

Sorrel, M. A., de la Torre, J., Abad, F. J., & Olea, J. (2017). Two-Step Likelihood Ratio Test for Item-Level Model Comparison in Cognitive Diagnosis Models. Methodology, 13, 39-47. Xu, X., & von Davier, M. (2008). Fitting the structured general diagnostic model to NAEP data. ETS research report, RR-08-27.

See Also

CDM for estimating G-DINA model and a set of other CDMs; ACTCD and NPCD for nonparametric CDMs; dina for DINA model in Bayesian framework

---

Generate hierarchical attribute structures

Description

This function can be used to generate hierarchical attributes structures, and to provide prior joint attribute distribution with hierarchical structures.

Usage

att.structure(hierarchy.list = NULL, K, Q, att.prob = "uniform")

Arguments

hierarchy.list	a list specifying the hierarchical structure between attributes. Each element in this list specifies a DIRECT prerequisite relation between two or more attributes. See example for more information.
K	the number of attributes involved in the assessment
Q	Q-matrix
att.prob	How are the probabilities for latent classes simulated? It can be "random" or "uniform".

Value

att.str reduced latent classes under the specified hierarchical structure

impossible.latentclass impossible latent classes under the specified hierarchical structure

att.prob probabilities for all latent classes; 0 for impossible latent classes

Author(s)

Wenchao Ma, The University of Alabama, [email protected]
Jimmy de la Torre, The University of Hong Kong

See Also

GDINA, autoGDINA

Examples

## Not run: 
#################
#
# Leighton et al. (2004, p.210)
#
##################
# linear structure A1->A2->A3->A4->A5->A6
K <- 6
linear=list(c(1,2),c(2,3),c(3,4),c(4,5),c(5,6))
att.structure(linear,K)

# convergent structure A1->A2->A3->A5->A6;A1->A2->A4->A5->A6
K <- 6
converg <- list(c(1,2),c(2,3),c(2,4),
               c(3,4,5), #this is how to show that either A3 or A4 is a prerequisite to A5
               c(5,6))
att.structure(converg,K)

# convergent structure [the difference between this one and the previous one is that
#                       A3 and A4 are both needed in order to master A5]
K <- 6
converg2 <- list(c(1,2),c(2,3),c(2,4),
               c(3,5), #this is how to specify that both A3 and A4 are needed for A5
               c(4,5), #this is how to specify that both A3 and A4 are needed for A5
               c(5,6))
att.structure(converg2,K)

# divergent structure A1->A2->A3;A1->A4->A5;A1->A4->A6
diverg <- list(c(1,2),
               c(2,3),
               c(1,4),
               c(4,5),
               c(4,6))
att.structure(diverg,K)

# unstructured A1->A2;A1->A3;A1->A4;A1->A5;A1->A6
unstru <- list(c(1,2),c(1,3),c(1,4),c(1,5),c(1,6))
att.structure(unstru,K)

## See Example 4 and 5 in GDINA function

## End(Not run)

---

Generate all possible attribute patterns

Description

This function generates all possible attribute patterns. The Q-matrix needs to be specified for polytomous attributes.

Usage

attributepattern(K, Q)

Arguments

K	number of attributes
Q	Q-matrix; required when Q-matrix is polytomous

Value

A $2^K\times K$ matrix consisting of attribute profiles for $2^K$ latent classes

Author(s)

Wenchao Ma, The University of Alabama, [email protected]
Jimmy de la Torre, The University of Hong Kong

Examples

attributepattern(3)

q <- matrix(scan(text = "0 1 2 1 0 1 1 2 0"),ncol = 3)
q
attributepattern(Q=q)

q <- matrix(scan(text = "0 1 1 1 0 1 1 1 0"),ncol = 3)
q
attributepattern(K=ncol(q),Q=q)

---

Q-matrix validation, model selection and calibration in one run

Description

autoGDINA conducts a series of CDM analyses within the G-DINA framework. Particularly, the GDINA model is fitted to the data first using the GDINA function; then, the Q-matrix is validated using the function Qval. Based on the suggested Q-matrix, the data is fitted by the G-DINA model again, followed by an item level model selection via the Wald test using modelcomp. Lastly, the selected models are calibrated based on the suggested Q-matrix using the GDINA function. The Q-matrix validation and item-level model selection can be disabled by the users. Possible reduced CDMs for Wald test include the DINA model, the DINO model, A-CDM, LLM and RRUM. See Details for the rules of item-level model selection.

Usage

autoGDINA(
  dat,
  Q,
  modelselection = TRUE,
  modelselectionrule = "simpler",
  alpha.level = 0.05,
  modelselection.args = list(),
  Qvalid = TRUE,
  Qvalid.args = list(),
  GDINA1.args = list(),
  GDINA2.args = list(),
  CDM.args = list()
)

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

Arguments

dat	A required $N \times J$ matrix or data.frame consisting of the responses of $N$ individuals to $J$ items. Missing values need to be coded as NA.
Q	A required matrix; The number of rows occupied by a single-strategy dichotomous item is 1, by a polytomous item is the number of nonzero categories, and by a mutiple-strategy dichotomous item is the number of strategies. The number of column is equal to the number of attributes if all items are single-strategy dichotomous items, but the number of attributes + 2 if any items are polytomous or have multiple strategies. For a polytomous item, the first column represents the item number and the second column indicates the nonzero category number. For a multiple-strategy dichotomous item, the first column represents the item number and the second column indicates the strategy number. For binary attributes, 1 denotes the attributes are measured by the items and 0 means the attributes are not measured. For polytomous attributes, non-zero elements indicate which level of attributes are needed (see Chen, & de la Torre, 2013). See Examples.
modelselection	logical; conducting model selection or not?
modelselectionrule	how to conducted model selection? Possible options include simpler, largestp and DS. See Details.
alpha.level	nominal level for the Wald test. The default is 0.05.
modelselection.args	arguments passed to modelcomp
Qvalid	logical; validate Q-matrix or not? TRUE is the default.
Qvalid.args	arguments passed to Qval
GDINA1.args	arguments passed to GDINA function for initial G-DINA calibration
GDINA2.args	arguments passed to GDINA function for the second G-DINA calibration
CDM.args	arguments passed to GDINA function for final calibration
object	GDINA object for various S3 methods
...	additional arguments

Details

After the Wald statistics for each reduced CDM were calculated for each item, the reduced models with p values less than the pre-specified alpha level were rejected. If all reduced models were rejected for an item, the G-DINA model was used as the best model; if at least one reduced model was retained, three diferent rules can be implemented for selecting the best model:

When modelselectionrule is simpler:

If (a) the DINA or DINO model was one of the retained models, then the DINA or DINO model with the larger p value was selected as the best model; but if (b) both DINA and DINO were rejected, the reduced model with the largest p value was selected as the best model for this item. Note that when the p-values of several reduced models were greater than 0.05, the DINA and DINO models were preferred over the A-CDM, LLM, and R-RUM because of their simplicity. This procedure is originally proposed by Ma, Iaconangelo, and de la Torre (2016).

When modelselectionrule is largestp:

The reduced model with the largest p-values is selected as the most appropriate model.

When modelselectionrule is DS:

The reduced model with non-significant p-values but the smallest dissimilarity index is selected as the most appropriate model. Dissimilarity index can be viewed as an effect size measure, which quatifies how dis-similar the reduced model is from the G-DINA model (See Ma, Iaconangelo, and de la Torre, 2016 for details).

Value

a list consisting of the following elements:

GDINA1.obj

initial GDINA calibration of class GDINA

GDINA2.obj

second GDINA calibration of class GDINA

Qval.obj

Q validation object of class Qval

Wald.obj

model comparison object of class modelcomp

CDM.obj

Final CDM calibration of class GDINA

Methods (by generic)

• summary(autoGDINA): print summary information

Note

Returned GDINA1.obj, GDINA2.obj and CDM.obj are objects of class GDINA, and all S3 methods suitable for GDINA objects can be applied. See GDINA and extract. Similarly, returned Qval.obj and Wald.obj are objects of class Qval and modelcomp.

Author(s)

Wenchao Ma, The University of Alabama, [email protected]

References

Ma, W., & de la Torre, J. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Ma, W., Iaconangelo, C., & de la Torre, J. (2016). Model similarity, model selection and attribute classification. Applied Psychological Measurement, 40, 200-217.

See Also

GDINA, modelcomp, Qval

Examples

## Not run: 
# simulated responses
Q <- sim10GDINA$simQ
dat <- sim10GDINA$simdat

#misspecified Q
misQ <- Q
misQ[10,] <- c(0,1,0)
out1 <- autoGDINA(dat,misQ,modelselectionrule="largestp")
out1
summary(out1)
AIC(out1$CDM.obj)

# simulated responses
Q <- sim30GDINA$simQ
dat <- sim30GDINA$simdat

#misspecified Q
misQ <- Q
misQ[1,] <- c(1,1,0,1,0)
auto <- autoGDINA(dat,misQ,Qvalid = TRUE, Qvalid.args = list(method = "wald"),
                  modelselectionrule="simpler")
auto
summary(auto)
AIC(auto$CDM.obj)

#using the other selection rule
out11 <- autoGDINA(dat,misQ,modelselectionrule="simpler",
                   modelselection.args = list(models = c("DINO","DINA")))
out11
summary(out11)

# disable model selection function
out12 <- autoGDINA(dat,misQ,modelselection=FALSE)
out12
summary(out12)


# Disable Q-matrix validation
out3 <- autoGDINA(dat = dat, Q = misQ, Qvalid = FALSE)
out3
summary(out3)

## End(Not run)

---

Create a block diagonal matrix

Description

Create a block diagonal matrix

Usage

bdiagMatrix(mlist, fill = 0)

Arguments

mlist	a list of matrices
fill	value to fill the non-diagnoal elements

Value

a block diagonal matrix

See Also

bdiag in Matrix

Examples

m1 <- bdiagMatrix(list(matrix(1:4, 2), diag(3)))
m2 <- bdiagMatrix(list(matrix(1:4, 2), diag(3)),fill = NA)

---

Calculating standard errors and variance-covariance matrix using bootstrap methods

Description

This function conducts nonparametric and parametric bootstrap to calculate standard errors of model parameters. Parametric bootstrap is only applicable to single group models.

Usage

bootSE(GDINA.obj, bootsample = 50, type = "nonparametric", randomseed = 12345)

Arguments

GDINA.obj	an object of class GDINA
bootsample	the number of bootstrap samples
type	type of bootstrap method. Can be parametric or nonparametric
randomseed	random seed for resampling

Value

itemparm.se standard errors for item probability of success in list format

delta.se standard errors for delta parameters in list format

lambda.se standard errors for structural parameters of joint attribute distribution

boot.est resample estimates

Author(s)

Wenchao Ma, The University of Alabama, [email protected]

References

Ma, W., & de la Torre, J. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Examples

## Not run: 
# For illustration, only 5 resamples are run
# results are definitely not reliable

dat <- sim30GDINA$simdat
Q <- sim30GDINA$simQ
fit <- GDINA(dat = dat, Q = Q, model = "GDINA",att.dist = "higher.order")
boot.fit <- bootSE(fit,bootsample = 5,randomseed=123)
boot.fit$delta.se
boot.fit$lambda.se

## End(Not run)

---

Calculate classification accuracy

Description

This function calculate test-, pattern- and attribute-level classification accuracy indices based on GDINA estimates from the GDINA function using approaches in Iaconangelo (2017) and Wang, Song, Chen, Meng, and Ding (2015). It is only applicable for dichotomous attributes.

Usage

CA(GDINA.obj, what = "MAP")

Arguments

GDINA.obj	estimated GDINA object returned from GDINA
what	what attribute estimates are used? Default is "MAP".

Value

a list with elements

tau

estimated test-level classification accuracy, see Iaconangelo (2017, Eq 2.2)

tau_l

estimated pattern-level classification accuracy, see Iaconangelo (2017, p. 13)

tau_k

estimated attribute-level classification accuracy, see Wang, et al (2015, p. 461 Eq 6)

CCM

Conditional classification matrix, see Iaconangelo (2017, p. 13)

Author(s)

Wenchao Ma, The University of Alabama, [email protected]

References

Iaconangelo, C.(2017). Uses of Classification Error Probabilities in the Three-Step Approach to Estimating Cognitive Diagnosis Models. (Unpublished doctoral dissertation). New Brunswick, NJ: Rutgers University.

Ma, W., & de la Torre, J. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Wang, W., Song, L., Chen, P., Meng, Y., & Ding, S. (2015). Attribute-Level and Pattern-Level Classification Consistency and Accuracy Indices for Cognitive Diagnostic Assessment. Journal of Educational Measurement, 52 , 457-476.

Examples

## Not run: 
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
fit <- GDINA(dat = dat, Q = Q, model = "GDINA")
fit
CA(fit)

## End(Not run)

---

Combine R Objects by Columns

Description

Combine a sequence of vector, matrix or data-frame arguments by columns. Vector is treated as a column matrix.

Usage

cjoint(..., fill = NA)

Arguments

...	vectors or matrices
fill	a scalar used when these objects have different number of rows.

Value

a data frame

See Also

cbind

Examples

cjoint(2,c(1,2,3,4),matrix(1:6,2,3))
cjoint(v1 = 2, v2 = c(3,2), v3 = matrix(1:6,3,2),
       v4 = data.frame(c(3,4,5,6,7),rep("x",5)),fill = 99)

---

Classification Rate Evaluation

Description

This function evaluates the classification rates for two sets of attribute profiles

Usage

ClassRate(att1, att2)

Arguments

att1	a matrix or data frame of attribute profiles
att2	a matrix or data frame of attribute profiles

Value

a list with the following components:

PCA

the proportion of correctly classified attributes (i.e., attribute level classification rate)

PCV

a vector giving the proportions of correctly classified attribute vectors (i.e., vector level classification rate). The fist element is the proportion of at least one attribute in the vector are correctly identified; the second element is the proportion of at least two attributes in the vector are correctly identified; and so forth. The last element is the proportion of all elements in the vector are correctly identified.

Author(s)

Wenchao Ma, The University of Alabama, [email protected]

References

Ma, W., & de la Torre, J. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Examples

## Not run: 
N <- 2000
# model does not matter if item parameter is probability of success
Q <- sim30GDINA$simQ
J <- nrow(Q)
gs <- matrix(0.1,J,2)

set.seed(12345)
sim <- simGDINA(N,Q,gs.parm = gs)
GDINA.est <- GDINA(sim$dat,Q)

CR <- ClassRate(sim$attribute,personparm(GDINA.est))
CR

## End(Not run)

---

Generate design matrix

Description

This function generates the design matrix for an item

Usage

designmatrix(Kj = NULL, model = "GDINA", Qj = NULL, no.bugs = 0)

Arguments

Kj	Required except for the MS-DINA model; The number of attributes required for item j
model	the model associated with the design matrix; It can be "GDINA","DINA","DINO", "ACDM","LLM", "RRUM", "MSDINA", "BUGDINO", and "SISM". The default is "GDINA". Note that models "LLM" and "RRUM" have the same design matrix as the "ACDM".
Qj	the Q-matrix for item j; This is required for "MSDINA", and "SISM" models; The number of rows is equal to the number of strategies for "MSDINA", and the number of columns is equal to the number of attributes.
no.bugs	the number of bugs (or misconceptions). Note that bugs must be given in the last no.bugs columns.

Value

a design matrix (Mj). See de la Torre (2011) for details.

References

de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179-199.

Examples

## Not run: 
designmatrix(Kj = 2, model = "GDINA")
designmatrix(Kj = 3, model = "DINA")
msQj <- matrix(c(1,0,0,1,
                 1,1,0,0),nrow=2,byrow=TRUE)
designmatrix(model = "MSDINA",Qj = msQj)

## End(Not run)

---

Differential item functioning for cognitive diagnosis models

Description

This function is used to detect differential item functioning using the Wald test (Hou, de la Torre, & Nandakumar, 2014; Ma, Terzi, & de la Torre, 2021) and the likelihood ratio test (Ma, Terzi, & de la Torre, 2021). The forward anchor item search procedure developed in Ma, Terzi, and de la Torre (2021) was implemented. Note that it can only detect DIF for two groups currently.

Usage

dif(
  dat,
  Q,
  group,
  model = "GDINA",
  method = "wald",
  anchor.items = NULL,
  dif.items = "all",
  p.adjust.methods = "holm",
  approx = FALSE,
  SE.type = 2,
  FS.args = list(on = FALSE, alpha.level = 0.05, maxit = 10, verbose = FALSE),
  ...
)

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

Arguments

dat	item responses from two groups; missing data need to be coded as NA
Q	Q-matrix specifying the association between items and attributes
group	a factor or a vector indicating the group each individual belongs to. Its length must be equal to the number of individuals.
model	model for each item.
method	DIF detection method; It can be "wald" for Hou, de la Torre, and Nandakumar's (2014) Wald test method, and "LR" for likelihood ratio test (Ma, Terzi, Lee,& de la Torre, 2017).
anchor.items	which items will be used as anchors? Default is NULL, which means none of the items are used as anchors. For LR method, it can also be an integer vector giving the item numbers for anchors or "all", which means all items are treated as anchor items.
dif.items	which items are subject to DIF detection? Default is "all". It can also be an integer vector giving the item numbers.
p.adjust.methods	adjusted p-values for multiple hypothesis tests. This is conducted using p.adjust function in stats, and therefore all adjustment methods supported by p.adjust can be used, including "holm", "hochberg", "hommel", "bonferroni", "BH" and "BY". See p.adjust for more details. "holm" is the default.
approx	Whether an approximated LR test is implemented? If TRUE, parameters of items except the studied one will not be re-estimated.
SE.type	Type of standard error estimation methods for the Wald test.
FS.args	arguments for the forward anchor item search procedure developed in Ma, Terzi, and de la Torre (2021). A list with the following elements: on - logical; TRUE if activate the forward anchor item search procedure. Default = FALSE. alpha.level - nominal level for Wald or LR test. Default = .05. maxit - maximum number of iterations allowed. Default = 10. verbose - logical; print information for each iteration or not? Default = FALSE.
...	arguments passed to GDINA function for model calibration
object	dif object for S3 method

Value

A data frame giving the Wald statistics and associated p-values.

Methods (by generic)

• summary(dif): print summary information

Author(s)

Wenchao Ma, The University of Alabama, [email protected]
Jimmy de la Torre, The University of Hong Kong

References

Hou, L., de la Torre, J., & Nandakumar, R. (2014). Differential item functioning assessment in cognitive diagnostic modeling: Application of the Wald test to investigate DIF in the DINA model. Journal of Educational Measurement, 51, 98-125.

Ma, W., Terzi, R., & de la Torre, J. (2021). Detecting differential item functioning using multiple-group cognitive diagnosis models. Applied Psychological Measurement.

See Also

GDINA

Examples

## Not run: 
set.seed(123456)
N <- 3000
Q <- sim30GDINA$simQ
gs <- matrix(.2,ncol = 2, nrow = nrow(Q))
# By default, individuals are simulated from uniform distribution
# and deltas are simulated randomly
sim1 <- simGDINA(N,Q,gs.parm = gs,model="DINA")
sim2 <- simGDINA(N,Q,gs.parm = gs,model=c(rep("DINA",nrow(Q)-1),"DINO"))
dat <- rbind(extract(sim1,"dat"),extract(sim2,"dat"))
gr <- rep(c("G1","G2"),each=N)

# DIF using Wald test
dif.wald <- dif(dat, Q, group=gr, method = "Wald")
dif.wald
# DIF using LR test
dif.LR <- dif(dat, Q, group=gr, method="LR")
dif.LR
# DIF using Wald test + forward search algorithm
dif.wald.FS <- dif(dat, Q, group=gr, method = "Wald", FS.args = list(on = TRUE, verbose = TRUE))
dif.wald.FS
# DIF using LR test + forward search algorithm
dif.LR.FS <- dif(dat, Q, group=gr, method = "LR", FS.args = list(on = TRUE, verbose = TRUE))
dif.LR.FS

## End(Not run)

---

Experimental function for diagnostic multiple-strategy CDMs

Description

This function estimates the diagnostic tree model (Ma, 2018) for polytomous responses with multiple strategies. It is an experimental function, and will be further optimized.

Usage

DTM(
  dat,
  Qc,
  delta = NULL,
  Tmatrix = NULL,
  conv.crit = 0.001,
  conv.type = "pr",
  maxitr = 1000
)

Arguments

dat	A required $N \times J$ data matrix of N examinees to J items. Missing values are currently not allowed.
Qc	A required $J \times K+2$ category and attribute association matrix, where J represents the number of items or nonzero categories and K represents the number of attributes. Entry 1 indicates that the attribute is measured by the item, and 0 otherwise. The first column gives the item number, which must be numeric and match the number of column in the data. The second column indicates the category number.
delta	initial item parameters
Tmatrix	The mapping matrix showing the relation between the OBSERVED responses (rows) and the PSEDUO items (columns); The first column gives the observed responses.
conv.crit	The convergence criterion for max absolute change in item parameters.
conv.type	convergence criteria; Can be pr,LL and delta, indicating category response function, log-likelihood and delta parameters,respectively.
maxitr	The maximum iterations allowed.

Author(s)

Wenchao Ma, The University of Alabama, [email protected]

References

Ma, W. (2018). A Diagnostic Tree Model for Polytomous Responses with Multiple Strategies. British Journal of Mathematical and Statistical Psychology.

See Also

GDINA for MS-DINA model and single strategy CDMs, and GMSCDM for generalized multiple strategies CDMs for dichotomous response data

Examples

## Not run: 
K=5
g=0.2
item.no <- rep(1:6,each=4)
# the first node has three response categories: 0, 1 and 2
node.no <- rep(c(1,1,2,3),6)
Q1 <- matrix(0,length(item.no),K)
Q2 <- cbind(7:(7+K-1),rep(1,K),diag(K))
for(j in 1:length(item.no)) {
  Q1[j,sample(1:K,sample(3,1))] <- 1
}
Qc <- rbind(cbind(item.no,node.no,Q1),Q2)
Tmatrix.set <- list(cbind(c(0,1,2,3,3),c(0,1,2,1,2),c(NA,0,NA,1,NA),c(NA,NA,0,NA,1)),
cbind(c(0,1,2,3,4),c(0,1,2,1,2),c(NA,0,NA,1,NA),c(NA,NA,0,NA,1)),
cbind(c(0,1),c(0,1)))
Tmatrix <- Tmatrix.set[c(1,1,1,1,1,1,rep(3,K))]
sim <- simDTM(N=2000,Qc=Qc,gs.parm=matrix(0.2,nrow(Qc),2),Tmatrix=Tmatrix)
est <- DTM(dat=sim$dat,Qc=Qc,Tmatrix = Tmatrix)

## End(Not run)

---

Examination for the Certificate of Proficiency in English (ECPE) data

Description

Examination for the Certificate of Proficiency in English (ECPE) data (the grammar section) has been used in Henson and Templin (2007), Templin and Hoffman (2013), Feng, Habing, and Huebner (2014), and Templin and Bradshaw (2014), among others.

Usage

ecpe

Format

A list of responses and Q-matrix with components:

dat

Responses of 2922 examinees to 28 items.

Q

The $28 \times 3$ Q-matrix.

Details

The data consists of responses of 2922 examinees to 28 items involving 3 attributes. Attribute 1 is morphosyntactic rules, Attribute 2 is cohesive rules and Attribute 3 is lexical rules.

Author(s)

Wenchao Ma, The University of Alabama, [email protected]

References

Ma, W., & de la Torre, J. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Feng, Y., Habing, B. T., & Huebner, A. (2014). Parameter estimation of the reduced RUM using the EM algorithm. Applied Psychological Measurement, 38, 137-150.

Henson, R. A., & Templin, J. (2007, April). Large-scale language assessment using cognitive diagnosis models. Paper presented at the annual meeting of the National Council for Measurement in Education in Chicago, Illinois.

Templin, J., & Bradshaw, L. (2014). Hierarchical diagnostic classification models: A family of models for estimating and testing attribute hierarchies. Psychometrika, 79, 317-339.

Templin, J., & Hoffman, L. (2013). Obtaining diagnostic classification model estimates using Mplus. Educational Measurement: Issues and Practice, 32, 37-50.

Examples

## Not run: 
mod1 <- GDINA(ecpe$dat,ecpe$Q)
mod1
summary(mod1)

mod2 <- GDINA(ecpe$dat,ecpe$Q,model="RRUM")
mod2
anova(mod1,mod2)
# You may compare the following results with Feng, Habing, and Huebner (2014)
coef(mod2,"rrum")

# G-DINA with hierarchical structure
# see Templin & Bradshaw, 2014
ast <- att.structure(list(c(3,2),c(2,1)),K=3)

est.gdina2 <- GDINA(ecpe$dat,ecpe$Q,model = "GDINA",
                   control = list(conv.crit = 1e-6),
                   att.str = list(c(3,2),c(2,1)))
# see Table 7 in Templin & Bradshaw, 2014
summary(est.gdina2)

## End(Not run)

---

extract elements from objects of various classes

Description

A generic function to extract elements from objects of class GDINA, itemfit, modelcomp, Qval or simGDINA. This page gives the elements that can be extracted from the class GDINA. To see what can be extracted from itemfit, modelcomp, and Qval, go to the corresponding function help page.

Objects which can be extracted from GDINA objects include:

AIC

AIC

att.prior

attribute prior weights for calculating marginalized likelihood in the last EM iteration

attributepattern

all attribute patterns involved in the current calibration

BIC

BIC

CAIC

Consistent AIC

catprob.cov

covariance matrix of item probability parameter estimates; Need to specify SE.type

catprob.parm

item parameter estimates

catprob.se

standard error of item probability parameter estimates; Need to specify SE.type

convergence

TRUE if the calibration is converged.

dat

raw data

del.ind

deleted observation number

delta.cov

covariance matrix of delta parameter estimates; Need to specify SE.type

delta.parm

delta parameter estimates

delta.se

standard error of delta parameter estimates; Need to specify SE.type

designmatrix

A list of design matrices for each item/category

deviance

deviance, or negative two times observed marginal log likelihood

discrim

GDINA discrimination index

expectedCorrect

expected # of examinees in each latent group answering item correctly

expectedTotal

expected # of examinees in each latent group

higher.order

higher-order model specifications

LCprob.parm

success probabilities for all latent classes

logLik

observed marginal log likelihood

linkfunc

link functions for each item

initial.catprob

initial item category probability parameters

natt

number of attributes

ncat

number of categories

ngroup

number of groups

nitem

number of items

nitr

number of EM iterations

nobs

number of observations, or sample size

nLC

number of latent classes

prevalence

prevalence of each attribute

posterior.prob

posterior weights for each latent class

reduced.LG

Reduced latent group for each item

SABIC

Sample size Adusted BIC

sequential

is a sequential model fitted?

Usage

extract(object, what, ...)

Arguments

object	objects from class GDINA,itemfit, modelcomp, Qval or simGDINA
what	what to extract
...	additional arguments

Examples

## Not run: 
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
fit <- GDINA(dat = dat, Q = Q, model = "GDINA")
extract(fit,"discrim")
extract(fit,"designmatrix")

## End(Not run)

---

Tatsuoka's fraction subtraction data

Description

Fraction Subtraction data (Tatsuoka, 2002) consists of responses of 536 examinees to 20 items measuring 8 attributes.

Usage

frac20

Format

A list of responses and Q-matrix with components:

dat

responses of 536 examinees to 20 items

Q

The $20 \times 8$ Q-matrix

Author(s)

Wenchao Ma, The University of Alabama, [email protected]

References

Ma, W., & de la Torre, J. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Tatsuoka, C. (2002). Data analytic methods for latent partially ordered classification models. Journal of the Royal Statistical Society, Series C, Applied Statistics, 51, 337-350.

Examples

## Not run: 
mod1 <- GDINA(frac20$dat,frac20$Q,model="DINA")
mod1
summary(mod1)
# Higher order model
mod2 <- GDINA(frac20$dat,frac20$Q,model="DINA",att.dist="higher.order")
mod2
anova(mod1,mod2)

## End(Not run)

---

CDM calibration under the G-DINA model framework

Description

GDINA calibrates the generalized deterministic inputs, noisy and gate (G-DINA; de la Torre, 2011) model for dichotomous responses, and its extension, the sequential G-DINA model (Ma, & de la Torre, 2016a; Ma, 2017) for ordinal and nominal responses. By setting appropriate constraints, the deterministic inputs, noisy and gate (DINA; de la Torre, 2009; Junker & Sijtsma, 2001) model, the deterministic inputs, noisy or gate (DINO; Templin & Henson, 2006) model, the reduced reparametrized unified model (R-RUM; Hartz, 2002), the additive CDM (A-CDM; de la Torre, 2011), the linear logistic model (LLM; Maris, 1999), and the multiple-strategy DINA model (MSDINA; de la Torre & Douglas, 2008; Huo & de la Torre, 2014) can also be calibrated. Note that the LLM is equivalent to the C-RUM (Hartz, 2002), a special case of the GDM (von Davier, 2008), and that the R-RUM is also known as a special case of the generalized NIDA model (de la Torre, 2011).

In addition, users are allowed to specify design matrix and link function for each item, and distinct models may be used in a single test for different items. The attributes can be either dichotomous or polytomous (Chen & de la Torre, 2013). Joint attribute distribution may be modelled using independent or saturated model, structured model, higher-order model (de la Torre & Douglas, 2004), or loglinear model (Xu & von Davier, 2008). Marginal maximum likelihood method with Expectation-Maximization (MMLE/EM) alogrithm is used for item parameter estimation.

To compare two or more GDINA objects, use method anova.

To calculate structural parameters for item and joint attribute distributions, use method coef.

To calculate lower-order incidental (person) parameters use method personparm. To extract other components returned, use extract. To plot item/category response function, use plot. To check whether monotonicity is violated, use monocheck. To conduct anaysis in graphical user interface, use startGDINA.

Usage

GDINA(
  dat,
  Q,
  model = "GDINA",
  sequential = FALSE,
  att.dist = "saturated",
  mono.constraint = FALSE,
  group = NULL,
  linkfunc = NULL,
  design.matrix = NULL,
  no.bugs = 0,
  att.prior = NULL,
  att.str = NULL,
  verbose = 1,
  higher.order = list(),
  loglinear = 2,
  catprob.parm = NULL,
  control = list(),
  item.names = NULL,
  solver = NULL,
  nloptr.args = list(),
  auglag.args = list(),
  solnp.args = list(),
  ...
)

## S3 method for class 'GDINA'
anova(object, ...)

## S3 method for class 'GDINA'
coef(
  object,
  what = c("catprob", "delta", "gs", "itemprob", "LCprob", "rrum", "lambda"),
  withSE = FALSE,
  SE.type = 2,
  digits = 4,
  ...
)

## S3 method for class 'GDINA'
extract(object, what, SE.type = 2, ...)

## S3 method for class 'GDINA'
personparm(object, what = c("EAP", "MAP", "MLE", "mp", "HO"), digits = 4, ...)

## S3 method for class 'GDINA'
logLik(object, ...)

## S3 method for class 'GDINA'
deviance(object, ...)

## S3 method for class 'GDINA'
nobs(object, ...)

## S3 method for class 'GDINA'
vcov(object, ...)

## S3 method for class 'GDINA'
npar(object, ...)

## S3 method for class 'GDINA'
indlogLik(object, ...)

## S3 method for class 'GDINA'
indlogPost(object, ...)

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

Arguments

dat	A required $N \times J$ matrix or data.frame consisting of the responses of $N$ individuals to $J$ items. Missing values need to be coded as NA.
Q	A required matrix; The number of rows occupied by a single-strategy dichotomous item is 1, by a polytomous item is the number of nonzero categories, and by a mutiple-strategy dichotomous item is the number of strategies. The number of column is equal to the number of attributes if all items are single-strategy dichotomous items, but the number of attributes + 2 if any items are polytomous or have multiple strategies. For a polytomous item, the first column represents the item number and the second column indicates the nonzero category number. For a multiple-strategy dichotomous item, the first column represents the item number and the second column indicates the strategy number. For binary attributes, 1 denotes the attributes are measured by the items and 0 means the attributes are not measured. For polytomous attributes, non-zero elements indicate which level of attributes are needed (see Chen, & de la Torre, 2013). See Examples.
model	A vector for each item or nonzero category, or a scalar which will be used for all items or nonzero categories to specify the CDMs fitted. The possible options include "GDINA","DINA","DINO","ACDM","LLM", "RRUM", "MSDINA", "BUGDINO", "SISM", and "UDF". Note that model can also be "logitGDINA" and "logGDINA", indicating the saturated G-DINA model in logit and log link functions. They are equivalent to the identity link saturated G-DINA model. The logit G-DINA model is identical to the log-linear CDM. When "UDF", indicating user defined function, is specified for any item, arguments design.matrix and linkfunc need to be specified.
sequential	logical; TRUE if the sequential model is fitted for polytomous responses.
att.dist	How is the joint attribute distribution estimated? It can be (1) saturated, which is the default, indicating that the proportion parameter for each permissible latent class is estimated separately; (2) higher.order, indicating that a higher-order joint attribute distribution is assumed (higher-order model can be specified in higher.order argument); (3) fixed, indicating that the weights specified in att.prior argument are fixed in the estimation process. If att.prior is not specified, a uniform joint attribute distribution is employed initially; (4) independent, indicating that all attributes are assumed to be independent; and (5) loglinear, indicating a loglinear model is employed. If different groups have different joint attribute distributions, specify att.dist as a character vector with the same number of elements as the number of groups. However, if a higher-order model is used for any group, it must be used for all groups.
mono.constraint	logical; TRUE indicates that $P(\boldsymbol{\alpha}_1) <=P(\boldsymbol{\alpha}_2)$ if for all $k$, $\alpha_{1k} < \alpha_{2k}$. Can be a vector for each item or nonzero category or a scalar which will be used for all items to specify whether monotonicity constraint should be added.
group	a factor or a vector indicating the group each individual belongs to. Its length must be equal to the number of individuals.
linkfunc	a vector of link functions for each item/category; It can be "identity","log" or "logit". Only applicable when, for some items, model="UDF".
design.matrix	a list of design matrices; Its length must be equal to the number of items (or nonzero categories for sequential models). If CDM for item j is specified as "UDF" in argument model, the corresponding design matrix must be provided; otherwise, the design matrix can be NULL, which will be generated automatically.
no.bugs	A numeric scalar (whole numbers only) indicating the number of bugs or misconceptions in the Q-matrix. The bugs must be included in the last no.bugs columns. It can be used along with the BUGDINO and SISM models (see, Kuo, Chen, Yang & Mok, 2016; Kuo, Chen, & de la Torre, 2018). This argument will be ignored if the model is not specified in model argument. Note that the BUGDINO and SISM models are reparametrized - see Details below. By default, no.bugs=0, implying that there is no bugs/misconceptions.
att.prior	A vector of length $2^K$ for single group model, or a matrix of dimension $2^K\times$ no. of groups to specify attribute prior distribution for $2^K$ latent classes for all groups under a multiple group model. Only applicable for dichotomous attributes. The sum of all elements does not have to be equal to 1; however, it will be normalized so that the sum is equal to 1 before calibration. The label for each latent class can be obtained by calling attributepattern(K). See examples for more info.
att.str	Specify attribute structures. NULL, by default, means there is no structure. Attribute structure needs be specified as a list - which will be internally handled by att.structure function. See examples. It can also be a matrix giving all permissible attribute profiles.
verbose	How to print calibration information after each EM iteration? Can be 0, 1 or 2, indicating to print no information, information for current iteration, or information for all iterations.
higher.order	A list specifying the higher-order joint attribute distribution with the following components: model - a number indicating the model for higher-order joint attribute distribution. Can be 1, 2 or 3, representing the intercept only approach, common slope approach and varied slope approach (see Details). nquad - a scalar specifying the number of integral nodes. Default = 25. SlopeRange - a vector of length two specifying the range of slope parameters. Default = [0.1, 5]. InterceptRange - a vector of length two specifying the range of intercept parameters. Default = [-4, 4]. SlopePrior - a vector of length two specifying the mean and variance of log(slope) parameters, which are assumed normally distributed. Default: mean = 0 and sd = 0.25. InterceptPrior - a vector of length two specifying the mean and variance of intercept parameters, which are assumed normally distributed. Default: mean = 0 and sd = 1. Prior - logical; indicating whether prior distributions should be imposed to slope and intercept parameters. Default is FALSE.
loglinear	the order of loglinear smooth for attribute space. It can be either 1 or 2 indicating the loglinear model with main effect only and with main effect and first-order interaction; It can also be a matrix, representing the design matrix for the loglinear model.
catprob.parm	A list of initial success probability parameters for each nonzero category.
control	A list of control parameters with elements: maxitr A vector for each item or nonzero category, or a scalar which will be used for all items or nonzero categories to specify the maximum number of EM cycles allowed. Default = 2000. conv.crit The convergence criterion. Default = 0.0001. conv.type How is the convergence criterion evaluated? A vector with possible elements: "ip", indicating the maximum absolute change in item success probabilities, "mp", representing the maximum absolute change in mixing proportion parameters, "delta", indicating the maximum absolute change in delta parameters, neg2LL indicating the absolute change in negative two times loglikeihood, or neg2LL indicating the relative absolute change in negative two times loglikeihood (i.e., the absolute change divided by -2LL of the previous iteration). Multiple criteria can be specified. If so, all criteria need to be met. Default = c("ip", "mp"). nstarts how many sets of starting values? Default = 3. lower.p A vector for each item or nonzero category, or a scalar which will be used for all items or nonzero categories to specify the lower bound for success probabilities. Default = .0001. upper.p A vector for each item or nonzero category, or a scalar which will be used for all items or nonzero categories to specify the upper bound for success probabilities. Default = .9999. lower.prior The lower bound for mixing proportion parameters (latent class sizes). Default = .Machine$double.eps. randomseed Random seed for generating initial item parameters. Default = 123456. smallNcorrection A numeric vector with two elements specifying the corrections applied when the expected number of individuals in some latent groups are too small. If the expected no. of examinees is less than the second element, the first element and two times the first element will be added to the numerator and denominator of the closed-form solution of probabilities of success. Only applicable for the G-DINA, DINA and DINO model estimation without monotonic constraints. MstepMessage Integer; Larger number prints more information from Mstep optimizer. Default = 1.
item.names	A vector giving the item names. By default, items are named as "Item 1", "Item 2", etc.
solver	A string indicating which solver should be used in M-step. By default, the solver is automatically chosen according to the models specified. Possible options include slsqp, nloptr, solnp and auglag.
nloptr.args	a list of control parameters to be passed to opts argument of nloptr function.
auglag.args	a list of control parameters to be passed to the alabama::auglag() function. It can contain two elements: control.outer and control.optim. See auglag.
solnp.args	a list of control parameters to be passed to control argument of solnp function.
...	additional arguments
object	GDINA object for various S3 methods
what	argument for various S3 methods; For calculating structural parameters using coef, what can be itemprob - item success probabilities of each reduced attribute pattern. catprob - category success probabilities of each reduced attribute pattern; the same as itemprob for dichtomous response data. LCprob - item success probabilities of each attribute pattern. gs - guessing and slip parameters of each item/category. delta - delta parameters of each item/category, see G-DINA formula in details. rrum - RRUM parameters when items are estimated using RRUM. lambda - structural parameters for joint attribute distribution. For calculating incidental parameters using personparm, what can be EAP - EAP estimates of attribute pattern. MAP - MAP estimates of attribute pattern. MLE - MLE estimates of attribute pattern. mp - marginal mastery probabilities. HO - EAP estimates of higher-order ability if a higher-order is fitted.
withSE	argument for method coef; estimate standard errors or not?
SE.type	type of standard errors. For now, SEs are calculated based on outper-product of gradient. It can be 1 based on item-wise information, 2 based on incomplete information and 3 based on complete information.
digits	How many decimal places in each number? The default is 4.

Value

GDINA returns an object of class GDINA. Methods for GDINA objects include extract for extracting various components, coef for extracting structural parameters, personparm for calculating incidental (person) parameters, summary for summary information. AIC, BIC,logLik, deviance and npar can also be used to calculate AIC, BIC, observed log-likelihood, deviance and number of parameters.

Methods (by generic)

• anova(GDINA): Model comparison using likelihood ratio test

• coef(GDINA): extract structural parameter estimates

• extract(GDINA): extract various elements of GDINA estimates

• personparm(GDINA): calculate person attribute patterns and higher-order ability

• logLik(GDINA): calculate log-likelihood

• deviance(GDINA): calculate deviance

• nobs(GDINA): calculate number of observations

• vcov(GDINA): calculate covariance-matrix for delta parameters

• npar(GDINA): calculate the number of parameters

• indlogLik(GDINA): extract log-likelihood for each individual

• indlogPost(GDINA): extract log posterior for each individual

• summary(GDINA): print summary information

The G-DINA model

The generalized DINA model (G-DINA; de la Torre, 2011) is an extension of the DINA model. Unlike the DINA model, which collaspes all latent classes into two latent groups for each item, if item $j$ requires $K_j^*$ attributes, the G-DINA model collapses $2^K$ latent classes into $2^{K_j^*}$ latent groups with unique success probabilities on item $j$, where $K_j^*=\sum_{k=1}^{K}q_{jk}$.

Let $\boldsymbol{\alpha}_{lj}^*$ be the reduced attribute pattern consisting of the columns of the attributes required by item $j$, where $l=1,\ldots,2^{K_j^*}$. For example, if only the first and the last attributes are required, $\boldsymbol{\alpha}_{lj}^*=(\alpha_{l1},\alpha_{lK})$. For notational convenience, the first $K_j^*$ attributes can be assumed to be the required attributes for item $j$ as in de la Torre (2011). The probability of success $P(X_{j}=1|\boldsymbol{\alpha}_{lj}^*)$ is denoted by $P(\boldsymbol{\alpha}_{lj}^*)$. To model this probability of success, different link functions as in the generalized linear models are used in the G-DINA model. The item response function of the G-DINA model using the identity link can be written as

$f[P(\boldsymbol{\alpha}_{lj}^*)]=\delta_{j0}+\sum_{k=1}^{K_j^*}\delta_{jk}\alpha_{lk}+ \sum_{k'=k+1}^{K_j^*}\sum_{k=1}^{K_j^*-1}\delta_{jkk'}\alpha_{lk}\alpha_{lk'}+\cdots+ \delta_{j12{\cdots}K_j^*}\prod_{k=1}^{K_j^*}\alpha_{lk},$

or in matrix form,

$f[\boldsymbol{P}_j]=\boldsymbol{M}_j\boldsymbol{\delta}_j,$

where $\delta_{j0}$ is the intercept for item $j$, $\delta_{jk}$ is the main effect due to $\alpha_{lk}$, $\delta_{jkk'}$ is the interaction effect due to $\alpha_{lk}$ and $\alpha_{lk'}$, $\delta_{j12{\ldots}K_j^*}$ is the interaction effect due to $\alpha_{l1}, \cdots,\alpha_{lK_j^*}$. The log and logit links can also be employed.

Other CDMs as special cases

Several widely used CDMs can be obtained by setting appropriate constraints to the G-DINA model. This section introduces the parameterization of different CDMs within the G-DINA model framework very breifly. Readers interested in this please refer to de la Torre(2011) for details.

DINA model

In DINA model, each item has two item parameters - guessing ($g$) and slip ($s$). In traditional parameterization of the DINA model, a latent variable $\eta$ for person $i$ and item $j$ is defined as

$\eta_{ij}=\prod_{k=1}^K\alpha_{ik}^{q_{jk}}$

Briefly speaking, if individual $i$ master all attributes required by item $j$, $\eta_{ij}=1$; otherwise, $\eta_{ij}=0$. Item response function of the DINA model can be written by

$P(X_{ij}=1|\eta_{ij})=(1-s_j)^{\eta_{ij}}g_j^{1-\eta_{ij}}$

To obtain the DINA model from the G-DINA model, all terms in identity link G-DINA model except $\delta_0$ and $\delta_{12{\ldots}K_j^*}$ need to be fixed to zero, that is,

$P(\boldsymbol{\alpha}_{lj}^*)= \delta_{j0}+\delta_{j12{\cdots}K_j^*}\prod_{k=1}^{K_j^*}\alpha_{lk}$

In this parameterization, $\delta_{j0}=g_j$ and $\delta_{j0}+\delta_{j12{\cdots}K_j^*}=1-s_j$.

DINO model

The DINO model can be given by

$P(\boldsymbol{\alpha}_{lj}^{*})= \delta_{j0}+\delta_{j1}I(\boldsymbol{\alpha}_{lj}^*\neq \boldsymbol{0})$

where $I(\cdot)$ is an indicator variable. The DINO model is also a constrained identity link G-DINA model. As shown by de la Torre (2011), the appropriate constraint is

$\delta_{jk}=-\delta_{jk^{'}k^{''}}=\cdots=(-1)^{K_j^*+1}\delta_{j12{\cdots}K_j^*},$

for $k=1,\cdots,K_j^*, k^{'}=1,\cdots,K_j^*-1$, and $k^{''}>k^{'},\cdots,K_j^*$.

Additive models with different link functions

The A-CDM, LLM and R-RUM can be obtained by setting all interactions to be zero in identity, logit and log link G-DINA model, respectively. Specifically, the A-CDM can be formulated as

$P(\boldsymbol{\alpha}_{lj}^*)= \delta_{j0}+\sum_{k=1}^{K_j^*}\delta_{jk}\alpha_{lk}$

The item response function for LLM can be given by

$logit[P(\boldsymbol{\alpha}_{lj}^*)]=\delta_{j0}+\sum_{k=1}^{K_j^*}\delta_{jk}\alpha_{lk},$

and lastly, the RRUM, can be written as

$log[P(\boldsymbol{\alpha}_{lj}^*)]=\delta_{j0}+\sum_{k=1}^{K_j^*}\delta_{jk}\alpha_{lk}.$

It should be noted that the LLM is equivalent to the compensatory RUM, which is subsumed by the GDM, and that the RRUM is a special case of the generalized noisy inputs, deterministic “And" gate model (G-NIDA).

Simultaneously identifying skills and misconceptions (SISM)

The SISM can be can be reformulated as

$P(\boldsymbol{\alpha}_{lj}^*)= \delta_{j0}+\delta_{j1}I[{mastering all skills}]+ \delta_{j2}I[{having no bugs}]+\delta_{j12}I[{mastering all skills and having no bugs}].$

As a result,the success probability of students who have mastered all the measured skills and possess none of the measured misconceptions ($h_j$ in Equation 4 of Kuo, et al, 2018) is $\delta_{j0}+\delta_{j1}+\delta_{j2}+\delta_{j12}$, the success probability of students who have mastered all the measured skills but possess some of the measured misconceptions ($\omega_j$) is $\delta_{j0}+\delta_{j1}$, the success probability of students who have not mastered all the measured skills and possess none of the measured misconceptions ($g_j$) is $\delta_{j0}+\delta_{j2}$ and success probability of students who have not mastered all the measured skills and possess at least one of the measured misconceptions($\epsilon_j$) is $\delta_{j0}$.

By specifying no.bugs being equal to the number of attributes, the Bug-DINO is obtained, as in

$P(\boldsymbol{\alpha}_{lj}^*)=\delta_{j0}+\delta_{j1}I[{having no bugs}].$

Joint Attribute Distribution

The joint attribute distribution can be modeled using various methods. This section mainly focuses on the so-called higher-order approach, which was originally proposed by de la Torre and Douglas (2004) for the DINA model. It has been extended in this package for all condensation rules. Particularly, three approaches are available for the higher-order attribute structure: intercept only approach, common slope approach and varied slope approach. For the intercept only approach, the probability of mastering attribute $k$ for individual $i$ is defined as

$P(\alpha_k=1|\theta_i,\lambda_{0k})=\frac{exp(\theta_i+\lambda_{0k})}{1+exp(\theta_i+\lambda_{0k})}$

For the common slope approach, the probability of mastering attribute $k$ for individual $i$ is defined as

$P(\alpha_k=1|\theta_i,\lambda_{0k},\lambda_{1})=\frac{exp(\lambda_{1}\theta_i+\lambda_{0k})}{1+exp(\lambda_{1}\theta_i+\lambda_{0k})}$

For the varied slope approach, the probability of mastering attribute $k$ for individual $i$ is defined as

$P(\alpha_k=1|\theta_i,\lambda_{0k},\lambda_{1k})=\frac{exp(\lambda_{1k}\theta_i+\lambda_{0k})}{1+exp(\lambda_{1k}\theta_i+\lambda_{0k})}$

where $\theta_i$ is the ability of examinee $i$. $\lambda_{0k}$ and $\lambda_{1k}$ are the intercept and slope parameters for attribute $k$, respectively. The probability of joint attributes can be written as

$P(\boldsymbol{\alpha}|\theta_i,\boldsymbol{\lambda})=\prod_k P(\alpha_k|\theta_i,\boldsymbol{\lambda})$

.

Model Estimation

The MMLE/EM algorithm is implemented in this package. For G-DINA, DINA and DINO models, closed-form solutions exist. See de la Torre (2009) and de la Torre (2011) for details. For ACDM, LLM and RRUM, closed-form solutions do not exist, and therefore some general optimization techniques are adopted in M-step (Ma, Iaconangelo & de la Torre, 2016). The selection of optimization techniques mainly depends on whether some specific constraints need to be added.

The sequential G-DINA model is a special case of the diagnostic tree model (DTM; Ma, 2019) and estimated using the mapping matrix accordingly (See Tutz, 1997; Ma, 2019).

The Number of Parameters

For dichotomous response models: Assume a test measures $K$ attributes and item $j$ requires $K_j^*$ attributes: The DINA and DINO model has 2 item parameters for each item; if item $j$ is ACDM, LLM or RRUM, it has $K_j^*+1$ item parameters; if it is G-DINA model, it has $2^{K_j^*}$ item parameters. Apart from item parameters, the parameters involved in the estimation of joint attribute distribution need to be estimated as well. When using the saturated attribute structure, there are $2^K-1$ parameters for joint attribute distribution estimation; when using a higher-order attribute structure, there are $K$, $K+1$, and $2\times K$ parameters for the intercept only approach, common slope approach and varied slope approach, respectively. For polytomous response data using the sequential G-DINA model, the number of item parameters are counted at category level.

Note

anova function does NOT check whether models compared are nested or not.

Author(s)

Wenchao Ma, The University of Alabama, [email protected]
Jimmy de la Torre, The University of Hong Kong

References

Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443-459.

Bock, R. D., & Lieberman, M. (1970). Fitting a response model forn dichotomously scored items. Psychometrika, 35, 179-197.

Bor-Chen Kuo, Chun-Hua Chen, Chih-Wei Yang, & Magdalena Mo Ching Mok. (2016). Cognitive diagnostic models for tests with multiple-choice and constructed-response items. Educational Psychology, 36, 1115-1133.

Carlin, B. P., & Louis, T. A. (2000). Bayes and empirical bayes methods for data analysis. New York, NY: Chapman & Hall

de la Torre, J., & Douglas, J. A. (2008). Model evaluation and multiple strategies in cognitive diagnosis: An analysis of fraction subtraction data. Psychometrika, 73, 595-624.

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

de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179-199.

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

de la Torre, J., & Lee, Y. S. (2013). Evaluating the wald test for item-level comparison of saturated and reduced models in cognitive diagnosis. Journal of Educational Measurement, 50, 355-373.

Haertel, E. H. (1989). Using restricted latent class models to map the skill structure of achievement items. Journal of Educational Measurement, 26, 301-321.

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.

Huo, Y., & de la Torre, J. (2014). Estimating a Cognitive Diagnostic Model for Multiple Strategies via the EM Algorithm. Applied Psychological Measurement, 38, 464-485.

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

Kuo, B.-C., Chen.-H., & de la Torre,J. (2018). A cognitive diagnosis model for identifying coexisting skills and misconceptions.Applied Psychological Measuremet, 179–191.

Ma, W., & de la Torre, J. (2016). A sequential cognitive diagnosis model for polytomous responses. British Journal of Mathematical and Statistical Psychology. 69, 253-275.

Ma, W., & de la Torre, J. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Ma, W. (2019). A diagnostic tree model for polytomous responses with multiple strategies. British Journal of Mathematical and Statistical Psychology, 72, 61-82.

Ma, W., Iaconangelo, C., & de la Torre, J. (2016). Model similarity, model selection and attribute classification. Applied Psychological Measurement, 40, 200-217.

Ma, W. (2017). A Sequential Cognitive Diagnosis Model for Graded Response: Model Development, Q-Matrix Validation,and Model Comparison. Unpublished doctoral dissertation. New Brunswick, NJ: Rutgers University.

Maris, E. (1999). Estimating multiple classification latent class models. Psychometrika, 64, 187-212.

Tatsuoka, K. K. (1983). Rule space: An approach for dealing with misconceptions based on item response theory. Journal of Educational Measurement, 20, 345-354.

Templin, J. L., & Henson, R. A. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11, 287-305.

Tutz, G. (1997). Sequential models for ordered responses. In W.J. van der Linden & R. K. Hambleton (Eds.), Handbook of modern item response theory p. 139-152). New York, NY: Springer.

Xu, X., & von Davier, M. (2008). Fitting the structured general diagnostic model to NAEP data. ETS research report, RR-08-27.

See Also

See autoGDINA for Q-matrix validation, item-level model comparison and model calibration in one run; See modelfit and itemfit for model and item fit analysis, Qval for Q-matrix validation, modelcomp for item level model comparison and simGDINA for data simulation. GMSCDM for a series of multiple strategy CDMs for dichotomous data, and DTM for diagnostic tree model for multiple strategies in polytomous response data Also see gdina in CDM package for the G-DINA model estimation.

Examples

## Not run: 
####################################
#        Example 1.                #
#     GDINA, DINA, DINO            #
#    ACDM, LLM and RRUM            #
# estimation and comparison        #
#                                  #
####################################

dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ

#--------GDINA model --------#

mod1 <- GDINA(dat = dat, Q = Q, model = "GDINA")
mod1
# summary information
summary(mod1)

AIC(mod1) #AIC
BIC(mod1) #BIC
logLik(mod1) #log-likelihood value
deviance(mod1) # deviance: -2 log-likelihood
npar(mod1) # number of parameters


head(indlogLik(mod1)) # individual log-likelihood
head(indlogPost(mod1)) # individual log-posterior

# structural parameters
# see ?coef
coef(mod1) # item probabilities of success for each latent group
coef(mod1, withSE = TRUE) # item probabilities of success & standard errors
coef(mod1, what = "delta") # delta parameters
coef(mod1, what = "delta",withSE=TRUE) # delta parameters
coef(mod1, what = "gs") # guessing and slip parameters
coef(mod1, what = "gs",withSE = TRUE) # guessing and slip parameters & standard errors

# person parameters
# see ?personparm
personparm(mod1) # EAP estimates of attribute profiles
personparm(mod1, what = "MAP") # MAP estimates of attribute profiles
personparm(mod1, what = "MLE") # MLE estimates of attribute profiles

#plot item response functions for item 10
plot(mod1,item = 10)
plot(mod1,item = 10,withSE = TRUE) # with error bars
#plot mastery probability for individuals 1, 20 and 50
plot(mod1,what = "mp", person =c(1,20,50))

# Use extract function to extract more components
# See ?extract

# ------- DINA model --------#
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
mod2 <- GDINA(dat = dat, Q = Q, model = "DINA")
mod2
coef(mod2, what = "gs") # guess and slip parameters
coef(mod2, what = "gs",withSE = TRUE) # guess and slip parameters and standard errors

# Model comparison at the test level via likelihood ratio test
anova(mod1,mod2)

# -------- DINO model -------#
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
mod3 <- GDINA(dat = dat, Q = Q, model = "DINO")
#slip and guessing
coef(mod3, what = "gs") # guess and slip parameters
coef(mod3, what = "gs",withSE = TRUE) # guess and slip parameters + standard errors

# Model comparison at test level via likelihood ratio test
anova(mod1,mod2,mod3)

# --------- ACDM model -------#
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
mod4 <- GDINA(dat = dat, Q = Q, model = "ACDM")
mod4
# --------- LLM model -------#
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
mod4b <- GDINA(dat = dat, Q = Q, model = "LLM")
mod4b
# --------- RRUM model -------#
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
mod4c <- GDINA(dat = dat, Q = Q, model = "RRUM")
mod4c

# --- Different CDMs for different items --- #

dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
models <- c(rep("GDINA",3),"LLM","DINA","DINO","ACDM","RRUM","LLM","RRUM")
mod5 <- GDINA(dat = dat, Q = Q, model = models)
anova(mod1,mod2,mod3,mod4,mod4b,mod4c,mod5)


####################################
#        Example 2.                #
#        Model estimations         #
# With monotonocity constraints    #
####################################

dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
# for item 10 only
mod11 <- GDINA(dat = dat, Q = Q, model = "GDINA",mono.constraint = c(rep(FALSE,9),TRUE))
mod11
mod11a <- GDINA(dat = dat, Q = Q, model = "DINA",mono.constraint = TRUE)
mod11a
mod11b <- GDINA(dat = dat, Q = Q, model = "ACDM",mono.constraint = TRUE)
mod11b
mod11c <- GDINA(dat = dat, Q = Q, model = "LLM",mono.constraint = TRUE)
mod11c
mod11d <- GDINA(dat = dat, Q = Q, model = "RRUM",mono.constraint = TRUE)
mod11d
coef(mod11d,"delta")
coef(mod11d,"rrum")

####################################
#           Example 3a.            #
#        Model estimations         #
# With Higher-order att structure  #
####################################

dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
# --- Higher order G-DINA model ---#
mod12 <- GDINA(dat = dat, Q = Q, model = "DINA",
               att.dist="higher.order",higher.order=list(nquad=31,model = "2PL"))
personparm(mod12,"HO") # higher-order ability
# structural parameters
# first column is slope and the second column is intercept
coef(mod12,"lambda")
# --- Higher order DINA model ---#
mod22 <- GDINA(dat = dat, Q = Q, model = "DINA", att.dist="higher.order",
               higher.order=list(model = "2PL",Prior=TRUE))

####################################
#           Example 3b.            #
#        Model estimations         #
#   With log-linear att structure  #
####################################

# --- DINA model with loglinear smoothed attribute space ---#
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
mod23 <- GDINA(dat = dat, Q = Q, model = "DINA",att.dist="loglinear",loglinear=1)
coef(mod23,"lambda") # intercept and three main effects

####################################
#           Example 3c.            #
#        Model estimations         #
#  With independent att structure  #
####################################

# --- GDINA model with independent attribute space ---#
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
mod33 <- GDINA(dat = dat, Q = Q, att.dist="independent")
coef(mod33,"lambda") # mastery probability for each attribute

####################################
#          Example 4.              #
#        Model estimations         #
#    With fixed att structure      #
####################################

# --- User-specified attribute priors ----#
# prior distribution is fixed during calibration
# Assume each of 000,100,010 and 001 has probability of 0.1
# and each of 110, 101,011 and 111 has probability of 0.15
# Note that the sum is equal to 1
#
prior <- c(0.1,0.1,0.1,0.1,0.15,0.15,0.15,0.15)
# fit GDINA model  with fixed prior dist.
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
modp1 <- GDINA(dat = dat, Q = Q, att.prior = prior, att.dist = "fixed")
extract(modp1, what = "att.prior")

####################################
#          Example 5a.             #
#             G-DINA               #
# with hierarchical att structure  #
####################################

# --- User-specified attribute structure ----#
Q <- sim30GDINA$simQ
K <- ncol(Q)
# divergent structure A1->A2->A3;A1->A4->A5
diverg <- list(c(1,2),
               c(2,3),
               c(1,4),
               c(4,5))
struc <- att.structure(diverg,K)
set.seed(123)
# data simulation
N <- 1000
true.lc <- sample(c(1:2^K),N,replace=TRUE,prob=struc$att.prob)
table(true.lc) #check the sample
true.att <- attributepattern(K)[true.lc,]
 gs <- matrix(rep(0.1,2*nrow(Q)),ncol=2)
 # data simulation
 simD <- simGDINA(N,Q,gs.parm = gs, model = "GDINA",attribute = true.att)
 dat <- extract(simD,"dat")

modp1 <- GDINA(dat = dat, Q = Q, att.str = diverg, att.dist = "saturated")
modp1
coef(modp1,"lambda")

####################################
#          Example 5b.             #
#   Reduced model (e.g.,ACDM)      #
# with hierarchical att structure  #
####################################

# --- User-specified attribute structure ----#
Q <- sim30GDINA$simQ
K <- ncol(Q)
# linear structure A1->A2->A3->A4->A5
linear <- list(c(1,2),
               c(2,3),
               c(3,4),
               c(4,5))
struc <- att.structure(linear,K)
set.seed(123)
# data simulation
N <- 1000
true.lc <- sample(c(1:2^K),N,replace=TRUE,prob=struc$att.prob)
table(true.lc) #check the sample
true.att <- attributepattern(K)[true.lc,]
 gs <- matrix(rep(0.1,2*nrow(Q)),ncol=2)
 # data simulation
 simD <- simGDINA(N,Q,gs.parm = gs, model = "ACDM",attribute = true.att)
 dat <- extract(simD,"dat")

modp1 <- GDINA(dat = dat, Q = Q, model = "ACDM",
               att.str = linear, att.dist = "saturated")
coef(modp1)
coef(modp1,"lambda")

####################################
#           Example 6.             #
# Specify initial values for item  #
# parameters                       #
####################################

 # check initials to see the format for initial item parameters
 initials <- sim10GDINA$simItempar

 dat <- sim10GDINA$simdat
 Q <- sim10GDINA$simQ
 mod.initial <- GDINA(dat,Q,catprob.parm = initials)

 # compare initial item parameters
 Map(rbind, initials,extract(mod.initial,"initial.catprob"))

####################################
#           Example 7a.            #
# Fix item and structure parameters#
# Estimate person attribute profile#
####################################

 # check initials to see the format for initial item parameters
 initials <- sim10GDINA$simItempar
 prior <- c(0.1,0.1,0.1,0.1,0.15,0.15,0.15,0.15)
 dat <- sim10GDINA$simdat
 Q <- sim10GDINA$simQ
 mod.ini <- GDINA(dat,Q,catprob.parm = initials,att.prior = prior,
                  att.dist = "fixed",control=list(maxitr = 0))
 personparm(mod.ini)
 # compare item parameters
 Map(rbind, initials,coef(mod.ini))

####################################
#           Example 7b.            #
#   Fix parameters for some items  #
# Estimate person attribute profile#
####################################

 # check initials to see the format for initial item parameters
 initials <- sim10GDINA$simItempar
 prior <- c(0.1,0.1,0.1,0.1,0.15,0.15,0.15,0.15)
 dat <- sim10GDINA$simdat
 Q <- sim10GDINA$simQ
 # fix parameters of the first 5 items; do not fix mixing proportion parameters
 mod.ini <- GDINA(dat,Q,catprob.parm = initials,
                  att.dist = "saturated",control=list(maxitr = c(rep(0,5),rep(2000,5))))
 personparm(mod.ini)
 # compare item parameters
 Map(rbind, initials,coef(mod.ini))

####################################
#           Example 8.             #
#        polytomous attribute      #
#          model estimation        #
#    see Chen, de la Torre 2013    #
####################################


# --- polytomous attribute G-DINA model --- #
dat <- sim30pGDINA$simdat
Q <- sim30pGDINA$simQ
#polytomous G-DINA model
pout <- GDINA(dat,Q)

# ----- polymous DINA model --------#
pout2 <- GDINA(dat,Q,model="DINA")
anova(pout,pout2)

####################################
#           Example 9.             #
#        Sequential G-DINA model   #
#    see Ma, & de la Torre 2016    #
####################################

# --- polytomous attribute G-DINA model --- #
dat <- sim20seqGDINA$simdat
Q <- sim20seqGDINA$simQ
Q
#    Item Cat A1 A2 A3 A4 A5
#       1   1  1  0  0  0  0
#       1   2  0  1  0  0  0
#       2   1  0  0  1  0  0
#       2   2  0  0  0  1  0
#       3   1  0  0  0  0  1
#       3   2  1  0  0  0  0
#       4   1  0  0  0  0  1
#       ...

#sequential G-DINA model
sGDINA <- GDINA(dat,Q,sequential = TRUE)
sDINA <- GDINA(dat,Q,sequential = TRUE,model = "DINA")
anova(sGDINA,sDINA)
coef(sDINA) # processing function
coef(sDINA,"itemprob") # success probabilities for each item
coef(sDINA,"LCprob") # success probabilities for each category for all latent classes

####################################
#           Example 10a.           #
#    Multiple-Group G-DINA model   #
####################################

Q <- sim10GDINA$simQ
K <- ncol(Q)
# parameter simulation
# Group 1 - female
N1 <- 3000
gs1 <- matrix(rep(0.1,2*nrow(Q)),ncol=2)
# Group 2 - male
N2 <- 3000
gs2 <- matrix(rep(0.2,2*nrow(Q)),ncol=2)

# data simulation for each group
sim1 <- simGDINA(N1,Q,gs.parm = gs1,model = "DINA",att.dist = "higher.order",
                 higher.order.parm = list(theta = rnorm(N1),
                 lambda = data.frame(a=rep(1.5,K),b=seq(-1,1,length.out=K))))
sim2 <- simGDINA(N2,Q,gs.parm = gs2,model = "DINO",att.dist = "higher.order",
                 higher.order.parm = list(theta = rnorm(N2),
                 lambda = data.frame(a=rep(1,K),b=seq(-2,2,length.out=K))))

# combine data - all items have the same item parameters
dat <- rbind(extract(sim1,"dat"),extract(sim2,"dat"))
gr <- rep(c(1,2),c(3000,3000))
# Fit G-DINA model
mg.est <- GDINA(dat = dat,Q = Q,group = gr)
summary(mg.est)
extract(mg.est,"posterior.prob")
coef(mg.est,"lambda")


####################################
#           Example 10b.           #
#    Multiple-Group G-DINA model   #
####################################

Q <- sim30GDINA$simQ
K <- ncol(Q)
# parameter simulation
N1 <- 3000
gs1 <- matrix(rep(0.1,2*nrow(Q)),ncol=2)
N2 <- 3000
gs2 <- matrix(rep(0.2,2*nrow(Q)),ncol=2)

# data simulation for each group
# two groups have different theta distributions
sim1 <- simGDINA(N1,Q,gs.parm = gs1,model = "DINA",att.dist = "higher.order",
                 higher.order.parm = list(theta = rnorm(N1),
                 lambda = data.frame(a=rep(1,K),b=seq(-2,2,length.out=K))))
sim2 <- simGDINA(N2,Q,gs.parm = gs2,model = "DINO",att.dist = "higher.order",
                 higher.order.parm = list(theta = rnorm(N2,1,1),
                 lambda = data.frame(a=rep(1,K),b=seq(-2,2,length.out=K))))

# combine data - different groups have distinct item parameters
# see ?bdiagMatrix
dat <- bdiagMatrix(list(extract(sim1,"dat"),extract(sim2,"dat")),fill=NA)
Q <- rbind(Q,Q)
gr <- rep(c(1,2),c(3000,3000))
mg.est <- GDINA(dat = dat,Q = Q,group = gr)
# Fit G-DINA model
mg.est <- GDINA(dat = dat,Q = Q,group = gr,att.dist="higher.order",
higher.order=list(model = "Rasch"))
summary(mg.est)
coef(mg.est,"lambda")
personparm(mg.est)
personparm(mg.est,"HO")
extract(mg.est,"posterior.prob")


####################################
#           Example 11.            #
#           Bug DINO model         #
####################################

set.seed(123)
Q <- sim10GDINA$simQ # 1 represents misconceptions/bugs
N <- 1000
J <- nrow(Q)

gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J))

sim <- simGDINA(N,Q,gs.parm = gs,model = "BUGDINO")
dat <- extract(sim,"dat")
est <- GDINA(dat=dat,Q=Q,model = "BUGDINO")
coef(est)

####################################
#           Example 12.            #
#           SISM model             #
####################################

# The Q-matrix used in Kuo, et al (2018)
# The first four columns are for Attributes 1-4
# The last three columns are for Bugs 1-3
Q <- matrix(c(1,0,0,0,0,0,0,
0,1,0,0,0,0,0,
0,0,1,0,0,0,0,
0,0,0,1,0,0,0,
0,0,0,0,1,0,0,
0,0,0,0,0,1,0,
0,0,0,0,0,0,1,
1,0,0,0,1,0,0,
0,1,0,0,1,0,0,
0,0,1,0,0,0,1,
0,0,0,1,0,1,0,
1,1,0,0,1,0,0,
1,0,1,0,0,0,1,
1,0,0,1,0,0,1,
0,1,1,0,0,0,1,
0,1,0,1,0,1,1,
0,0,1,1,0,1,1,
1,0,1,0,1,1,0,
1,1,0,1,1,1,0,
0,1,1,1,1,1,0),ncol = 7,byrow = TRUE)

J <- nrow(Q)
N <- 1000
gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J))

sim <- simGDINA(N,Q,gs.parm = gs,model = "SISM",no.bugs=3)
dat <- extract(sim,"dat")
est <- GDINA(dat=dat,Q=Q,model="SISM",no.bugs=3)
coef(est,"delta")

####################################
#           Example 13a.           #
#     user specified design matrix #
#        LCDM (logit G-DINA)       #
####################################

dat <- sim30GDINA$simdat
Q <- sim30GDINA$simQ

# LCDM
lcdm <- GDINA(dat = dat, Q = Q, model = "logitGDINA", control=list(conv.type="neg2LL"))

#Another way is to find design matrix for each item first => must be a list
D <- lapply(rowSums(Q),designmatrix,model="GDINA")
# for comparison, use change in -2LL as convergence criterion
# LCDM
lcdm2 <- GDINA(dat = dat, Q = Q, model = "UDF", design.matrix = D,
linkfunc = "logit", control=list(conv.type="neg2LL"),solver="slsqp")

# identity link GDINA
iGDINA <- GDINA(dat = dat, Q = Q, model = "GDINA",
control=list(conv.type="neg2LL"),solver="slsqp")

# compare all three models => identical
anova(lcdm,lcdm2,iGDINA)

####################################
#           Example 13b.           #
#     user specified design matrix #
#            RRUM                  #
####################################

dat <- sim30GDINA$simdat
Q <- sim30GDINA$simQ

# specify design matrix for each item => must be a list
# D can be defined by the user
D <- lapply(rowSums(Q),designmatrix,model="ACDM")
# for comparison, use change in -2LL as convergence criterion
# RRUM
logACDM <- GDINA(dat = dat, Q = Q, model = "UDF", design.matrix = D,
linkfunc = "log", control=list(conv.type="neg2LL"),solver="slsqp")

# identity link GDINA
RRUM <- GDINA(dat = dat, Q = Q, model = "RRUM",
              control=list(conv.type="neg2LL"),solver="slsqp")

# compare two models => identical
anova(logACDM,RRUM)

####################################
#           Example 14.            #
#     Multiple-strategy DINA model #
####################################

Q <- matrix(c(1,1,1,1,0,
1,2,0,1,1,
2,1,1,0,0,
3,1,0,1,0,
4,1,0,0,1,
5,1,1,0,0,
5,2,0,0,1),ncol = 5,byrow = TRUE)
d <- list(
  item1=c(0.2,0.7),
  item2=c(0.1,0.6),
  item3=c(0.2,0.6),
  item4=c(0.2,0.7),
  item5=c(0.1,0.8))

  set.seed(12345)
sim <- simGDINA(N=1000,Q = Q, delta.parm = d,
               model = c("MSDINA","MSDINA","DINA",
                         "DINA","DINA","MSDINA","MSDINA"))

# simulated data
dat <- extract(sim,what = "dat")
# estimation
# MSDINA need to be specified for each strategy
est <- GDINA(dat,Q,model = c("MSDINA","MSDINA","DINA",
                             "DINA","DINA","MSDINA","MSDINA"))
coef(est,"delta")

## End(Not run)

---

Estimating multiple-strategy cognitive diagnosis models

Description

An (experimental) function for calibrating the multiple-strategy CDMs for dichotomous response data (Ma & Guo, 2019)

Usage

GMSCDM(
  dat,
  msQ,
  model = "ACDM",
  s = 1,
  att.prior = NULL,
  delta = NULL,
  control = list()
)

Arguments

dat	A required binary item response matrix
msQ	A multiple-strategy Q-matrix; the first column gives item numbers and the second column gives the strategy number. See examples.
model	CDM used; can be "DINA","DINO","ACDM","LLM", and "RRUM", representing the GMS-DINA, GMS-DINO, GMS-ACDM, GMS-LLM and GMS-RRUM in Ma & Guo (2019), respectively. It can also be "rDINA" and "rDINO", representing restricted GMS-DINA and GMS-DINO models where delta_jm1 are equal for all strategies. Note that only a single model can be used for the whole test.
s	strategy selection parameter. It is equal to 1 by default.
att.prior	mixing proportion parameters.
delta	delta parameters in list format.
control	a list of control arguments

Value

an object of class GMSCDM with the following components:

IRF

A matrix of success probabilities for each latent class on each item (IRF)

delta

A list of delta parameters

attribute

A list of estimated attribute profiles including EAP, MLE and MAP estimates.

testfit

A list of test fit statistics including deviance, number of parameters, AIC and BIC

sIRF

strategy-specific item response function

pjmc

Probability of adopting each strategy on each item for each latent class

sprv

Strategy pravelence

Author(s)

Wenchao Ma, The University of Alabama, [email protected]

References

Ma, W., & de la Torre, J. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Ma, W., & Guo, W. (2019). Cognitive Diagnosis Models for Multiple Strategies. British Journal of Mathematical and Statistical Psychology.

See Also

GDINA for MS-DINA model and single strategy CDMs, and DTM for diagnostic tree model for multiple strategies in polytomous response data

Examples

## Not run: 
##################
#
# data simulation
#
##################
set.seed(123)
msQ <- matrix(
c(1,1,0,1,
1,2,1,0,
2,1,1,0,
3,1,0,1,
4,1,1,1,
5,1,1,1),6,4,byrow = T)
# J x L - 00,10,01,11
LC.prob <- matrix(c(
0.2,0.7727,0.5889,0.8125,
0.1,0.9,0.1,0.9,
0.1,0.1,0.8,0.8,
0.2,0.5,0.4,0.7,
0.2,0.4,0.7,0.9),5,4,byrow=TRUE)
N <- 10000
att <- sample(1:4,N,replace=TRUE)
dat <- 1*(t(LC.prob[,att])>matrix(runif(N*5),N,5))


est <- GMSCDM(dat,msQ)
# item response function
est$IRF
# strategy specific IRF
est$sIRF


################################
#
# Example 14 from GDINA function
#
################################
Q <- matrix(c(1,1,1,1,0,
1,2,0,1,1,
2,1,1,0,0,
3,1,0,1,0,
4,1,0,0,1,
5,1,1,0,0,
5,2,0,0,1),ncol = 5,byrow = TRUE)
d <- list(
  item1=c(0.2,0.7),
  item2=c(0.1,0.6),
  item3=c(0.2,0.6),
  item4=c(0.2,0.7),
  item5=c(0.1,0.8))

  set.seed(123)
sim <- simGDINA(N=1000,Q = Q, delta.parm = d,
               model = c("MSDINA","MSDINA","DINA",
                         "DINA","DINA","MSDINA","MSDINA"))

# simulated data
dat <- extract(sim,what = "dat")
# estimation
# MSDINA need to be specified for each strategy
est <- GDINA(dat,Q,model = c("MSDINA","MSDINA","DINA",
                             "DINA","DINA","MSDINA","MSDINA"),
             control = list(conv.type = "neg2LL",conv.crit = .01))

# Approximate the MS-DINA model using GMS DINA model
est2 <- GMSCDM(dat, Q, model = "rDINA", s = 10,
               control = list(conv.type = "neg2LL",conv.crit = .01))

## End(Not run)

---

Iterative latent-class analysis

Description

This function implements an iterative latent class analysis (ILCA; Jiang, 2019) approach to estimating attributes for cognitive diagnosis.

Usage

ILCA(dat, Q, seed.num = 5)

Arguments

dat	A required binary item response matrix.
Q	A required binary item and attribute association matrix.
seed.num	seed number; Default = 5.

Value

Estimated attribute profiles.

Author(s)

Zhehan Jiang, The University of Alabama

References

Jiang, Z. (2019). Using the iterative latent-class analysis approach to improve attribute accuracy in diagnostic classification models. Behavior research methods, 1-10.

Examples

## Not run: 
ILCA(sim10GDINA$simdat, sim10GDINA$simQ)

## End(Not run)

---

Extract log-likelihood for each individual

Description

Extract individual log-likelihood.

Usage

indlogLik(object, ...)

Arguments

object	GDINA object
...	additional arguments

Examples

## Not run: 
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ

fit <- GDINA(dat = dat, Q = Q, model = "GDINA")
iL <- indlogLik(fit)
iL[1:6,]

## End(Not run)

---

Extract log posterior for each individual

Description

Extract individual log posterior.

Usage

indlogPost(object, ...)

Arguments

object	GDINA object
...	additional arguments

Examples

## Not run: 
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
fit <- GDINA(dat = dat, Q = Q, model = "GDINA")
iP <- indlogPost(fit)
iP[1:6,]

## End(Not run)

---

Item fit statistics

Description

Calculate item fit statistics (Chen, de la Torre, & Zhang, 2013) and draw heatmap plot for item pairs

Usage

itemfit(
  GDINA.obj,
  person.sim = "post",
  p.adjust.methods = "holm",
  cor.use = "pairwise.complete.obs",
  digits = 4,
  N.resampling = NULL,
  randomseed = 123456
)

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

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

Arguments

GDINA.obj	An estimated model object of class GDINA
person.sim	Simulate expected responses from the posterior or based on EAP, MAP and MLE estimates.
p.adjust.methods	p-values for the proportion correct, transformed correlation, and log-odds ratio can be adjusted for multiple comparisons at test and item level. This is conducted using p.adjust function in stats, and therefore all adjustment methods supported by p.adjust can be used, including "holm", "hochberg", "hommel", "bonferroni", "BH" and "BY". See p.adjust for more details. "holm" is the default.
cor.use	how to deal with missing values when calculating correlations? This argument will be passed to use when calling stats::cor.
digits	How many decimal places in each number? The default is 4.
N.resampling	the sample size of resampling. By default, it is the maximum of 1e+5 and ten times of current sample size.
randomseed	random seed; This is used to make sure the results are replicable. The default random seed is 123456.
object	objects of class itemfit for various S3 methods
what	argument for S3 method extract indicating what to extract; It can be "p" for proportion correct statistics, "r" for transformed correlations, logOR for log odds ratios and "maxitemfit" for maximum statistics for each item.
...	additional arguments

Value

an object of class itemfit consisting of several elements that can be extracted using method extract. Components that can be extracted include:

p

the proportion correct statistics, adjusted and unadjusted p values for each item

r

the transformed correlations, adjusted and unadjusted p values for each item pair

logOR

the log odds ratios, adjusted and unadjusted p values for each item pair

maxitemfit

the maximum proportion correct, transformed correlation, and log-odds ratio for each item with associated item-level adjusted p-values

Methods (by generic)

• extract(itemfit): extract various elements from itemfit objects

• summary(itemfit): print summary information

Author(s)

Wenchao Ma, The University of Alabama, [email protected]
Jimmy de la Torre, The University of Hong Kong

References

Chen, J., de la Torre, J., & Zhang, Z. (2013). Relative and Absolute Fit Evaluation in Cognitive Diagnosis Modeling. Journal of Educational Measurement, 50, 123-140.

Ma, W., & de la Torre, J. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Examples

## Not run: 
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ

mod1 <- GDINA(dat = dat, Q = Q, model = "GDINA")
mod1
itmfit <- itemfit(mod1)

# Print "test-level" item fit statistics
# p-values are adjusted for multiple comparisons
# for proportion correct, there are J comparisons
# for log odds ratio and transformed correlation,
# there are J*(J-1)/2 comparisons

itmfit

# The following gives maximum item fit statistics for
# each item with item level p-value adjustment
# For each item, there are J-1 comparisons for each of
# log odds ratio and transformed correlation
summary(itmfit)

# use extract to extract various components
extract(itmfit,"r")

mod2 <- GDINA(dat,Q,model="DINA")
itmfit2 <- itemfit(mod2)
#misfit heatmap
plot(itmfit2)
itmfit2

## End(Not run)

---