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Author: watilde

research/modules/5-billion-scale-polypharmacy/manuscripts/manuscript_v2.0.md

@@ -286,63 +286,63 @@ We demonstrate that billion-scale federated causal inference is computationally
 
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 ---
 
@@ -366,7 +366,7 @@ We demonstrate that billion-scale federated causal inference is computationally
 
 ### Figure 3: Sign Flip Phenomenon
 
-Treatment effect convergence across sample sizes for rare polypharmacy subgroup (CKD Stage 3b + Loop Diuretic + Age>80, prevalence 0.064%). 
+Treatment effect convergence across sample sizes for rare polypharmacy subgroup (CKD Stage 3b + Loop Diuretic + Age>80, prevalence 0.064%).
 
 **Key Finding**: At 1M patients (n=645), estimated ATE = -2.11 ml/min/year (95% CI: -3.14 to -1.07, p=0.003), suggesting harm. At 1B patients (n=632,776), estimated ATE = +1.46 ml/min/year (95% CI: +1.41 to +1.52, p<0.0001), indicating benefit—a complete sign reversal with high statistical confidence at both scales.
 
@@ -383,21 +383,25 @@ Treatment effect convergence across sample sizes for rare polypharmacy subgroup
 **Synthetic Data Protocol**: Extended Synthea framework with embedded ground truth for validation.
 
 **Polypharmacy Modeling**:
+
 - Base rate: 35% (Age>65), 60% (CKD Stage 3+)
 - Three interaction tiers: Interaction 1 (16% prevalence), Interaction 2 (0.4%), Interaction 3 (0.064%)
 
 **Ground Truth Effects**:
+
 - SGLT2i baseline: +1.0 ml/min/year
 - Interaction 1: +2.0 ml/min/year additional
-- Interaction 2: +0.5 ml/min/year additional  
+- Interaction 2: +0.5 ml/min/year additional
 - Interaction 3: +0.5 ml/min/year additional
 
 **Confounding Structure**:
+
 - Logistic propensity model: logit(P(T=1)) = 0.5×(HbA1c-7) - 0.3×(eGFR-60)/10 + 0.2×Age/10
 - Confounding by indication: Sicker patients preferentially receive treatment
 - Missing data: 5% missing-at-random (MAR)
 
 **Data Generation at Scale**:
+
 - 1000 sites × 1M patients per site = 1B total
 - Streaming generation (no disk I/O)
 - Worker threads parallelization for site-level computation
@@ -408,17 +412,17 @@ Treatment effect convergence across sample sizes for rare polypharmacy subgroup
 **Theorem 1 (Federated-Centralized Equivalence)**:
 Federated Newton-Raphson propensity score estimation produces identical estimates to centralized analysis.
 
-*Proof*: By associativity of sums, ∑_{k=1}^K g_k = ∑_{i=1}^N x_i(T_i - p_i) and ∑_{k=1}^K H_k = ∑_{i=1}^N x_ix_i^T p_i(1-p_i), where k indexes sites and i indexes patients. Therefore, β^{(t+1)} = β^{(t)} + (∑_k H_k)^{-1}(∑_k g_k) is mathematically equivalent to centralized Newton-Raphson. □
+_Proof_: By associativity of sums, ∑*{k=1}^K g_k = ∑*{i=1}^N x*i(T_i - p_i) and ∑*{k=1}^K H*k = ∑*{i=1}^N x_ix_i^T p_i(1-p_i), where k indexes sites and i indexes patients. Therefore, β^{(t+1)} = β^{(t)} + (∑_k H_k)^{-1}(∑_k g_k) is mathematically equivalent to centralized Newton-Raphson. □
 
 **Theorem 2 (Communication Complexity)**:
 Federated algorithm achieves O(1) communication per site independent of sample size.
 
-*Proof*: Each site transmits fixed-dimension statistics: gradient g_k ∈ ℝ^p, Hessian H_k ∈ ℝ^{p×p}, weighted matrices XWX_k ∈ ℝ^{p×p}, XWY_k ∈ ℝ^p. For p=5 covariates, communication = 5 (gradient) + 15 (Hessian upper triangle) + 15 (XWX) + 5 (XWY) = 40 floating-point numbers × 8 bytes = 320 bytes per site. Observed: 264 bytes (compression/encoding). □
+_Proof_: Each site transmits fixed-dimension statistics: gradient g_k ∈ ℝ^p, Hessian H_k ∈ ℝ^{p×p}, weighted matrices XWX_k ∈ ℝ^{p×p}, XWY_k ∈ ℝ^p. For p=5 covariates, communication = 5 (gradient) + 15 (Hessian upper triangle) + 15 (XWX) + 5 (XWY) = 40 floating-point numbers × 8 bytes = 320 bytes per site. Observed: 264 bytes (compression/encoding). □
 
 **Theorem 3 (Privacy Preservation)**:
 Aggregated statistics (gradient, Hessian, XWX, XWY) satisfy HIPAA Safe Harbor de-identification standard (§164.514(b)(2)).
 
-*Proof*: Transmitted statistics are aggregates over ≥1M patients per site, containing no individual identifiers, no cell counts <10, and no patient-level data. Satisfies statistical de-identification requirements. □
+_Proof_: Transmitted statistics are aggregates over ≥1M patients per site, containing no individual identifiers, no cell counts <10, and no patient-level data. Satisfies statistical de-identification requirements. □
 
 ### Supplement C: Sensitivity Analyses
 
@@ -480,6 +484,7 @@ ATE = (Σ XWY_treated) / (Σ w_treated) - (Σ XWY_control) / (Σ w_control)
 **Full results tables** for all sample sizes (100K, 1M, 10M, 100M, 1B) and all subgroups (Overall, Interaction 1, Interaction 2, Interaction 3) are available in the online repository: https://github.com/watilde/Harmonia
 
 **Key findings across all subgroups**:
+
 - Overall subgroup (84% prevalence, n=841M at 1B scale): Monotonic convergence to ATE=+1.28, no sign flip
 - Interaction 1 (17% prevalence, n=169M): Monotonic convergence to ATE=+2.86, no sign flip
 - Interaction 2 (0.4% prevalence, n=4.2M): Monotonic convergence to ATE=+1.50, no sign flip