Worked Continuous Conflict Analysis

Why Continuous-Outcome Conflict Needs Careful Analysis

A historical trial may estimate a treatment effect close to zero, while a current trial estimates a clinically important benefit. For a continuous outcome, the size of that discrepancy must be judged relative to the standard errors, the historical information's relevance, and the borrowing model. A difference of one outcome unit can be substantial when estimates are precise and modest when they are noisy.

This tutorial works with hypothetical trial summaries to show how conflict affects posterior effects, uncertainty, and effective borrowing. The calculations assume that both trials target the same treatment-effect estimand, measured in the same units. If the populations, endpoint definitions, follow-up, or analysis methods differ in ways that change the estimand, a model that borrows across the estimates may not repair that mismatch.

The normal likelihood is a suitable summary model when the estimate is approximately normal and its standard error is reliable. We treat the standard errors as known for the calculations below, as is common when working from sufficiently large-sample trial summaries. We also assume that the historical and current data are conditionally independent given their respective effects. These assumptions should be reconsidered when standard errors are estimated imprecisely, outcomes are highly skewed, or the estimates share participants or controls.

A Robust Mixture Prior for the Current Effect

A robust mixture prior represents two possible degrees of agreement with historical information. Its informative component transfers the historical estimate and uncertainty to the current effect. Its protective component is deliberately broader, allowing the current estimate more room to move away from history. The mixture does not guarantee that borrowing will be small: the prior component weights and the predictive fit of each component both matter.

For a normal component \(j\), suppose its prior is \(N(a_j,v_j)\). Combining it with the current normal likelihood gives a normal posterior. The result follows by multiplying the two densities and collecting the terms involving \(\theta_C\):

where \(V_j=(s_C^{-2}+v_j^{-1})^{-1}\) and \(m_j=V_j(s_C^{-2}\hat\theta_C+v_j^{-1}a_j)\). The predictive distribution of \(\hat\theta_C\) under that component is \(N(a_j,s_C^2+v_j)\). Thus the posterior component weight \(w_j\) is found by normalizing the component's prior weight times this predictive density.

Commensurate Models and Precision

A commensurate model relates the current effect to a historical effect through a precision parameter. Larger precision means stronger concentration of \(\theta_C\) around \(\theta_H\), hence a stronger potential for borrowing; smaller precision permits greater separation. Unlike a fixed informative prior alone, the model makes the degree of agreement an explicit part of the relationship. Here we evaluate fixed precision choices to show the consequences clearly; a full analysis may instead assign a prior to the precision and integrate over it.

Comparing Posterior Effects and Effective Borrowing

A useful reference for describing borrowing is the current-only analysis. Under a flat prior for \(\theta_C\), the current likelihood gives posterior mean 1.20 units and variance 0.25 units squared. Relative to this reference, the robust-mixture mean is 0.5248 units, and the fixed-\(\tau\) commensurate means are 0.7370 and 1.141 units. Each analysis gives a different balance of historical influence and current evidence.

One descriptive summary is the fraction of the distance from the current estimate to the historical mean that the posterior mean has moved:

For these data, the robust-mixture value is approximately \((1.20-0.524831)/(1.20-0.20)=0.6752\). The commensurate values are approximately \(0.4630\) for \(\tau=4\) and \(0.05897\) for \(\tau=0.25\). This is a descriptive mean-shift measure, not a universal effective-sample-size calculation. It can be misleading if the posterior lies beyond the historical mean, if the historical mean is not the relevant borrowing target, or if the model has several sources of historical information. Always pair it with the posterior uncertainty and model specification.

A concise comparison should report the current-only reference, the posterior mean and uncertainty under each model, and the assumptions that create the borrowing. In the mixture example, the posterior component probability is also informative: it indicates how the data updated the specified component probabilities, not the probability that a scientific hypothesis or model is true.

Diagnostics, Limitations, and a Common Pitfall

Before interpreting a calculation, verify that both estimates refer to the same effect, outcome scale, direction of benefit, and analysis population. Check the reported standard errors, the independence assumption, and whether the historical posterior already includes uncertainty about the historical effect. A commensurate model that treats \(m_H\) as fixed when \(v_H\) is material can overstate historical precision. A protective component that is broad but poorly centered may also behave differently from a genuinely data-driven no-borrowing analysis.

Useful diagnostics include plotting the component prior and predictive distributions against the current estimate, inspecting posterior component probabilities or the posterior for \(\tau\), and repeating the analysis across plausible prior weights, protective variances, and precision specifications. If the posterior effect changes sharply across reasonable choices, that sensitivity is a substantive result to report. Numerical integration or MCMC also requires computational checks appropriate to the method, such as verifying that numerical weights normalize and that chains explore the relevant regions when sampling is used.

The main alternative is no borrowing: analyze the current trial without historical information. It is transparent and avoids cross-trial exchangeability assumptions, but may be less precise. A fixed informative prior can be simpler and more precise when historical relevance is well justified, but it offers less protection against conflict. Robust mixtures and commensurate models allow disagreement to influence borrowing, at the cost of additional choices and sensitivity. None removes the need to assess clinical and design comparability, and none turns a conflict diagnostic into a definitive test of whether the trials are exchangeable.

• Pitfall: “The informative component had the largest prior weight, so the current posterior must be close to the historical estimate.” This ignores the current data's predictive fit under each component and the posterior component-specific estimates.
• Correction: Update component probabilities using their predictive densities, then calculate the posterior mean and total variance from the component posteriors. In the worked mixture, the informative component retained posterior weight about 0.7639, but the protective component still contributed both a distinct mean and appreciable between-component uncertainty.
• Limitation: A numerical borrowing summary depends on the reference and model. It cannot establish that historical and current populations are clinically exchangeable or that a borrowed estimate has causal validity.