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, wenchao.ma@ua.edu
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

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, wenchao.ma@ua.edu
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

Value

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

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu
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

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, wenchao.ma@ua.edu

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

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

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, wenchao.ma@ua.edu

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

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, wenchao.ma@ua.edu

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

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

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, wenchao.ma@ua.edu

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

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

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, wenchao.ma@ua.edu
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

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu

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, wenchao.ma@ua.edu

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

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, wenchao.ma@ua.edu

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

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, wenchao.ma@ua.edu
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

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, wenchao.ma@ua.edu

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

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

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

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

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, wenchao.ma@ua.edu
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)

extract item parameters (deprecated)

Description

This function has been deprecated; use coef instead.

Usage

itemparm(
  object,
  what = c("catprob", "gs", "delta", "rrum", "itemprob", "LCprob"),
  withSE = FALSE,
  SE.type = 2,
  digits = 4,
  ...
)

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

Arguments

References

Philipp, M., Strobl, C., de la Torre, J., & Zeileis, A.(2017). On the estimation of standard errors in cognitive diagnosis models. Journal of Educational and Behavioral Statistics, 43, 88-115.

Examples

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

## End(Not run)

Transformation between latent classes and latent groups

Description

This function gives the equivalent latent classes which have the same category success probabilities for each category or item.

Usage

LC2LG(Q, sequential = FALSE, att.str = NULL)

Arguments

Value

An item or category by latent class matrix. In the G-DINA model, if item j measures Kj attributes, 2^K latent classes can be combined into 2^{Kj} latent groups. This matrix gives which latent group each of 2^K latent classes belongs to for each item.

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu
Jimmy de la Torre, The University of Hong Kong

Examples

attributepattern(3)

q <- matrix(scan(text = "0 1 0 1 0 1 1 1 0"),ncol = 3)
q
LC2LG(Q = q)

Multiple-choice models

Description

This function estimates the multiple-choice DINA model (de la Torre, 2009).

Usage

MCmodel(
  dat,
  Qc,
  model = "MCDINA",
  key = NULL,
  conv.crit = 0.001,
  maxitr = 2000,
  conv.type = "pr",
  SE = FALSE
)

Arguments

Value

an object of class MCmodel with the following components:

prob.parm

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

prob.se

A list of standard errors of item 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

R

expected # of individuals in each latent group choosing each option

lik

posterior probability

itr

Total # of iterations

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu

References

De La Torre, J. (2009). A cognitive diagnosis model for cognitively based multiple-choice options. Applied Psychological Measurement, 33, 163–183.

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

See Also

GDINA for G-DINA model

Examples

## Not run: 
 # check the format of the data
 # Entry 0 is not allowed
 head(sim10MCDINA$simdat)

 #---------------------------------
 # check the format of the Q-matrix
 #---------------------------------
 # Take item 1 as an example:
 # category 2 has a q-vector (1,0,0)
 # category 1 has a q-vector (0,1,0)
 # category 4 has a q-vector (1,1,0)
 # category 3 is not included in the Q-matrix because it is not coded
 # the order of the coded categories in the Q-matrix doesn't matter

 sim10MCDINA$simQ
  #     Item coded cat A1 A2 A3
  #        1         2  1  0  0
  #        1         1  0  1  0
  #        1         4  1  1  0
  #...
 est <- MCmodel(sim10MCDINA$simdat,sim10MCDINA$simQ)
 est
 est$testfit

 #--------------------------------------
 # Distractors involving more attributes
 #--------------------------------------
 # some distractors may involve attributes that are not invovled by the key option
 # this is not allowed by the "original" MC-DINA (de la Torre, 2009) but is allowed
 # in the current implementation

 # Users need to specify the key for each item to appropriate handle such an issue
 # Note item 1 below: category 1 is the key (as indicated in the key argument below)
 # The distractor (category 4) involves an attribute not included by the key option

 Qc <- matrix(c(1,	1,	1,	1,	0,
                1,	2,	0,	1,	0,
                1,	3,	1,	0,	0,
                1,	4,	1,	0,	1,
                2,	1,	1,	0,	0,
                2,	3,	1,	1,	0,
                2,	2,	1,	1,	1,
                3,	4,	1,	1,	1,
                3,	2,	1,	1,	0,
                3,	3,	0,	1,	1,
                4,	1,	0,	1,	1,
                4,	2,	0,	0,	1,
                5,	1,	1,	0,	0,
                6,	3,	0,	1,	0,
                7,	2,	0,	0,	1,
                8,	4,	1,	0,	0,
                9,	1,	0,	1,	0,
                10, 4,	0,	0,	1),ncol = 5,byrow = TRUE)

 est2 <- MCmodel(sim10MCDINA2$simdat,Qc, key = c(1,2,4,1,1,3,2,4,1,4))
 est2
 est2$prob.parm
 est2$testfit
 est2$attribute

## End(Not run)

Item-level model comparison using Wald, LR or LM tests

Description

This function evaluates whether the saturated G-DINA model can be replaced by reduced CDMs without significant loss in model data fit for each item using the Wald test, likelihood ratio (LR) test or Lagrange multiplier (LM) test. For Wald test, see de la Torre (2011), de la Torre and Lee (2013), Ma, Iaconangelo and de la Torre (2016) and Ma & de la Torre (2018) for details. For LR test and a two-step LR approximation procedure, see Sorrel, de la Torre, Abad, and Olea (2017), Ma (2017) and Ma & de la Torre (2019). For LM test, which is only applicable for DINA, DINO and ACDM, see Sorrel, Abad, Olea, de la Torre, and Barrada (2017). This function also calculates the dissimilarity between the reduced models and the G-DINA model, which can be viewed as a measure of effect size (Ma, Iaconangelo & de la Torre, 2016).

Usage

modelcomp(
  GDINA.obj = NULL,
  method = "Wald",
  items = "all",
  p.adjust.methods = "holm",
  models = c("DINA", "DINO", "ACDM", "LLM", "RRUM"),
  decision.args = list(rule = "largestp", alpha.level = 0.05, adjusted = FALSE),
  DS = FALSE,
  Wald.args = list(SE.type = 2, varcov = NULL),
  LR.args = list(LR.approx = FALSE),
  LM.args = list(reducedMDINA = NULL, reducedMDINO = NULL, reducedMACDM = NULL, SE.type =
    2)
)

## S3 method for class 'modelcomp'
extract(
  object,
  what = c("stats", "pvalues", "adj.pvalues", "df", "DS", "selected.model"),
  digits = 4,
  ...
)

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

Arguments

Details

After the test 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, two diferent rules can be implemented for selecting the best model specified in argument decision.args:

(1) when rule="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.

(2) When rule="largestp" (default),

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

Value

an object of class modelcomp. Elements that can be extracted using extract method include

stats

Wald or LR statistics

pvalues

p-values associated with the test statistics

adj.pvalues

adjusted p-values

df

degrees of freedom

DS

dissimilarity between G-DINA and other CDMs

Methods (by generic)

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

• summary(modelcomp): print summary information

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu
Miguel A. Sorrel, Universidad Autonoma de Madrid
Jimmy de la Torre, The University of Hong Kong

References

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.

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.

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. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

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.

See Also

GDINA, autoGDINA

Examples

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

###################
#
# Wald test
#
###################

w <- modelcomp(fit)
w
# wald statistics
extract(w,"stats")
#p values
extract(w,"pvalues")
# selected models
extract(w,"selected.model")
##########################
#
# LR and Two-step LR test
#
##########################

lr <- modelcomp(fit,method = "LR")
lr
TwostepLR <- modelcomp(fit,items =c(6:10),method = "LR",LR.args = list(LR.approx = TRUE))
TwostepLR

##########################
#
# LM test
#
##########################

dina <- GDINA(dat = dat, Q = Q, model = "DINA")
dino <- GDINA(dat = dat, Q = Q, model = "DINO")
acdm <- GDINA(dat = dat, Q = Q, model = "ACDM")
lm <- modelcomp(method = "LM",LM.args=list(reducedMDINA = dina,
reducedMDINO = dino, reducedMACDM  = acdm))
lm



## End(Not run)

Model fit statistics

Description

Calculate various absolute model-data fit statistics

Usage

modelfit(GDINA.obj, CI = 0.9, ItemOnly = FALSE)

Arguments

Details

Various model-data fit statistics including M2 statistic for G-DINA model with dichotmous responses (Liu, Tian, & Xin, 2016; Hansen, Cai, Monroe, & Li, 2016) and for sequential G-DINA model with graded responses (Ma, 2020). It also calculates SRMSR and RMSEA2.

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu

References

Hansen, M., Cai, L., Monroe, S., & Li, Z. (2016). Limited-information goodness-of-fit testing of diagnostic classification item response models. British Journal of Mathematical and Statistical Psychology. 69, 225–252.

Liu, Y., Tian, W., & Xin, T. (2016). An Application of M2 Statistic to Evaluate the Fit of Cognitive Diagnostic Models. Journal of Educational and Behavioral Statistics, 41, 3-26.

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. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Maydeu-Olivares, A. (2013). Goodness-of-Fit Assessment of Item Response Theory Models. Measurement, 11, 71-101.

Examples

## Not run: 
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
mod1 <- GDINA(dat = dat, Q = Q, model = "DINA")
modelfit(mod1)

## End(Not run)

This function checks if monotonicity is violated

Description

If mastering an additional attribute lead to a lower probabilities of success, the monotonicity is violated.

Usage

monocheck(object, strict = FALSE)

Arguments

Value

a logical vector for each item or category indicating whether the monotonicity is violated (TRUE) or not (FALSE)

Examples

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


mod1 <- GDINA(dat = dat, Q = Q, model = "GDINA")
check <- monocheck(mod1)
check
mod2 <- GDINA(dat = dat, Q = Q, model = "GDINA",mono.constraint = check)
check2 <- monocheck(mod2)
check2

## End(Not run)

Calculate the number of parameters

Description

Calculate the number of parameters for GDINA estimates. Returned the total number of parameters, the number of item parameters and the number parameters of joint attribute distribution.

Usage

npar(object, ...)

Arguments

Examples

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

fit <- GDINA(dat = dat, Q = Q, model = "GDINA")
npar(fit)

## End(Not run)

calculate person (incidental) parameters

Description

Function to calculate various person attribute parameters, including "EAP", "MAP", and "MLE", for EAP, MAP and MLE estimates of attribute patterns (see Huebner & Wang, 2011), "mp" for marginal mastery probabilities, and "HO" for higher-order ability estimates if a higher-order model is fitted. See GDINA for examples.

Usage

personparm(object, what = c("EAP", "MAP", "MLE", "mp", "HO"), digits = 4, ...)

Arguments

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu
Jimmy de la Torre, The University of Hong Kong

References

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

Examples

## Not run: 
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
fit <- GDINA(dat = dat, Q = Q, model = "GDINA")
# EAP
head(personparm(fit))
# MAP
head(personparm(fit, what = "MAP"))

## End(Not run)

Create plots for GDINA estimates

Description

Create various plots for GDINA estimates

Usage

## S3 method for class 'GDINA'
plot(
  x,
  what = "IRF",
  item = "all",
  withSE = FALSE,
  SE.type = 2,
  person = 1,
  att.names = NULL,
  ...
)

Arguments

See Also

GDINA, autoGDINA

Examples

## Not run: 
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
mod1 <- GDINA(dat = dat, Q = Q, model = "GDINA")
#plot item response functions for item 10
plot(mod1, item = 10)
plot(mod1, what = "IRF", item = 10,withSE = TRUE)

# plot mastery probabilities for individuals 4 and 10
plot(mod1, what = "mp", person = c(4,10))
plot(mod1, what = "mp", person = c(4,10,15),
att.names = c("addition","subtraction","multiplication"))

## End(Not run)

Item fit plots

Description

Create plots of bivariate heatmap for item fit

Usage

## S3 method for class 'itemfit'
plot(x, type = "all", adjusted = TRUE, ...)

Arguments

See Also

GDINA, itemfit

Examples

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

fit <- GDINA(dat = dat, Q = Q, model = "GDINA")
ift <- itemfit(fit)
# plot the adjusted p values for log odds or transformed correlation
plot(ift)
# plot unadjusted p values for log odds
plot(ift,adjusted = FALSE, type = "logOR")

## End(Not run)

Mesa plot for Q-matrix validation

Description

The mesa plot was first proposed by de la Torre and Ma (2016) for graphically illustrating the best q-vector(s) for each item. The q-vector on the edge of the mesa is likely to be the best q-vector.

Usage

## S3 method for class 'Qval'
plot(
  x,
  item,
  type = "best",
  no.qvector = 10,
  data.label = TRUE,
  eps = "auto",
  original.q.label = FALSE,
  auto.ylim = TRUE,
  ...
)

Arguments

References

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.

See Also

Qval, autoGDINA

Examples

## Not run: 
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
Q[1,] <- c(0,1,0)
mod1 <- GDINA(dat = dat, Q = Q, model = "GDINA")
out <- Qval(mod1,eps = 0.9)
item <- c(1,2,10)
plot(out,item=item,data.label=FALSE,type="all")
plot(out,item=10,type="best",eps=0.95)
plot(out,item=10,type="all",no.qvector=6)

## End(Not run)

Q-matrix validation

Description

Q-matrix validation for the (sequential) G-DINA model based on PVAF (de la Torre & Chiu, 2016; Najera, Sorrel, & Abad, 2019; Najera et al., 2020), stepwise Wald test (Ma & de la Torre, 2020) or mesa plot (de la Torre & Ma, 2016). All these methods are suitable for dichotomous and ordinal response data. If too many modifications are suggested based on the default PVAF method, you are suggested to try the stepwise Wald test method, iterative procedures or predicted cutoffs. You should always check the mesa plots for further examination.

Usage

Qval(
  GDINA.obj,
  method = "PVAF",
  iter = "none",
  eps = 0.95,
  digits = 4,
  wald.args = list(),
  iter.args = list(empty.att = FALSE, max.iter = 150, verbose = FALSE)
)

## S3 method for class 'Qval'
extract(object, what = c("sug.Q", "varsigma", "PVAF", "eps", "Q"), ...)

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

Arguments

Value

An object of class Qval. Elements that can be extracted using extract method include:

sug.Q

suggested Q-matrix

Q

original Q-matrix

varsigma

varsigma index

PVAF

PVAF

Methods (by generic)

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

• summary(Qval): print summary information

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu,
Miguel A. Sorrel, Universidad Autónoma de Madrid,
Jimmy de la Torre, The University of Hong Kong

References

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

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.

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

Najera, P., Sorrel, M. A., & Abad, F.J. (2019). Reconsidering cutoff points in the general method of empirical Q-matrix validation. Educational and Psychological Measurement, 79, 727-753.

Najera, P., Sorrel, M. A., de la Torre, J., & Abad, F. J. (2020). Improving robustness in Q-matrix validation using an iterative and dynamic procedure. Applied Psychological Measurement.

See Also

GDINA

Examples

## Not run: 
################################
#
# Binary response
#
################################
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
Q[10,] <- c(0,1,0)

# Fit the G-DINA model
mod1 <- GDINA(dat = dat, Q = Q, model = "GDINA")

# Q-validation using de la Torre and Chiu's method
pvaf <- Qval(mod1,method = "PVAF",eps = 0.95)
pvaf
extract(pvaf,what = "PVAF")
#See also:
extract(pvaf,what = "varsigma")
extract(pvaf,what = "sug.Q")

# Draw mesa plots using the function plot

plot(pvaf,item=10)

#The stepwise Wald test
stepwise <- Qval(mod1,method = "wald")
stepwise
extract(stepwise,what = "PVAF")
#See also:
extract(stepwise,what = "varsigma")
extract(stepwise,what = "sug.Q")

#Set eps = -1 to determine the cutoff empirically
pvaf2 <- Qval(mod1,method = "PVAF",eps = -1)
pvaf2

#Iterative procedure (test-attribute level)
pvaf3 <- Qval(mod1, method = "PVAF", eps = -1,
              iter = "test.att", iter.args = list(verbose = 1))
pvaf3

################################
#
# Ordinal response
#
################################
seq.est <- GDINA(sim20seqGDINA$simdat,sim20seqGDINA$simQ, sequential = TRUE)
stepwise <- Qval(seq.est, method = "wald")

## End(Not run)

Count the frequency of a row vector in a data frame

Description

Count the frequency of a row vector in a data frame

Usage

rowMatch(df, vec = NULL)

Arguments

Value

count the number of vector vec in the data frame

row.no row numbers of the vector vec in the data frame

Examples

df <- data.frame(V1=c(1L,2L),V2=LETTERS[1:3],V3=rep(1,12))
rowMatch(df,c(2,"B",1))

Score function

Description

Calculate score function for each dichotomous item or each nonzero category for polytomous items Only applicable to saturated model ofr joint attribute distribution

Usage

score(object, parm = "delta")

Arguments

Value

a list where elements give the score functions for each item or category

Examples

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

## End(Not run)

Simulated data (10 items, G-DINA model)

Description

Simulated data, Q-matrix and item parameters for a 10-item test with 3 attributes.

Usage

sim10GDINA

Format

A list with components:

simdat

simulated responses of 1000 examinees

simQ

artificial Q-matrix

simItempar

artificial item parameters (probability of success for each latent group)

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu

References

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

Simulated data (10 items, MC-DINA model)

Description

Simulated data, Q-matrix and item parameters for a 10-item test measuring 3 attributes.

Usage

sim10MCDINA

Format

A list with components:

simdat

simulated responses of 3000 examinees

simQ

artificial Q-matrix

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu

References

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

Simulated data (10 items, MC-DINA model)

Description

Simulated data, Q-matrix and item parameters for a 10-item test measuring 5 attributes.

Usage

sim10MCDINA2

Format

A list with components:

simdat

simulated responses of 3000 examinees

simQ

artificial Q-matrix

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu

References

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

Simulated data (20 items, sequential G-DINA model)

Description

Simulated data, Qc-matrix and item parameters for a 20-item test measuring 5 attributes.

Usage

sim20seqGDINA

Format

A list with components:

simdat

simulated polytomous responses of 2000 examinees

simQ

artificial Qc-matrix

simItempar

artificial item parameters (category level probability of success for each latent group)

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu

References

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

Simulated data (21 items, sequential DINA model)

Description

Simulated data, and Qc-matrix for a 21-item test measuring 5 attributes.

Usage

sim21seqDINA

Format

A list with components:

simdat

simulated responses of 2000 examinees

simQ

artificial Qc-matrix

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu

References

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

Simulated data (30 items, DINA model)

Description

Simulated data, Q-matrix and item parameters for a 30-item test measuring 5 attributes.

Usage

sim30DINA

Format

A list with components:

simdat

simulated responses of 1000 examinees

simQ

artificial Q-matrix

simItempar

artificial item parameters (probability of success for each latent group)

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu

References

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

Simulated data (30 items, G-DINA model)

Description

Simulated data, Q-matrix and item parameters for a 30-item test measuring 5 attributes.

Usage

sim30GDINA

Format

A list with components:

simdat

simulated responses of 1000 examinees

simQ

artificial 30 \times 5 Q-matrix

simItempar

artificial item parameters(probability of success for each latent group)

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu

References

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

Simulated data (30 items, polytomous G-DINA model)

Description

Simulated data, Q-matrix and item parameters for a 30-item test measuring 5 attributes.

Usage

sim30pGDINA

Format

A list with components:

simdat

simulated responses of 3000 examinees

simQ

artificial Q-matrix

simItempar

artificial item parameters(probability of success for each latent group)

Simulating data for diagnostic tree model

Description

Data generation for diagnostic tree model

Usage

simDTM(N, Qc, gs.parm, Tmatrix, red.delta = NULL, att.gr = NULL)

Arguments

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)

Data simulation based on the G-DINA models

Description

Simulate responses based on the G-DINA model (de la Torre, 2011) and sequential G-DINA model (Ma & de la Torre, 2016), or CDMs subsumed by them, including the DINA model, DINO model, ACDM, LLM and R-RUM. Attributes can be simulated from uniform, higher-order or multivariate normal distributions, or be supplied by users. See Examples and Details for how item parameter specifications. See the help page of GDINA for model parameterizations.

Usage

simGDINA(
  N,
  Q,
  gs.parm = NULL,
  delta.parm = NULL,
  catprob.parm = NULL,
  model = "GDINA",
  sequential = FALSE,
  no.bugs = 0,
  gs.args = list(type = "random", mono.constraint = TRUE),
  design.matrix = NULL,
  linkfunc = NULL,
  att.str = NULL,
  attribute = NULL,
  att.dist = "uniform",
  item.names = NULL,
  higher.order.parm = list(theta = NULL, lambda = NULL),
  mvnorm.parm = list(mean = NULL, sigma = NULL, cutoffs = NULL),
  att.prior = NULL,
  digits = 4
)

## S3 method for class 'simGDINA'
extract(
  object,
  what = c("dat", "Q", "attribute", "catprob.parm", "delta.parm", "higher.order.parm",
    "mvnorm.parm", "LCprob.parm"),
  ...
)

Arguments

Details

Item parameter specifications in simGDINA:

Item parameters can be specified in one of three different ways.

The first and probably the easiest way is to specify the guessing and slip parameters for each item or nonzero category using gs.parm, which is a matrix or data frame for P(\bm{\alpha}_{lj}^*=0) and 1-P(\bm{\alpha}_{lj}^*=1) for all items for dichotomous items and S(\bm{\alpha}_{ljh}^*=0) and 1-S(\bm{\alpha}_{ljh}^*=1) for all nonzero categories for polytomous items. Note that 1-P(\bm{\alpha}_{lj}^*=0)-P(\bm{\alpha}_{lj}^*=1) or 1-S(\bm{\alpha}_{lj}^*=0)-S(\bm{\alpha}_{lj}^*=1) must be greater than 0. For generating ACDM, LLM, and RRUM, delta parameters are generated randomly if type="random", or in a way that each required attribute contributes equally, as in Ma, Iaconangelo, & de la Torre (2016) if type="equal". For ACDM, LLM and RRUM, generated delta parameters are always positive, which implies that monotonicity constraints are always satisfied. If the generating model is the G-DINA model, mono.constraint can be used to specify whether monotonicity constraints should be satisfied.

The second way of simulating responses is to specify success probabilities (i.e., P(\bm{\alpha}_{lj}^*) or S(\bm{\alpha}_{ljh}^*)) for each nonzero category of each item directly using the argument catprob.parm. If an item or category requires K_j^* attributes, 2^{K_j^*} success probabilities need to be provided. catprob.parm must be a list, where each element gives the success probabilities for nonzero category of each item. Note that success probabilities cannot be negative or greater than one.

The third way is to specify delta parameters for data simulation. For DINA and DINO model, each nonzero category requires two delta parameters. For ACDM, LLM and RRUM, if a nonzero category requires K_j^* attributes, K_j^*+1 delta parameters need to be specified. For the G-DINA model, a nonzero category requiring K_j^* attributes has 2^{K_j^*} delta parameters. It should be noted that specifying delta parameters needs to ascertain the derived success probabilities are within the [0,1] interval.

Please note that you need to specify item parameters in ONLY one of these three ways. If gs.parm is specified, it will be used regardless of the inputs in catprob.parm and delta.parm. If gs.parm is not specified, simGDINA will check if delta.parm is specified; if yes, it will be used for data generation. if both gs.parm and delta.parm are not specified, catprob.parm is used for data generation.

Value

an object of class simGDINA. Elements that can be extracted using method extract include:

dat

simulated item response matrix

Q

Q-matrix

attribute

A N \times K matrix for inviduals' attribute patterns

catprob.parm

a list of non-zero category success probabilities for each latent group

delta.parm

a list of delta parameters

higher.order.parm

Higher-order parameters

mvnorm.parm

multivariate normal distribution parameters

LCprob.parm

A matrix of item/category success probabilities for each latent class

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu
Jimmy de la Torre, The University of Hong Kong

References

Chiu, C.-Y., Douglas, J. A., & Li, X. (2009). Cluster analysis for cognitive diagnosis: Theory and applications. Psychometrika, 74, 633-665.

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.

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.

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

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

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

Examples

## Not run: 
####################################################
#                     Example 1                    #
#             Data simulation (DINA)               #
####################################################
N <- 500
Q <- sim30GDINA$simQ
J <- nrow(Q)
gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J))

# Simulated DINA model; to simulate G-DINA model
# and other CDMs, change model argument accordingly

sim <- simGDINA(N,Q,gs.parm = gs,model = "DINA")

# True item success probabilities
extract(sim,what = "catprob.parm")

# True delta parameters
extract(sim,what = "delta.parm")

# simulated data
extract(sim,what = "dat")

# simulated attributes
extract(sim,what = "attribute")

####################################################
#                     Example 2                    #
#             Data simulation (RRUM)               #
####################################################
N <- 500
Q <- sim30GDINA$simQ
J <- nrow(Q)
gs <- data.frame(guess=rep(0.2,J),slip=rep(0.2,J))
# Simulated RRUM
# deltas except delta0 for each item will be simulated
# randomly subject to the constraints of RRUM
sim <- simGDINA(N,Q,gs.parm = gs,model = "RRUM")

# simulated data
extract(sim,what = "dat")

# simulated attributes
extract(sim,what = "attribute")

####################################################
#                     Example 3                    #
#             Data simulation (LLM)                #
####################################################
N <- 500
Q <- sim30GDINA$simQ
J <- nrow(Q)
gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J))
# Simulated LLM
# By specifying type="equal", each required attribute is
# assumed to contribute to logit(P) equally
sim <- simGDINA(N,Q,gs.parm = gs,model = "LLM",gs.args = list (type="equal"))
#check below for what the equal contribution means
extract(sim,what = "delta.parm")

# simulated data
extract(sim,what = "dat")

# simulated attributes
extract(sim,what = "attribute")

####################################################
#                   Example 4                      #
#          Data simulation (all CDMs)              #
####################################################

set.seed(12345)

N <- 500
Q <- sim10GDINA$simQ
J <- nrow(Q)
gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J))
# Simulated different CDMs for different items
models <- c("GDINA","DINO","DINA","ACDM","LLM","RRUM","GDINA","LLM","RRUM","DINA")
sim <- simGDINA(N,Q,gs.parm = gs,model = models,gs.args = list(type="random"))

# simulated data
extract(sim,what = "dat")

# simulated attributes
extract(sim,what = "attribute")

####################################################
#                   Example 5a                     #
#          Data simulation (all CDMs)              #
#  using probability of success in list format     #
####################################################

# success probabilities for each item need to be provided in list format as follows:
# if item j requires Kj attributes, 2^Kj success probabilities
# need to be specified
# e.g., item 1 only requires 1 attribute
# therefore P(0) and P(1) should be specified;
# similarly, item 10 requires 3 attributes,
# P(000),P(100),P(010)...,P(111) should be specified;
# the latent class represented by each element can be obtained
# by calling attributepattern(Kj)
itemparm.list <- list(item1=c(0.2,0.9),
                    item2=c(0.1,0.8),
                    item3=c(0.1,0.9),
                    item4=c(0.1,0.3,0.5,0.9),
                    item5=c(0.1,0.1,0.1,0.8),
                    item6=c(0.2,0.9,0.9,0.9),
                    item7=c(0.1,0.45,0.45,0.8),
                    item8=c(0.1,0.28,0.28,0.8),
                    item9=c(0.1,0.4,0.4,0.8),
                    item10=c(0.1,0.2,0.3,0.4,0.4,0.5,0.7,0.9))
set.seed(12345)
N <- 500
Q <- sim10GDINA$simQ
# When simulating data using catprob.parm argument,
# it is not necessary to specify model and type
sim <- simGDINA(N,Q,catprob.parm = itemparm.list)

####################################################
#                   Example 5b                     #
#          Data simulation (all CDMs)              #
#  using probability of success in list format     #
#  attribute has a linear structure                #
####################################################

est <- GDINA(sim10GDINA$simdat,sim10GDINA$simQ,att.str = list(c(1,2),c(2,3)))
# design matrix
# link function
# item probabilities
ip <- extract(est,"itemprob.parm")
sim <- simGDINA(N=500,sim10GDINA$simQ,catprob.parm = ip,
design.matrix = dm,linkfunc = lf,att.str = list(c(1,2),c(2,3)))


####################################################
#                   Example 6a                     #
#            Data simulation (all CDMs)            #
#      using delta parameters in list format       #
####################################################

delta.list <- list(c(0.2,0.7),
                    c(0.1,0.7),
                    c(0.1,0.8),
                    c(0.1,0.7),
                    c(0.1,0.8),
                    c(0.2,0.3,0.2,0.1),
                    c(0.1,0.35,0.35),
                    c(-1.386294,0.9808293,1.791759),
                    c(-1.609438,0.6931472,0.6),
                    c(0.1,0.1,0.2,0.3,0.0,0.0,0.1,0.1))

model <- c("GDINA","GDINA","GDINA","DINA","DINO","GDINA","ACDM","LLM","RRUM","GDINA")
N <- 500
Q <- sim10GDINA$simQ
sim <- simGDINA(N,Q,delta.parm = delta.list, model = model)
####################################################
#                   Example 6b                     #
#          Data simulation (all CDMs)              #
#  using delta parameters in list format           #
#  attribute has a linear structure                #
####################################################

est <- GDINA(sim10GDINA$simdat,sim10GDINA$simQ,att.str = list(c(1,2),c(2,3)))
# design matrix
# link function
# item probabilities
ip <- extract(est,"delta.parm")
sim <- simGDINA(N=500,sim10GDINA$simQ,delta.parm =  d,
design.matrix = dm,linkfunc = lf,att.str = list(c(1,2),c(2,3)))

####################################################
#                   Example 7                      #
#      Data simulation (higher order DINA model)   #
####################################################

Q <- sim30GDINA$simQ
gs <- matrix(0.1,nrow(Q),2)
N <- 500
set.seed(12345)
theta <- rnorm(N)
K <- ncol(Q)
lambda <- data.frame(a=rep(1,K),b=seq(-2,2,length.out=K))
sim <- simGDINA(N,Q,gs.parm = gs, model="DINA", att.dist = "higher.order",
                 higher.order.parm = list(theta = theta,lambda = lambda))

####################################################
#                   Example 8                      #
#      Data simulation (higher-order CDMs)         #
####################################################

Q <- sim30GDINA$simQ
gs <- matrix(0.1,nrow(Q),2)
models <- c(rep("GDINA",5),
            rep("DINO",5),
            rep("DINA",5),
            rep("ACDM",5),
            rep("LLM",5),
            rep("RRUM",5))
N <- 500
set.seed(12345)
theta <- rnorm(N)
K <- ncol(Q)
lambda <- data.frame(a=runif(K,0.7,1.3),b=seq(-2,2,length.out=K))
sim <- simGDINA(N,Q,gs.parm = gs, model=models, att.dist = "higher.order",
                 higher.order.parm = list(theta = theta,lambda = lambda))


####################################################
#                   Example 9                      #
#      Data simulation (higher-order model)        #
#  using the multivariate normal threshold model   #
####################################################


# See Chiu et al., (2009)

N <- 500
Q <- sim10GDINA$simQ
K <- ncol(Q)
gs <- matrix(0.1,nrow(Q),2)
cutoffs <- qnorm(c(1:K)/(K+1))
m <- rep(0,K)
vcov <- matrix(0.5,K,K)
diag(vcov) <- 1
simMV <- simGDINA(N,Q,gs.parm = gs, att.dist = "mvnorm",
                 mvnorm.parm=list(mean = m, sigma = vcov,cutoffs = cutoffs))

####################################
#          Example 10              #
#        Simulation using          #
#      user-specified att structure#
####################################

# --- User-specified attribute structure ----#
Q <- sim30GDINA$simQ
K <- ncol(Q)
# 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))
struc <- att.structure(diverg,K)

# data simulation
N <- 1000
# data simulation
gs <- matrix(0.1,nrow(Q),2)
simD <- simGDINA(N,Q,gs.parm = gs,
                   model = "DINA",att.dist = "categorical",att.prior = struc$att.prob)


####################################################
#                   Example 11                     #
#                Data simulation                   #
#  (GDINA with monotonicity constraints)           #
####################################################

set.seed(12345)

N <- 500
Q <- sim30GDINA$simQ
J <- nrow(Q)
gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J))
# Simulated different CDMs for different items
sim <- simGDINA(N,Q,gs.parm = gs,model = "GDINA",gs.args=list(mono.constraint=TRUE))

# True item success probabilities
extract(sim,what = "catprob.parm")

# True delta parameters
extract(sim,what = "delta.parm")

# simulated data
extract(sim,what = "dat")

# simulated attributes
extract(sim,what = "attribute")

####################################################
#                   Example 12                     #
#                Data simulation                   #
# (Sequential G-DINA model - polytomous responses) #
####################################################

set.seed(12345)

N <- 2000
# restricted Qc matrix
Qc <- sim20seqGDINA$simQ
#total number of categories
J <- nrow(Qc)
gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J))
# simulate sequential DINA model
simseq <- simGDINA(N, Qc, sequential = TRUE, gs.parm = gs, model = "GDINA")

# True item success probabilities
extract(simseq,what = "catprob.parm")

# True delta parameters
extract(simseq,what = "delta.parm")

# simulated data
extract(simseq,what = "dat")

# simulated attributes
extract(simseq,what = "attribute")


####################################################
#                   Example 13
#         DINA model Attribute generated using
#             categorical distribution
####################################################

Q <- sim10GDINA$simQ
gs <- matrix(0.1,nrow(Q),2)
N <- 5000
set.seed(12345)
prior <- c(0.1,0.2,0,0,0.2,0,0,0.5)
sim <- simGDINA(N,Q,gs.parm = gs, model="DINA", att.dist = "categorical",att.prior = prior)
# check latent class sizes
table(sim$att.group)/N

####################################################
#                   Example 14
#                  MS-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
extract(sim,what = "dat")

# simulated attributes
extract(sim,what = "attribute")


##############################################################
#                   Example 15
#  reparameterized SISM model (Kuo, Chen, & de la Torre, 2018)
#  see GDINA function for more details
###############################################################

# 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 <- 500
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)

# True item success probabilities
extract(sim,what = "catprob.parm")

# True delta parameters
extract(sim,what = "delta.parm")

# simulated data
extract(sim,what = "dat")

# simulated attributes
extract(sim,what = "attribute")

## End(Not run)

Graphical user interface of the GDINA function

Description

An interactive Shiny application for running GDINA function. See Ma and de la Torre (2019) and de la Torre and Akbay (2019) for tutorials.

Usage

startGDINA()

Author(s)

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu

References

de la Torre, J & Akbay, L. (2019). Implementation of Cognitive Diagnosis Modeling using the GDINA R Package. Eurasian Journal of Educational Research, 80, 171-192.

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. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Examples

## Not run: 
library(shiny)
library(shinydashboard)
startGDINA()

## End(Not run)

Unique values in a vector

Description

Unique values in a vector

Usage

unique_only(vec)

Arguments

Value

sorted unique values

See Also

unique

Examples

vec <- c(4,2,3,5,4,4,4)
unique_only(vec)
# see the difference from unique
unique(vec)

vec <- letters[1:5]
unique_only(vec)

Generate unrestricted Qc matrix from an restricted Qc matrix

Description

Generate unrestricted Qc matrix from an restricted Qc matrix

Usage

unrestrQ(Qc)

Arguments

Value

an unrestricted Qc matrix

Examples

Qc <- sim21seqDINA$simQc
Qc
unrestrQ(Qc)