49-Error Budget Card Template v1.0 | Chapter 7 — Sample Size, Power & Robust Options

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Chapter 7 — Sample Size, Power & Robust Options

I. Purpose & Scope

II. Prerequisites & Inputs

• Control equations & explicit path: when T_arr/Phi are involved, show both gamma(ell) and d ell in text; record delta_form ∈ {general, factored} in data/metadata.
• Power target & test settings: preregister α, target power 1−β, alternative effect size (d, r_target, etc.), and multiplicity control (FDR).
• Covariance & correlation structure: kernels and parameters (Σ) per Chapter 5; stratify/segment if needed.
• Gate alignment: confirm mapping to τ_T, r_phi_min, ε_flux guard, Q_res band.

III. Continuous Endpoints: Difference in Means

• Two-sided, independent samples, approx. homoscedastic:
  N_per_group ≈ 2 · ( z_{1−α/2} + z_{1−β} )^2 / d^2, where d = |μ_1 − μ_0| / σ_pooled.
• Paired design (e.g., same record_id pre/post):
  Replace σ_pooled with the std. of paired differences σ_Δ; effect size d = |Δ̄| / σ_Δ.
• Heteroscedastic correction: use Welch df approximation, or replace σ_pooled with a robust variance (Huber/HC family).

IV. Correlation Endpoints: r (e.g., Phase Consistency)

• Fisher-z approximation: z = atanh(r);
  N ≈ ( z_{1−α/2} + z_{1−β} )^2 / ( atanh(r_target) )^2 + 3.
• Intervals & decisions: align phases in the reference window first; report Fisher-z intervals and back-transform to r.

V. Proportions/Pass-Rate Endpoints

• Two-proportion comparison:
  N_per_group ≈ ( z_{1−α/2}√(2 p̄(1−p̄)) + z_{1−β}√( p_1(1−p_1)+p_0(1−p_0) ) )^2 / (p_1−p_0)^2, with p̄=(p_1+p_0)/2.
• Single proportion vs threshold: invert Clopper–Pearson or Wilson CI for required N; for heavy tails consider a Bayesian Beta–Binomial prior.

VI. Correlation Structure & Design Effects

• Cluster/channel correlation (design effect): N_eff = N / DEFF, DEFF ≈ 1 + (m−1)ρ (cluster size m, intra-cluster corr. ρ).
• Path correlation (along ℓ): per Chapter 5 kernels, use effective dof ν_eff ≈ L_total / (2 L_c); replace N with ν_eff in power formulae.
• Stratification & pairing: stratify by {batch/device/region}; pair/block within strata to reduce variance.

VII. Robust Options (Heavy Tail / Heteroskedasticity / Outliers)

• Robust effect size: use median difference and quantile spread (IQR/MAD) to define d_robust, and plug d ← d_robust into formulas.
• Loss & estimation: Huber/quantile regression; report equivalent second-order surrogates for Delta-method propagation.
• Conservative inflation: when evidence indicates heavy tails/non-normality, inflate sample size by ×(1.15–1.35); and/or increase B (Bootstrap/MC) for robust evaluation.

VIII. Non-Parametric Power via Bootstrap/Permutation

• Bootstrap power: simulate B ≥ 10^4; in each run, resample within train/validation strata; compute positive rate π̂_power and ensure it ≥ target power.
• Permutation power: under H0, permute labels to obtain the statistic distribution and threshold; for phase metrics, align within the reference window before permutation.

IX. Sequential & Adaptive Plans

• Group-sequential: O’Brien–Fleming / Pocock spending; at interim analyses, adjust α spending and update remaining N.
• Adaptive re-estimation: update N using interim estimates of σ, ρ, effect size; follow preregistered rules to maintain type-I error control.

X. Alignment with Gate Thresholds

• Compare U = k·u_c for ΔT_arr against τ_T; ensure the lower bound of r_phi ≥ r_phi_min; the upper quantile of ε_flux lies within the conservation band; Q_res is within the admissible band (G6).
• If any core gate fails, tag [Restricted] and publish qualitative trends and diagnostics only.

XI. Machine-Readable Templates

A. power_plan.yaml

version: "1.0.0"

targets:

- name: "DeltaT_arr"

type: "means_diff"

alpha: 0.01

power: 0.90

effect_size_d: 0.35

design: { paired: true, deff: 1.00 }

- name: "r_phi"

type: "correlation"

alpha: 0.01

power: 0.90

r_target: 0.60

fisher_z: true

- name: "epsilon_flux"

type: "threshold_rate"

alpha: 0.05

power: 0.80

p0: 0.15

p1: 0.07

robust:

loss: "huber"

inflate_factor: 1.20

stratification: ["batch","device","region"]

correlation:

path: { kernel: "exp", L_c_m: 25.0 }

sequential:

scheme: "obrien_fleming"

looks: 2

see:

- "EFT.WP.Core.Metrology v1.0:check_dim"

- "Data.Benchmarks v1.0:PROTO"

B. sample_size_report.md (minimal items)

# Sample Size & Power Report

- Targets: DeltaT_arr (d=0.35), r_phi (r=0.60), epsilon_flux (p0→p1).

- Assumptions: alpha=0.01/0.05, power=0.90/0.80, DEFF=1.10, L_c=25 m.

- Methods: z-approx / Fisher-z / exact proportion; robust inflate ×1.20.

- Outputs: N per group/stratum, sequential spending, bootstrap power curve.

XII. Cross-References

XIII. Checklist

• Preregistered α, 1−β, effect sizes, and multiplicity plan; aligned with gate thresholds.
• Correlation structure and design effects specified; N_eff computed; stratification/pairing defined.
• Robust options enabled for heavy tails/heteroskedasticity (loss, surrogates, inflation factor); MC/Bootstrap with B ≥ 10^4.
• Sequential/adaptive rules recorded with type-I error control; interim update strategy traceable.
• Exported power_plan.yaml / sample_size_report.md; see[]/references[]/version compliant and consistent.