Priors on Risk Differences and Relative Risks

Why the Effect Scale Matters

A prior on a treatment effect expresses beliefs about how treatment changes the event probability. But “change” can mean an absolute difference or a multiplicative comparison, and those are not interchangeable descriptions. A 25% relative reduction, for example, corresponds to a larger absolute reduction when the control risk is high than when it is low. The effect scale therefore affects both the clinical interpretation of a prior and the distribution it implies for treatment-arm risks.

Let \(p_C\) denote the event probability under control and \(p_T\) the event probability under treatment. Both are probabilities, so \(0\leq p_C\leq1\) and \(0\leq p_T\leq1\). This tutorial compares priors on the risk difference and the relative risk, focusing on their support and on the dependence between an effect and the baseline risk.

Feasible Support Is Part of the Prior

The probability constraints determine which values of an effect are possible. Because \(p_T\) and \(p_C\) both lie between 0 and 1, the risk difference always lies in \([-1,1]\). But once a particular control risk is specified, its feasible range is narrower:

The lower endpoint corresponds to \(p_T=0\), and the upper endpoint to \(p_T=1\). Thus, a prior for \(\Delta\) cannot be considered separately from its relationship to \(p_C\) if it is to define valid treatment risks. For example, at \(p_C=0.8\), a risk difference of \(0.4\) would imply \(p_T=1.2\), which is impossible.

For \(p_C>0\), the relative risk satisfies \(p_T=Rp_C\). Since \(p_T\) may equal zero, the probability-scale constraints give \(0\leq R\leq1/p_C\). If strictly positive treatment risk is assumed, the lower bound can instead be strict. The upper bound depends on the control risk: when \(p_C=0.2\), \(R\) can be as large as 5, whereas when \(p_C=0.8\), it cannot exceed 1.25. A relative-risk prior independent of baseline risk must therefore be checked against the range of baseline risks it allows.

This dependence is not just a technicality. A prior on \(\Delta\) that is symmetric around zero may require truncation at different points for different values of \(p_C\); a prior on \(R\) centered at 1 also has different feasible upper limits at different control risks. The resulting prior over valid pairs \((p_C,p_T)\) expresses beliefs about both baseline risk and treatment effect.

Translating Between Effect Scales

For a given positive control risk, the two scales are linked by

These equations show why an effect-scale prior cannot generally be translated without specifying or modeling the baseline risk. A fixed relative risk maps to different risk differences as \(p_C\) changes. Conversely, a fixed risk difference maps to different relative risks and can become impossible near the boundaries of the probability scale.

The transformation also changes densities. If \(p_C\) is fixed and a continuous prior has density \(f_R(r)\), then \(\Delta=p_C(R-1)\) has density

The factor \(1/p_C\) is the absolute derivative \(\left|dR/d\Delta\right|\). It ensures that the transformed density integrates to one on the corresponding support. When \(p_C\) is itself uncertain, the transformation is joint: a prior for \(p_C\) and \(R\) induces a prior for \(p_C\) and \(p_T\), subject to \(p_T=p_CR\leq1\). In the map \((p_C,R)\mapsto(p_C,p_T)\), the absolute Jacobian determinant is \(p_C\); consequently, for \(p_C>0\),

Thus, choosing a prior density on one parameterization and transforming it is not the same as choosing a density with the same algebraic form on another parameterization. This is the general parameterization issue discussed in the earlier material on uniform priors.

Building a Prior From Arm-Specific Risks

Another option is to specify a joint prior on \(p_C\) and \(p_T\), then derive the prior distribution of \(\Delta\) or \(R\). This makes the probability constraints explicit if each arm probability is supported on \([0,1]\). But it does not automatically encode a desired treatment-effect belief: the induced effect distribution depends on both marginal priors and their dependence.

For example, independent beta priors on the two event probabilities produce a risk difference whose mean and variance can be calculated directly. If \(p_C\sim\mathrm{Beta}(a_C,b_C)\), its mean is \(a_C/(a_C+b_C)\) and variance is \(a_Cb_C/\{(a_C+b_C)^2(a_C+b_C+1)\}\); the same formulas apply to \(p_T\). With independent arm probabilities, the risk-difference variance is the sum of the two variances. Independence is a modeling assumption, not a consequence of using beta distributions.

Choosing and Checking an Effect Scale

The risk difference and relative risk answer distinct clinical questions. The risk difference describes the absolute change in event probability and is often directly connected to numbers of events prevented or caused. The relative risk describes a proportional comparison and may align better with beliefs that a treatment has a similar proportional effect across settings. Neither scale is universally preferable: the scientifically meaningful prior depends on the clinical context and on what experts can reasonably judge.

Before using either effect prior, check its implied distribution on the probability scale. Examine representative baseline risks, the implied \(p_T\), and prior probabilities of clinically important benefit or harm. If a prior is truncated to enforce feasibility, inspect the resulting distribution after truncation rather than relying on its untruncated description. For a joint prior, check dependence as well as each marginal distribution: dependence can substantially alter the spread of \(p_T-p_C\).

Common Mistakes and Limitations

• Using \(R>0\) as the only relative-risk constraint. If \(p_C>0\), \(R=0\) is possible when \(p_T=0\), and \(R\) cannot exceed \(1/p_C\). Check both bounds.
• Treating an effect prior as baseline-free. Converting between \(R\) and \(\Delta\) requires \(p_C\). If baseline risk is uncertain, the induced prior on the other scale depends on its distribution and its relationship to the effect.
• Assuming equal-looking priors encode equal beliefs. A uniform prior on one scale does not generally transform to a uniform prior on another. Compare the implied quantities on the clinical probability scale.
• Truncating without reassessing. Restricting an infeasible prior changes its mass and may shift its mean or asymmetry. Quantify those changes before treating truncation as a harmless adjustment.

A limitation of either single-scale specification is that it emphasizes one type of comparison. If clinicians can articulate beliefs about absolute changes but the analysis uses a relative-risk prior, translation requires baseline-risk assumptions that may add uncertainty. Conversely, an absolute-effect prior may not reflect beliefs that proportional effects are stable across populations. The appropriate response is not to hide this choice, but to state it and assess the prior implications on both scales.