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

simGDINA(N, Q, gs.parm = NULL, delta.parm = NULL,
catprob.parm = NULL, model = "GDINA", sequential = FALSE,
gs.args = list(type = "random", mono.constraint = TRUE),
delta.args = list(design.matrix = NULL, linkfunc = NULL),
attribute = NULL, att.dist = "uniform", item.names = NULL,
higher.order.parm = list(theta = NULL, lambda = NULL),
mvnorm.parm = list(mean = NULL, sigma = NULL, cutoffs = NULL),
att.prior = NULL, digits = 4)

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

## Arguments

N Sample size. A required matrix; The number of rows occupied by a single-strategy dichotomous item is 1, by a polytomous item is the number of nonzero categories, and by a mutiple-strategy dichotomous item is the number of strategies. The number of column is equal to the number of attributes if all items are single-strategy dichotomous items, but the number of attributes + 2 if any items are polytomous or have multiple strategies. For a polytomous item, the first column represents the item number and the second column indicates the nonzero category number. For a multiple-strategy dichotomous item, the first column represents the item number and the second column indicates the strategy number. For binary attributes, 1 denotes the attributes are measured by the items and 0 means the attributes are not measured. For polytomous attributes, non-zero elements indicate which level of attributes are needed. See Examples. A matrix or data frame for guessing and slip parameters. The number of rows occupied by a dichotomous item is 1, and by a polytomous item is the number of nonzero categories. The number of columns must be 2, where the first column represents the guessing parameters (or $$P(0)$$), and the second column represents slip parameters (or $$1-P(1)$$). This may need to be used in conjunction with the argument gs.args. A list of delta parameters of each latent group for each item or category. This may need to be used in conjunction with the argument delta.args. A list of success probabilities of each latent group for each non-zero category of each item. See Examples and Details for more information. A character vector for each item or nonzero category, or a scalar which will be used for all items or nonzero categories to specify the CDMs. The possible options include "GDINA","DINA","DINO","ACDM","LLM", "RRUM", "MSDINA" and "UDF". When "UDF", indicating user defined function, is specified for any item, delta.parm must be specified, as well as options design.matrix and linkfunc in argument delta.args. logical; TRUE if the sequential model is used for polytomous responses simulation, and FALSE if there is no polytomously scored items. a list of options when gs.parm is specified. It consists of two components: type How are the delta parameters for ACDM, LLM, RRUM generated? It can be either "random" or "equal". "random" means the delta parameters are simulated randomly, while "equal" means that each required attribute contributes equally to the probability of success (P), logit(P) or log(P) for ACDM, LLM and RRUM, respectively. See Details for more information. mono.constraint A vector for each item/category or a scalar which will be used for all items/categories to specify whether monotonicity constraints should be satisfied if the generating model is the G-DINA model. Note that this is applicable only for the G-DINA model when gs.parm is used. For ACDM, LLM and RRUM, monotonicity constraints are always satisfied and therefore this argument is ignored. a list of options when delta.parm is specified. It consists of two components: linkfunc a vector of link functions for each item/category; It can be "identity","log" or "logit". Only necessary when, for some items, model="UDF". design.matrix a list of design matrices; Its length must be equal to the number of items (or nonzero categories for sequential models). If CDM for item j is specified as "UDF" in argument model, the corresponding design matrix must be provided; otherwise, the design matrix can be NULL, which will be generated automatically. optional user-specified person attributes. It is a $$N\times K$$ matrix or data frame. If this is not supplied, attributes are simulated from a distribution specified in att.dist. A string indicating the distribution for attribute simulation. It can be "uniform", "higher.order", "mvnorm" or "categorical" for uniform, higher-order, multivariate normal and categorical distributions, respectively. The default is the uniform distribution. To specify structural parameters for the higher-order and multivariate normal distributions, see higher.order.parm and mvnorm.parm, respectively. To specify the probabilities for the categorical distribution, use att.prior argument. A vector giving the name of items or categories. If it is NULL (default), items are named as "Item 1", "Item 2", etc. A list specifying parameters for higher-order distribution for attributes if att.dist=higher.order. Particularly, theta is a vector of length $$N$$ representing the higher-order ability for each examinee. and lambda is a $$K \times 2$$ matrix. Column 1 gives the slopes for the higher-order model and column 2 gives the intercepts. See GDINA for the formulations of the higher-order models. a list of parameters for multivariate normal attribute distribution. mean is a vector of length $$K$$ specifying the mean of multivariate normal distribution; and sigma is a positive-definite symmetric matrix specifying the variance-covariance matrix. cutoffs is a vector giving the cutoff for each attribute. See Examples. probability for each attribute pattern. Order is the same as that returned from attributepattern(Q = Q). This is only applicable when att.dist="categorical". How many decimal places in each number? The default is 4. object of class simGDINA for method extract argument for S3 method extract indicating what to extract additional arguments

## Value

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

dat

simulated item response matrix

Q

Q-matrix

attribute

A $$N \times K$$ matrix for inviduals' attribute patterns

catprob.parm

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

delta.parm

a list of delta parameters

higher.order.parm

Higher-order parameters

mvnorm.parm

multivariate normal distribution parameters

LCprob.parm

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

## Details

Item parameter specifications in simGDINA:

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

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

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

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

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

## References

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

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

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

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

Hartz, S. M. (2002). A bayesian framework for the unified model for assessing cognitive abilities: Blending theory with practicality (Unpublished doctoral dissertation). University of Illinois at Urbana-Champaign.

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

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

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

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

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

## Examples

if (FALSE) {
####################################################
#                     Example 1                    #
#             Data simulation (DINA)               #
####################################################
N <- 500
Q <- sim30GDINA$simQ J <- nrow(Q) gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J)) # Simulated DINA model; to simulate G-DINA model # and other CDMs, change model argument accordingly sim <- simGDINA(N,Q,gs.parm = gs,model = "DINA") # True item success probabilities extract(sim,what = "catprob.parm") # True delta parameters extract(sim,what = "delta.parm") # simulated data extract(sim,what = "dat") # simulated attributes extract(sim,what = "attribute") #################################################### # Example 2 # # Data simulation (RRUM) # #################################################### N <- 500 Q <- sim30GDINA$simQ
J <- nrow(Q)
gs <- data.frame(guess=rep(0.2,J),slip=rep(0.2,J))
# Simulated RRUM
# deltas except delta0 for each item will be simulated
# randomly subject to the constraints of RRUM
sim <- simGDINA(N,Q,gs.parm = gs,model = "RRUM")

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

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

####################################################
#                     Example 3                    #
#             Data simulation (LLM)                #
####################################################
N <- 500
Q <- sim30GDINA$simQ J <- nrow(Q) gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J)) # Simulated LLM # By specifying type="equal", each required attribute is # assumed to contribute to logit(P) equally sim <- simGDINA(N,Q,gs.parm = gs,model = "LLM",gs.args = list (type="equal")) #check below for what the equal contribution means extract(sim,what = "delta.parm") # simulated data extract(sim,what = "dat") # simulated attributes extract(sim,what = "attribute") #################################################### # Example 4 # # Data simulation (all CDMs) # #################################################### set.seed(12345) N <- 500 Q <- sim10GDINA$simQ
J <- nrow(Q)
gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J))
# Simulated different CDMs for different items
models <- c("GDINA","DINO","DINA","ACDM","LLM","RRUM","GDINA","LLM","RRUM","DINA")
sim <- simGDINA(N,Q,gs.parm = gs,model = models,gs.args = list(type="random"))

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

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

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

# success probabilities for each item need to be provided in list format as follows:
# if item j requires Kj attributes, 2^Kj success probabilities
# need to be specified
# e.g., item 1 only requires 1 attribute
# therefore P(0) and P(1) should be specified;
# similarly, item 10 requires 3 attributes,
# P(000),P(100),P(010)...,P(111) should be specified;
# the latent class represented by each element can be obtained
# by calling attributepattern(Kj)
itemparm.list <- list(item1=c(0.2,0.9),
item2=c(0.1,0.8),
item3=c(0.1,0.9),
item4=c(0.1,0.3,0.5,0.9),
item5=c(0.1,0.1,0.1,0.8),
item6=c(0.2,0.9,0.9,0.9),
item7=c(0.1,0.45,0.45,0.8),
item8=c(0.1,0.28,0.28,0.8),
item9=c(0.1,0.4,0.4,0.8),
item10=c(0.1,0.2,0.3,0.4,0.4,0.5,0.7,0.9))
set.seed(12345)
N <- 500
Q <- sim10GDINA$simQ # When simulating data using catprob.parm argument, # it is not necessary to specify model and type sim <- simGDINA(N,Q,catprob.parm = itemparm.list) #################################################### # Example 6 # # Data simulation (all CDMs) # # using delta parameters in list format # #################################################### delta.list <- list(c(0.2,0.7), c(0.1,0.7), c(0.1,0.8), c(0.1,0.7), c(0.1,0.8), c(0.2,0.3,0.2,0.1), c(0.1,0.35,0.35), c(-1.386294,0.9808293,1.791759), c(-1.609438,0.6931472,0.6), c(0.1,0.1,0.2,0.3,0.0,0.0,0.1,0.1)) model <- c("GDINA","GDINA","GDINA","DINA","DINO","GDINA","ACDM","LLM","RRUM","GDINA") N <- 500 Q <- sim10GDINA$simQ
# When simulating using delta.parm argument, model needs to be
# specified
sim <- simGDINA(N,Q,delta.parm = delta.list, model = model)

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

Q <- sim30GDINA$simQ gs <- matrix(0.1,nrow(Q),2) N <- 500 set.seed(12345) theta <- rnorm(N) K <- ncol(Q) lambda <- data.frame(a=rep(1,K),b=seq(-2,2,length.out=K)) sim <- simGDINA(N,Q,gs.parm = gs, model="DINA", att.dist = "higher.order", higher.order.parm = list(theta = theta,lambda = lambda)) #################################################### # Example 8 # # Data simulation (higher-order CDMs) # #################################################### Q <- sim30GDINA$simQ
gs <- matrix(0.1,nrow(Q),2)
models <- c(rep("GDINA",5),
rep("DINO",5),
rep("DINA",5),
rep("ACDM",5),
rep("LLM",5),
rep("RRUM",5))
N <- 500
set.seed(12345)
theta <- rnorm(N)
K <- ncol(Q)
lambda <- data.frame(a=runif(K,0.7,1.3),b=seq(-2,2,length.out=K))
sim <- simGDINA(N,Q,gs.parm = gs, model=models, att.dist = "higher.order",
higher.order.parm = list(theta = theta,lambda = lambda))

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

# See Chiu et al., (2009)

N <- 500
Q <- sim10GDINA$simQ K <- ncol(Q) gs <- matrix(0.1,nrow(Q),2) cutoffs <- qnorm(c(1:K)/(K+1)) m <- rep(0,K) vcov <- matrix(0.5,K,K) diag(vcov) <- 1 simMV <- simGDINA(N,Q,gs.parm = gs, att.dist = "mvnorm", mvnorm.parm=list(mean = m, sigma = vcov,cutoffs = cutoffs)) #################################### # Example 10 # # Simulation using # # user-specified att structure# #################################### # --- User-specified attribute structure ----# Q <- sim30GDINA$simQ
K <- ncol(Q)
# divergent structure A1->A2->A3;A1->A4->A5;A1->A4->A6
diverg <- list(c(1,2),
c(2,3),
c(1,4),
c(4,5))
struc <- att.structure(diverg,K)

# data simulation
N <- 1000
# data simulation
gs <- matrix(0.1,nrow(Q),2)
simD <- simGDINA(N,Q,gs.parm = gs,
model = "DINA",att.dist = "categorical",att.prior = struc$att.prob) #################################################### # Example 11 # # Data simulation # # (GDINA with monotonicity constraints) # #################################################### set.seed(12345) N <- 500 Q <- sim30GDINA$simQ
J <- nrow(Q)
gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J))
# Simulated different CDMs for different items
sim <- simGDINA(N,Q,gs.parm = gs,model = "GDINA",gs.args=list(mono.constraint=TRUE))

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

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

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

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

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

set.seed(12345)

N <- 2000
# restricted Qc matrix
Qc <- sim20seqGDINA$simQ #total number of categories J <- nrow(Qc) gs <- data.frame(guess=rep(0.1,J),slip=rep(0.1,J)) # simulate sequential DINA model simseq <- simGDINA(N, Qc, sequential = TRUE, gs.parm = gs, model = "GDINA") # True item success probabilities extract(simseq,what = "catprob.parm") # True delta parameters extract(simseq,what = "delta.parm") # simulated data extract(simseq,what = "dat") # simulated attributes extract(simseq,what = "attribute") #################################################### # Example 13 # DINA model Attribute generated using # categorical distribution #################################################### Q <- sim10GDINA$simQ
gs <- matrix(0.1,nrow(Q),2)
N <- 5000
set.seed(12345)
prior <- c(0.1,0.2,0,0,0.2,0,0,0.5)
sim <- simGDINA(N,Q,gs.parm = gs, model="DINA", att.dist = "categorical",att.prior = prior)
# check latent class sizes
table(sim\$att.group)/N

####################################################
#                   Example 14
#                  MS-DINA model
####################################################

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

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

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

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