Determination of a Statistically Equivalent DINA Model

This function determines a statistically equivalent DINA model given a Q-matrix using the method of von Davier (2014). Thereby, the dimension of the skill space is expanded, but in the reparameterized version, the Q-matrix has a simple structure or the IRT model is no longer be conjuctive (like in DINA) due to a redefinition of the skill space.

equivalent.dina(q.matrix, reparameterization="B")

Arguments

q.matrix

The Q-matrix (see din)

reparameterization

The used reparameterization (see von Davier, 2014). A and B are possible reparameterizations.

Value

A list with following entries

q.matrix

Original Q-matrix

q.matrix.ast

Reparameterized Q-matrix

alpha

Original skill space

alpha.ast

Reparameterized skill space

References

von Davier, M. (2014). The DINA model as a constrained general diagnostic model: Two variants of a model equivalency. British Journal of Mathematical and Statistical Psychology, 67, 49-71.

Examples

#############################################################################
# EXAMPLE 1: Toy example
#############################################################################

# define a Q-matrix
Q <- matrix( c( 1,0,0,  0,1,0,
        0,0,1,   1,0,1,  1,1,1 ), byrow=TRUE, ncol=3 )
Q <- Q[ rep(1:(nrow(Q)),each=2), ]

# equivalent DINA model (using the default reparameterization B)
res1 <- CDM::equivalent.dina( q.matrix=Q )
res1

# equivalent DINA model (reparametrization A)
res2 <- CDM::equivalent.dina( q.matrix=Q, reparameterization="A")
res2

if (FALSE) {
#############################################################################
# EXAMPLE 2: Estimation with two equivalent DINA models
#############################################################################

# simulate data
set.seed(789)
D <- ncol(Q)
mean.alpha <- c( -.5, .5, 0  )
r1 <- .5
Sigma.alpha <- matrix( r1, D, D ) + diag(1-r1,D)
dat1 <- CDM::sim.din( N=2000, q.matrix=Q, mean=mean.alpha, Sigma=Sigma.alpha )

# estimate DINA model
mod1 <- CDM::din( dat1$dat, q.matrix=Q )

# estimate equivalent DINA model
mod2 <- CDM::din( dat1$dat, q.matrix=res1$q.matrix.ast, skillclasses=res1$alpha.ast)
# restricted skill space must be defined by using the argument 'skillclasses'

# compare model summaries
summary(mod2)
summary(mod1)

# compare estimated item parameters
cbind( mod2$coef, mod1$coef )

# compare estimated skill class probabilities
round( cbind( mod2$attribute.patt, mod1$attribute.patt ), 4 )


#############################################################################
# EXAMPLE 3: Examples from von Davier (2014)
#############################################################################

# define Q-matrix
Q <- matrix( 0, nrow=8, ncol=3 )
Q[2, ] <- c(1,0,0)
Q[3, ] <- c(0,1,0)
Q[4, ] <- c(1,1,0)
Q[5, ] <- c(0,0,1)
# Q[6, ] <- c(1,0,1)
Q[6, ] <- c(0,0,1)
Q[7, ] <- c(0,1,1)
Q[8, ] <- c(1,1,1)

#- parametrization A
res1 <- CDM::equivalent.dina(q.matrix=Q, reparameterization="A")
res1

#- parametrization B
res2 <- CDM::equivalent.dina(q.matrix=Q, reparameterization="B")
res2
}