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, attr.prob.fixed=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,  ...)

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 $P(X_i=1 | \bm{\alpha})$. 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 $\delta$ 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 $\delta$ for skill class representation
covbeta: Standard errors of $\delta$ 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

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 $\delta$ parameters
delta.fixed: List with fixed $\delta$ 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.
attr.prob.fixed: Vector or matrix with fixed 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 ($\le$ 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 $\lambda$
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 $\alpha$ (applicable for elastic net or SCAD-L2.
regular_tau: Regularization parameter $\tau$ 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 $j$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, $Q_{ik}=l$ means that the $i$th item requires the mastery (at least) of level $l$ of attribute $k$.

Assume that two skills $\alpha_1$ and $\alpha_2$ are required for mastering item $j$. Then the GDINA model can be written as $$ g [ P( X_{nj}=1 | \alpha_n ) ]=\delta_{j0} + \delta_{j1} \alpha_{n1} + \delta_{j2} \alpha_{n2} + \delta_{j12} \alpha_{n1} \alpha_{n2} $$ 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 $\delta_{j12}=0$. The DINA model (rule="DINA") assumes $ \delta_{j1}=\delta_{j2}=0$.

For the reduced RUM model (rule="RRUM"), the item response model is $$P(X_{nj}=1 | \alpha_n )=\pi_i^\ast \cdot r_{i1}^{1-\alpha_{i1} } \cdot r_{i2}^{1-\alpha_{i2} } $$ 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: $$ \log P[ ( \alpha_{n1}, \alpha_{n2}, \ldots, \alpha_{nK} ) ] =\gamma_0 + \sum_k \gamma_k \alpha_{nk} + \sum_{k < l} \gamma_{kl} \alpha_{nk} \alpha_{nl} $$

If a higher order DINA model is assumed (HOGDINA=1), then a higher order factor $\theta_n$ for the attributes is assumed: $$P( \alpha_{nk}=1 | \theta_n )=\Phi ( a_k \theta_n + b_k ) $$

For HOGDINA=0, all attributes $\alpha_{nk}$ are assumed to be independent of each other: $$ P[ ( \alpha_{n1}, \alpha_{n2}, \ldots, \alpha_{nK} ) ] =\prod_k P( \alpha_{nk} ) $$

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 $$ g [ P( X_{nj}=1 | \alpha_n ) ]=\bm{M}_j ( \alpha _n ) \bm{\delta}_j $$ with item-specific design matrices $\bm{M}_j (\alpha _n ) $ and item parameters $\bm{\delta}_j$. 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.