Why Partial Pooling Does Not Imply Equal Effects

Borrowing Information Is Not Imposing Equality

In a basket trial, estimates from different disease subgroups may be informed by a shared hierarchical model. This can improve estimation when some subgroups have limited data, but it is easy to misread the resulting estimates: if the model pulls them toward a common value, does it assume every subgroup has the same treatment effect? Not necessarily. Partial pooling means that subgroup estimates are informed both by their own data and by a shared distribution; it does not ordinarily set the subgroup effects equal.

The distinction matters clinically. If the true effects differ, a model that forces equality can hide meaningful heterogeneity. Conversely, estimating every subgroup in isolation can leave estimates unstable when data are sparse. A hierarchical model provides an intermediate approach, conditional on its assumptions about how subgroup effects are related.

Definition: In a partially pooled model, each subgroup has its own effect parameter, while the parameters are linked through a shared probability distribution or shared hyperparameters. The model allows subgroup effects to differ; the degree of borrowing depends on the data and the assumed distribution of effects.

A Normal Hierarchy and Its Subgroup-Specific Posterior

Let \(y_j\) be an estimated treatment effect for subgroup \(j\), measured in clinically meaningful units, with known standard error \(s_j\). For the normal-normal model, assume the sampling model and hierarchy

y_j\mid\theta_j\sim N(\theta_j,s_j^2),\qquad \theta_j\mid\mu,\tau\sim N(\mu,\tau^2).

Here \(\theta_j\) is the true effect for subgroup \(j\), \(\mu\) is the center of the distribution of subgroup effects, and \(\tau\) describes their between-subgroup variation. The \(y_j\) are assumed conditionally independent given their effects; the subgroup effects are conditionally independent given \(\mu\) and \(\tau\). This likelihood assumes the estimated effects are approximately normal, their standard errors adequately represent sampling uncertainty, and the measurement scales and effect definitions are comparable across subgroups. In a full Bayesian analysis, priors are also assigned to \(\mu\) and \(\tau\), with \(\tau\) restricted to be nonnegative.

First consider \(\mu\) and \(\tau\) as fixed. The posterior distribution for each \(\theta_j\) is normal. Its mean is a weighted average of the subgroup estimate and the shared center:

E(\theta_j\mid y_j,\mu,\tau) = \frac{\tau^2}{\tau^2+s_j^2}y_j + \frac{s_j^2}{\tau^2+s_j^2}\mu, \qquad \operatorname{Var}(\theta_j\mid y_j,\mu,\tau) = \frac{\tau^2s_j^2}{\tau^2+s_j^2}.

The weight on the subgroup estimate is larger when its sampling variance \(s_j^2\) is small relative to the between-subgroup variance \(\tau^2\). The posterior mean is therefore generally specific to that subgroup: it depends on its own \(y_j\) and \(s_j\). A finite, positive \(\tau\) does not make the posterior means identical. It moves them toward \(\mu\), with the amount of movement determined by the relative precision.

Why the Formula Produces Shrinkage, Not Equality

The conditional posterior formula follows by multiplying the normal likelihood and normal prior densities and collecting the terms involving \(\theta_j\). Ignoring factors that do not depend on \(\theta_j\), the log posterior kernel is

-\frac{1}{2}\left\{\frac{(y_j-\theta_j)^2}{s_j^2} +\frac{(\theta_j-\mu)^2}{\tau^2}\right\}.

The coefficient of \(\theta_j^2\) inside the resulting quadratic is \(1/s_j^2+1/\tau^2\). Thus the posterior variance is the reciprocal of this precision, and the coefficient determining the posterior mean gives the weighted-average expression above:

\left(\frac{1}{s_j^2}+\frac{1}{\tau^2}\right)^{-1} = \frac{\tau^2s_j^2}{\tau^2+s_j^2}, \qquad \frac{y_j/s_j^2+\mu/\tau^2}{1/s_j^2+1/\tau^2} = \frac{\tau^2y_j+s_j^2\mu}{\tau^2+s_j^2}.

For two subgroups \(j\) and \(k\), their conditional posterior mean difference is

E(\theta_j\mid y_j,\mu,\tau)-E(\theta_k\mid y_k,\mu,\tau) = \frac{\tau^2}{\tau^2+s_j^2}(y_j-\mu) - \frac{\tau^2}{\tau^2+s_k^2}(y_k-\mu).

There is no general reason for this expression to be zero. It is zero in special cases, such as when both data estimates equal the shared center, or at the complete-pooling boundary \(\tau=0\). But when \(\tau>0\), different data and different standard errors usually yield different posterior means.

Worked Example: Two Subgroups With Distinct Posterior Effects

Suppose a hypothetical trial estimates mean blood-pressure reductions in two subgroups. Positive values represent a reduction, measured in mmHg. The subgroup estimates are \(y_1=108\) and \(y_2=122\), each with standard error \(4\) mmHg. For illustration, take the shared center as \(\mu=115\) mmHg and the between-subgroup variance as \(\tau^2=36\;(\text{mmHg})^2\). Assume independent normal sampling errors and the stated normal hierarchy.

The sampling variance is \(s_j^2=16\;(\text{mmHg})^2\), so the weight on each subgroup estimate is \(36/(36+16)=9/13\), and the weight on the shared center is \(4/13\). For subgroup 1:

E(\theta_1\mid y_1,\mu,\tau) = \frac{36(108)+16(115)}{52} = \frac{5728}{52} \approx 110.154\text{ mmHg}.

For subgroup 2:

E(\theta_2\mid y_2,\mu,\tau) = \frac{36(122)+16(115)}{52} = \frac{6232}{52} \approx 119.846\text{ mmHg}.

The estimates have moved toward 115 mmHg, but they remain distinct. Each conditional posterior variance is \(576/52=144/13\approx11.0769\;(\text{mmHg})^2\); using the unrounded variance, the posterior standard deviation is \(\sqrt{576/52}\approx3.328\) mmHg. These are conditional results for the specified \(\mu\) and \(\tau\), not a complete analysis that accounts for uncertainty in those hyperparameters.

Precision and Between-Subgroup Variation Control the Pull

The same hierarchical structure can produce different amounts of shrinkage for different subgroups. A precise estimate is relatively influential; an imprecise estimate is pulled more strongly toward the shared center. Likewise, when the model supports larger between-subgroup variation, it permits effects to differ more and shrinks less. These features are consequences of the model’s variance structure, not a rule that all effects must be equal.

Worked Example: Different Standard Errors, Different Shrinkage

Consider a hypothetical study of symptom improvement, where effects are measured in points and larger values indicate greater improvement. Suppose two subgroup estimates are \(y_1=4\) points with \(s_1=1\) point and \(y_2=8\) points with \(s_2=2\) points. For an illustrative conditional calculation, take \(\mu=6\) points and \(\tau=3\) points, so \(\tau^2=9\;(\text{points})^2\). Assume the normal sampling model and hierarchy are adequate.

For subgroup 1, \(s_1^2=1\), so its weight on its own estimate is \(9/(9+1)=0.9\):

E(\theta_1\mid y_1,\mu,\tau) = 0.9(4)+0.1(6) = 4.2\text{ points}.

For subgroup 2, \(s_2^2=4\), so its weight on its own estimate is \(9/(9+4)=9/13\):

E(\theta_2\mid y_2,\mu,\tau) = \frac{9}{13}(8)+\frac{4}{13}(6) = \frac{96}{13} \approx7.385\text{ points}.

The less precise second estimate receives more shrinkage toward 6 points. The conditional posterior variances are \(9(1)/(10)=0.900\;(\text{points})^2\) and \(9(4)/(13)=36/13\approx2.769\;(\text{points})^2\), respectively. Their standard deviations are approximately \(0.949\) and \(1.664\) points. The subgroup estimates and their uncertainty are not made identical by sharing \(\mu\) and \(\tau\).

Uncertainty in Shared Hyperparameters

In practice, \(\mu\) and \(\tau\) are usually unknown. A full hierarchical analysis assigns them priors and averages over their joint posterior distribution. This integration adds uncertainty; it does not replace all subgroup effects by one common effect. Formally, the marginal posterior for subgroup \(j\) is a mixture of the conditional posteriors:

p(\theta_j\mid\mathbf y) = \int p(\theta_j\mid y_j,\mu,\tau)\, p(\mu,\tau\mid\mathbf y)\,d\mu\,d\tau.

Here \(\mathbf y\) denotes the collection of subgroup estimates. Because all subgroups share uncertain hyperparameters, their posterior distributions can be dependent. Their marginal means, variances, and credible intervals are nevertheless subgroup-specific in general. Conditional formulae with fixed hyperparameters are useful for understanding the mechanism, but a reported analysis should account for hyperparameter uncertainty when it is material.

Worked Example: Averaging Over Uncertain Between-Subgroup Variation

To illustrate hyperparameter averaging, suppose a hypothetical analysis has two equally weighted representative posterior draws for \((\mu,\tau^2)\): \((5,4)\) and \((5,16)\), with effects measured in a common unit. This is a deliberately simple discrete approximation, not a claim about a particular trial. Let subgroup estimates be \(y_1=2\), \(y_2=8\), each with standard error \(2\), so \(s_j^2=4\). The normal sampling model and hierarchy are assumed.

At the first draw, \(\tau^2=4\), so the weight on each estimate is \(4/(4+4)=0.5\). The conditional posterior means are \(3.5\) and \(6.5\), and each conditional variance is \(4(4)/(8)=2\). At the second draw, \(\tau^2=16\), the estimate weight is \(16/(16+4)=0.8\). The conditional means are \(2.6\) and \(7.4\), and each conditional variance is \(16(4)/(20)=3.2\).

With equal draw weights, the marginal posterior mean for subgroup 1 is \((3.5+2.6)/2=3.05\); for subgroup 2 it is \((6.5+7.4)/2=6.95\). The mixture variance for subgroup 1 is the average conditional variance plus the variance of the conditional means:

\frac{2+3.2}{2} + \frac{(3.5-3.05)^2+(2.6-3.05)^2}{2} = 2.6+0.2025 = 2.8025.

The same variance applies to subgroup 2 in this symmetric illustration, giving a marginal standard deviation of \(\sqrt{2.8025}\approx1.674\) units. Averaging over uncertainty in \(\tau^2\) preserves two distinct marginal posterior means and incorporates the variation between conditional means into uncertainty.

Recognizing the Alternatives and the Boundaries

The limiting cases clarify what partial pooling is—and is not. If \(\tau=0\), the hierarchy places all subgroup effects at \(\mu\); this is complete pooling and imposes a common effect. If \(\tau\) is very large relative to the subgroup sampling errors, each estimate receives nearly all the weight and the model approaches no pooling. For intermediate values, the estimates are pulled toward one another without generally becoming identical.

Approach	How subgroup effects are treated	Main implication
Complete pooling	One common effect is used for all subgroups.	Efficient if equality is appropriate, but does not represent genuine subgroup variation.
No pooling	Each subgroup is estimated independently.	Allows unrestricted differences, but does not borrow information and can be unstable with sparse data.
Partial pooling	Each subgroup has its own effect, linked through a shared distribution.	Shares information while retaining subgroup-specific estimates and uncertainty.

These approaches answer different modeling questions. Partial pooling does not guarantee that subgroup differences are clinically important, nor does it establish that effects are exchangeable in the first place. The normal hierarchy assumes subgroup effects are plausibly drawn from a common distribution, after accounting for any modeled differences. If subgroups have systematic distinctions that the model omits, a single shared distribution may be inappropriate. More generally, conclusions can be sensitive to the likelihood, the prior on \(\tau\), and the number and precision of subgroups.

Diagnostics and Common Misinterpretations

A posterior estimate close to the shared center is not by itself evidence that the model has forced equality. Check the estimated posterior distribution of \(\tau\), subgroup-specific posterior summaries, and sensitivity to reasonable prior choices. If the posterior for \(\tau\) concentrates near zero, the data and prior together support little between-subgroup variation; shrinkage may then be strong. If \(\tau\) is poorly identified, especially with few subgroups, the amount of borrowing may depend substantially on its prior. Posterior predictive checks can also reveal whether the assumed distribution of effects reproduces important features of the observed subgroup estimates.

A further diagnostic is to compare each subgroup’s raw estimate and standard error with its posterior estimate and uncertainty. A pattern of substantial movement for a precise subgroup, or little movement for a very imprecise subgroup despite a well-supported small \(\tau\), merits investigation of the data, model specification, and computation. Such comparisons are checks, not automatic rules for rejecting a model.

Misconception: “All estimates move toward one center, so the model assumes equal effects.” Correction: A shared center is not a shared subgroup parameter. With \(\tau>0\), each subgroup retains an effect parameter and typically a distinct posterior distribution.
Misconception: “Different subgroup estimates prove that the effects differ.” Correction: Estimates are uncertain. Distinct point estimates alone do not establish a meaningful difference.
Misconception: “Partial pooling is always a compromise that the data determine completely.” Correction: Borrowing depends on the likelihood, the hierarchical structure, and the prior, and may be sensitive to them when subgroup information is limited.

The introductory material on hierarchical borrowing describes the general role of shared structure; the key interpretive point here is that borrowing changes the amount and direction of information in subgroup estimates, not the number of subgroup effects in the model. A common effect is a separate, stronger assumption.

Key takeaway: In a partially pooled model, subgroup effects remain subgroup-specific. Shrinkage pulls uncertain estimates toward a shared distribution, while the estimated between-subgroup variation and each subgroup’s precision determine how strongly they are pulled.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In the normal-normal model, what happens to the subgroup posterior mean as \(\tau^2\) approaches zero?
Suppose \(\mu=10\), \(y_j=14\), \(s_j^2=4\), and \(\tau^2=12\). Find the conditional posterior mean for subgroup \(j\).
Why can two subgroups with different standard errors have different amounts of shrinkage even when they share the same \(\mu\) and \(\tau\)?
Does a posterior estimate close to the shared center prove that subgroup effects are equal? Give one diagnostic relevant to assessing the amount of shrinkage.

Solutions. (1) The weight on \(y_j\) is \(\tau^2/(\tau^2+s_j^2)\), which approaches zero as \(\tau^2\) approaches zero, while the weight on \(\mu\) approaches one. This is the complete-pooling boundary. (2) The conditional posterior mean is \(\{12(14)+4(10)\}/(12+4)=208/16=13\), in the same units as the effects. (3) The estimate weight depends on \(s_j^2\): it is \(\tau^2/(\tau^2+s_j^2)\). A larger sampling variance gives less weight to the subgroup’s own estimate and more to \(\mu\). (4) No. A subgroup-specific estimate can be near the center because its own data are imprecise or because the fitted between-subgroup variation is small; it does not alone establish equality. Inspecting the posterior for \(\tau\), and checking sensitivity to its prior, helps assess how much pooling the model supports.