18-EFT.WP.Methods.CrossStats v1.0 | Chapter 3 — Sampling Design and Survey Weights (SRS/STRAT/CLUSTER)

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Chapter 3 — Sampling Design and Survey Weights (SRS/STRAT/CLUSTER)

One-Line Objective

I. Scope and Objects

1. Scope
   • Applies to probability sampling on finite and streaming populations, weighted estimation, design-based variance, and weight calibration.
   • Supports multistage designs (PSU/SSU), post-stratification and iterative proportional fitting (raking/IPF), and is compatible with event streams (Poisson/Bernoulli/reservoir) and time windows Delta_t.
2. Objects
   • Population size N (possibly unknown), sample size n, stratum h ∈ {1..H}, cluster c ∈ {1..C}, inclusion probability pi(i), survey weight w_i = 1 / pi(i), normalization factor W_norm = ( ∑ w_i ) / N_hat.
   • Calibration matrix (control totals) X_cal, constraint vector t_cal (e.g., marginal totals).
   • Time semantics: perform windowing and online sampling on tau_mono; publish on ts. For any statistic involving arrival time, record both T_arr conventions and delta_form in parallel.

II. Terms and Variables

1. Estimators
   • Horvitz–Thompson: hat{T}_HT = ∑_i ( y_i / pi(i) ); Hájek ratio: hat{Y}_Hajek = ( ∑ w_i y_i ) / ( ∑ w_i ).
   • Stratified mean: hat{Y} = ∑_h ( N_h / N ) * hat{Y}_h, where hat{Y}_h = ( ∑_{i ∈ h} w_i y_i ) / ( ∑_{i ∈ h} w_i ).
2. Variance and design effects
   • DEFF = Var_complex( hat{Y} ) / Var_SRS( hat{Y} ); weight-induced DEFF_w ≈ 1 + CV(w)^2.
   • Intra-cluster correlation rho_icc; cluster effect DEFF_c ≈ 1 + ( m_bar - 1 ) * rho_icc.
3. Weight calibration
   Calibrated weights w_i* satisfy ∑ w_i* x_i = t_cal; raking applies multiplicative updates over multiple margins.
4. Replicate weights
   Replication scheme R ∈ {1..R_rep} (JK, BRR, Bootstrap), with replicate weights w_i^(r).
5. Units and dimensions
   unit(w_i) = 1, dim(w_i) = []; estimators inherit the unit of y; run check_dim before release.

III. Axioms P303-*

• P303-1 (Explicit probability sampling): any published statistic derived from a sample must provide pi(i) or an equivalent, versioned weight-generation convention.
• P303-2 (Normalization and extremum control): default normalization targets W_norm ≈ 1; define cap_w_max and trimming policy, and record the trimmed set.
• P303-3 (Exchangeability within strata and clusters): assume exchangeability within strata; handle intra-cluster correlation via rho_icc or replication.
• P303-4 (Consistent control totals): when using calibrated weights, disclose the provenance and timestamp of t_cal.
• P303-5 (Unified timebase and streaming): execute online sampling on tau_mono with explicit Delta_t; for T_arr-dependent statistics, compute both conventions and constrain delta_form.
• P303-6 (Auditable replicate weights): replication scheme, random seeds, and R_rep must be traceable.
• P303-7 (Dimensional conservation): any ratio estimator must declare a positive denominator with matching units; failures are contract violations.

IV. Minimal Equations S303-*

• S303-1 (SRSWOR inclusion probability): pi(i) = n / N, w_i = N / n.
• S303-2 (Stratified sampling): pi_h(i) = n_h / N_h, w_i = N_h / n_h; Neyman optimal allocation
  n_h* = n * ( N_h * S_h ) / ( ∑_k N_k * S_k ), where S_h is the within-stratum standard deviation.
• S303-3 (Two-stage cluster PPS):
  First select cluster c with pi_c ∝ M_c (measure of size); then SRSWOR within cluster with pi(i | c) = n_c / M_c; overall pi(i) = pi_c * pi(i | c); weight w_i = 1 / pi(i).
• S303-4 (HT and Hájek variance approximations):
  Var( hat{T}_HT ) ≈ ∑_i ∑_j ( ( pi_ij - pi_i pi_j ) / ( pi_ij ) ) * ( y_i / pi_i ) * ( y_j / pi_j ) (requires second-order inclusion probabilities);
  ratio (Hájek) linearization: Var( hat{Y}_Hajek ) ≈ Var( ( ∑ w_i ( y_i - hat{Y}_Hajek ) ) / ( ∑ w_i ) ).
• S303-5 (Replicate-weight variance):
  Var_rep( hat{theta} ) = c_rep * ( 1 / R_rep ) * ∑_{r=1}^{R_rep} ( hat{theta}^{(r)} - hat{theta} )^2, with scheme factor c_rep (JK/BRR/Bootstrap).
• S303-6 (Weight calibration — quadratic program):
  Minimize ∑ g( w_i*, w_i ) subject to X_cal^T w* = t_cal; a common g is the quadratic distance ( w_i* - w_i )^2 / ( w_i * q_i ); the solution satisfies
  w* = w ⊙ h( X_cal, t_cal ) (⊙ element-wise product), with h determined by the Lagrangian subproblem.
• S303-7 (Raking / iterative proportional fitting):
  Initialize w(0) = w; over each margin m, update w(k+1) = w(k) * ( t_m / ( ∑_{i∈m} w(k) ) ) until || X_cal^T w - t_cal || ≤ tol_rake.
• S303-8 (Streaming Poisson/Bernoulli sampling):
  For event e, include independently with probability p_e, so pi(e) = p_e, w_e = 1 / p_e; sliding-window mean
  hat{mu}_w( t; Delta_t ) = ( ∑_{e: |tau_e - t| ≤ Delta_t/2} w_e y_e ) / ( ∑_{e: |tau_e - t| ≤ Delta_t/2} w_e ).
• S303-9 (Reservoir sampling of size K):
  Maintain an equal-probability reservoir with pi(i) = K / n_seen; weight w_i = n_seen / K; record the sequence n_seen(t) for publication.
• S303-10 (Finite-population correction FPC):
  FPC = sqrt( ( N - n ) / ( N - 1 ) ); SRS mean variance Var( hat{Y} ) ≈ ( S^2 / n ) * ( 1 - n / N ).

V. Statistical Process M30-3 (Design → Sampling → Weights → Variance → Calibration → Release)

• Readiness
  Specify the population frame and stratification/cluster structure; set sample size and power targets; load control totals t_cal and window Delta_t; run check_dim.
• Sampling
  Generate inclusion probabilities pi(i); for streams, apply p_e or reservoir; record random seeds and versions.
• Weights
  Compute initial weights w_i = 1 / pi(i); normalize to W_norm ≈ 1; apply extremum limits or smoothing; emit a weight log.
• Variance
  Choose linearization or a replicate-weight scheme; for multistage designs, estimate rho_icc and report DEFF_c; synthesize total variance.
• Calibration
  Run raking or quadratic programming to match t_cal; after calibration, re-check extremes and W_norm; record convergence diagnostics.
• Release
  Compute target estimates and intervals; when latency applies, record both T_arr conventions and delta_form in parallel; persist manifest.stats.sampling.* and sign for freeze.

VI. Contracts and Assertions (Examples C30-31x)

• C30-311 (Normalization): | W_norm - 1 | ≤ tol_w_norm.
• C30-312 (Weight extremes): max(w_i) / median(w_i) ≤ cap_w_max, CV(w) ≤ cap_cv_w.
• C30-313 (Calibration convergence): || X_cal^T w* - t_cal ||_1 ≤ tol_cal, iters_rake ≤ cap_iters.
• C30-314 (Replication sufficiency): R_rep ≥ R_min, replicate-vs-linearization variance gap ≤ tol_var_gap.
• C30-315 (Cluster correlation reporting): for cluster designs, report an interval for rho_icc and DEFF_c; if unavailable, mark as unknown and downgrade the release.
• C30-316 (Arrival-time discrepancy): for T_arr-dependent statistics, assert delta_form ≤ tol_Tarr.
• C30-317 (Streaming coverage): window coverage cov_rate(Delta_t) ≥ tol_cover, missing capture estimate ≤ tol_miss_rate.

VII. Implementation Bindings I30-*

1. I30-31 compute_weights(ds, scheme) -> w
   • scheme ∈ {SRSWOR, STRAT, CLUSTER_PPS, POISSON, BERNOULLI, RESERVOIR}; outputs w, W_norm, logs, and extremum report.
   • Invariants: sum(w)/N_hat ≈ 1, seed/version persisted.
2. I30-32 estimate_variance(ds, method) -> var_report
   method ∈ {LINEARIZATION, JK, BRR, BOOT}; outputs SE, DEFF, rho_icc (if applicable).
3. I30-33 calibrate_weights(w, X_cal, t_cal, method) -> w*
   method ∈ {RAKING, QP_CAL}; outputs convergence curves and residuals.
4. I30-34 stream_sampler(stream, policy) -> sample, w
   policy ∈ {POISSON(p), RESERVOIR(K), WINDOW(Delta_t)}; outputs sample and time metadata.
5. I30-35 emit_sampling_manifest(design, weights, variance) -> manifest.stats.sampling
   Writes TraceID, design summary, parameters, contract evaluation, and signature.

VIII. Cross-References

• Schema and field registration: Methods.Cleaning v1.0, Ch. 3.
• Units, dimensions, and checks: Methods.Cleaning v1.0, Ch. 4.
• Timeline, windowing, and the two T_arr conventions: Methods.Cleaning v1.0, Ch. 5–6.
• Resampling and cross-validation: this volume, Ch. 5; error control and sequential methods: this volume, Ch. 6.

IX. Quality Metrics and Risk Control

• SLI/SLO
  Weight stability CV(w), rate of cap_w_max violations; variance-method consistency var_gap; DEFF trend; window coverage and miss rate; latency_ms_p99 (compute latency).
• Risk control and rollback
  Triggers: failures of C30-312/313/316; Actions: rollback to uncalibrated weights or the previous signed manifest, downgrade release tier, and raise alerts.

Summary