18-EFT.WP.Methods.CrossStats v1.0 | Chapter 13 — Statistical Synthesis and Meta-Analysis (Fixed/Random Effects)

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Chapter 13 — Statistical Synthesis and Meta-Analysis (Fixed/Random Effects)

One-line objective: Establish an end-to-end meta-analytic methodology—from effect-size extraction through heterogeneity modeling, prediction intervals, and bias auditing—aligned with cross-modal metrology and unified time-base conventions.

I. Scope & Targets

1. Scope
   • Fixed-effect and random-effects meta-analysis (continuous, binary, and rate outcomes).
   • Univariate and multivariate (correlated effects) synthesis, meta-regression, and small-study effect diagnostics.
   • Frequentist pipelines (DL, REML, HK) and Bayesian publication harmonization.
2. Targets
   • Inputs: D = { study i : (effect e_i, SE_i or var v_i, n_i, x_i, ts_i) }, or raw contrast data to convert into standardized effect sizes.
   • Outputs: pooled effect hat{theta}, confidence/posterior intervals, prediction intervals, heterogeneity {Q, tau^2, I^2}, bias & sensitivity reports, and manifest.stats.meta.*.
   • Time-base: compute windowed statistics on tau_mono, publish on ts with offset/skew/J; if a study includes arrival-time T_arr, record both formulations and delta_form.

II. Terms & Symbols

• Effect size: e_i (e.g., Hedges g, log OR, log RR, Fisher z), variance v_i = SE_i^2, weights w_i.
• Overall effect: theta; estimator hat{theta}.
• Heterogeneity: Cochran Q, between-study variance tau^2, proportion I^2 = max( 0, ( Q - (k - 1) ) / Q ).
• Prediction interval: PI = hat{theta} ± t_{k-2,1-α/2} * sqrt( V_rand + tau^2 ).
• Meta-regression: covariate vector x_i, coefficients beta.
• Small-study effects: funnel asymmetry statistics and regression intercepts.

*III. Axioms P313- **

• P313-1 (Unified Effect Domain): Before synthesis, map all studies into a common effect domain and unit; enforce check_dim( e_i ) and direction harmonization.
• P313-2 (Positive-Definite Variances): v_i > 0, tau^2 ≥ 0; multivariate covariance matrices are positive semidefinite.
• P313-3 (Model Exchangeability): Under random-effects, given theta_i ~ N( mu, tau^2 ), studies are conditionally independent.
• P313-4 (Time-Base Consistency): Windowing, alignment, and the two T_arr formulations follow Methods.Cleaning Chapters 5–6.
• P313-5 (Parallel Publication): Report fixed and random estimates in parallel; discrepancies feed the audit and strategy cards. If T_arr arises, record delta_form.

*IV. Minimal Equations S313- **

1. S313-1 (Fixed-Effect Synthesis)
   • Weights: w_i = 1 / v_i; pooled estimate: hat{theta}_FE = ( ∑ w_i e_i ) / ( ∑ w_i ).
   • Variance: Var( hat{theta}_FE ) = 1 / ( ∑ w_i ).
2. S313-2 (Heterogeneity Statistics)
   • Q = ∑ w_i ( e_i - hat{theta}_FE )^2, df = k - 1.
   • I^2 = max( 0, ( Q - df ) / Q ).
3. S313-3 (Random Effects: DL and REML)
   • DL estimator: tau^2_DL = max( 0, ( Q - df ) / ( ∑ w_i - ( ∑ w_i^2 ) / ( ∑ w_i ) ) ).
   • REML: numerically maximize logL_REML( tau^2 ).
   • Random weights: w_i* = 1 / ( v_i + tau^2 ); pooled estimate: hat{theta}_RE = ( ∑ w_i* e_i ) / ( ∑ w_i* ).
   • Variance: Var( hat{theta}_RE ) = 1 / ( ∑ w_i* ); Hartung–Knapp uses t_{df} in place of Normal and adjusts variance.
4. S313-4 (Prediction Interval)
   PI = hat{theta}_RE ± t_{k-2,1-α/2} * sqrt( Var( hat{theta}_RE ) + tau^2 ).
5. S313-5 (Meta-Regression)
   • Model: e_i = x_i' beta + u_i + eps_i, u_i ~ N(0, tau^2), eps_i ~ N(0, v_i).
   • Estimation: weighted GLS or REML; center covariates to stabilize numerics.
6. S313-6 (Multivariate Meta-Analysis)
   e_i ∈ R^p, Cov(e_i) = V_i + tau^2 Σ; solve via GLS/REML or Bayesian hierarchies.
7. S313-7 (Combining p-Values)
   Fisher: X = -2 ∑ log p_i ~ chi2_{2k}; Stouffer: Z = ( ∑ w_i z_i ) / sqrt( ∑ w_i^2 ).

V. Metrology Flow M30-13 (Ready → Convert → Estimate → Diagnose → Publish)

• Ready
  Source data checks, time-base alignment, missingness and unit validation; unify contrast direction (positivize).
• Effect-Size & Variance Conversion
  Continuous: g, SE(g); binary: log OR/log RR; rates: log rate with exposure.
• Model Fitting
  Compute FE and RE (DL, REML, HK) in parallel; run meta-regression as needed; fit multivariate models for correlated endpoints.
• Diagnostics & Sensitivity
  Heterogeneity {Q, I^2, tau^2}; influence and leave-one-out; small-study effects (regression intercepts, funnel plots, trim-and-fill).
• Prediction & Uncertainty
  Report pooled intervals and prediction intervals; optionally publish Bayesian posterior quantiles and U = k * u_c mapping.
• Persist & Publish
  Emit manifest.stats.meta.*: calibrations, conversions, model & hyperparameters, diagnostics, contract evaluation, TraceID and signature.

VI. Contracts & Assertions C30-131x

• C30-1311 (Positive-Definite Variances): min(v_i) > 0, tau^2 ≥ 0, and multivariate eigmin(V_i) ≥ 0.
• C30-1312 (Complete Heterogeneity Report): Q, df, I^2, and tau^2 are all computed and persisted.
• C30-1313 (Parallel Formulations): Both fixed and random pipelines succeed; abs( hat{theta}_RE - hat{theta}_FE ) ≤ tol_theta or flag flag_diff.
• C30-1314 (Prediction Interval Presence): If k ≥ 3, PI must be reported with PI_width ≥ PI_min.
• C30-1315 (Small-Study Effects Check): When k ≥ k_egger_min, provide regression test and decisioning; if p < alpha_bias, flag and trigger sensitivity procedures.
• C30-1316 (Time-Base & Arrival Time): If studies involve T_arr, record delta_form and assert delta_form ≤ tol_Tarr.
• C30-1317 (Unit Consistency): unit(e_i) matches unit( hat{theta} ); cross-study conversion matrices are persisted for auditability.

*VII. Implementation Bindings I30- **

• I30-131 compute_effect_sizes(raw, spec) -> {e, v, unit}
• I30-132 meta_fixed(e, v) -> {theta, SE, Q, I2}
• I30-133 meta_random_dl(e, v) -> {theta, SE, tau2, I2}
• I30-134 meta_random_reml(e, v) -> {theta, SE, tau2, I2}
• I30-135 hartung_knapp(e, v, tau2) -> {theta, SE_adj}
• I30-136 meta_regression(e, V, X, method) -> {beta, tau2}
• I30-137 meta_multivariate(E, V_list, method) -> model
• I30-138 bias_diagnostics(e, v) -> {egger_p, trimfill_delta}
• I30-139 leave_one_out(e, v, method) -> influence_report
• I30-13A combine_pvalues(pvals, scheme) -> p_combined
• I30-13B emit_meta_manifest(results, policy) -> manifest.stats.meta
• Invariants: sum(w_i)/k ≈ mean(1/v_i) (spot-check); theta and intervals reproducible; manifest.stats.meta.signature is signed.

VIII. Cross-References

• Dimension & unit harmonization: Methods.Cleaning v1.0, Chapter 4.
• Timelines & synchronization: Methods.Cleaning v1.0, Chapter 5.
• Multilevel partial pooling: this volume, Chapter 12.
• Multiple comparisons & error control (multi-endpoint/subgroup): this volume, Chapter 6.
• Imaging/physical-domain calibration transforms: Methods.Imaging v1.0, Chapters 4 & 14.

IX. Quality & Risk Control

1. SLI/SLO
   • contract_pass_rate ≥ 0.99; latency_ms_p99 ≤ 1000; consistent reproducibility_hash.
   • Report completeness: meta_report_completeness ≥ 0.98 (mandatory-field coverage).
2. Risks & Fallbacks
   • Small k or extreme I^2: publish PI and robust estimators (HK/Bayesian); if necessary, release descriptive statistics only and label “insufficient studies”.
   • Severe funnel asymmetry: trigger sensitivity (trim-and-fill, log/square-root transforms, strong-prior constraints) and a re-analysis plan.
   • Extreme weights: cap w_i, use robust re-weighting (e.g., Huber), and record the strategy card.

Summary