Binomial Approximation for Count Variance Method

A Binomial Approximation for Count Variance Method is a variance approximation method that models classification counts as binomial random variables with variance n*p*(1-p) for finite samples.

• AKA: Binomial Count Variance Method, Bernoulli Trial Variance Method, Finite Sample Variance Approximation, Binomial Proportion Variance Method.
• Context:
  • It can typically assume fixed sample size n with success probability p.
  • It can typically compute variance as n*p*(1-p) for binomial counts.
  • It can typically provide exact variance for binary outcomes with known n.
  • It can often differ from Poisson Approximation for Count Variance Method which assumes variance equals mean.
  • It can often be more appropriate for small sample sizes than Poisson approximation.
  • It can often converge to normal approximation for large n via central limit theorem.
  • It can range from being a Simple Binomial Approximation for Count Variance Method to being a Overdispersed Binomial Approximation for Count Variance Method, depending on its dispersion parameter.
  • It can range from being an Independent Binomial Approximation for Count Variance Method to being a Correlated Binomial Approximation for Count Variance Method, depending on its trial dependence.
  • It can range from being a Fixed-p Binomial Approximation for Count Variance Method to being a Beta-Binomial Approximation for Count Variance Method, depending on its probability model.
  • It can range from being an Exact Binomial Approximation for Count Variance Method to being a Normal-Approximated Binomial Approximation for Count Variance Method, depending on its sample size.
  • ...
• Example(s):
  • True Positive Count Variances, such as:
    • n=100 positives, p=0.9 detection rate → Var(TP)=100*0.9*0.1=9.
    • n=1000, p=0.7 → Var=1000*0.7*0.3=210.
  • False Positive Count Variances, such as:
    • n=500 negatives, p=0.05 false alarm rate → Var(FP)=500*0.05*0.95=23.75.
    • Specificity=0.95 → FP rate=0.05 for variance calculation.
  • Confidence Interval Applications, such as:
    • Wilson score interval using binomial variance.
    • Clopper-Pearson exact binomial CI.
  • ...
• Counter-Example(s):
  • Poisson Approximation for Count Variance Method, which assumes variance=mean.
  • Hypergeometric Variance Method, which models sampling without replacement.
  • Multinomial Variance Method, which handles multiple categories.
• See: Variance Approximation Method, Binomial Distribution, Bernoulli Trial, Poisson Approximation for Count Variance Method, Variance Estimation, Proportion Estimation, Wilson Score Interval, Finite Sample Theory, Success Probability, Binary Classification Count.

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