Priors for Baseline Event Probabilities

Baseline risk is a separate quantity to specify

In a trial with a binary endpoint, the control-arm event probability is the risk of that event under the control strategy over a stated follow-up period. It is not the treatment effect: it describes the reference arm, while a treatment effect compares outcomes between arms. Specifying baseline risk matters because it sets the expected number of control events and can affect the absolute size of a treatment benefit, even when the relative treatment effect is unchanged.

Here \(p_C\) denotes the control-arm event probability. The definition depends on the endpoint, the control strategy, the population, and the follow-up period. For example, a 30-day event probability and a one-year event probability are different quantities and should not be assigned the same prior without a substantive reason.

The beta distribution is useful because it has support on probabilities and is conjugate to the binomial likelihood. Its two parameters can describe many shapes: mass near low risks, near high risks, around an interior value, or spread broadly across the unit interval. A beta prior is not automatically suitable merely because the parameter is a probability. The chosen distribution must also represent plausible uncertainty for this particular control strategy and endpoint.

Choose the prior by its clinical implications

Start by making the target precise. Name the event, control arm, population, and time horizon. Then identify a plausible central risk and a defensible degree of uncertainty. A prior mean and standard deviation are one convenient way to express those judgments. If the intended prior mean is \(m\) and prior variance is \(v\), define \(k=a_C+b_C\). Since \(m=a_C/k\) and \(v=m(1-m)/(k+1)\), solving gives:

This parameterization is possible for a nondegenerate beta distribution when \(0<m<1\) and \(0<v<m(1-m)\). The upper bound is the variance of a Bernoulli variable with mean \(m\); a beta distribution with positive parameters has variance strictly below it. If elicited summaries violate these conditions, they cannot be matched exactly by a beta prior.

The sum \(a_C+b_C\) controls concentration: for a fixed mean, a larger sum gives a tighter distribution. The parameters are sometimes informally described as prior event and non-event counts, but they need not represent actual observed patients. In particular, external data may differ in population, follow-up, or control care, so treating them as literal counts can overstate how directly they inform the trial.

Moment matching is a starting point, not a complete validation. Inspect the implied distribution, especially where its tails lie relative to clinically plausible risks. If judgments are instead expressed as quantiles or ranges, a beta distribution can be calibrated numerically, but the resulting fit still needs to be checked against the intended beliefs. A convenient parameterization cannot rescue summaries that are internally inconsistent or a distribution shape that puts implausible probability on important regions.

Check implications through prediction

A useful diagnostic is the prior predictive distribution: what outcomes does the model imply before trial data are observed? For a control group of size \(n_C\), assume conditionally independent binary outcomes with a common probability \(p_C\):

Here \(Y_C\) is the number of control-arm events. Averaging over the prior gives a beta-binomial predictive distribution. For example, the probability of observing no events is the expectation of \((1-p_C)^{n_C}\) under the beta prior. Checking such probabilities can expose a prior that seems reasonable by its mean but assigns too much probability to outcomes that would be surprising clinically or operationally.

The derivation also gives the general conjugate update. If \(y_C\) events are observed among \(n_C\) conditionally independent control participants, the likelihood is proportional to \(p_C^{y_C}(1-p_C)^{n_C-y_C}\). Multiplying by the beta prior kernel yields:

This result depends on the stated binomial model: the endpoint is binary, the number followed is defined, and participants have a common event probability conditional on \(p_C\), with outcomes conditionally independent. If risks vary systematically across participants, sites, or follow-up times, a single binomial probability may be an inadequate description. The prior-predictive check should therefore consider whether the model as well as the prior is plausible.

Baseline uncertainty is not treatment-effect uncertainty

A prior for \(p_C\) describes uncertainty about the control risk. A prior for a treatment contrast describes uncertainty about how the treatment changes risk relative to control. These are distinct quantities, even though they jointly determine the risks in a two-arm trial. A narrowly specified treatment effect does not imply a precisely known control risk; likewise, a well-known baseline risk does not settle the treatment effect.

This distinction matters for design and interpretation. Baseline risk influences expected control events and, for a given relative effect, the absolute event difference. Treatment-effect uncertainty concerns the contrast itself. When both are uncertain, a trial analysis should not silently substitute one for the other or present a baseline prior as though it were a prior on efficacy.

Diagnostics, alternatives, and limitations

Before using a baseline beta prior, check the assumptions and implications systematically:

• Confirm the estimand. The event definition, arm, population, and follow-up horizon must match the probability being assigned a prior.
• Inspect support and tails. A beta prior assigns probability only between 0 and 1, but it may still assign too much probability near zero or one, or in clinically implausible regions.
• Check predictive outcomes. Consider plausible control event counts at the planned sample size, including zero or unusually high counts when relevant.
• Question transportability and homogeneity. Differences between the planned trial and the sources informing prior beliefs may make a single exchangeable control risk questionable.
• Check the model-data fit. If control outcomes show substantial variation across sites or patient subgroups beyond what a common-risk binomial model can represent, changing the beta prior alone does not resolve the model mismatch.

A fixed baseline risk is a simpler alternative when the risk can reasonably be treated as known, but it ignores uncertainty and can make downstream quantities appear too precise. A beta prior allows uncertainty while preserving the probability bounds and yields a simple update under the binomial model. Other probability-scale priors are possible; earlier material on Jeffreys priors and parameterization explains why a uniform density is not a parameterization-free default. Choosing among these approaches should be guided by the intended prior beliefs and model, not by convenience alone.

A beta prior is limited by its shape and by the assumptions of a single probability. It does not by itself account for participant-level risk predictors, site variation, changing hazards, or unequal follow-up. Those features require an appropriate model, not merely a different choice of \(a_C\) and \(b_C\). Nor should a prior be made artificially precise simply to represent a large external data set: differences in setting can make those data less informative for the trial’s target risk.