7.9 Limited information statistics
Limited-information statistics focus on marginal proportions or probabilities. The primary idea of the limited information statistics is to utilize r-th order moments (\(\pi_r\)) instead of using proportions of all possible response patterns.
Hansen et al. (2014) explored the utility of limited information matrix for CDMs, and suggested that \(M_2\) is more consistent in detecting test-level model misspecification. Hence, let us explore about \(M_2\) statistic to assess the test lelvel model data fit.
We consider \(M_2\) statistic, which only focuses on the first two marginals.
The first marginal proportions are the proportions of students who answer each item correctly:
\[\mathbf{p}_1=\big[P(Y_{1}=1),\ldots,P(Y_{j}=1),\ldots,P(Y_{J}=1)\big]\]
The second marginal proportions are the proportions of students who answer a pair of items correctly:
\[\mathbf{p}_2=\big[P(Y_{1}=1,Y_{2}=1),\ldots,P(Y_{j}=1,Y_{j'}=1),\ldots,P(Y_{J-1}=1,Y_{J}=1)\big]\] Let us try to understand the values of \(\mathbf{p}_1\) and \(\mathbf{p}_2\)
## Item1 Item2 Item3 Item4 Item5
## 1 1 1 1 1 1
## 2 1 1 1 1 0
## 3 1 0 1 0 1
## 4 1 0 1 0 1
## 5 1 0 0 0 0
## 6 1 0 1 1 1
## 7 1 1 0 1 0
## 8 1 1 1 1 1
## 9 1 1 0 1 0
## 10 1 1 1 1 01st order probabilities,
- \(P(Y_{1}=1) = 10/10\)
- \(P(Y_{2}=1) = 6/10\) and so on.
Second order probabilities,
- \(P(Y_{1}=1,Y_{2}=1)= 6/10=0.6\)
- \(P(Y_{2}=1,Y_{3}=1)= 4/10=0.4\)
and so on.
We can also find the model-predicted counterparts (i.e., \(\hat{\pi_1}\) and \(\hat{\pi_2}\)) in similar way.