gdina {CDM} R Documentation

## Estimating the Generalized DINA (GDINA) Model

### Description

This function implements the generalized DINA model for dichotomous attributes (GDINA; de la Torre, 2011) and polytomous attributes (pGDINA; Chen & de la Torre, 2013, 2018). In addition, multiple group estimation is also possible using the gdina function. This function also allows for the estimation of a higher order GDINA model (de la Torre & Douglas, 2004). Polytomous item responses are treated by specifying a sequential GDINA model (Ma & de la Torre, 2016; Tutz, 1997). The simulataneous modeling of skills and misconceptions (bugs) can be also estimated within the GDINA framework (see Kuo, Chen & de la Torre, 2018; see argument rule).

The estimation can also be conducted by posing monotonocity constraints (Hong, Chang, & Tsai, 2016) using the argument mono.constr. Moreover, regularization methods SCAD, lasso, ridge, SCAD-L2 and truncated L_1 penalty (TLP) for item parameters can be employed (Xu & Shang, 2018).

Normally distributed priors can be specified for item parameters (item intercepts and item slopes). Note that (for convenience) the prior specification holds simultaneously for all items.

### Usage

gdina(data, q.matrix, skillclasses=NULL, conv.crit=0.0001, dev.crit=.1,  maxit=1000,
delta.init=NULL, delta.fixed=NULL, delta.designmatrix=NULL,
delta.basispar.lower=NULL, delta.basispar.upper=NULL, delta.basispar.init=NULL,
zeroprob.skillclasses=NULL, attr.prob.init=NULL, reduced.skillspace=NULL,
reduced.skillspace.method=2, HOGDINA=-1, Z.skillspace=NULL,
weights=rep(1, nrow(data)), rule="GDINA", bugs=NULL, regular_lam=0,
regular_type="none", regular_alpha=NA, regular_tau=NA, regular_weights=NULL,
mono.constr=FALSE, prior_intercepts=NULL, prior_slopes=NULL, progress=TRUE,
progress.item=FALSE, mstep_iter=10, mstep_conv=1E-4, increment.factor=1.01,
fac.oldxsi=0, max.increment=.3, avoid.zeroprobs=FALSE, seed=0,
save.devmin=TRUE, calc.se=TRUE, se_version=1, PEM=TRUE, PEM_itermax=maxit,
cd=FALSE, cd_steps=1, mono_maxiter=10, freq_weights=FALSE, optimizer="CDM", ...)

## S3 method for class 'gdina'
summary(object, digits=4, file=NULL,  ...)

## S3 method for class 'gdina'

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


### Arguments

 data A required N \times J data matrix containing integer responses, 0, 1, ..., K. Polytomous item responses are treated by the sequential GDINA model. NA values are allowed. q.matrix A required integer J \times K matrix containing attributes not required or required, 0 or 1, to master the items in case of dichotomous attributes or integers in case of polytomous attributes. For polytomous item responses the Q-matrix must also include the item name and item category, see Example 11. skillclasses An optional matrix for determining the skill space. The argument can be used if a user wants less than 2^K skill classes.
 conv.crit Convergence criterion for maximum absolute change in item parameters dev.crit Convergence criterion for maximum absolute change in deviance maxit Maximum number of iterations linkfct A string which indicates the link function for the GDINA model. Options are "identity" (identity link), "logit" (logit link) and "log" (log link). The default is the "identity" link. Note that the link function is chosen for the whole model (i.e. for all items). Mj A list of design matrices and labels for each item. The definition of Mj follows the definition of M_j in de la Torre (2011). Please study the value Mj of the function in default analysis. See Example 3. group A vector of group identifiers for multiple group estimation. Default is NULL (no multiple group estimation). invariance Logical indicating whether invariance of item parameters is assumed for multiple group models. If a subset of items should be treated as noninvariant, then invariance can be a vector of item names. method Estimation method for item parameters (see) (de la Torre, 2011). The default "WLS" weights probabilities attribute classes by a weighting matrix W_j of expected frequencies, whereas the method "ULS" perform unweighted least squares estimation on expected frequencies. The method "ML" directly maximizes the log-likelihood function. The "ML" method is a bit slower but can be much more stable, especially in the case of the RRUM model. Only for the RRUM model, the default is changed to method="ML" if not specified otherwise. delta.init List with initial δ parameters delta.fixed List with fixed δ parameters. For free estimated parameters NA must be declared. delta.designmatrix A design matrix for restrictions on delta. See Example 4. delta.basispar.lower Lower bounds for delta basis parameters. delta.basispar.upper Upper bounds for delta basis parameters. delta.basispar.init An optional vector of starting values for the basis parameters of delta. This argument only applies when using a designmatrix for delta, i.e. delta.designmatrix is not NULL. zeroprob.skillclasses An optional vector of integers which indicates which skill classes should have zero probability. Default is NULL (no skill classes with zero probability). attr.prob.init Initial probabilities of skill distribution. reduced.skillspace A logical which indicates if the latent class skill space dimension should be reduced (see Xu & von Davier, 2008). The default is NULL which applies skill space reduction for more than four skills. The dimensional reduction is only well defined for more than three skills. If the argument zeroprob.skillclasses is not NULL, then reduced.skillspace is set to FALSE. reduced.skillspace.method Computation method for skill space reduction in case of reduced.skillspace=TRUE. The default is 2 which is computationally more efficient but introduced in CDM 2.6. For reasons of compatibility of former CDM versions (≤ 2.5), reduced.skillspace.method=1 uses the older implemented method. In case of non-convergence with the new method, please try the older method. HOGDINA Values of -1, 0 or 1 indicating if a higher order GDINA model (see Details) should be estimated. The default value of -1 corresponds to the case that no higher order factor is assumed to exist. A value of 0 corresponds to independent attributes. A value of 1 assumes the existence of a higher order factor. Z.skillspace A user specified design matrix for the skill space reduction as described in Xu and von Davier (2008). See in the Examples section for applications. See Example 6. weights An optional vector of sample weights. rule A string or a vector of itemwise condensation rules. Allowed entries are GDINA, DINA, DINO, ACDM (additive cognitive diagnostic model) and RRUM (reduced reparametrized unified model, RRUM, see Details). The rule GDINA1 applies only main effects in the GDINA model which is equivalent to ACDM. The rule GDINA2 applies to all main effects and second-order interactions of the attributes. If some item is specified as RRUM, then for all the items the reduced RUM will be estimated which means that the log link function and the ACDM condensation rule is used. In the output, the entry rrum.params contains the parameters transformed in the RUM parametrization. If rule is a string, the condensation rule applies to all items. If rule is a vector, condensation rules can be specified itemwise. The default is GDINA for all items. bugs Character vector indicating which columns in the Q-matrix refer to bugs (misconceptions). This is only available if some rule is set to "SISM". Note that bugs must be included as last columns in the Q-matrix. regular_lam Regularization parameter λ regular_type Type of regularization. Can be scad (SCAD penalty), lasso (lasso penalty), ridge (ridge penalty), elnet (elastic net), scadL2 (SCAD-L_2; Zeng & Xie, 2014), tlp (truncated L_1 penalty; Xu & Shang, 2018; Shen, Pan, & Zhu, 2012), mcp (MCP penalty; Zhang, 2010) or none (no regularization). regular_alpha Regularization parameter α (applicable for elastic net or SCAD-L2. regular_tau Regularization parameter τ for truncated L_1 penalty. regular_weights Optional list of item parameter weights used for penalties in regularized estimation (see Example 13) mono.constr Logical indicating whether monotonicity constraints should be fulfilled in estimation (implemented by the increasing penalty method; see Nash, 2014, p. 156). prior_intercepts Vector with mean and standard deviation for prior of random intercepts (applies to all items) prior_slopes Vector with mean and standard deviation for prior of random slopes (applies to all items and all parameters) progress An optional logical indicating whether the function should print the progress of iteration in the estimation process. progress.item An optional logical indicating whether item wise progress should be displayed mstep_iter Number of iterations in M-step if method="ML". mstep_conv Convergence criterion in M-step if method="ML". increment.factor A factor larger than 1 (say 1.1) to control maximum increments in item parameters. This parameter can be used in case of nonconvergence. fac.oldxsi A convergence acceleration factor between 0 and 1 which defines the weight of previously estimated values in current parameter updates. max.increment Maximum size of change in increments in M steps of EM algorithm when method="ML" is used. avoid.zeroprobs An optional logical indicating whether for estimating item parameters probabilities occur. Especially if not a skill classes are used, it is recommended to switch the argument to TRUE. seed Simulation seed for initial parameters. A value of zero corresponds to deterministic starting values, an integer value different from zero to random initial values with set.seed(seed). save.devmin An optional logical indicating whether intermediate estimates should be saved corresponding to minimal deviance. Setting the argument to FALSE could help for preventing working memory overflow. calc.se Optional logical indicating whether standard errors should be calculated. se_version Integer for calculation method of standard errors. se_version=1 is based on the observed log likelihood and included since CDM 5.1 and is the default. Comparability with previous CDM versions can be obtained with se_version=0. PEM Logical indicating whether the P-EM acceleration should be applied (Berlinet & Roland, 2012). PEM_itermax Number of iterations in which the P-EM method should be applied. cd Logical indicating whether coordinate descent algorithm should be used. cd_steps Number of steps for each parameter in coordinate descent algorithm mono_maxiter Maximum number of iterations for fulfilling the monotonicity constraint freq_weights Logical indicating whether frequency weights should be used. Default is FALSE. optimizer String indicating which optimizer should be used in M-step estimation in case of method="ML". The internal optimizer of CDM can be requested by optimizer="CDM". The optimization with stats::optim can be requested by optimizer="optim". For the RRUM model, it is always chosen optimizer="optim". object A required object of class gdina, obtained from a call to the function gdina. digits Number of digits after decimal separator to display. file Optional file name for a file in which summary should be sinked. x A required object of class gdina ask A logical indicating whether every separate item should be displayed in plot.gdina ... Optional parameters to be passed to or from other methods will be ignored.

### Details

The estimation is based on an EM algorithm as described in de la Torre (2011). Item parameters are contained in the delta vector which is a list where the jth entry corresponds to item parameters of the jth item.

The following description refers to the case of dichotomous attributes. For using polytomous attributes see Chen and de la Torre (2013) and Example 7 for a definition of the Q-matrix. In this case, Q_{ik}=l means that the ith item requires the mastery (at least) of level l of attribute k.

Assume that two skills α_1 and α_2 are required for mastering item j. Then the GDINA model can be written as

g [ P( X_{nj}=1 | α_n ) ]=δ_{j0} + δ_{j1} α_{n1} + δ_{j2} α_{n2} + δ_{j12} α_{n1} α_{n2}

which is a two-way GDINA-model (the rule="GDINA2" specification) with a link function g (which can be the identity, logit or logarithmic link). If the specification ACDM is chosen, then δ_{j12}=0. The DINA model (rule="DINA") assumes δ_{j1}=δ_{j2}=0.

For the reduced RUM model (rule="RRUM"), the item response model is

P(X_{nj}=1 | α_n )=π_i^\ast \cdot r_{i1}^{1-α_{i1} } \cdot r_{i2}^{1-α_{i2} }

From this equation, it is obvious, that this model is equivalent to an additive model (rule="ACDM") with a logarithmic link function (linkfct="log").

If a reduced skillspace (reduced.skillspace=TRUE) is employed, then the logarithm of probability distribution of the attributes is modeled as a log-linear model:

\log P[ ( α_{n1}, α_{n2}, …, α_{nK} ) ] =γ_0 + ∑_k γ_k α_{nk} + ∑_{k < l} γ_{kl} α_{nk} α_{nl}

If a higher order DINA model is assumed (HOGDINA=1), then a higher order factor θ_n for the attributes is assumed:

P( α_{nk}=1 | θ_n )=Φ ( a_k θ_n + b_k )

For HOGDINA=0, all attributes α_{nk} are assumed to be independent of each other:

P[ ( α_{n1}, α_{n2}, …, α_{nK} ) ] =∏_k P( α_{nk} )

Note that the noncompensatory reduced RUM (NC-RRUM) according to Rupp and Templin (2008) is the GDINA model with the arguments rule="ACDM" and linkfct="log". NC-RRUM can also be obtained by choosing rule="RRUM".

The compensatory RUM (C-RRUM) can be obtained by using the arguments rule="ACDM" and linkfct="logit".

The cognitive diagnosis model for identifying skills and misconceptions (SISM; Kuo, Chen & de la Torre, 2018) can be estimated with rule="SISM" (see Example 12).

The gdina function internally parameterizes the GDINA model as

g [ P( X_{nj}=1 | α_n ) ]=\boldmath{M}_j ( α _n ) \boldmath{δ}_j

with item-specific design matrices \boldmath{M}_j (α _n ) and item parameters \boldmath{δ}_j. Only those attributes are modelled which correspond to non-zero entries in the Q-matrix. Because the Q-matrix (in q.matrix) and the design matrices (in M_j; see Example 3) can be specified by the user, several cognitive diagnosis models can be estimated. Therefore, some additional extensions of the DINA model can also be estimated using the gdina function. These models include the DINA model with multiple strategies (Huo & de la Torre, 2014)

### Value

An object of class gdina with following entries

 coef Data frame of item parameters delta List with basis item parameters se.delta Standard errors of basis item parameters probitem Data frame with model implied conditional item probabilities P(X_i=1 | \bold{α}). These probabilities are displayed in plot.gdina. itemfit.rmsea The RMSEA item fit index (see itemfit.rmsea). mean.rmsea Mean of RMSEA item fit indexes. loglike Log-likelihood deviance Deviance G Number of groups N Sample size AIC AIC BIC BIC CAIC CAIC Npars Total number of parameters Nipar Number of item parameters Nskillpar Number of parameters for skill class distribution Nskillclasses Number of skill classes varmat.delta Covariance matrix of δ item parameters posterior Individual posterior distribution like Individual likelihood data Original data q.matrix Used Q-matrix pattern Individual patterns, individual MLE and MAP classifications and their corresponding probabilities attribute.patt Probabilities of skill classes skill.patt Marginal skill probabilities subj.pattern Individual subject pattern attribute.patt.splitted Splitted attribute pattern pjk Array of item response probabilities Mj Design matrix M_j in GDINA algorithm (see de la Torre, 2011) Aj Design matrix A_j in GDINA algorithm (see de la Torre, 2011) rule Used condensation rules linkfct Used link function delta.designmatrix Designmatrix for item parameters reduced.skillspace A logical if skillspace reduction was performed Z.skillspace Design matrix for skillspace reduction beta Parameters δ for skill class representation covbeta Standard errors of δ parameters iter Number of iterations rrum.params Parameters in the parametrization of the reduced RUM model if rule="RRUM". group.stat Group statistics (sample sizes, group labels) HOGDINA The used value of HOGDINA mono.constr Monotonicity constraint regularization Logical indicating whether regularization is used regular_lam Regularization parameter numb_bound_mono Number of items with parameters at boundary of monotonicity constraints numb_regular_pars Number of regularized item parameters delta_regularized List indicating which item parameters are regularized cd_algorithm Logical indicating whether coordinate descent algorithm is used cd_steps Number of steps for each parameter in coordinate descent algorithm seed Used simulation seed a.attr Attribute parameters a_k in case of HOGDINA>=0 b.attr Attribute parameters b_k in case of HOGDINA>=0 attr.rf Attribute response functions. This matrix contains all a_k and b_k parameters converged Logical indicating whether convergence was achieved. control Optimization parameters used in estimation partable Parameter table for gdina function polychor Group-wise matrices with polychoric correlations sequential Logical indicating whether a sequential GDINA model is applied for polytomous item responses ... Further values

### Note

The function din does not allow for multiple group estimation. Use this gdina function instead and choose the appropriate rule="DINA" as an argument.

Standard error calculation in analyses which use sample weights or designmatrix for delta parameters (delta.designmatrix!=NULL) is not yet correctly implemented. Please use replication methods instead.

### References

Berlinet, A. F., & Roland, C. (2012). Acceleration of the EM algorithm: P-EM versus epsilon algorithm. Computational Statistics & Data Analysis, 56(12), 4122-4137.

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. (2018). Introducing the general polytomous diagnosis modeling framework. Frontiers in Psychology | Quantitative Psychology and Measurement, 9(1474).

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

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

Hong, C. Y., Chang, Y. W., & Tsai, R. C. (2016). Estimation of generalized DINA model with order restrictions. Journal of Classification, 33(3), 460-484.

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.

Kuo, B.-C., Chen, C.-H., & de la Torre, J. (2018). A cognitive diagnosis model for identifying coexisting skills and misconceptions. Applied Psychological Measurement, 42(3), 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(3), 253-275.

Nash, J. C. (2014). Nonlinear parameter optimization using R tools. West Sussex: Wiley.

Rupp, A. A., & Templin, J. (2008). Unique characteristics of diagnostic classification models: A comprehensive review of the current state-of-the-art. Measurement: Interdisciplinary Research and Perspectives, 6, 219-262.

Shen, X., Pan, W., & Zhu, Y. (2012). Likelihood-based selection and sharp parameter estimation. Journal of the American Statistical Association, 107, 223-232.

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

Xu, G., & Shang, Z. (2018). Identifying latent structures in restricted latent class models. Journal of the American Statistical Association, 523, 1284-1295.

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

Zeng, L., & Xie, J. (2014). Group variable selection via SCAD-L_2. Statistics, 48, 49-66.

Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. Annals of Statistics, 38, 894-942.

See also the din function (for DINA and DINO estimation).

For assessment of model fit see modelfit.cor.din and anova.gdina.

See itemfit.sx2 for item fit statistics.

See sim.gdina for simulating the GDINA model.

See gdina.wald for a Wald test for testing the DINA and ACDM rules at the item-level.

See gdina.dif for assessing differential item functioning.

See discrim.index for computing discrimination indices.

See the GDINA::GDINA function in the GDINA package for similar functionality.

### Examples

#############################################################################
# EXAMPLE 1: Simulated DINA data | different condensation rules
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

dat <- sim.dina
Q <- sim.qmatrix

#***
# Model 1: estimation of the GDINA model (identity link)
mod1 <- CDM::gdina( data=dat,  q.matrix=Q)
summary(mod1)
plot(mod1) # apply plot function

## Not run:
# Model 1a: estimate model with different simulation seed
mod1a <- CDM::gdina( data=dat,  q.matrix=Q, seed=9089)
summary(mod1a)

# Model 1b: estimate model with some fixed delta parameters
delta.fixed <- as.list( rep(NA,9) )        # List for parameters of 9 items
delta.fixed[[2]] <- c( 0, .15, .15, .45 )
delta.fixed[[6]] <- c( .25, .25 )
mod1b <- CDM::gdina( data=dat,  q.matrix=Q, delta.fixed=delta.fixed)
summary(mod1b)

# Model 1c: fix all delta parameters to previously fitted model
mod1c <- CDM::gdina( data=dat,  q.matrix=Q, delta.fixed=mod1$delta) summary(mod1c) # Model 1d: estimate GDINA model with GDINA package mod1d <- GDINA::GDINA( dat=dat, Q=Q, model="GDINA" ) summary(mod1d) # extract item parameters GDINA::itemparm(mod1d) GDINA::itemparm(mod1d, what="delta") # compare likelihood logLik(mod1) logLik(mod1d) #*** # Model 2: estimation of the DINA model with gdina function mod2 <- CDM::gdina( data=dat, q.matrix=Q, rule="DINA") summary(mod2) plot(mod2) #*** # Model 2b: compare results with din function mod2b <- CDM::din( data=dat, q.matrix=Q, rule="DINA") summary(mod2b) # Model 2: estimation of the DINO model with gdina function mod3 <- CDM::gdina( data=dat, q.matrix=Q, rule="DINO") summary(mod3) #*** # Model 4: DINA model with logit link mod4 <- CDM::gdina( data=dat, q.matrix=Q, rule="DINA", linkfct="logit" ) summary(mod4) #*** # Model 5: DINA model log link mod5 <- CDM::gdina( data=dat, q.matrix=Q, rule="DINA", linkfct="log") summary(mod5) #*** # Model 6: RRUM model mod6 <- CDM::gdina( data=dat, q.matrix=Q, rule="RRUM") summary(mod6) #*** # Model 7: Higher order GDINA model mod7 <- CDM::gdina( data=dat, q.matrix=Q, HOGDINA=1) summary(mod7) #*** # Model 8: GDINA model with independent attributes mod8 <- CDM::gdina( data=dat, q.matrix=Q, HOGDINA=0) summary(mod8) #*** # Model 9: Estimating the GDINA model with monotonicity constraints mod9 <- CDM::gdina( data=dat, q.matrix=Q, rule="GDINA", mono.constr=TRUE, linkfct="logit") summary(mod9) #*** # Model 10: Estimating the ACDM model with SCAD penalty and regularization # parameter of .05 mod10 <- CDM::gdina( data=dat, q.matrix=Q, rule="ACDM", linkfct="logit", regular_type="scad", regular_lam=.05 ) summary(mod10) #*** # Model 11: Estimation of GDINA model with prior distributions # N(0,10^2) prior for item intercepts prior_intercepts <- c(0,10) # N(0,1^2) prior for item slopes prior_slopes <- c(0,1) # estimate model mod11 <- CDM::gdina( data=dat, q.matrix=Q, rule="GDINA", prior_intercepts=prior_intercepts, prior_slopes=prior_slopes) summary(mod11) ############################################################################# # EXAMPLE 2: Simulated DINO data # additive cognitive diagnosis model with different link functions ############################################################################# data(sim.dino, package="CDM") data(sim.matrix, package="CDM") dat <- sim.dino Q <- sim.qmatrix #*** # Model 1: additive cognitive diagnosis model (ACDM; identity link) mod1 <- CDM::gdina( data=dat, q.matrix=Q, rule="ACDM") summary(mod1) #*** # Model 2: ACDM logit link mod2 <- CDM::gdina( data=dat, q.matrix=Q, rule="ACDM", linkfct="logit") summary(mod2) #*** # Model 3: ACDM log link mod3 <- CDM::gdina( data=dat, q.matrix=Q, rule="ACDM", linkfct="log") summary(mod3) #*** # Model 4: Different condensation rules per item I <- 9 # number of items rule <- rep( "GDINA", I ) rule[1] <- "DINO" # 1st item: DINO model rule[7] <- "GDINA2" # 7th item: GDINA model with first- and second-order interactions rule[8] <- "ACDM" # 8ht item: additive CDM rule[9] <- "DINA" # 9th item: DINA model mod4 <- CDM::gdina( data=dat, q.matrix=Q, rule=rule ) summary(mod4) ############################################################################# # EXAMPLE 3: Model with user-specified design matrices ############################################################################# data(sim.dino, package="CDM") data(sim.qmatrix, package="CDM") dat <- sim.dino Q <- sim.qmatrix # do a preliminary analysis and modify obtained design matrices mod0 <- CDM::gdina( data=dat, q.matrix=Q, maxit=1) # extract default design matrices Mj <- mod0$Mj
Mj.user <- Mj   # these user defined design matrices are modified.
#~~~  For the second item, the following model should hold
#     X1 ~ V2 + V2*V3
mj <- Mj[[2]][[1]]
mj.lab <- Mj[[2]][[2]]
mj <- mj[,-3]
mj.lab <- mj.lab[-3]
Mj.user[[2]] <- list( mj, mj.lab )
#    [[1]]
#        [,1] [,2] [,3]
#    [1,]    1    0    0
#    [2,]    1    1    0
#    [3,]    1    0    0
#    [4,]    1    1    1
#    [[2]]
#    [1] "0"   "1"   "1-2"
#~~~  For the eight item an equality constraint should hold
#     X8 ~ a*V2 + a*V3 + V2*V3
mj <- Mj[[8]][[1]]
mj.lab <- Mj[[8]][[2]]
mj[,2] <- mj[,2] + mj[,3]
mj <- mj[,-3]
mj.lab <- c("0", "1=2", "1-2" )
Mj.user[[8]] <- list( mj, mj.lab )
Mj.user[[8]]
##   [[1]]
##        [,1] [,2] [,3]
##   [1,]    1    0    0
##   [2,]    1    1    0
##   [3,]    1    1    0
##   [4,]    1    2    1
##
##   [[2]]
##   [1] "0"   "1=2" "1-2"
mod <- CDM::gdina( data=dat,  q.matrix=Q,
Mj=Mj.user,  maxit=200 )
summary(mod)

#############################################################################
# EXAMPLE 4: Design matrix for delta parameters
#############################################################################

data(sim.dino, package="CDM")
data(sim.qmatrix, package="CDM")

#~~~ estimate an initial model
mod0 <- CDM::gdina( data=dat,  q.matrix=Q, rule="ACDM", maxit=1)
# extract coefficients
c0 <- mod0$coef I <- 9 # number of items delta.designmatrix <- matrix( 0, nrow=nrow(c0), ncol=nrow(c0) ) diag( delta.designmatrix) <- 1 # set intercept of item 1 and item 3 equal to each other delta.designmatrix[ 7, 1 ] <- 1 ; delta.designmatrix[,7] <- 0 # set loading of V1 of item1 and item 3 equal delta.designmatrix[ 8, 2 ] <- 1 ; delta.designmatrix[,8] <- 0 delta.designmatrix <- delta.designmatrix[, -c(7:8) ] # exclude original parameters with indices 7 and 8 #*** # Model 1: ACDM with designmatrix mod1 <- CDM::gdina( data=dat, q.matrix=Q, rule="ACDM", delta.designmatrix=delta.designmatrix ) summary(mod1) #*** # Model 2: Same model, but with logit link instead of identity link function mod2 <- CDM::gdina( data=dat, q.matrix=Q, rule="ACDM", delta.designmatrix=delta.designmatrix, linkfct="logit") summary(mod2) ############################################################################# # EXAMPLE 5: Multiple group estimation ############################################################################# # simulate data set.seed(9279) N1 <- 200 ; N2 <- 100 # group sizes I <- 10 # number of items q.matrix <- matrix(0,I,2) # create Q-matrix q.matrix[1:7,1] <- 1 ; q.matrix[ 5:10,2] <- 1 # simulate first group dat1 <- CDM::sim.din(N1, q.matrix=q.matrix, mean=c(0,0) )$dat
# simulate second group
dat2 <- CDM::sim.din(N2, q.matrix=q.matrix, mean=c(-.3, -.7) )$dat # merge data dat <- rbind( dat1, dat2 ) # group indicator group <- c( rep(1,N1), rep(2,N2) ) # estimate GDINA model with multiple groups assuming invariant item parameters mod1 <- CDM::gdina( data=dat, q.matrix=q.matrix, group=group) summary(mod1) # estimate DINA model with multiple groups assuming invariant item parameters mod2 <- CDM::gdina( data=dat, q.matrix=q.matrix, group=group, rule="DINA") summary(mod2) # estimate GDINA model with noninvariant item parameters mod3 <- CDM::gdina( data=dat, q.matrix=q.matrix, group=group, invariance=FALSE) summary(mod3) # estimate GDINA model with some invariant item parameters (I001, I006, I008) mod4 <- CDM::gdina( data=dat, q.matrix=q.matrix, group=group, invariance=c("I001", "I006","I008") ) #--- model comparison IRT.compareModels(mod1,mod2,mod3,mod4) # estimate GDINA model with non-invariant item parameters except for the # items I001, I006, I008 mod5 <- CDM::gdina( data=dat, q.matrix=q.matrix, group=group, invariance=setdiff( colnames(dat), c("I001", "I006","I008") ) ) ############################################################################# # EXAMPLE 6: User specified reduced skill space ############################################################################# # Some correlations between attributes should be set to zero. q.matrix <- expand.grid( c(0,1), c(0,1), c(0,1), c(0,1) ) colnames(q.matrix) <- colnames( paste("Attr", 1:4,sep="")) q.matrix <- q.matrix[ -1, ] Sigma <- matrix( .5, nrow=4, ncol=4 ) diag(Sigma) <- 1 Sigma[3,2] <- Sigma[2,3] <- 0 # set correlation of attribute A2 and A3 to zero dat <- CDM::sim.din( N=1000, q.matrix=q.matrix, Sigma=Sigma)$dat

#~~~ Step 1: initial estimation
mod1a <- CDM::gdina( data=dat, q.matrix=q.matrix, maxit=1, rule="DINA")
# estimate also "full" model
mod1 <- CDM::gdina( data=dat, q.matrix=q.matrix, rule="DINA")

#~~~ Step 2: modify designmatrix for reduced skillspace
Z.skillspace <- data.frame( mod1a$Z.skillspace ) # set correlations of A2/A4 and A3/A4 to zero vars <- c("A2_A3","A2_A4") for (vv in vars){ Z.skillspace[,vv] <- NULL } #~~~ Step 3: estimate model with reduced skillspace mod2 <- CDM::gdina( data=dat, q.matrix=q.matrix, Z.skillspace=Z.skillspace, rule="DINA") #~~~ eliminate all covariances Z.skillspace <- data.frame( mod1$Z.skillspace )
colnames(Z.skillspace)
Z.skillspace <- Z.skillspace[, -grep( "_", colnames(Z.skillspace),fixed=TRUE)]
colnames(Z.skillspace)

mod3 <- CDM::gdina( data=dat, q.matrix=q.matrix,
Z.skillspace=Z.skillspace, rule="DINA")
summary(mod1)
summary(mod2)
summary(mod3)

#############################################################################
# EXAMPLE 7: Polytomous GDINA model (Chen & de la Torre, 2013)
#############################################################################

data(data.pgdina, package="CDM")

dat <- data.pgdina$dat q.matrix <- data.pgdina$q.matrix

# pGDINA model with "DINA rule"
mod1 <- CDM::gdina( dat, q.matrix=q.matrix, rule="DINA")
summary(mod1)
# no reduced skill space
mod1a <- CDM::gdina( dat, q.matrix=q.matrix, rule="DINA",reduced.skillspace=FALSE)
summary(mod1)

# pGDINA model with "GDINA rule"
mod2 <- CDM::gdina( dat, q.matrix=q.matrix, rule="GDINA")
summary(mod2)

#############################################################################
# EXAMPLE 8: Fraction subtraction data: DINA and HO-DINA model
#############################################################################

data(fraction.subtraction.data, package="CDM")
data(fraction.subtraction.qmatrix, package="CDM")

dat <- fraction.subtraction.data
Q <- fraction.subtraction.qmatrix

# Model 1: DINA model
mod1 <- CDM::gdina( dat, q.matrix=Q, rule="DINA")
summary(mod1)

# Model 2: HO-DINA model
mod2 <- CDM::gdina( dat, q.matrix=Q, HOGDINA=1, rule="DINA")
summary(mod2)

#############################################################################
# EXAMPLE 9: Skill space approximation data.jang
#############################################################################

data(data.jang, package="CDM")

data <- data.jang$data q.matrix <- data.jang$q.matrix

#*** Model 1: Reduced RUM model
mod1 <- CDM::gdina( data, q.matrix, rule="RRUM", conv.crit=.001, maxit=500 )

#*** Model 2: Reduced RUM model with skill space approximation
# use 300 instead of 2^9=512 skill classes
skillspace <- CDM::skillspace.approximation( L=300, K=ncol(q.matrix) )
mod2 <- CDM::gdina( data, q.matrix, rule="RRUM", conv.crit=.001,
skillclasses=skillspace )
##   > logLik(mod1)
##   'log Lik.' -30318.08 (df=153)
##   > logLik(mod2)
##   'log Lik.' -30326.52 (df=153)

#############################################################################
# EXAMPLE 10: CDM with a linear hierarchy
#############################################################################
# This model is equivalent to a unidimensional IRT model with an ordered
# ordinal latent trait and is actually a probabilistic Guttman model.
set.seed(789)

# define 3 competency levels
alpha <- scan()
0 0 0   1 0 0   1 1 0   1 1 1

# define skill class distribution
K <- 3
skillspace <- alpha <- matrix( alpha, K + 1,  K, byrow=TRUE )
alpha <- alpha[ rep(  1:4,  c(300,300,200,200) ), ]
# P(000)=P(100)=.3, P(110)=P(111)=.2
# define Q-matrix
Q <- scan()
1 0 0   1 1 0   1 1 1

Q <- matrix( Q, nrow=K,  ncol=K, byrow=TRUE )
Q <- Q[ rep(1:K, each=4 ), ]
colnames(skillspace) <- colnames(Q) <- paste0("A",1:K)
I <- nrow(Q)

# define guessing and slipping parameters
guess <- stats::runif( I, 0, .3 )
slip <- stats::runif( I, 0, .2 )
# simulate data
dat <- CDM::sim.din( q.matrix=Q, alpha=alpha, slip=slip, guess=guess )$dat #*** Model 1: DINA model with linear hierarchy mod1 <- CDM::din( dat, q.matrix=Q, rule="DINA", skillclasses=skillspace ) summary(mod1) #*** Model 2: pGDINA model with 3 levels # The multidimensional CDM with a linear hierarchy is a unidimensional # polytomous GDINA model. Q2 <- matrix( rowSums(Q), nrow=I, ncol=1 ) mod2 <- CDM::gdina( dat, q.matrix=Q2, rule="DINA" ) summary(mod2) #*** Model 3: estimate probabilistic Guttman model in sirt # Proctor, C. H. (1970). A probabilistic formulation and statistical # analysis for Guttman scaling. Psychometrika, 35, 73-78. library(sirt) mod3 <- sirt::prob.guttman( dat, itemlevel=Q2[,1] ) summary(mod3) # -> The three models result in nearly equivalent fit. ############################################################################# # EXAMPLE 11: Sequential GDINA model (Ma & de la Torre, 2016) ############################################################################# data(data.cdm04, package="CDM") #** attach dataset dat <- data.cdm04$data    # polytomous item responses
q.matrix1 <- data.cdm04$q.matrix1 q.matrix2 <- data.cdm04$q.matrix2

#-- DINA model with first Q-matrix
mod1 <- CDM::gdina( dat, q.matrix=q.matrix1, rule="DINA")
summary(mod1)
#-- DINA model with second Q-matrix
mod2 <- CDM::gdina( dat, q.matrix=q.matrix2, rule="DINA")
#-- GDINA model
mod3 <- CDM::gdina( dat, q.matrix=q.matrix2, rule="GDINA")

#** model comparison
IRT.compareModels(mod1,mod2,mod3)

#############################################################################
# EXAMPLE 12: Simulataneous modeling of skills and misconceptions (Kuo et al., 2018)
#############################################################################

data(data.cdm08, package="CDM")
dat <- data.cdm08$data q.matrix <- data.cdm08$q.matrix

#*** estimate model
mod <- CDM::gdina( dat0, q.matrix, rule="SISM", bugs=colnames(q.matrix)[5:7] )
summary(mod)

#############################################################################
# EXAMPLE 13: Regularized estimation in GDINA model data.dtmr
#############################################################################

data(data.dtmr, package="CDM")
dat <- data.dtmr$data q.matrix <- data.dtmr$q.matrix

#***** LASSO regularization with lambda parameter of .02
mod1 <- CDM::gdina(dat, q.matrix=q.matrix, rule="GDINA", regular_lam=.02,
regular_type="lasso")
summary(mod1)
mod$delta_regularized #***** using starting values from previuos estimation delta.init <- mod1$delta
attr.prob.init <- mod1$attr.prob mod2 <- CDM::gdina(dat, q.matrix=q.matrix, rule="GDINA", regular_lam=.02, regular_type="lasso", delta.init=delta.init, attr.prob.init=attr.prob.init) summary(mod2) #***** final estimation fixing regularized estimates to zero and estimate all other #***** item parameters unregularized regular_weights <- mod2$delta_regularized
delta.init <- mod2$delta attr.prob.init <- mod2$attr.prob

mod3 <- CDM::gdina(dat, q.matrix=q.matrix, rule="GDINA", regular_lam=1E5, regular_type="lasso",
delta.init=delta.init, attr.prob.init=attr.prob.init,
regular_weights=regular_weights)
summary(mod3)

## End(Not run)


[Package CDM version 7.5-15 Index]