Prior Distributions for Arm-Specific Effects

Why Specify Priors Arm by Arm?

In a multi-arm trial, each treatment arm can have its own effect relative to a defined reference. Before observing outcomes, a Bayesian analysis represents uncertainty about those effects with prior distributions. These distributions encode assumptions about plausible effect sizes and uncertainty; they are not estimates from the trial data. A clear prior specification therefore begins by defining the effect, its scale, and which direction represents benefit.

This tutorial considers independent normal priors for arm-specific effects. “Independent” describes the joint prior assumptions about the effects, while “normal” describes each effect’s marginal distribution. The assumptions are useful to state explicitly: using the same normal distribution for several arms does not assert that their true effects are equal.

The meaning of \(\theta_j\) depends on the chosen outcome and contrast. For example, if the outcome is a score and positive values represent improvement, \(\theta_j=2\) means an average improvement of two score points relative to the reference. The prior mean \(m_j\) is the center of prior belief about that effect. The variance \(v_j\) measures prior uncertainty in squared effect units: if effects are measured in score points, the variance is in score-points squared. A larger variance spreads more prior probability over values farther from the mean.

Normal priors are defined over the whole real line. They may be reasonable for an additive effect when values on both sides of zero are plausible and the prior places negligible mass on implausible extremes. If the effect scale or scientific constraints make this inappropriate, a normal prior may be a poor choice. The specification also assumes that all arms’ effects are defined comparably enough for their prior locations and scales to be interpreted.

Calibrating a Normal Prior

One practical way to assess a proposed prior is to translate it into probabilities for effects that have a direct scientific interpretation. Let \(\Phi(z)\) denote the cumulative distribution function of a standard normal random variable. For \(\theta_j\sim N(m_j,v_j)\), standardization gives

These expressions help check whether a prior matches the intended beliefs. The threshold \(c\), or interval \((a,b)\), should be stated in the effect’s units. For example, a prior mean of zero does not by itself mean that effects are expected to be practically negligible: a large variance can still assign substantial probability to large benefits and harms.

Arm-Specific Priors and a Common Prior

There are two common ways to specify the marginal distributions. With arm-specific priors, one may choose different means or variances, such as \(N(m_1,v_1)\) for one arm and \(N(m_2,v_2)\) for another. This can represent a priori differences in plausibility or uncertainty, but those differences should have a scientific or design-based rationale. With a common prior, every arm receives the same marginal distribution \(N(m,v)\). If the effects are still specified as independent, the joint prior density is the product of those identical densities.

For independent priors, the joint density factors as follows. If \(f_j\) is the normal density with mean \(m_j\) and variance \(v_j\), then independence means that the joint density at \((\theta_1,\ldots,\theta_K)\) is the product of the marginal densities. This factorization also justifies multiplying probabilities for events involving separate effects.

The probability identity follows because the joint event is a Cartesian product of sets for the separate effects, and the joint density factors; integrating that density over the product set separates into the product of the marginal integrals. This is a prior statement only. It does not say that estimates based on future data will be independent.

Assumptions, Diagnostics, and Alternatives

A normal-prior specification should be checked at the level of both the scientific estimand and the numerical distribution. The following checks are useful before proceeding to any data model:

• Confirm the effect definition and direction. State what the contrast is, the population and outcome it concerns, and whether positive values mean benefit or harm.
• Check the units. A prior mean must be in effect units and a prior variance in squared effect units. Confusing a standard deviation with a variance changes the distribution substantially.
• Translate the prior into probabilities. Calculate probabilities above clinically meaningful benefit or harm thresholds, and over plausible ranges. Ask whether those probabilities reflect the intended prior beliefs.
• Assess the independence assumption. Independence is a substantive joint assumption, not a consequence of having one prior distribution per arm. If prior information suggests related effects, a joint model that represents that dependence may be more appropriate.
• Check comparability across arms. A common prior is easiest to defend when the effects have the same definition and scale. Different indications, doses, populations, or prior evidence may motivate different marginal priors.

A simple diagnostic is to compare the prior’s central range and tail probabilities with scientifically plausible effects. For example, a prior can have a plausible mean but implausibly wide tails, or a narrow variance that rules out effects the evidence makes credible. Conversely, assigning different priors to arms without a clear rationale can encode an unintended preference for one arm. These checks are assessments of prior assumptions; they do not establish that a prior is correct.

The main alternatives answer different modeling needs. Arm-specific independent normals allow different prior means and variances while retaining independence. A common independent normal prior imposes the same marginal beliefs across arms but still allows different effect values. A model with a single shared effect parameter instead asserts equality of the modeled effects. A model that introduces dependence among effects requires an explicit joint distribution. These choices should not be conflated: they encode different beliefs, and none is justified merely by the fact that a trial has multiple arms.

Normal priors also have limitations. They assign positive probability to every real value, including extreme values and values in a direction considered implausible. Whether that is negligible depends on the chosen mean, variance, and effect scale. For example, a normal prior on a raw effect can behave poorly if the scientific parameter is constrained. The model’s scale should therefore be chosen before specifying prior numbers, and any alternative distribution should be calibrated to the same substantive beliefs.