Choosing GEE Correlation aka. Repeated Measures: ind = Independence, exc = exchangeable, ar1 = Autoregressive Order 1, sta1 = Stationary m-dependent (m=1), uns = Unstructured

Marginal Model-Based Correlation Structures in GEE (Generalized Estimating Equations)

Marginal models (via GEE) assume correlated outcomes within subjects/clusters and require specification of a "working correlation structure".

While coefficient estimates (β) are consistent regardless of structure, the efficiency of SE estimates depends heavily on choosing an appropriate correlation form.

---

🚫 ind — Independent

• Assumes: No within-subject correlation (ρ = 0)
• 🔥 Not appropriate in repeated measures — violates core assumption of within-subject dependence
• ❌ Use only if you're absolutely sure responses are uncorrelated (which defeats the purpose of GEE)
• ✅ Use in clustered cross-section, NOT in longitudinal/repeated measures

Conclusion: Do not use ind if you're modeling intra-subject dependency — it contradicts the purpose of GEE in this context.

---

✅ exchangeable — Compound Symmetry (CS)

• Assumes: All pairwise correlations are equal
• Use when:
  • Time spacing is not relevant
  • Repeated measures are irregular or few
  • Clinical example: BP at admission, discharge, follow-up (no decay expected)

xtgee y x1 x2, family(gaussian) link(identity) corr(exchangeable) i(id) vce(robust)

Advantage: Efficient if assumption holds; still robust if not (with vce(robust))

---

✅ ar1 — Autoregressive Order 1

• Assumes: Correlation decays exponentially with time lagCorr(yt,yt+k)=ρk\text{Corr}(y_t, y_{t+k}) = \rho^k
• Use when:
  • Equally spaced time points (e.g., every week/month)
  • There's temporal ordering
  • Clinical example: Daily symptom ratings, serial biomarkers

xtgee y x1 x2, family(gaussian) link(identity) corr(ar1) i(id) vce(robust)

Warning: Requires proper time index and spacing — not for irregular intervals

---

✅ sta1 — Stationary m-dependent (m=1)

• Assumes:
  • Observations ≤ m steps apart have non-zero correlation
  • All others = 0
• sta1: Correlation only among immediate neighbors
• Use when:
  • You expect correlation only between adjacent time points
  • Clinical example: Hourly measurements where only consecutive readings are related
• Syntax:

xtgee y x1 x2, family(gaussian) link(identity) corr(sta1) i(id) vce(robust)

Note: Underused but powerful in short, dense, equally spaced panels

---

✅ uns — Unstructured

• Allows: Each pair of observations to have its own unique correlation
• Most flexible — no assumptions
• BUT: Requires large sample sizes and many observations per subject
• Use when:
  • N is large (hundreds of subjects)
  • Each subject has many (≥5–6) repeated measures
  • You want to let the data dictate the correlation

xtgee y x1 x2, family(gaussian) link(identity) corr(uns) i(id) vce(robust)

Caution: Fails or overfits with sparse/imbalanced data. Huge computational cost.

---

🧭 Summary Table: GEE Correlation Structures for Clinical Repeated Measures

💡 Clinical Decision Logic

• Sparse, irregular follow-ups → Use exchangeable
• Time-ordered, equally spaced → Use ar1
• Immediate-neighbor dependence only → Use sta1
• Dense + high-frequency + large N → uns
• Never use ind unless modeling cross-sectional clusters (not repeated measures)

---

Correlation Matrix Simulation — GEE Working Correlation Structures

Data: Repeated measures on 5 time points (T1–T5)

🔴 corr(independent)

Assumes no within-subject correlation (ρ = 0) — each time point is independent.

🟢 corr(exchangeable)

Assumes constant correlation between all pairs (ρ = 0.80)

🟡 corr(ar1)

Assumes correlation decays by lag (ρ = 0.90, then ρ² = 0.81, ρ³ = 0.73…)

🔵 corr(sta1)

Only adjacent timepoints are correlated (lag-1), ρ = 0.80

🟣 corr(uns)

No assumptions — each pair has its own unique correlation