6.2 Global identifiability of the DINA and DINO model

When guessing and slip parameters are unknown, the following conditions are sufficient and necessary for model identification (Xu, 2019).

Identifiability Condition #1(C1)

  1. The Q-matrix has the following forms, where \(I_K\) is an identity matrix. \[ Q=\begin{pmatrix} I_K \\ Q' \end{pmatrix} \]
  2. Each attribute is measured by at least three items.
  3. Any two different columns of \(Q'\) are distinct.

Now let’s check how nonidentifiability affects DINA model analysis.

Based on the identifiability conditions above, the Q-matrix below is not identified.

Code 
Q <- matrix(c(1,0,
              0,1,
              1,1,
              1,1,
              1,1),ncol = 2,byrow = TRUE)
Attribute 1 Attribute 2
Item 1 1 0
Item 2 0 1
Item 3 1 1
Item 4 1 1
Item 5 1 1

The data in DINA_idf.csv was simulated based on the following parameters:

Code 
g <- c(.05,.1,.15,.2,.25)
s <- c(.05,.1,.15,.2,.25)
p <- c(.25,.25,.25,.25)

Let us fit the DINA model to the data using the code below. Please change the randomseed used to generate initial item parameters and see how that will affect the estimates.

Code 
library(GDINA)
df <- read.csv("data/DINA_idf.csv")
DINA.est <- GDINA(df,Q,model = "DINA",
                  control = list(conv.crit=1e-6,
                                 randomseed=12566),
                  verbose = FALSE)
coef(DINA.est,"gs")
##        guessing  slip
## Item 1    0.123 0.048
## Item 2    0.082 0.135
## Item 3    0.157 0.152
## Item 4    0.202 0.151
## Item 5    0.211 0.256
Code 
coef(DINA.est,"lambda")
## p(00) p(10) p(01) p(11) 
##  0.25  0.22  0.29  0.24

References

Xu, G. (2019). Identifiability and Cognitive Diagnosis Models (M. von Davier & Y.-S. Lee, Eds.; pp. 333–357). Springer International Publishing. http://link.springer.com/10.1007/978-3-030-05584-4_16