Nested latent structure
Separating within- and between-cluster covariance so classroom, learner, and item effects are not confounded. The total covariance decomposes into level-specific structures, each with its own measurement and structural model. Applications include cross-cultural measurement invariance, school effectiveness research, and longitudinal growth modeling with nested observations.
\( \boldsymbol{\Sigma}_T=\boldsymbol{\Sigma}_W+\boldsymbol{\Sigma}_B \)
\( \text{ICC}=\dfrac{\sigma^2_B}{\sigma^2_B+\sigma^2_W} \)
\( \text{Design effect}=1+(n_c-1)\cdot\text{ICC} \)
Random-coefficient models
Slopes and intercepts that vary across higher-level units, capturing context as a source of systematic variation. Shrinkage pulls extreme estimates toward the grand mean, borrowing strength across clusters. We apply HLM to model learner growth trajectories, classroom-level effects on achievement, and cross-classified designs where students are nested in both schools and neighborhoods.
\( y_{ij}=(\beta_0+u_{0j})+(\beta_1+u_{1j})x_{ij}+\varepsilon_{ij} \)
\( \mathbf{u}_j\sim\mathcal{N}(\mathbf{0},\boldsymbol{\Psi}) \)
\( \text{Reliability}=\dfrac{\tau^2}{\tau^2+\sigma^2/n_j} \)
Adaptive measurement
Each item is chosen to maximize Fisher information at the provisional ability estimate, shortening tests without losing precision. Item characteristic curves govern response probabilities; test information aggregates item contributions. We develop CAT systems for language proficiency assessment, diagnostic testing with cognitive models, and multidimensional adaptive testing for complex constructs.
\( P_i(\theta)=\dfrac{1}{1+e^{-a_i(\theta-b_i)}} \)
\( I(\theta)=\sum_i a_i^2 P_i Q_i \)
\( \text{SE}(\hat\theta)=1/\sqrt{I(\hat\theta)} \)
Synthesis across studies
Random-effects models that treat each study's true effect as drawn from a distribution, capturing heterogeneity rather than averaging it away. Forest plots visualize individual effects; funnel plots diagnose publication bias. Our meta-analyses examine language learning interventions, motivation-achievement relationships, and the effectiveness of technology-enhanced instruction.
\( \hat{\delta}_i=\delta+u_i+e_i \)
\( I^2=\dfrac{Q-(k-1)}{Q}\times 100\% \)
\( \tau^2=\dfrac{Q-(k-1)}{\sum w_i - \sum w_i^2/\sum w_i} \)
Learning representations
Dimension reduction, topic models, and neural encoders that turn corpora and behavioural logs into interpretable coordinates. From bag-of-words to transformer embeddings, we extract latent semantic structure from learner writing, open-ended responses, and textual feedback.
\( p(w|d)=\sum_k p(w|z_k)p(z_k|d) \)
\( \text{TF-IDF}_{t,d}=\text{tf}_{t,d}\cdot\log\dfrac{N}{n_t} \)
\( \cos(\mathbf{u},\mathbf{v})=\dfrac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|} \)
Affect trajectories over time
Differential-equation models of how motivation, engagement, and anxiety drive one another along a learning trajectory. Coupled ODEs let us trace regime shifts that snapshots miss, mapping how learner states settle into attractors and escape from local minima. Phase portraits reveal stability; bifurcation diagrams show how parameter changes flip system behavior.
\( \dot{\mathbf{x}}=\mathbf{f}(\mathbf{x};\boldsymbol{\theta}) \)
\( \mathbf{J}=\dfrac{\partial\mathbf{f}}{\partial\mathbf{x}}\big|_{\mathbf{x}^*} \)
\( \lambda(\mathbf{J})<0 \Rightarrow \text{stable} \)