gdina: 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,
    linkfct="identity", Mj=NULL, group=NULL, invariance=TRUE,method=NULL,
    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 gdina
summary(object, digits=4, file=NULL,  …)
# S3 method for gdina
plot(x, ask=FALSE,  …)
# S3 method for 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).

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 ( $\leq$ 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.

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.

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.

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

Number of groups

Sample size

AIC

BIC

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

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

Design matrix $M_{j}$ in GDINA algorithm (see de la Torre, 2011)

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

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 $j$ th entry corresponds to item parameters of the $j$ th 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, $Q_{i k} = l$ means that the $i$ th 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 $g [P (X_{n j} = 1 | α_{n})] = δ_{j 0} + δ_{j 1} α_{n 1} + δ_{j 2} α_{n 2} + δ_{j 12} α_{n 1} α_{n 2}$ 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 $δ_{j 12} = 0$ . The DINA model (rule="DINA") assumes $δ_{j 1} = δ_{j 2} = 0$ .

For the reduced RUM model (rule="RRUM"), the item response model is $P (X_{n j} = 1 | α_{n}) = π_{i}^{*} \cdot r_{i 1}^{1 - α_{i 1}} \cdot r_{i 2}^{1 - α_{i 2}}$ 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: $\log P [(α_{n 1}, α_{n 2}, \dots, α_{n K})] = γ_{0} + \sum_{k} γ_{k} α_{n k} + \sum_{k < l} γ_{k l} α_{n k} α_{n l}$

If a higher order DINA model is assumed (HOGDINA=1), then a higher order factor $θ_{n}$ for the attributes is assumed: $P (α_{n k} = 1 | θ_{n}) = Φ (a_{k} θ_{n} + b_{k})$

For HOGDINA=0, all attributes $α_{n k}$ are assumed to be independent of each other: $P [(α_{n 1}, α_{n 2}, \dots, α_{n K})] = \prod_{k} P (α_{n k})$

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 $g [P (X_{n j} = 1 | α_{n})] = \boldmath M_{j} (α_{n}) \boldmath δ_{j}$ 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)

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.

Examples

Run this code

# NOT RUN {
#############################################################################
# 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)
# }

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