Effective Sample Size Wilson F1 Method

An Effective Sample Size Wilson F1 Method is a Wilson score variance method that uses effective sample size n_eff = n + z² as the key mechanism providing implicit second-order corrections for F1 confidence intervals.

• AKA: n+z² Wilson F1 Method, Adjusted Sample Size F1 CI Method, Implicit Second-Order Wilson Method, Effective n Wilson F1 Method.
• Context:
  • It can typically compute effective sample size as n_eff = n + z² where z ≈ 1.96 for 95% CI.
  • It can typically provide implicit second-order correction achieving O(1/n) accuracy vs O(1/√n) for Wald.
  • It can typically ensure minimum coverage never falls far below 95% across parameter space.
  • It can often add approximately 3.84 to sample size for 95% intervals (z² ≈ 3.84).
  • It can often have larger relative effect when n is small (e.g., 38% increase for n=10).
  • It can often be mathematically equivalent to solving score test inversion.
  • It can range from being a Small-n Effective Sample Size Wilson F1 Method to being a Large-n Effective Sample Size Wilson F1 Method, depending on its relative adjustment.
  • It can range from being a Conservative Effective Sample Size Wilson F1 Method to being a Exact Effective Sample Size Wilson F1 Method, depending on its coverage achievement.
  • It can range from being a Analytical Effective Sample Size Wilson F1 Method to being a Numerical Effective Sample Size Wilson F1 Method, depending on its computation approach.
  • It can range from being a Fixed-z Effective Sample Size Wilson F1 Method to being a Adaptive-z Effective Sample Size Wilson F1 Method, depending on its critical value.
  • ...
• Example(s):
  • Small Sample Adjustments, such as:
    • n=10: n_eff = 10 + 3.84 = 13.84 (38.4% increase).
    • n=20: n_eff = 20 + 3.84 = 23.84 (19.2% increase).
    • n=100: n_eff = 100 + 3.84 = 103.84 (3.84% increase).
  • Coverage Achievements, such as:
    • Minimum coverage 94.8% even at extreme F1 values.
    • Average coverage 95.3% across all parameter combinations.
    • Never drops to 0% like Wald at boundaries.
  • Second-Order Behaviors, such as:
    • F1=0 in sample: Positive upper bound due to effective n.
    • F1=1 in sample: Lower bound < 1 from denominator adjustment.
    • Automatic variance inflation stronger near boundaries.
  • ...
• Counter-Example(s):
  • Raw Sample Size Method, which uses unadjusted n.
  • Fixed Inflation Method, which adds constant regardless of n.
  • First-Order Delta Method, which lacks second-order terms.
• See: Wilson Score F1 Confidence Interval Method, Effective Sample Size, Second-Order Correction, Score Test Inversion, Implicit Variance Inflation, Coverage Probability, O(1/n) Accuracy, Boundary Behavior, F1 Confidence Interval Construction Method, Statistical Order Theory.

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