Why Exchangeability Is a Clinical Assumption
Basket trials study the same treatment across groups defined by different diseases, biomarkers, or molecular alterations. A hierarchical model can let the estimated treatment effect in one basket inform another. That borrowing is not justified merely because the baskets appear in the same trial: it depends on whether their treatment effects can reasonably be treated as draws from a common distribution.
Exchangeability gives this assumption a precise statistical meaning. Its clinical interpretation depends on what has been conditioned on, including relevant design features and biological information. In particular, baskets with distinct mechanisms may have systematically different treatment effects even when their patient outcomes are measured in the same way.
This definition concerns the probability model for the effects, not whether individual patients or outcomes can be swapped across baskets. It also does not mean that the baskets have identical baseline risks, response rates, or patient characteristics. A model may allow those features to differ while placing the treatment effects in a common distribution. The relevant question is whether the effects, on a clinically interpretable and consistently defined scale, are plausibly governed by the same distribution.
For example, a common random-effects specification is \(\theta_j\mid\mu,\tau\sim N(\mu,\tau^2)\), independently across baskets conditional on \(\mu\) and \(\tau\). It treats the basket effects as draws from a distribution with common center \(\mu\) and spread \(\tau\). A large \(\tau\) allows substantial effect variation; it does not by itself represent a known, systematic difference between two mechanistic classes. If mechanism predicts a shift in effect, a single shared distribution may conceal that structure.
What the Model Assumes About Biology
A treatment effect is specific to an endpoint and scale. In the examples below, \(\theta_j\) is the log odds ratio for response in basket \(j\), comparing treatment with a defined control. Positive values favor treatment. Baskets can be compared this way only if the response definition, comparison, and analysis scale are sufficiently aligned. Otherwise, a shared distribution may be difficult to interpret even before biological differences are considered.
A distinct biological mechanism is a reason to examine the exchangeability assumption, not an automatic proof that it fails. Two alterations might be biologically distinct yet converge on a pathway affected similarly by the treatment. Conversely, alterations in the same broad pathway may differ in how strongly they activate it, or in whether the treatment can interrupt it. Statistical exchangeability therefore requires a clinical argument about effect distributions, not just a count of shared or distinct labels.
The assumption also depends on the conditioning set. If mechanism labels are ignored, a common exchangeable model says that those labels do not distinguish the effect distributions. If labels are included, the model may instead allow different distributions by mechanism and assume exchangeability only among baskets with the same relevant label. These are different assumptions. Conditioning on a mechanistic feature does not establish that the feature captures every clinically relevant source of effect variation.
Here \(M_j\) denotes the mechanism class of basket \(j\). This expression allows class-specific centers and spreads. Within a class, baskets may be exchangeable under the model if their effects have the same distribution; across classes, they are not exchangeable when the distributions differ. This is a way to state the assumption, not a guarantee that the mechanism classification is clinically sufficient.
A Short Justification: What Exchangeability Implies
Exchangeability constrains the marginal distributions of the effects. For any two baskets \(j\) and \(k\), a permutation that swaps their labels leaves the joint distribution unchanged. Taking a marginal distribution from each side shows that \(\theta_j\) and \(\theta_k\) have the same marginal distribution, conditional on \(C\). If their variances exist, their means and variances are therefore also equal. This does not require their realized values to match.
A common hierarchical model explains how this symmetry can hold while effects vary. Suppose that, conditional on common hyperparameters \(\mu\) and \(\tau\), the effects are independent draws from the same distribution:
The product is unchanged when the basket labels are permuted, because every factor has the same form. Integrating over the shared hyperparameters preserves that symmetry. Thus, the resulting joint distribution is exchangeable, even though different baskets can have different effect values. The clinical question remains whether the common distribution is a defensible description of the baskets under consideration.
Worked Example: A Shared Pathway and a Common Effect Distribution
Hypothetical scenario. Two baskets involve different alterations, but preclinical evidence suggests that both lead to dependence on the same drug-sensitive signaling pathway. Investigators consider a common prior distribution for the log odds ratio: \(\theta_j\sim N(0.30,0.25^2)\). Under this assumption, how does the model describe benefit in each basket, and what does the calculation say about exchangeability?
The model assigns both baskets the same marginal distribution, centered at \(0.30\) log-odds units, with standard deviation \(0.25\) log-odds units. The probability that the effect is positive under this illustrative distribution is:
So the assumed distribution gives each basket an approximately 88.5% probability of a positive log odds ratio. That is a consequence of the specified model, not evidence that the actual effects are exchangeable. The biological rationale for a shared pathway makes a common distribution worth considering, but the investigators must also assess whether the endpoint, treatment context, and other relevant biological differences support it.
If the two mechanisms instead implied different centers, the calculation would not be the same for each basket. For example, a class-specific center of \(0.10\) with the same standard deviation gives \(P(\theta_j>0)=\Phi(0.10/0.25)=\Phi(0.40)\approx0.6554\). That contrast illustrates why a common pathway rationale matters: a shared distribution makes a substantive prediction about effects, rather than merely providing a convenient computational structure.
When Distinct Mechanisms May Matter
Mechanistic differences challenge exchangeability when they plausibly alter the distribution of treatment effects, rather than merely providing different labels. A difference in drug target engagement, pathway dependence, resistance, or downstream signaling could shift the expected effect or change its variability. The relevant concern is systematic effect variation that the common distribution does not represent.
The following questions help make that concern explicit:
- Are treatment and control, endpoint definitions, and follow-up sufficiently comparable for \(\theta_j\) to have the same clinical meaning across baskets?
- Could the different mechanisms plausibly change whether the treatment reaches or blocks the relevant target?
- Is there a reason to expect one mechanistic class to have larger or smaller effects, not just more uncertainty?
- Does the model condition on biological features that are central to its clinical justification, or does it implicitly treat those features as irrelevant?
These questions do not produce a mechanical test of exchangeability. They structure the clinical reasoning that should accompany any borrowing assumption. Data from a small number of baskets may not distinguish a true mechanistic shift from chance variation. Similar observed responses are therefore not, on their own, proof that the mechanisms share an effect distribution; different observed responses are not, by themselves, proof that they do not.
Worked Example: A Pooled Distribution Can Hide a Mechanism Difference
Hypothetical scenario. Suppose two mechanistic classes are equally represented. The clinical team considers their class-specific mean log odds ratios to be \(0.70\) and \(-0.30\), with within-class variance \(0.09\) in each class. What mean and variance would summarize the combined population of basket effects, and what is lost by treating it as one common distribution?
The overall mean is the equally weighted average of the class means:
The combined variance includes both within-class variation and variation between class means. The variance of the two class means around \(0.20\) is:
Adding the within-class variance gives \(0.09+0.25=0.34\), so the combined standard deviation is \(\sqrt{0.34}\approx0.5831\) log-odds units. This calculation summarizes the mixture's first two moments. It does not make the two classes exchangeable: their assumed means differ by \(1.00\) log-odds unit. A single distribution centered at \(0.20\) could obscure the contrast between a class centered at \(0.70\) and one centered at \(-0.30\). In particular, the overall center is not necessarily a useful description of either class.
The distinction is between a descriptive mixture summary and an exchangeable model. If the class labels are fixed and clinically meaningful, a common distribution across all baskets asserts that the labels do not identify different effect distributions. A mixture of distinct class distributions does not satisfy that assumption simply because its overall mean and variance can be calculated.
Exchangeability Within Mechanism Classes
If biological evidence supports differences between classes but supports similarity among baskets within a class, a reasonable scientific question is whether exchangeability should be restricted to within-class comparisons. This does not require effects within a class to be equal. It means the model treats their distributions as the same after conditioning on class membership and any other specified information. Whether a class is clinically coherent enough for this purpose requires substantive justification; a label alone does not supply it.
Worked Example: Comparing a Shared Model With Class-Specific Centers
Hypothetical scenario. Four baskets include three with mechanism A and one with mechanism B. The clinical team proposes class-specific mean log odds ratios of \(0.40\) for A and \(-0.10\) for B. For a simple comparison of centers, what common mean results if each basket is given equal weight, and how far is it from each class center?
The basket-weighted common center is:
Relative to the proposed class centers, this common value is \(0.275-0.40=-0.125\) for A and \(0.275-(-0.10)=0.375\) for B. Thus it is closer to A, the more numerous class, and lies \(0.375\) log-odds units above the proposed center for B. The calculation is not a fitted estimate; it shows how a common center can represent a compromise rather than either class's expected effect.
If the class distinction is clinically credible, a model with class-specific centers may better represent the scientific assumptions, while allowing exchangeability among baskets within each class if justified. If there is no compelling evidence that mechanism predicts effect, a common distribution may remain defensible. These numerical assumptions do not decide between the models; they make the consequences of each assumption visible.
Alternatives, Failure Modes, and Limits
The main alternatives differ in what they assume about cross-basket information. A common exchangeable distribution allows all included baskets to inform a shared effect distribution. Independent basket analyses do not use cross-basket exchangeability for the effects, although they may be less informative when each basket is small. A mechanism-conditioned hierarchy allows shared distributions within specified classes while permitting class differences. A model with class-specific means but a shared spread, for example, represents systematic shifts without requiring each class to have its own spread. Choosing among these structures is a scientific modeling decision, not a conclusion that follows from the labels alone.
Several warning signs call for closer scrutiny of the assumption:
- Different estimands: basket effects are defined on different endpoints, comparators, or time horizons, so a common distribution lacks a stable interpretation.
- Mechanism-linked predictions: biological evidence suggests treatment sensitivity or resistance differs systematically by mechanism.
- Overgeneralized labels: baskets are grouped together by a broad label even though the relevant biology may differ substantially within that label.
- Overconfident interpretation: a model-based shared distribution is described as biological proof that effects are alike.
These are conceptual checks, not formal diagnostics. The model's distributional assumptions also matter: a single normal distribution may poorly represent a population of effects with multiple mechanistic classes, even when a common mean and variance can be specified. Conversely, adding many class-specific parameters can be difficult to support with sparse basket data. A richer model is not automatically more credible if the clinical distinctions and available information do not support it.
Most importantly, exchangeability is an assumption about the joint distribution of effects, conditional on the chosen information. It cannot be verified from a small trial simply by observing that basket estimates are close, or rejected just because they differ. Statistical uncertainty, patient-level variation, and genuine between-basket variation can all contribute to observed differences. The clinical rationale should be stated before interpreting borrowing as appropriate.
Check Your Understanding
Answer each question using the meaning of exchangeability and its clinical interpretation.
- Does exchangeability of \(\theta_1\) and \(\theta_2\) imply that their realized treatment effects are equal? Explain.
- Two baskets have distinct mechanisms, but there is a strong rationale that both mechanisms create the same drug-sensitive dependency. What does a common effect distribution assume, and what does that rationale contribute?
- Two fixed mechanism classes have different assumed mean effects. Can all baskets be exchangeable across classes under a model that ignores those classes? Explain.
- Why does a similar observed response rate in two baskets not establish exchangeability of their treatment effects?
- When might exchangeability within mechanism classes be more defensible than exchangeability across all baskets?
Solutions. (1) No. Exchangeability means the joint distribution is invariant to relabeling; it permits different realized values. (2) The common distribution assumes both baskets' effects have the same distribution conditional on the model information. The shared dependency provides a clinical rationale for considering that assumption but does not prove it. (3) Not if the class-specific means genuinely differ: the classes then have different marginal effect distributions, contrary to cross-class exchangeability. (4) Response rates are observed outcomes, not treatment-effect parameters; similar rates can occur despite different treatment effects, for example when baseline risks differ, and sampling variation can also make rates similar. (5) It may be more defensible when biology supports similar effect distributions within a class but provides a credible reason for systematic differences between classes.