Terminology
Respondents (r): The individuals from whom behavioral data are collected
- For today, this is dichotomous assessment item responses
- Not limited to only item responses in practice
Items (i): Assessment questions used to classify/diagnose respondents
Attributes (a): Unobserved latent categorical characteristics underlying the behaviors (i.e., diagnostic status)
Diagnostic Assessment: The method used to elicit behavioral data
Attribute profiles
[0, 0]
[1, 0]
[0, 1]
[1, 1]
DCMs as latent class models
\[
\color{#D55E00}{P(X_r=x_r)} = \sum_{c=1}^C\color{#009E73}{\nu_c} \prod_{i=1}^I\color{#56B4E9}{\pi_{ic}^{x_{ir}}(1-\pi_{ic})^{1 - x_{ir}}}
\]
Observed data: Probability of observing examinee r's item reponses
Structural component: Proportion of examinees in each class
Measurement component: Product of item response probabilities
Item response probabilities
Non-compensatory DCMs
Must be proficient in all attributes measured by the item to provide a correct response
Deterministic inputs, noisy “and” gate (DINA; de la Torre & Douglas, 2004)
Compensatory DCMs
Must be proficient in at least 1 attribute measured by the item to provide a correct response
Deterministic inputs, noisy “or” gate (DINO; Templin & Henson, 2006)
General DCMs
Different response probabilities for each class (partially compensatory)
Log-linear cognitive diagnostic model (LCDM; Henson et al., 2009)
This will be our focus
Simple structure LCDM
Item measures only 1 attribute
\[
\text{logit}(X_i = 1) = \color{#D7263D}{\lambda_{i,0}} + \color{#219EBC}{\lambda_{i,1(1)}}\color{#009E73}{\alpha}
\]
λi,0: Log-odds when not proficient
λi,1(1): Increase in log-odds when proficient
α: Attribute proficiency status (either 0 or 1)
Subscript notation
- i = The item to which the parameter belongs
- e = The level of the effect
- 0 = intercept
- 1 = main effect
- 2 = two-way interaction
- 3 = three-way interaction
- Etc.
- \((\alpha_1,...)\) = The attributes to which the effect applies
- The same number of attributes as listed in subscript 2
Complex structure LCDM
Item measures multiple attributes
\[
\text{logit}(X_i = 1) = \color{#D7263D}{\lambda_{i,0}} + \color{#4B3F72}{\lambda_{i,1(1)}\alpha_1} + \color{#9589BE}{\lambda_{i,1(2)}\alpha_2} +
\color{#219EBC}{\lambda_{i,2(1,2)}\alpha_1\alpha_2}
\]
Log-odds when proficient in neither attribute
Increase in log-odds when proficient in attribute 1
Increase in log-odds when proficient in attribute 2
Change in log-odds when proficient in both attributes
The LCDM as a general DCM
So called “general” DCM because the LCDM subsumes other DCMs
Constraints on item parameters make LCDM equivalent to other DCMs (e.g., DINA and DINO)
- Interactive Shiny app: https://atlas-aai.shinyapps.io/dcm-probs/
- DINA
- Only the intercept and highest-order interaction are non-0
- DINO
- All main effects are equal
- All two-way interactions are -1 \(\times\) main effect
- All three-way interactions are -1 \(\times\) two-way interaction (i.e., equal to main effects)
- Etc.
