Baseline-Risk And Treatment-Effect Dependence

Why Dependence Matters

A prior for a treatment effect is not always independent of the prior for baseline risk. For a binary clinical endpoint, the expected event rate in the treatment arm depends jointly on the control-arm event risk and the treatment effect. If a prior makes these quantities dependent, the resulting joint predictions can differ substantially from those under independent priors—even when the marginal prior distributions for each quantity are the same.

This matters because clinical judgments often concern combinations: for example, whether a large treatment benefit is plausible when untreated risk is high, or whether treatment is allowed to increase risk in patients with low baseline risk. Checking the two marginals separately cannot answer those questions. The joint prior and the trial outcomes it generates must be examined together.

For this tutorial, consider a binary endpoint measured over a specified follow-up period. Conditional on the arm-specific event probabilities, the event counts in two randomized arms are modeled as independent binomial counts: \(Y_C\mid p_C\sim\operatorname{Binomial}(n_C,p_C)\) and \(Y_T\mid p_T\sim\operatorname{Binomial}(n_T,p_T)\). A prior predictive check draws the parameters from their joint prior and then generates outcomes from these sampling models. The arm outcomes are independent conditional on the parameters, but they need not be independent after averaging over prior uncertainty.

Support and Dependence on the Risk-Difference Scale

Because probabilities must lie between zero and one, a risk-difference model requires \(-p_C\leq\Delta\leq1-p_C\). This is a joint constraint: the allowable treatment effect depends on the baseline risk. A prior that gives plausible separate values to \(p_C\) and \(\Delta\) can still assign positive probability to impossible pairs.

Dependence is a property of the joint distribution, not simply a verbal belief that the parameters are “related.” For example, the prior correlation between \(p_C\) and \(\Delta\) summarizes linear association on those scales, but it does not fully describe nonlinear dependence or determine whether the joint support is valid. Also, a correlation can change when the effect is expressed on another scale. Clinical interpretation should therefore accompany any correlation summary.

What the Dependence Does to Prior Predictive Outcomes

Dependence can affect the spread and pairing of potential arm outcomes without changing either arm’s prior mean. Since \(p_T=p_C+\Delta\), linearity of expectation gives \(E(p_T)=E(p_C)+E(\Delta)\) whenever the expectations exist. But the treatment-risk variance includes a covariance term:

This identity follows by expanding the variance of a sum. It shows directly why the sign and size of the dependence matter. A negative covariance reduces variation in treatment risk; a positive covariance increases it. Neither direction is automatically clinically appropriate.

The same issue appears in paired arm predictions. By \(p_T=p_C+\Delta\),

Consequently, a negative association between baseline risk and treatment effect does not necessarily imply a negative association between the two arm risks. In the example above, \(\operatorname{Cov}(p_C,p_T)=0.0225-0.0045=0.018\), which is positive. High-risk settings still tend to have higher treatment-arm risks, even though the risk difference becomes more negative as baseline risk rises.

Checking a Joint Prior Through Trial Replicates

A prior predictive check translates a joint prior into quantities that can be discussed in trial terms: arm event rates, event counts, their difference, and the way those quantities co-occur. The following derivation shows how both parameter uncertainty and binomial sampling variability contribute to predictive count variation. For a control count \(Y_C\), the law of total variance gives

The first term is average binomial variation conditional on the event probability; the second is variation across prior values of that probability. If the two arm counts are conditionally independent given the parameters, the law of total covariance similarly yields \(\operatorname{Cov}(Y_C,Y_T)=n_Cn_T\operatorname{Cov}(p_C,p_T)\). Shared uncertainty about baseline risk can therefore make the predictive arm counts strongly associated.

Choosing a Scale and Assessing Clinical Plausibility

Risk differences are often directly interpretable, but their support depends on baseline risk. A different option is to specify a treatment effect as a log odds ratio, \(\theta=\operatorname{logit}(p_T)-\operatorname{logit}(p_C)\), where \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). Then \(p_T=\operatorname{logit}^{-1}\{\operatorname{logit}(p_C)+\theta\}\), which lies between zero and one for finite values. This avoids impossible probabilities from an unconstrained log-odds effect, but it does not remove the need to check dependence. It also does not make effects on the risk-difference scale constant across baseline risks.

For instance, if the odds ratio is 0.5, then \(p_T=p_C/(2-p_C)\). At baseline risk \(p_C=0.10\), this gives \(p_T=0.0526\), a risk difference of \(-0.0474\); at \(p_C=0.40\), it gives \(p_T=0.25\), a risk difference of \(-0.15\). These are both reductions in event risk, but the absolute reductions differ. A prior that is independent on the log-odds scale does not imply independence between baseline risk and risk difference.

When checking a candidate joint prior, examine simulated pairs \((p_C,\Delta)\) or \((p_C,\theta)\) as well as the induced arm outcomes. Ask whether the pairings reflect clinical knowledge, whether the signs and magnitudes of effects are plausible at different baseline risks, and whether trial-sized counts include credible ranges and combinations. For a risk-difference prior, verify that all support lies within the probability constraints. For either scale, a marginal summary or a single correlation coefficient is not a substitute for examining the joint pattern.

An independent-prior specification is a useful comparison when independence is clinically defensible and the joint support is valid. A dependent joint prior can instead encode a believed relationship, for example through a specified conditional distribution of treatment effect given baseline risk or a joint distribution with stated dependence structure. These are modeling choices, not diagnostics that prove a prior is correct. The relevant comparison is whether each choice yields credible joint prior predictions for the intended trial.

Common Pitfalls and Limitations

• Checking only marginal priors. Plausible values for \(p_C\) and \(\Delta\) separately do not guarantee plausible or even valid combinations. Check their joint support and the induced \(p_T\).
• Assuming independence is neutral. Independence is a substantive restriction, and on the risk-difference scale it may generate invalid combinations. Compare its joint predictions with clinically defensible alternatives.
• Interpreting correlation as the whole relationship. Correlation summarizes linear association on a chosen scale. It does not determine tail behavior, nonlinear patterns, or clinical plausibility.
• Confusing outcome association with treatment-effect dependence. The arm risks can be positively associated because baseline risk varies, even when baseline risk and the risk difference are negatively associated.
• Assuming a valid parameterization ensures a credible prior. A log-odds model prevents probabilities outside zero and one, but its implied absolute effects and dependence patterns may still be clinically implausible.

Prior predictive checks can reveal consequences of assumptions; they cannot establish that a dependence structure is true. The result also depends on the chosen endpoint, follow-up period, treatment-effect scale, and trial design. Those features must match the clinical question for the resulting predictions to be useful.