Signature
p_ik = exp(mu_i + delta_ik) / (1 + exp(mu_i + delta_ik))
| Inputs | Definition | Unit |
|---|---|---|
mu_i | Log odds of the event on the control treatment in trial i | log odds |
delta_ik | Log odds ratio for arm k against the control arm of trial i, zero for the control arm | log odds ratio |
p_ik | Probability of the event in arm k of trial i | 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
Binomial arm probability from log odds in one cell
With named cells BaselineLogOdds and LogOddsRatio, the formula returns the arm probability.
=EXP(BaselineLogOdds+LogOddsRatio)/(1+EXP(BaselineLogOdds+LogOddsRatio))
Assumptions
Binomial arm counts with a logit link in TSD 2
Each arm's count is binomial with a common probability for its patients, and treatment effects are additive on the log odds scale. When trials report outcomes at different follow-up times, TSD 2 keeps the binomial likelihood but uses a complementary log-log link, which assumes a constant hazard.
Worked examples
Control arm with a baseline log odds of minus 1.7346
Illustrative figures: a baseline log odds of minus 1.7346 corresponds to odds of about 0.1765 and a control arm probability of 0.15.
mu_i = -1.7346; delta_ik = 0; p_ik = 0.15
Treatment arm with an odds ratio of 0.5
Illustrative figures: a log odds ratio of minus 0.6931, an odds ratio of 0.5, halves the odds to about 0.0882 and gives an arm probability of about 0.0811.
mu_i = -1.7346; delta_ik = -0.6931; p_ik = 0.0811
Common errors
Adding 0.5 to zero cells before pooling binomial data
Some frequentist approaches add an arbitrary constant, usually 0.5, to zero cells. TSD 2 states that this biases the estimated effect, whereas the binomial likelihood allows a zero count and needs no special precaution for the occasional zero cell. It also warns that the models can become numerically unstable when several small trials have zero cells.
Odds ratio applied to a binomial probability instead of the odds
Multiplying the control probability of 0.15 by an odds ratio of 0.5 gives 0.075 rather than about 0.0811. The two agree closely only when the event is rare.
Sources
Binomial likelihood and logit link in NICE DSU TSD 2
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. Section 2.1.1, equations 2 and 3, which model arm counts as binomial with a logit link, the trial baselines mu_i as control log odds and delta as trial-specific log odds ratios; section 3.2 on the complementary log-log link; and sections 6.3 and 7.2 on zero cells and the bias from adding 0.5.
Canonical Identity
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