Figures · L-DAHS

§ 03 Figures · three 3D scenes

§ 03

Interactive 3D illustrations

Seeing in three dimensions

Three interactive figures, each built from a simulated 3D dataset that mirrors the structures the lab studies. Rotate, zoom, and hover to read the geometry.

Figure 1

The latent trait space

Three hundred simulated learners, each placed by three correlated latent factors: Aptitude, Motivation, and Anxiety, drawn from a structured covariance matrix with three latent profiles. Point colour encodes a fourth derived dimension (achievement), so the figure shows a 4D structure projected into a rotatable cube.

\[ \boldsymbol{\xi}_i=\boldsymbol{\mu}_{g(i)}+\mathbf{L}\mathbf{z}_i,\quad \mathbf{z}_i\sim\mathcal{N}(\mathbf{0},\mathbf{I}),\ \ \mathbf{L}\mathbf{L}^{\!\top}=\boldsymbol{\Sigma} \]

x — Aptitude (ξ₁) y — Motivation (ξ₂) z — Anxiety (ξ₃) colour — Achievement (η)

Figure 2

A two-dimensional response surface

The probability of item mastery as a smooth surface over two latent abilities, in the spirit of a multidimensional item-response model. The ridge running diagonally is the compensatory region where strength on one trait offsets weakness on the other: the kind of interaction a flat table of coefficients hides but a surface makes obvious.

\[ P(\theta_1,\theta_2)=\dfrac{1}{1+\exp\!\big[-(a_1\theta_1+a_2\theta_2-d)\big]} \]

x — Ability θ₁ y — Ability θ₂ height — P(mastery)

Figure 3

Affect dynamics in phase space

Several simulated learners released from nearly-identical starting states, each trajectory integrating a coupled nonlinear system of motivation, engagement, and anxiety. Tiny differences in initial affect produce divergent paths around a shared attractor: a visual argument for why static models miss regime shifts that dynamical ones catch.

\[ \dot{M}=\sigma(E-M),\quad \dot{E}=M(\rho-A)-E,\quad \dot{A}=ME-\beta A \]

x — Motivation y — Engagement z — Anxiety colour — time t

§ 04 Gallery · eight more figures

§ 04

Additional figures · one or two per research focus

A gallery of figures.

Eight publication-style figures spanning the lab's research foci. Each is tagged by the methodology it serves and whether it is 2D or 3D.

Fig A

Test information surface

CAT · IRT3-D

Where a multidimensional test measures most precisely. Each item contributes a ridge of Fisher information; their sum is the landscape an adaptive algorithm climbs when choosing the next item.

\[ \mathcal{I}(\boldsymbol{\theta})=\sum_i \big(a_{1i}^2+a_{2i}^2\big)\,P_i(\boldsymbol{\theta})\big[1-P_i(\boldsymbol{\theta})\big] \]

Fig B

CAT convergence

CAT

A simulated examinee. The ability estimate and its shrinking 95% band close in on the true value as items are administered.

\[ \mathrm{SE}=\mathcal{I}^{-1/2} \]

Fig C

Forest plot — meta-analytic synthesis

Meta-analysis2-D

Twelve simulated studies with marker size scaled to precision, and the random-effects pooled estimate as a diamond at the foot. The canonical synthesis figure.

\[ \hat{\delta}_{\text{RE}}=\frac{\sum_i w_i\,d_i}{\sum_i w_i},\qquad w_i=\frac{1}{\,s_i^2+\hat{\tau}^2\,} \]

Fig D

Funnel plot

Bias

Effect size against its standard error. Asymmetry at the base is the visual signature of small-study or publication bias.

\[ \hat{\delta}\pm 1.96\,s \]

Fig E

Random-slope spaghetti plot

HLM · multilevel2-D

Twenty classrooms, each its own regression line coloured by intercept, scattered around the fixed-effect line (white). Exactly the picture random intercepts and slopes are meant to capture.

\[ y_{ij}=(\beta_0+u_{0j})+(\beta_1+u_{1j})\,x_{ij}+\varepsilon_{ij} \]

Fig F

Correlation matrix heatmap

SEM2-D

The block structure that a measurement model assumes, made visible: high within-construct correlations on the diagonal blocks, low cross-construct correlations off them.

\[ \mathbf{R}=\mathbf{D}^{-1/2}\,\boldsymbol{\Sigma}\,\mathbf{D}^{-1/2} \]

Fig G

Gradient descent on a loss surface

Matrix calculus3-D

Estimation as a walk downhill. The path zig-zags down an elongated quadratic valley toward the minimum, the geometric reason ill-conditioned problems are slow to converge.

\[ \mathbf{w}_{t+1}=\mathbf{w}_t-\eta\,\nabla\mathcal{L}(\mathbf{w}_t) \]

Fig H

Phase portrait with vector field

Dynamical systems2-D

A two-state affect system: the arrows are the flow, the dashed lines the nullclines, and the bright curves are trajectories spiralling into a stable equilibrium, a planar companion to the 3D attractor.

\[ \dot{M}=-\gamma M+\omega A,\qquad \dot{A}=-\omega M-\gamma A \]