Binomial Proportion Confidence Interval Estimate

A Binomial Proportion Confidence Interval Estimate is a statistical interval estimate that provides a range of values within which the true probability of success in a Binomial Distribution is expected to fall (based on the outcomes of a finite number of Bernoulli trials).

AKA: BPCI, Binomial CI, Success Probability Interval.
Context:
- Input: Bernoulli Trial results (number of trials n, number of successes nₛ)
  - optional: confidence level
  - optional: estimation method
- Output: Confidence Interval (lower and upper bounds)
- Performance Measure: Coverage Probability, Interval Width
- ...
- It can (typically) assume that each trial in the experiment has only two outcomes (success and failure), the probability of success is constant across trials, and the trials are Statistically Independent
- It can (typically) require Sample Size considerations
- It can (typically) provide Interval Estimates
- It can (often) use Normal Approximations
- It can (often) need Continuity Corrections
- ...
- It can range from being an Exact Binomial Proportion Confidence Interval to being an Approximate Binomial Proportion Confidence Interval, depending on its calculation method
- It can range from being a Conservative Binomial Interval to being a Liberal Binomial Interval, depending on its coverage probability
- ...
- It can be constructed using various methods, such as the Wilson Score Interval, Clopper-Pearson Interval, and Agresti-Coull Interval
- It can provide insights into the uncertainty of proportion estimates in experiments with binary outcomes
- It can be influenced by factors like sample size, confidence level, and observed proportion
- It can be used to estimate the probability of success (p) from a series of Bernoulli Trials
- It can involve various methods of estimation due to the discrete and often complex nature of the binomial distribution
- ...
- It can support:
  - Drug Trial Analysis Tasks that estimate success rates in clinical trials
  - Opinion Poll Analysis Tasks that determine voter support in pre-election polling
  - Quality Control Tasks that assess product defect rates in manufacturing
  - Risk Assessment Tasks that evaluate failure probabilitys in system testing
  - Medical Diagnostic Tasks that analyze test accuracy in clinical settings
  - Market Research Tasks that measure customer preferences in surveys
  - ...
- ...
Examples:
- Wald Interval (1939) for quick approximations using normal distribution
- Wilson Score Interval (1927) for improved estimation with small samples
- Clopper-Pearson Interval (1934) for exact but conservative bounds
- Agresti-Coull Interval (1998) for better performance with small samples
- Jeffreys Interval (1961) using Bayesian approaches
- ...
Counter-Example(s):
- Confidence Interval for Means, which handle continuous data
- Poisson Rate Confidence Intervals, which address count data
- Prediction Intervals, which target future observations
- Tolerance Intervals, which contain population proportions
See: Coin Flipping, Confidence Interval, Bernoulli Trial, Binomial Distribution, Discrete Probability Distribution, Approximation Method.

References

2023

(Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval Retrieved:2023-11-26.
- In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). In other words, a binomial proportion confidence interval is an interval estimate of a success probability p when only the number of experiments n and the number of successes n_S are known.
  There are several formulas for a binomial confidence interval, but all of them rely on the assumption of a binomial distribution. In general, a binomial distribution applies when an experiment is repeated a fixed number of times, each trial of the experiment has two possible outcomes (success and failure), the probability of success is the same for each trial, and the trials are statistically independent. Because the binomial distribution is a discrete probability distribution (i.e., not continuous) and difficult to calculate for large numbers of trials, a variety of approximations are used to calculate this confidence interval, all with their own tradeoffs in accuracy and computational intensity.
  A simple example of a binomial distribution is the set of various possible outcomes, and their probabilities, for the number of heads observed when a coin is flipped ten times. The observed binomial proportion is the fraction of the flips that turn out to be heads. Given this observed proportion, the confidence interval for the true probability of the coin landing on heads is a range of possible proportions, which may or may not contain the true proportion. A 95% confidence interval for the proportion, for instance, will contain the true proportion 95% of the times that the procedure for constructing the confidence interval is employed.

1993

(Vollset, 1993) ⇒ Stein Emil Vollset. (1993). “Confidence intervals for a binomial proportion." In: Statistics in Medicine. Wiley Online Library.
- NOTE: This paper evaluates the probabilities of 95% confidence intervals computed by various methods for binomial proportions, providing coverage over a specific range and highlighting the challenges outside this range.

2002

(Brown et al., 2002) ⇒ Lawrence D. Brown, T. Tony Cai, Anirban DasGupta. (2002). “Confidence intervals for a binomial proportion and asymptotic expansions." In: The Annals of Statistics. Projecteuclid.org.
- NOTE: This study addresses the classic problem of interval estimation for a binomial proportion, critically evaluating the Wald interval and offering insights into alternative approaches.

2001

(Brown et al., 2001) ⇒ Lawrence D. Brown, T. Tony Cai, Anirban DasGupta. (2001). “Interval estimation for a binomial proportion." In: Statistical Science. Projecteuclid.org.
- NOTE: This paper revisits the problem of interval estimation for a binomial proportion, discussing the erratic behavior of the standard Wald confidence interval and exploring other methodologies.