Bayesian Borrowing Across Biomarker Strata

Why Borrow Across Biomarker Strata?

A trial may estimate treatment effects separately in marker-positive and marker-negative participants. If one stratum is small, its estimate can be imprecise. Bayesian borrowing offers a way to let information from the other stratum influence that estimate, while allowing the amount of influence to depend on how compatible the stratum effects appear. The central design question is not simply whether to borrow, but under which assumptions, and how much.

The earlier material on hierarchical models for treatment effects across subgroups introduces partial pooling. Here the focus is how to specify and examine that borrowing for two biomarker strata. Borrowing is a modeling assumption, not a consequence of randomization: randomization supports treatment comparisons within strata, but does not establish that treatment effects are exchangeable across them.

Definition: Let \(\theta_+\) and \(\theta_-\) denote the treatment effects in the marker-positive and marker-negative strata, respectively, on a prespecified scale (for example, a difference in mean outcome). Exchangeability means that, before seeing the stratum-specific treatment-effect data, the model treats the labels \(+\) and \(-\) symmetrically with respect to the effects. It does not require \(\theta_+=\theta_-\); a model can allow differences while assigning greater prior probability to modest differences than to large ones.

A Two-Stratum Model and Its Assumptions

Suppose each randomized stratum yields an approximately normal estimate \(Y_j\) of its treatment effect, where \(j\in\{+,-\}\), with known or suitably estimated standard error \(s_j\). A basic likelihood is

Y_j\mid\theta_j\sim N(\theta_j,s_j^2),\qquad j\in\{+,-\}.

This likelihood assumes the treatment-effect estimates are approximately unbiased for their stratum-specific effects, and that their sampling variances are correctly represented. Conditional independence is reasonable when the strata contain distinct participants and the estimates are formed separately; shared nuisance estimates or overlapping populations can induce dependence and require a joint likelihood instead. For binary or time-to-event outcomes, a normal likelihood for an effect estimate may be an approximation, and its adequacy should be checked against the trial’s analysis model.

A convenient parameterization separates the overall location from the between-stratum difference:

\beta=\frac{\theta_++\theta_-}{2},\qquad \delta=\theta_+-\theta_-, \qquad \theta_+=\beta+\frac{\delta}{2},\quad \theta_-=\beta-\frac{\delta}{2}.

A prior such as \(\delta\sim N(0,2\tau^2)\) expresses a presumption of exchangeability centered on no difference, with \(\tau\) controlling the expected heterogeneity. Small \(\tau\) concentrates prior mass near equal effects and encourages strong borrowing. Large \(\tau\) allows substantial differences and weakens borrowing. A prior on \(\beta\) expresses beliefs about the common location; it should be clinically plausible and specified on the chosen treatment-effect scale. One may assign a prior to \(\tau\) rather than fixing it, but then the posterior borrowing depends on that prior as well as on the data.

Exchangeability is substantive. Marker status may represent biology that modifies response, and a model centered on \(\delta=0\) can be inappropriate if strong prior evidence suggests systematic differences. Conversely, using independent priors for \(\theta_+\) and \(\theta_-\) avoids cross-stratum borrowing but gives up potential precision gains. The choice should be made before examining the trial’s stratum-specific outcomes, or justified by information external to those outcomes.

How the Difference Prior Produces Partial Borrowing

The borrowing mechanism can be seen directly when both strata have the same standard error \(s\). Define \(D=Y_+-Y_-\). Under the likelihood above,

D\mid\delta\sim N(\delta,2s^2).

Combining this likelihood with \(\delta\sim N(0,2\tau^2)\) gives a normal posterior. Completing the square in the product of the likelihood and prior yields

\delta\mid D\sim N\left( \frac{\tau^2}{\tau^2+s^2}D,\; \frac{2\tau^2s^2}{\tau^2+s^2} \right).

The posterior mean of the observed difference is shrunk toward zero by the factor \(\tau^2/(\tau^2+s^2)\). When \(\tau\) is large relative to \(s\), that factor is near one and little borrowing occurs. When \(\tau\) is small relative to \(s\), the difference is strongly shrunk, so each stratum’s estimate is pulled toward the other’s. This derivation makes clear that the data do not choose borrowing independently of the model: the assumed scale of heterogeneity governs how much observed difference is retained.

Worked Example: Separate Estimates, Complete Pooling, and a Corrected Posterior Probability

Consider hypothetical estimates on a treatment-effect scale measured in score points: \(Y_+=0.300\) with \(s_+=0.200\), and \(Y_-=0.0430\) with \(s_-=0.1857\). Assume independent normal sampling distributions. Compare separate estimation with complete pooling, using a diffuse prior on the common effect in the pooled analysis.

Separate estimates. With a diffuse prior within each stratum, the posterior for \(\theta_+\) is approximately \(N(0.300,0.200^2)\). Thus

P(\theta_+>0\mid Y_+) =\Phi\left(\frac{0.300}{0.200}\right) =\Phi(1.500) \approx 0.933.

This is a posterior probability for a positive effect under the stated model, not the probability that a null hypothesis is true.

Complete pooling. Setting \(\theta_+=\theta_-=\theta\), the common-effect estimate is the inverse-variance weighted mean. The precisions are approximately \(1/0.200^2=25.00\) and \(1/0.1857^2=28.998\). Therefore

\hat{\theta}_{\mathrm{pool}} =\frac{25.00(0.300)+28.998(0.0430)}{25.00+28.998} \approx 0.1620,\qquad \operatorname{Var}(\theta\mid Y_+,Y_-) \approx\frac{1}{53.998}=0.01852.

The posterior probability that the common effect is positive is

P(\theta>0\mid Y_+,Y_-) =\Phi\left(\frac{0.1620}{\sqrt{0.01852}}\right) =\Phi(1.1905) \approx 0.883.

The pooled result is less positive than the marker-positive-only result because the marker-negative estimate also informs the common effect. Complete pooling assumes no treatment-effect difference at all; it is not a neutral default. All values here are hypothetical, and the calculation depends on the normal likelihood and diffuse common-effect prior.

Partial Borrowing in a Symmetric Example

With equal standard errors, a diffuse prior for \(\beta\) makes the sample average \(B=(Y_++Y_-)/2\) informative about \(\beta\), while the difference \(D=Y_+-Y_-\) informs \(\delta\). For independent normal errors with common standard error \(s\), \(B\) and \(D\) are independent, with variances \(s^2/2\) and \(2s^2\), respectively. This makes a simple partial-pooling calculation possible.

Worked Example: Partial Borrowing With a Prespecified Heterogeneity Scale

Suppose hypothetical stratum estimates are \(Y_+=0.300\) and \(Y_-=-0.100\) score points, each with standard error \(s=0.200\). Assume independent normal sampling errors, a diffuse prior for \(\beta\), and the prespecified prior \(\delta\sim N(0,2\tau^2)\) with \(\tau=0.200\). These assumptions give \(B=0.100\), \(D=0.400\), \(\operatorname{Var}(B)=0.020\), and \(\operatorname{Var}(D)=0.080\).

The difference posterior has mean \(0.200\) and variance \(0.040\), since

E(\delta\mid D) =\frac{0.200^2}{0.200^2+0.200^2}(0.400) =0.200,\qquad \operatorname{Var}(\delta\mid D) =\frac{2(0.200^2)(0.200^2)}{0.200^2+0.200^2} =0.040.

The posterior for \(\beta\) is \(N(0.100,0.020)\). Because \(B\) and \(D\) are independent in this equal-standard-error setting, the posterior variances of \(\theta_+=\beta+\delta/2\) and \(\theta_-=\beta-\delta/2\) are each

0.020+\frac{0.040}{4}=0.030.

Their posterior means are \(0.100+0.200/2=0.200\) and \(0.100-0.200/2=0.000\), respectively. Hence

P(\theta_+>0\mid Y_+,Y_-) =\Phi\left(\frac{0.200}{\sqrt{0.030}}\right) =\Phi(1.155)\approx0.876.

For comparison, separate estimation gives \(P(\theta_+>0\mid Y_+)=\Phi(1.500)\approx0.933\), while complete pooling gives a common posterior \(N(0.100,0.020)\) and \(P(\theta>0\mid Y_+,Y_-)=\Phi(0.7071)\approx0.760\). The partial model lies between those analyses for this positive-stratum probability. The result is conditional on the chosen \(\tau\); it is not evidence that the strata are in fact exchangeable.

Conflict-Sensitive Borrowing

A single normal prior on \(\delta\) can still borrow too strongly if its heterogeneity scale is too small. A robust alternative is a mixture prior that assigns some probability to a tightly exchangeable component and some to a weakly informative component. The weak component allows a large difference and therefore protects against forcing agreement when the observed strata conflict. Such a mixture must be specified in advance; choosing its weights after seeing the treatment-effect estimates can undermine the interpretation of the posterior.

Worked Example: A Mixture Prior That Can Relax Borrowing

Use the hypothetical data \(Y_+=0.300\), \(Y_-=-0.100\), and \(s_+=s_-=0.200\), so \(D=0.400\) and \(D\mid\delta\sim N(\delta,0.080)\). Consider a prespecified prior with probability \(0.80\) on a tightly exchangeable component, \(\delta\sim N(0,0.080)\), and probability \(0.20\) on a weakly informative component, \(\delta\sim N(0,1.000)\). The marginal distributions of \(D\) in these components are \(N(0,0.160)\) and \(N(0,1.080)\).

At \(D=0.400\), their densities are approximately

f_{\mathrm{tight}}(0.400) =\frac{1}{\sqrt{2\pi(0.160)}}\exp\left(-\frac{0.400^2}{2(0.160)}\right) \approx0.6049,

f_{\mathrm{weak}}(0.400) =\frac{1}{\sqrt{2\pi(1.080)}}\exp\left(-\frac{0.400^2}{2(1.080)}\right) \approx0.3565.

Bayes’ rule gives posterior probability for the tight component of

\frac{0.80(0.6049)} {0.80(0.6049)+0.20(0.3565)} \approx0.872.

Thus the data retain substantial support for the exchangeable component, but the weak component’s posterior probability rises from its prior value of \(0.20\) to about \(0.128\). Conditional on the tight component, the posterior mean of \(\delta\) is \(0.200\); conditional on the weak component, it is \(0.400(1.000)/(1.080)=0.3704\). The mixture therefore allows more separation than a model that forces the tight component alone. This is a conflict-sensitive specification, not a guarantee against harmful borrowing: its behavior still depends on the chosen component scales and weights.

Checking the Assumptions and the Borrowing

Borrowing can reduce posterior uncertainty when the strata are sufficiently similar, but it can also pull an estimate away from its stratum-specific evidence. Before interpreting a borrowed estimate, check that the model’s likelihood and prior correspond to the trial and scientific question. Useful checks include:

Likelihood fit. Confirm that the effect scale, standard errors, and sampling distribution represent the planned analysis. Examine whether sparse events, skewness, or dependence make the normal summary likelihood questionable.
Prior scale. Translate plausible values of \(\tau\) into treatment-effect differences in clinical units. A prior that appears numerically diffuse may still strongly constrain a rare stratum if its estimate is very imprecise.
Conflict sensitivity. Refit under separate-stratum, partial-pooling, and robust-mixture specifications. Report whether the clinically relevant posterior summaries change materially.
Posterior predictive checks. Compare observed stratum estimates with replicated estimates generated from the fitted model. A model that rarely reproduces a discrepancy as large as the observed one may be imposing excessive similarity.
Operating characteristics. Simulate the planned analysis under equal effects and under clinically important heterogeneity, including a small stratum. Evaluate bias, interval coverage, decision error, and the probability of a misleading subgroup conclusion.

These checks are model diagnostics, not proof of exchangeability. With only two strata, heterogeneity is difficult to estimate from the trial data alone. If the data are weak, the prior can dominate the amount of borrowing, so sensitivity to plausible heterogeneity assumptions is especially important. Posterior predictive agreement also does not establish that the strata share a causal treatment effect.

False Borrowing and Alternative Approaches

A false-borrowing risk occurs when the model borrows across strata even though their true treatment effects differ in a clinically important way. For example, a marker may identify a responsive subgroup while the negative stratum has little benefit or harm. Strong shrinkage toward a common effect can then bias the subgroup estimate and make uncertainty appear smaller than it should under the true heterogeneity. The risk is most acute when a small stratum has little independent information and the prior strongly favors similarity.

The main alternatives make different assumptions:

Approach	Assumption about stratum effects	Practical implication
Separate analyses	No cross-stratum borrowing	Preserves stratum-specific estimates but may be imprecise in a small stratum.
Complete pooling	\(\theta_+=\theta_-\)	Maximizes shared information but cannot represent treatment-effect heterogeneity.
Hierarchical partial pooling	Effects vary around a shared distribution	Allows differences; borrowing depends on the heterogeneity prior and data.
Robust mixture borrowing	Similarity is plausible but not certain	Can reduce borrowing under conflict, but requires prespecified mixture components and weights.

No approach is uniformly best. Separate analyses may be preferable when biology strongly supports effect modification or when the consequences of false borrowing are serious. Complete pooling requires a particularly strong scientific justification. Hierarchical or robust models can be useful when some sharing is reasonable, provided the model is prespecified and its sensitivity is transparent. Bayesian borrowing also does not itself resolve multiplicity or validate a treatment claim; posterior evidence must be interpreted within the trial’s decision and testing framework.

Common pitfall: “The subgroup estimates are not significantly different, so they are exchangeable and should be pooled.” A failure to detect a difference is not evidence that the effects are identical, especially when a stratum is small. Specify the borrowing model using scientific rationale and clinically interpretable heterogeneity assumptions, then show how conclusions change under weaker or absent borrowing.

Check Your Understanding

Answer each question using the assumptions and notation in this tutorial. Brief solutions follow the questions.

In the model \(\delta\sim N(0,2\tau^2)\), what happens to borrowing as \(\tau\) becomes very small? What scientific assumption does this encode?
With equal standard errors \(s\), \(D=0.300\), and \(\tau=s\), what is the posterior mean of \(\delta\) under the normal difference prior?
Why does complete pooling not merely represent “strong evidence” that the strata are similar?
Name one situation in which conditional independence of the stratum estimates could fail.
What is a false-borrowing risk, and name one design-stage way to assess it.

Solutions. (1) The difference is shrunk strongly toward zero because its posterior-mean multiplier is \(\tau^2/(\tau^2+s^2)\), which approaches zero; this encodes a prior belief that large between-stratum differences are unlikely. (2) The multiplier is \(s^2/(s^2+s^2)=1/2\), so \(E(\delta\mid D)=0.300/2=0.150\) in the treatment-effect units. (3) Complete pooling imposes \(\theta_+=\theta_-\) as a constraint; it does not estimate similarity from the data or allow residual heterogeneity. (4) Dependence could arise if both estimates use a shared nuisance estimate or are based on overlapping participants. (5) It is bias or overconfidence from borrowing when true effects differ materially; simulate the planned analysis under clinically important heterogeneous-effect scenarios.