4.11 Assignment I

1. Monotonicity is an important concept in not only CDM but also IRT models. In CDM context, monotonicity means students who master additional required attributes should not have lower success probability on an item. For example, if item $j$ measures two attributes, then$$P(Y_j=1|{\alpha }_{lj}^{\ast }=10)\ge P(Y_j=1|{\alpha }_{lj}^{\ast }=00)$$Please discuss what constraints the parameters need to meet to satisify monotonicity for the DINA, DINO, A-CDM, LLM and R-RUM. (For example: For DINA model, the monotonicity is met when $1-s_j-g_j\ge 0$ )

2. When a test measures $K=10$ attributes, how many parameters do you have in the (1) saturated joint attribute distribution, (2) independent model, (3) 2PL higher-order model and (4) a linear structure model ($A1\to A2\to \dots \to A10$).

3. Please read Henson et al. (2009) and Davier (2008) and discuss how the loglinear CDM and general diagnostic model are related to the G-DINA model.

4. Please show that the following models can all be used for the DINA model and they are all equivalent.

Eq 1: $$P(Y_j=1|{\eta }_{jl})=\{\begin{array}{cc}{g}_j& \text{if}{\eta }_{jl}=0\\ 1-s_j& \text{if}{\eta }_{jl}=1\end{array}$$

where ${\eta }_{jl}=0$ if students in latent group $l$ does not master all required attributes and ${\eta }_{jl}=1$ if students in latent group $l$ master all required attributes.

Eq 2: $$P(Y_j=1|{\eta }_{jl})=g_j^{1-{\eta }_{jl}}(1-s_j{)}^{{\eta }_{jl}}$$ where ${\eta }_{jl}=0$ if students in latent group $l$ does not master all required attributes and ${\eta }_{jl}=1$ if students in latent group $l$ master all required attributes.

Eq. 3: $$P(Y_j=1|{\alpha }_{lj}^{\ast })=\{\begin{array}{cc}{g}_j& \text{if}{\alpha }_{lj}^{\ast }\ne 1\\ 1-s_j& \text{if}{\alpha }_{lj}^{\ast }=1\end{array}$$ where ${\alpha }_{lj}^{\ast }$ represents the $l$th reduced latent profile with required attributes of item $j$. $1$ is a vector with all elements being 1.

Eq. 4: $$P(Y_j=1|{\alpha }_{lj}^{\ast })={\delta }_{j0}+{\delta }_{j1}I({\alpha }_{lj}^{\ast }=1)$$ Eq. 5: $$P(Y_j=1|{\alpha }_{lj}^{\ast })={\delta }_{j0}+{\delta }_{j1}I({\alpha }_{lj}^{\ast }\ne 1)$$ Eq. 6: $$\text{logit}P(Y_j=1|{\alpha }_{lj}^{\ast })={\delta }_{j0}+{\delta }_{j1}I({\alpha }_{lj}^{\ast }=1)$$

where ${\alpha }_{lj}^{\ast }$ represents the $l$th reduced latent profile with required attributes of item $j$ and $I\left(x\right)=1$ when $x$ is true and 0 when $x$ is false.

Eq. 7: $$P(Y_{ij}=1|{\alpha }_c)=\{\begin{array}{cc}{g}_j& \text{if}{\alpha }_c^T{q}_j<q_j^T{q}_j\\ 1-s_j& \text{otherwise}\end{array}$$ where ${\alpha }_c$ is the $c$th attribute profile (a column vector) with all attributes measured by the test and $q_j$ is the q-vector of item $j$. Subscript T denotes vector transposition.

5. Please fit the G-DINA model to data2 with Q2 and answer the following questions

• For Item 15, the G-DINA model parameter estimates are P(100) = 0.4 57, P(101) = 0.6 96.

• The corresponding SEs for P(100) and P(101) are 0.0 37 and 0.0 30, respectively.

• The proportion of individuals having an attribute pattern of 111 in the population is estimated to be 0.3 12

• The proportion of individuals who master ${\alpha }_1$ in the population is estimated to be 0.80 7.

• Plot item success probabilities for Items 5, 8 and 15. Which item(s) appear to follow the DINO models?

• Find the estimated attribute pattern for the third student.

• Find the $\delta$ parameters of item 11 and manually calculated probabilities of success for students with different reduced attribute profiles.

6. Fit a higher-order LLM to data2 with Q2 (for the higher-order component, please use 2PL-type model) and answer the following questions

• What is the number of parameters .

• What is the number of parameters for the joint attribute distribution?

• The proportion of individuals having an attribute pattern of 111 in the population is estimated to be 0.2 40

• The proportion of individuals who master ${\alpha }_1$ in the population is estimated to be 0. 800.

• What are the estimated higher-order parameters? Please interpret them.

References

Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61(2), 287–307. https://doi.org/10.1348/000711007X193957

Henson, R. A., Templin, J. L., & Willse, J. T. (2009). Defining a Family of Cognitive Diagnosis Models Using Log-Linear Models with Latent Variables. Psychometrika, 74(2), 191–210. https://doi.org/10.1007/s11336-008-9089-5