Binomial Distribution

Binomial Distribution in Health Economic Models

Much of the evidence behind a decision model arrives as a count, such as the number of patients who responded, relapsed, died or had an adverse event out of the number treated. The binomial distribution is the standard model for such counts, and its likelihood turns "r events among n patients" into an estimate of the underlying probability with a measure of its precision.

Counting patients with an event

The binomial distribution applies when a fixed number of patients is followed for a set period and each patient either has the event of interest or does not. It also requires every patient to have the same probability of the event, with outcomes independent between patients. Under those conditions the number with the event has the following probability function.

$$P(R = r) = \binom{n}{r} p^{r} (1 - p)^{n - r}, \quad r = 0, 1, \ldots, n$$

where $R$ is the random number of patients with the event, $r$ is a particular count, $n$ is the number of patients at risk, $p$ is the probability that any one patient has the event within the period, and $\binom{n}{r}$ is the binomial coefficient $n!/(r!(n - r)!)$, the number of ways of choosing which $r$ of the $n$ patients have it. A single patient is the special case $n = 1$, a Bernoulli trial. The mean and variance of the count are given below.

$$E(R) = np, \qquad \operatorname{Var}(R) = np(1 - p)$$

where $E(R)$ is the expected number of patients with the event and $\operatorname{Var}(R)$ is its variance. The variance is largest when $p$ is one half and shrinks towards either bound. Zero counts depend on the expected count instead. The probability of no events is $(1 - p)^n$, which stays high when $np$ is small, so a rare event in a small study often gives a count of zero. The observed proportion $\hat{p} = r/n$ has variance $p(1 - p)/n$, which is why larger studies give more precise estimates. In Excel, BINOM.DIST returns the probability of exactly a given count, or of at most that count.

What a count says about a probability under the binomial likelihood

In an economic evaluation the count is observed and the probability is the unknown parameter. The binomial formula can therefore be turned around to ask which values of the probability are well supported by the data. Read as a function of $p$ for fixed data, the binomial probability becomes the likelihood.

$$L(p \mid r, n) \propto p^{r} (1 - p)^{n - r}$$

where $L(p \mid r, n)$ is the likelihood of the probability $p$ given $r$ events among $n$ patients; the binomial coefficient is dropped because it does not depend on $p$. The likelihood peaks at $\hat{p} = r/n$, the estimate given by maximum likelihood estimation, and its standard error is estimated as $\sqrt{\hat{p}(1 - \hat{p})/n}$.

In a Bayesian analysis, a beta distribution prior combined with this likelihood gives a beta posterior distribution, the property Gelman and colleagues call conjugacy, and the ISPOR-SMDM modelling task force on parameter estimation describes beta distributions as a natural match for binomial data in probabilistic sensitivity analysis; fitting the beta and choosing its prior are covered on the beta distribution page.

Binomial likelihoods in NICE DSU evidence synthesis

The NICE Decision Support Unit's Technical Support Document 2 (TSD 2) sets out a generalised linear modelling framework for pairwise and network meta-analysis of randomised controlled trials. Its core models start from binomial data. The number of events in each trial arm is modelled as binomial, and the arm probabilities are mapped to an unbounded scale by a link function, as follows.

$$r_{ik} \sim \text{Binomial}(p_{ik}, n_{ik}), \qquad \text{logit}(p_{ik}) = \mu_i + \delta_{i,1k}\, I_{\{k \neq 1\}}$$

where $r_{ik}$ is the number of events in arm $k$ of trial $i$, $n_{ik}$ is the number of patients in that arm, $p_{ik}$ is the arm's event probability, $\mu_i$ is the log odds of the event on the control treatment in trial $i$, $\delta_{i,1k}$ is the trial-specific log odds ratio for arm $k$ against the control in arm 1, and $I_{\{k \neq 1\}}$ equals one for arms other than the first and zero otherwise. The same generalised linear model structure accepts Poisson, normal and multinomial likelihoods with other links.

Two features of the binomial likelihood matter in appraisal. When trials report the proportion with an event at different follow-up times, TSD 2 keeps the binomial likelihood but uses a complementary log-log link, which assumes a constant hazard so that treatment effects become log hazard ratios. The second feature concerns zero counts. Because the binomial likelihood allows a count of zero, the document says the occasional trial with a zero cell needs no special precautions, whereas some frequentist approaches add an arbitrary constant, usually 0.5, and so bias the estimated effect. It also warns that the models can become numerically unstable when several small trials have zero cells.

The normal approximation and its limits with small counts

Many routine calculations for proportions replace the binomial with a normal distribution. When $n$ is large and $p$ is not close to zero or one, the binomial count is approximately normal with mean $np$ and variance $np(1 - p)$, a consequence of the central limit theorem. This approximation gives the familiar Wald confidence interval for a proportion, shown below.

$$\hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$$

where $\hat{p} = r/n$ is the observed proportion, $n$ is the number of patients and $z$ is the standard normal quantile, 1.96 for a 95% interval. Brown, Cai and DasGupta showed that the coverage of this interval behaves erratically, that the problem is far more persistent than had been appreciated, and that common textbook prescriptions about when it is safe cannot be trusted. For small samples they recommended the Wilson interval or the equal-tailed Jeffreys interval, which uses a Beta(0.5, 0.5) prior. The Wilson interval is shown below.

$$\frac{\hat{p} + \frac{z^2}{2n} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n} + \frac{z^2}{4n^2}}}{1 + \frac{z^2}{n}}$$

where the terms are as defined for the Wald interval. In an illustrative exact calculation over all 41 possible outcomes of a 40-patient study with a true event probability of 0.02, the nominal 95% Wald interval contains the true value in about 55% of samples and the Wilson interval in about 95%.

Binomial events in Markov cohort models and microsimulations

A Markov model moves patients between health states each cycle according to transition probabilities. In a cohort model the probability is applied to the share of the cohort in a state, so the number moving is the binomial mean, $Np$, for a cohort of $N$, and the trace shows expected values with no chance variation. In a Markov microsimulation, each simulated patient's transition is a Bernoulli draw, so for a fixed probability the number of $N$ patients who move in a cycle is binomial with variance $Np(1 - p)$.

The ISPOR-SMDM task force calls this chance variation stochastic or first-order uncertainty, the random variability in outcomes between identical patients, and separates it from parameter uncertainty in the probability itself. It notes that in patient-level simulations, assessing parameter uncertainty requires stochastic uncertainty to be eliminated, a quantity also called Monte Carlo error, whereas cohort models can address parameter uncertainty without that concern. A probabilistic microsimulation therefore samples at two levels, an outer draw of the probability, usually from a beta, and inner binomial outcomes for patients. When a state has more than two exits, the counts are multinomial and joint uncertainty in the probabilities is described by a Dirichlet distribution.

Worked example of an adverse event probability from a small study

The figures below are illustrative. A single-arm study of 40 patients reports 6 with a serious adverse event over one year, and a cohort model with a one-year cycle needs the annual probability of that event.

1. Estimate the probability. The maximum likelihood estimate is $\hat{p} = 6/40 = 0.15$. If the true probability were 0.15, the chance of seeing exactly 6 events would be as follows.

$$P(R = 6) = 3838380 \times 0.15^{6} \times 0.85^{34} = 0.174$$

where 3,838,380 is the binomial coefficient for choosing 6 patients from 40; Excel gives the same value with BINOM.DIST(6, 40, 0.15, FALSE).

2. Compare the Wald and Wilson intervals. The standard error is $\sqrt{0.15 \times 0.85 / 40} = 0.0565$, so the Wald interval is $\hat{p} \pm 1.96 \times 0.0565$, about 0.039 to 0.261. For the Wilson interval, with $z^2 = 3.8416$, the centre term is $\hat{p} + 3.8416/80 = 0.1980$, the half-width is $z \sqrt{0.0031875 + 0.00060025} = 0.1206$ and the denominator is $d = 1 + 3.8416/40 = 1.0960$. The interval runs from $(0.1980 - 0.1206)/1.0960 = 0.0706$ to $(0.1980 + 0.1206)/1.0960 = 0.2907$, shifted upwards and asymmetric because the estimate lies near the lower bound.

3. Repeat with zero events. With no events the Wald standard error is zero and the interval is the single point 0. The Wilson half-width is $z \sqrt{0.00060025} = 0.0480$, equal to the centre term, so the interval runs from 0 to $(0.0480 + 0.0480)/1.0960 = 0.0876$. No events in 40 patients is consistent with an annual probability of up to about 9%.

4. Describe uncertainty for the model. With a uniform Beta(1, 1) prior, the posterior for the original data is Beta(7, 35), with mean $E(p) = 7/42 = 0.167$. The uniform prior pulls the mean from 0.15 to 0.167, an effect of prior choice with small counts discussed on the beta distribution page.

5. Separate chance variation from parameter uncertainty. For 1,000 patients entering the state, a cohort model with $p = 0.15$ records $N p = 1000 \times 0.15 = 150$ events in the first cycle. A microsimulation of the same patients, with the probability held at 0.15, gives a binomial count with standard deviation $\sqrt{1000 \times 0.15 \times 0.85} = 11.29$; BINOM.INV(1000, 0.15, 0.025) returns 128 and BINOM.INV(1000, 0.15, 0.975) returns 172. That spread is stochastic noise, absent from the cohort model and averaged away in a microsimulation with enough patients. The uncertainty in the probability itself is far wider. The 2.5% and 97.5% points of Beta(7, 35) are 0.0715 and 0.2917, about 72 to 292 events per 1,000 patients, and it is this beta distribution that the probabilistic sensitivity analysis carries through to costs and QALYs.

Assumptions that decide whether the binomial fits

The binomial model rests on assumptions that trial and cohort data often strain. When one fails, a different model is usually needed.

• A fixed denominator. Every patient must be followed for the whole period; with incomplete follow-up or censoring, a simple proportion is biased and survival analysis is the usual route.
• Independence. Patients clustered within centres tend to resemble each other, inflating the variance beyond $np(1 - p)$; differences between trials in a synthesis are handled through random effects and between-study heterogeneity.
• One event per patient. Repeated events such as admissions are counts over person-time, which suit the Poisson distribution or, when overdispersed, the negative binomial; TSD 2 notes that the Poisson model for repeated events assumes the events are independent.

Where binomial reasoning goes wrong

One error is to trust the normal approximation with small or zero counts, for example reporting a Wald interval of zero width after a study with no events. A related habit is adding 0.5 to zero cells before pooling trials, which TSD 2 identifies as a source of bias that an exact binomial likelihood largely avoids.

A second group of errors mixes the two kinds of uncertainty. The variance $Np(1 - p)$ of a simulated count in the model population describes chance variation among patients rather than how well the probability is known, so it should not set the width of a parameter distribution; that width comes from the sampling variance of the study proportion, $p(1 - p)/n$, which shrinks as the study grows. The binomial and the beta are also sometimes presented as competing choices for the same input, when in Bayesian updating they work as a conjugate pair.

Sources

• Agresti A. Categorical Data Analysis. 3rd ed. Hoboken, NJ: Wiley; 2013.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford: Oxford University Press; 2006.
• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Medical Decision Making. 2012;32(5):722-732.
• Brown LD, Cai TT, DasGupta A. Interval estimation for a binomial proportion. Statistical Science. 2001;16(2):101-133.
• Casella G, Berger RL. Statistical Inference. 2nd ed. Pacific Grove, CA: Duxbury; 2002.
• Dias S, Welton NJ, Sutton AJ, Ades AE. NICE DSU Technical Support Document 2: A generalised linear modelling framework for pairwise and network meta-analysis of randomised controlled trials. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2011, last updated September 2016. Sections 2.1, 3.1, 3.2, 6.3 and 7.2.
• Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd ed. Boca Raton: CRC Press; 2013. Sections 2.1, 2.4 and 2.8, and Appendix A.
• Microsoft. BINOM.DIST function and BINOM.INV function. Microsoft Support; accessed 1 October 2026. https://support.microsoft.com/en-gb/excel/functions/binom-dist-function and https://support.microsoft.com/en-gb/excel/functions/binom-inv-function