The correlation between two occasions comes from two ingredients: a random intercept (people differ in level — the G side) plus a residual structure (how leftover fluctuation is related over time — the R side). You set the true (population) model on the left; the matrix and correlation-by-lag curve show exactly what it implies, while the bottom panel draws one random sample from it.
◆ Random intercept · G
In UN the matrix is the marginal V — the random intercept is absorbed into it, so τ² is not separately identified (inactive here).
▚ Residual structure · R
Each cell: correlation between two occasions. Darker = more correlated.
Total (marginal) correlation — the random intercept lifts every off-diagonal; this equals the pure R structure only when τ² = 0.
each cell is the correlation split: random intercept ICC = τ²/(τ²+σ²) (the same in every cell) + residual σ²·R (fades with lag). Bar height = total correlation.
τ² = σ² = 0 → no variance in the model. Every observation equals the grand mean, so correlation is undefined (0 / 0). Real data always have variance — nudge either knob up.
⚠ These residual correlations do not form a valid (positive-definite) covariance matrix, so the sampled lines below no longer match this matrix. Lower a lag value.
How the total correlation changes as occasions get further apart (lag).
Lag counts occasions, not clock time — so how often you sample shapes this curve: measure more frequently and each step spans less real time, so the correlation at each lag looks higher (the same process, denser sampling).
Ten people — one random sample from the model on the left. Drag a slider — the same people's lines re-shape.
τ² = 0, yet the person means still scatter (the SD of person means above). That is sampling over a finite number of occasions — positively-correlated residuals exaggerate it. Drag Occasions (n) up and the spread shrinks toward 0; a real random intercept would keep it up as n grows.