Why the Pooling Choice Matters
A multi-subgroup trial can produce very different conclusions depending on whether the analysis treats the subgroups as identical, unrelated, or related but not identical. These are not merely alternative computational techniques. Each encodes a different claim about how subgroup effects are generated and how much evidence from one subgroup should influence another.
The distinction matters when subgroup sample sizes are small. An independent analysis may leave an estimate highly uncertain, while complete pooling can make it look precise by treating all observations as if they came from one common response process. Partial pooling lies between these extremes: subgroup parameters remain distinct, but a model connects them so that evidence can be shared to an extent informed by the model and data.
These choices concern the model for the outcome and the subgroup parameters, not just the prior’s numerical strength. In a Bayesian analysis, a shared prior distribution applied separately to each subgroup is not, by itself, data borrowing: if the subgroup parameters and likelihoods remain independent, another subgroup’s observed outcomes do not update the parameter of interest. Borrowing requires a model link, such as a shared unknown hyperparameter or an explicit common parameter.
Assumptions Behind the Three Strategies
Let \(y_j\) denote the observed data in subgroup \(j\), and let \(\theta_j\) be its treatment-effect or response parameter. The precise likelihood depends on the outcome type, but the pooling distinction can be stated generally.
| Strategy | Parameter structure | What is shared? | Main assumption |
|---|---|---|---|
| Complete pooling | \(\theta_1=\cdots=\theta_J=\theta\) | One common parameter is estimated from all subgroups’ data. | Subgroup differences in the target parameter are absent or irrelevant to the estimand. |
| No pooling | \(\theta_1,\ldots,\theta_J\) are distinct and modeled independently | No subgroup outcome data are borrowed for another subgroup’s parameter. | Each subgroup is analyzed on its own; any prior information is not linked across subgroup parameters. |
| Partial pooling | Distinct \(\theta_j\) values arise from a shared distribution or other linking model. | Information about shared model features can inform subgroup parameters. | Subgroups are related in a way adequately represented by the linking model. |
Complete pooling is appropriate only if the common parameter is scientifically meaningful. For example, a common response probability assumes that the outcome has the same probability in every included subgroup under the modeled conditions. If meaningful effect modification is possible, a single parameter may answer the wrong clinical question even if it is estimated precisely.
No pooling avoids assumptions that subgroup parameters are similar, but it does not make the subgroups’ estimates automatically reliable. Sparse subgroup data can yield broad posterior distributions, and independent analyses do not stabilize one subgroup using evidence from the others. Partial pooling requires a defensible relationship among subgroups. The earlier material on exchangeability explains why exchangeability is a modeling assumption about the joint distribution, not proof that subgroup effects are equal.
Worked Example: Complete Pooling and No Pooling for Binary Outcomes
Worked Example: Complete Pooling and No Pooling for Binary Outcomes
Suppose a hypothetical basket trial observes responses in three subgroups. The data are \(y_1=8\) responses among \(n_1=10\), \(y_2=3\) among \(n_2=10\), and \(y_3=6\) among \(n_3=10\). For this comparison, use a binomial likelihood and a Beta(1,1) prior for each probability being estimated. Under complete pooling there is one common response probability \(p\); under no pooling each subgroup has its own \(p_j\) and independent Beta(1,1) prior.
Complete pooling. The shared parameter model is \(y_j\mid p\sim\operatorname{Binomial}(n_j,p)\). Conditional independence of the subgroup observations gives a likelihood proportional to \(p^{17}(1-p)^{13}\), since there are 17 responses and 13 nonresponses in all. Combining this with the uniform Beta(1,1) prior gives a Beta(18,14) posterior. Its mean is
This posterior summarizes a common probability across all three subgroups. It does not estimate a separate response probability for each subgroup.
No pooling. Each subgroup has posterior \(p_j\mid y_j\sim\operatorname{Beta}(y_j+1,n_j-y_j+1)\). The subgroup posterior means are
These estimates preserve the observed differences and use each subgroup’s data alone. The posterior mean under complete pooling is not the arithmetic mean of these three posterior means; it weights the total response and nonresponse counts under one common parameter. Neither strategy establishes that the subgroups truly have equal or unrelated response probabilities. Those are model assumptions.
Partial Pooling: Distinct Parameters, A Shared Link
A common partial-pooling construction gives each subgroup its own parameter while modeling those parameters as arising from a shared distribution. The shared distribution may have unknown features, so observations in one subgroup can inform those features and thereby affect the posterior for other subgroups. A standard example for continuous outcomes is a normal hierarchical model. To make the information-sharing mechanism transparent, the next example treats the between-subgroup variance as known rather than estimating it.
This construction does not assert \(\theta_j=\mu\). Rather, it says that subgroup effects are centered around a shared mean, with their differences represented by \(\tau^2\). When \(\tau^2\) is small relative to sampling uncertainty, the model favors closer subgroup values. When it is large, it permits greater differences. In practice, treating \(\tau^2\) as known is a simplifying assumption; uncertainty about heterogeneity can materially affect borrowing.
Worked Example: A Shared Mean Lets Subgroups Inform Each Other
Consider two hypothetical subgroups with sample means \(\bar{x}_A=6\) and \(\bar{x}_B=10\), each with known sampling variance \(v=4\), in outcome units squared. Use \(\theta_j\mid\mu\sim N(\mu,4)\) and \(\mu\sim N(8,100)\). The sampling model is \(\bar{x}_j\mid\theta_j\sim N(\theta_j,4)\). Thus subgroup parameters are distinct, the prior standard deviation around the shared mean is 2 outcome units, and the prior standard deviation for \(\mu\) is 10 outcome units.
Integrating over \(\theta_j\) gives \(\bar{x}_j\mid\mu\sim N(\mu,4+4)=N(\mu,8)\). Therefore the posterior precision for \(\mu\) is the prior precision plus the two subgroup likelihood precisions, and its posterior mean is
For a subgroup with sampling variance 4 and prior variance 4 conditional on \(\mu\), normal conjugate updating gives \(E(\theta_j\mid\bar{x}_j,\mu)=(\bar{x}_j+\mu)/2\). Averaging over the posterior for \(\mu\), the subgroup posterior means are \(E(\theta_A\mid\bar{x}_A,\bar{x}_B)=(6+8)/2=7\) and \(E(\theta_B\mid\bar{x}_A,\bar{x}_B)=(10+8)/2=9\). The estimates remain different, unlike complete pooling, but are closer to one another than the observed means, unlike no pooling.
To see the cross-subgroup link, change only subgroup B’s mean from 10 to 14. The posterior mean of \(\mu\) becomes
Subgroup A’s posterior mean is then \((6+9.9231)/2=7.9615\), rather than 7. The change in B’s data has affected A’s estimate through the shared \(\mu\). This is borrowing; it arises from the linking structure, not from declaring the subgroup effects identical.
Worked Example: A Sparse Subgroup and the Cost of Forced Equality
Worked Example: A Sparse Subgroup and the Cost of Forced Equality
Suppose, hypothetically, subgroup A has 1 response among 2 participants and subgroup B has 8 responses among 10. With independent Beta(1,1) priors, the no-pooling posterior mean for A is \((1+1)/(2+2)=0.5000\), while B’s is \((8+1)/(10+2)=0.7500\). These are posterior means under separate subgroup models, not evidence that the true probabilities differ by exactly 0.25.
Under complete pooling, total responses are \(1+8=9\), and total participants are \(2+10=12\). The common-probability posterior is Beta(10,4), with mean \(10/(10+4)=0.7143\). The sparse subgroup’s observed response proportion of 0.5000 is replaced by a common estimate of 0.7143 because the model assumes the same probability in both subgroups. That may be reasonable if the common probability is the intended estimand and the assumption is credible; it may obscure important subgroup differences otherwise.
A partial-pooling analysis would retain separate probabilities while linking them through a model. Its exact posterior estimates cannot be determined from these counts alone: they also depend on the specified linking structure and its prior or other assumptions about between-subgroup variation. That dependence is not a defect to conceal; it is part of the model and should be examined.
Derivation: Why Complete Pooling Uses the Combined Counts
For independent binomial subgroup outcomes with a common probability \(p\), the likelihood is
The terms involving \(p\) depend on the data only through total responses \(\sum_j y_j\) and total nonresponses \(\sum_j(n_j-y_j)\). This explains why, under the common-probability model, subgroup labels do not affect the posterior for \(p\) once the totals are known. It also exposes the assumption doing the work: if the subgroup probabilities are not common, that factorization into one parameter \(p\) is not the appropriate likelihood model.
Diagnostics, Comparisons, and Limitations
No single pooling strategy is best in every trial. The right choice depends on the estimand, the scientific plausibility of subgroup similarity, the amount of data, and the consequences of borrowing incorrectly. Pooling is not a substitute for defining the question: an overall response rate and subgroup-specific response rates are different targets.
- Check whether complete pooling is plausible. Compare subgroup outcome patterns and relevant clinical characteristics. Large observed differences may signal conflict with a common-parameter model, although small samples make such comparisons uncertain. Consider whether treatment effects, baseline risks, endpoint definitions, or follow-up differ across subgroups.
- Check whether partial pooling is robust. Examine how subgroup estimates and decisions change under reasonable alternatives for the between-subgroup variation and the shared structure. If a subgroup’s conclusion changes substantially, report that sensitivity rather than presenting one borrowing level as inevitable.
- Check no-pooling precision. Very broad subgroup posteriors can make subgroup decisions unstable. That is a consequence of limited subgroup data, not a reason to silently substitute a common effect.
- Check model fit and data comparability. Look for differences in outcome distributions, covariates, eligibility, treatment delivery, and missingness that the model does not represent. Apparent subgroup conflict may reflect these features rather than genuine effect variation, while an overly simple model may miss real variation.
Complete pooling is simple and can be efficient when a common parameter is defensible, but it cannot represent subgroup heterogeneity. No pooling makes fewer cross-subgroup assumptions and preserves separate targets, but may be imprecise with sparse data. Partial pooling offers a continuum of borrowing, but its behavior depends on the linking model and the assumptions about heterogeneity. Compared with these model-based choices, separate descriptive summaries can be useful for transparency, but they do not by themselves provide a coherent joint posterior or a borrowing rule.
A frequent mistake is to say, “The subgroups are exchangeable, so their parameters are equal.” Exchangeability supports a symmetric joint model in which subgroup labels do not determine the prior ordering; it does not imply equality of realized parameters. The correction is to state the model link explicitly, retain subgroup-specific parameters when that is the target, and describe how shared information can influence their estimates.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A model assigns every subgroup the same response probability. Which pooling strategy is this, and what assumption does it make?
- Two subgroup parameters have independent priors and separate likelihoods. Does observing more data in one subgroup directly update the other subgroup’s parameter?
- In the continuous-outcome example, why did subgroup A’s posterior mean change when subgroup B’s sample mean changed?
- For 2 responses among 4 participants, find the posterior mean under a Beta(1,1) prior when this subgroup is analyzed without pooling.
- Give one diagnostic concern for partial pooling and one limitation of no pooling.
Solutions. (1) Complete pooling; it assumes one common response probability across subgroups. (2) No. With independent subgroup parameters and separate likelihoods, the other subgroup’s data do not enter that parameter’s posterior. (3) Both subgroup data update the shared mean \(\mu\), which is linked to each \(\theta_j\); changing B’s data therefore changes the posterior distribution of \(\mu\) and A’s posterior. (4) The posterior is Beta(3,3), so the mean is \(3/(3+3)=0.5000\). (5) For partial pooling, check sensitivity to the heterogeneity assumptions and whether the shared structure fits the subgroups. A limitation of no pooling is that sparse subgroup data can yield highly uncertain, unstable estimates without borrowing.