Worked Survival Conflict Analysis

Survival Effects and Prior-Data Conflict

In a time-to-event trial, a treatment effect is often summarized by a hazard ratio. The hazard is the instantaneous event rate among individuals still event-free. Under a proportional-hazards model, the hazard ratio is constant over time; its logarithm is the treatment coefficient on the log-hazard scale. A negative log hazard ratio corresponds to a hazard ratio below one, which may indicate a lower event hazard for treatment when the event is undesirable.

Historical survival data can inform a current trial, but the historical and current treatment effects may differ. Differences in eligibility, follow-up, background care, endpoint definitions, or treatment implementation can create genuine prior-data conflict. Model and analysis differences can also create apparent conflict. Robust borrowing methods let the current estimate influence how much historical information contributes, rather than assuming that historical information is always fully commensurate.

Definition: Let $\theta_c$ denote the current treatment effect on the log-hazard-ratio scale, and let $\hat{\theta}_c$ be its estimated value. Let $\theta_h$ denote the corresponding historical effect or historical posterior mean. A normal summary likelihood represents the current estimate as $\hat{\theta}_c\mid\theta_c\sim N(\theta_c,s_c^2)$, where $s_c^2$ is its estimated sampling variance. The hazard ratio corresponding to $\theta_c$ is $\exp(\theta_c)$.

This summary likelihood is an approximation, not an automatic property of every survival analysis. It is reasonable when the estimator is approximately normal and its standard error is adequately estimated. Interpretation also depends on the underlying survival model: proportional hazards, a common estimand, suitable handling of censoring, and any covariate adjustment used in producing the estimate. Historical and current estimates should refer to comparable treatment contrasts and populations. If those conditions fail, numerical borrowing can be precise while answering the wrong clinical question.

Robust Mixture Borrowing on the Log-Hazard Scale

A robust mixture prior assigns prior probability to a borrowing component and to a weakly informative component. In a simple normal-summary analysis, write the prior as a mixture of two normal distributions. The borrowing component is centered on a historical estimate and has relatively small variance; the weak component has a much larger variance and is centered at a neutral value such as zero. After observing the current estimate, Bayes’ rule updates the probability of each component.

Definition: Let $B$ indicate the borrowing component, with prior probability $w_0=P(B)$. Suppose $\theta_c\mid B\sim N(m_B,v_B)$ and $\theta_c\mid \neg B\sim N(m_W,v_W)$. Under the current normal summary likelihood, the posterior borrowing probability is $$ w=P(B\mid\hat{\theta}_c)= \frac{w_0 f_B(\hat{\theta}_c)} {w_0 f_B(\hat{\theta}_c)+(1-w_0)f_W(\hat{\theta}_c)}, $$ where $f_B$ and $f_W$ are the normal predictive densities of $\hat{\theta}_c$ under the two prior components. Within either component $j$, the posterior variance and mean are $$ v_j^*=\left(\frac{1}{v_j}+\frac{1}{s_c^2}\right)^{-1}, \qquad m_j^*=v_j^*\left(\frac{m_j}{v_j}+\frac{\hat{\theta}_c}{s_c^2}\right). $$ The posterior distribution is a mixture of these component posteriors, weighted by $w$ and $1-w$.

The predictive density under component $j$ is normal with mean $m_j$ and variance $v_j+s_c^2$. Thus, a current estimate that is much more plausible under the weak component reduces the posterior probability of borrowing. This is conflict-sensitive borrowing: the data update the component probability, rather than requiring a preliminary conflict test followed by a fixed accept-or-reject decision.

Worked Example: Updating the Borrowing Probability

Suppose a hypothetical analysis assigns prior probability $w_0=0.70$ to borrowing. Under the borrowing component, the prior for the current log hazard ratio is $N(-0.4408,0.0108)$; under the weak component it is $N(0,2.2164)$. The current estimate is $\hat{\theta}_c=-0.30$ with sampling variance $s_c^2=0.04$. The two predictive densities at the observed estimate are approximately $f_B(-0.30)=1.4563$ and $f_W(-0.30)=0.2603$. For example, the weak-component predictive distribution is $N(0,2.2564)$, since $2.2164+0.04=2.2564$.

Applying Bayes’ rule gives $$ w=\frac{0.70(1.4563)} {0.70(1.4563)+0.30(0.2603)} =\frac{1.01941}{1.01941+0.07809} \approx 0.9288. $$

The current estimate is substantially more predictive under the borrowing component, so its posterior probability rises from 0.70 to about 0.9288. This is a probability of the component within the specified prior model, not the probability that historical information is objectively correct. Nor does it establish that the historical and current effects are identical.

Component Posteriors and the Treatment Effect

A posterior borrowing probability alone does not give the posterior treatment effect. Each component also updates the value of $\theta_c$ through the current likelihood. The overall posterior mean is the weighted average of component posterior means. Its variance includes both within-component uncertainty and between-component separation:

E(\theta_c\mid\hat{\theta}_c)=wm_B^*+(1-w)m_W^*, \qquad \operatorname{Var}(\theta_c\mid\hat{\theta}_c) =w\{v_B^*+(m_B^*)^2\}+(1-w)\{v_W^*+(m_W^*)^2\} -\{E(\theta_c\mid\hat{\theta}_c)\}^2.

The second term in the variance calculation matters when the component means differ. Reporting only the variance within the most probable component can understate posterior uncertainty. A mixture posterior also need not be normal, so a credible interval should be computed from the full mixture rather than by attaching a normal interval to its mean.

Worked Example: A Conflicting Current Survival Estimate

Consider a second hypothetical analysis with prior borrowing probability $w_0=0.70$, current estimate $\hat{\theta}_c=-0.05$, and $s_c^2=0.02$. Let the borrowing prior be $N(-0.60,0.0108)$, and the weak prior be $N(0,2.2164)$. The predictive distributions are $N(-0.60,0.0308)$ and $N(0,2.2364)$, respectively. The latter variance is $2.2164+0.02=2.2364$.

The predictive density under the borrowing component is about 0.0167: its normalizing factor is approximately $1/\sqrt{2\pi(0.0308)}=2.272$, and its exponential factor is $\exp\{-0.55^2/[2(0.0308)]\}\approx0.00736$. The weak-component density is approximately 0.2666. Therefore, $$ w=\frac{0.70(0.0167)} {0.70(0.0167)+0.30(0.2666)} \approx 0.1276. $$ The observed estimate is much less predictive under the historical borrowing component, and the posterior borrowing probability falls from 0.70 to about 0.128.

For the borrowing component, the posterior variance and mean are $$ v_B^*=\frac{(0.0108)(0.02)}{0.0108+0.02} =\frac{0.000216}{0.0308}\approx0.007013, \qquad m_B^*=\frac{(-0.60)(0.02)+(-0.05)(0.0108)}{0.0308} =\frac{-0.01254}{0.0308}\approx-0.4071. $$ For the weak component, $$ v_W^*=\frac{(2.2164)(0.02)}{2.2164+0.02} =\frac{0.044328}{2.2364}\approx0.01982, \qquad m_W^*=\frac{(-0.05)(2.2164)}{2.2364} =\frac{-0.11082}{2.2364}\approx-0.04955. $$

The overall posterior mean is therefore approximately $0.1276(-0.4071)+0.8724(-0.04955)=-0.0952$. On the hazard-ratio scale, exponentiating this posterior mean gives $\exp(-0.0952)\approx0.9092$; it is the hazard ratio corresponding to the posterior mean log effect, not generally the posterior mean hazard ratio. The posterior remains a mixture, with uncertainty reflecting both component variances and the difference between their means.

Commensurate Priors and Precision

A commensurate model expresses the relationship between current and historical effects through a precision parameter. Conditional on the historical effect, a common formulation is $\theta_c\mid\theta_h,\tau\sim N(\theta_h,1/\tau)$. Large $\tau$ means a tight link and potentially strong borrowing; small $\tau$ permits greater separation. If historical uncertainty is represented by $\theta_h\sim N(m_h,v_h)$, integrating over $\theta_h$ gives $\theta_c\mid\tau\sim N(m_h,v_h+1/\tau)$. This addition prevents uncertainty in the historical effect from disappearing simply because the commensurability precision is large.

The current estimate informs $\tau$ through its marginal likelihood. A prior on $\tau$ is required; it may be continuous or may, for illustration, put mass on a small number of precision values. Conditional on each value, the usual normal-normal update applies. The resulting posterior averages across precision values, rather than treating the selected precision as known.

Worked Example: A Discrete Commensurate Precision Prior

Use the hypothetical current estimate $\hat{\theta}_c=-0.05$ with sampling variance $0.02$, and a historical posterior $N(-0.60,0.0108)$. Suppose $\tau$ has prior probability $0.50$ at each of 1 and 25. For $\tau=1$, the marginal prior variance for the current effect is $0.0108+1=1.0108$, so the predictive variance of the estimate is $1.0308$. Its predictive density at $-0.05$ is approximately 0.339. For $\tau=25$, the marginal prior variance is $0.0108+0.04=0.0508$, and the predictive variance is $0.0708$; the density at $-0.05$ is approximately 0.177.

With equal prior probabilities, the posterior probability of $\tau=1$ is $$ P(\tau=1\mid\hat{\theta}_c) =\frac{0.50(0.339)}{0.50(0.339)+0.50(0.177)} \approx0.657. $$ The data favor the lower precision value, which permits more separation from the historical effect.

Conditional on $\tau=1$, the current-effect prior is $N(-0.60,1.0108)$. Its posterior mean is $$ \frac{(-0.60)(0.02)+(-0.05)(1.0108)}{1.0308} =\frac{-0.06254}{1.0308}\approx-0.0607. $$ Conditional on $\tau=25$, the posterior mean is $$ \frac{(-0.60)(0.02)+(-0.05)(0.0508)}{0.0708} =\frac{-0.01454}{0.0708}\approx-0.2054. $$ The posterior mean averaged over the two precision values is approximately $0.657(-0.0607)+0.343(-0.2054)=-0.1103$. This calculation illustrates how a commensurate model can respond to conflict by favoring lower precision, while still retaining some historical information. In a continuous-$\tau$ model, the corresponding posterior averaging is usually numerical.

Diagnostics, Alternatives, and Limitations

A conflict-sensitive model should be accompanied by checks of the survival analysis and the borrowing assumptions. Review the proportional-hazards diagnostics and censoring assumptions underlying the current estimate; compare endpoint definitions, follow-up, covariate adjustment, and eligibility across data sources; and examine whether the historical estimate is itself uncertain or based on a small effective sample. Sensitivity analyses over mixture weights, weak-component scales, and commensurability priors can show whether the treatment conclusion depends on a particular borrowing specification.

Apparent conflict from different estimands. If historical and current estimates refer to different treatment contrasts or populations, a prior model cannot make them comparable. Align the estimand or explain why the difference is clinically relevant.
Mixture sensitivity. The posterior borrowing probability depends on the prior mixture weight and on the weak component’s location and scale. A very broad component can behave differently from a moderately diffuse one.
Commensurate-prior sensitivity. The prior on precision determines how readily the model can reduce borrowing. Inspect the posterior for $\tau$ and check computation when it is estimated by simulation.
Model misspecification. A normal estimate with an inaccurate standard error, non-proportional hazards, or informative censoring can undermine the summary-likelihood calculation. Robust priors do not repair these failures.

No borrowing is a useful reference analysis, but it discards potentially relevant information. Complete pooling makes the strongest compatibility assumption and can be badly misleading under conflict. Robust mixtures instead represent discrete borrowing and weak-information possibilities; commensurate models express a graded relationship through precision. These approaches are not interchangeable, and their conclusions can differ because their prior structures differ. The next step in practice is sensitivity assessment, not interpreting a conflict probability as a formal model-selection verdict.

Common mistake: “The posterior borrowing probability is high, so the historical and current treatment effects must be equal.” That conclusion does not follow. The probability is conditional on the chosen mixture components and prior weight; it indicates relative predictive support within that model. Report the posterior treatment-effect distribution, the borrowing specification, and sensitivity to plausible alternatives.

A further interpretive caution is that a log hazard ratio is not a risk difference or a difference in survival probability. For example, $\exp(\theta_c)=0.91$ describes a hazard ratio under the proportional-hazards interpretation; it does not mean a 9-percentage-point reduction in event risk. Clinical importance also depends on baseline event risk, follow-up, and the outcome definition.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $\theta_c=-0.20$, what is the corresponding hazard ratio, and how should it be interpreted for an undesirable event?
Why can a current estimate far from the historical center reduce the posterior borrowing probability in a robust mixture?
In a commensurate model with $\theta_h\sim N(m_h,v_h)$, what is the marginal prior variance of $\theta_c$ conditional on $\tau$?
Name one reason historical and current estimates might appear to conflict even if the underlying treatment effect has not changed.

Solutions. (1) $\exp(-0.20)\approx0.8187$, a hazard about 18% lower under treatment, provided the event is undesirable and the proportional-hazards interpretation is appropriate. (2) Its predictive density under the borrowing component may be small relative to that under the weak component, so Bayes’ rule shifts posterior mass toward the weak component. (3) It is $v_h+1/\tau$, because integrating over uncertainty in $\theta_h$ adds its variance to the conditional variance. (4) Differences in covariate adjustment, endpoint definition, eligibility, follow-up, or survival-model assumptions can create apparent conflict.