Signature
P_0 = (1 - p)^n
| Inputs | Definition | Unit |
|---|---|---|
p | Probability that any one patient has the event within the period | probability |
n | Number of patients at risk, each followed for the whole period | patients |
P_0 | Probability that none of the n patients has the event within the period | probability |
|---|
Function
Binomial distribution of an event count among patients at risk
Gives the probability that exactly r of n patients have an event within a fixed period when each patient has the same probability p of the event and outcomes are independent. Read as a function of p for an observed count, the same expression is the binomial likelihood, proportional to p^r (1 minus p)^(n minus r), which peaks at the observed proportion r/n. The formulae below give the probability function, the chance of a zero count, the mean and variance of the count, the estimated probability with its standard error, the Wald and Wilson intervals and the logit model for binomial data in NICE DSU evidence synthesis. The conjugate beta update of a binomial probability is set out on the Beta Distribution page (HE-FM-BETA-002), and the general Wald interval for any estimate on the Asymptotic Normality page (HE-FM-AN-001).
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Implementations
Excel
Zero-count binomial probability in one cell
With named cells Prob and Patients, the formula returns the probability that no patient has the event.
=(1-Prob)^Patients
Assumptions
Binomial conditions for the zero-count probability
The conditions of HE-FM-BINOM-001 apply: a fixed number of patients followed for the whole period, a common probability and independent outcomes.
Worked examples
No events among 40 patients at a probability of 0.02
With an expected count of 0.8, a 40-patient study sees no events about 45% of the time although the probability is 0.02, the setting of the article's coverage comparison.
p = 0.02; n = 40; P_0 = 0.4457
No events among 40 patients at a probability of 0.15
With an expected count of 6, a zero count has a probability of only about 0.0015.
p = 0.15; n = 40; P_0 = 0.0015
Common errors
Reading a zero binomial event count as a zero probability
No events among 40 patients is consistent with a sizeable probability: at a true probability of 0.02 the chance of a zero count is about 0.45, and the Wilson interval for 0 of 40 reaches about 0.088. Entering zero as the model probability removes the event from the model.
Sources
Probability of no events from the binomial function in Bayesian Data Analysis
Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd edition. Boca Raton: Chapman & Hall/CRC; 2013. Appendix A, Table A.2, which gives the binomial probability function for counts 0 to n; at a count of zero it reduces to (1 minus p)^n.
Canonical Identity
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