Interpreting Information Borrowing Across Endpoints

Why Information Borrowing Needs Interpretation

A joint model can use observations from one endpoint to improve estimation or prediction for another. This is especially attractive when one endpoint is measured often and another is sparse. But a narrower interval for the sparse endpoint does not, by itself, show that borrowing was appropriate. The gain depends on the model’s link between endpoints: a shared parameter, a shared latent effect, or another specified dependence structure.

The practical question is therefore not simply whether the endpoints are associated. It is which quantities are shared, how strongly they are linked, and what assumptions make information transfer valid. If the link is too restrictive, precision can increase while estimates become less representative of the sparse endpoint.

Where the Borrowing Comes From

With a shared parameter, the model constrains two endpoint processes to depend on a common quantity. Data from either endpoint can then inform that quantity, and consequently affect estimates for both endpoints. Exact sharing is a strong assumption: if an exposure-effect parameter is identical across endpoints, the model does not allow the endpoints to have different values of that effect.

With partial pooling, endpoint-specific parameters are allowed to differ, but their differences are modeled as coming from a shared distribution. One endpoint informs the distribution, and thereby the estimate for another, without forcing equality. The strength of borrowing depends on the modeled between-endpoint variation: less variation implies tighter linkage and generally more borrowing.

With a shared latent effect, observations from one endpoint can inform an individual’s latent state or random effect, which also enters the other endpoint’s model. This can improve estimates of that latent quantity and predictions for a sparsely observed endpoint. The earlier material on shared random effects, latent disease states, and joint likelihoods provides the relevant model structures; here the emphasis is on interpreting the resulting information transfer.

These mechanisms are not interchangeable. Shared parameters link population-level effects; shared random effects can link individual-level departures; residual dependence links outcomes after modeled effects are accounted for. A model may use more than one mechanism, but each link needs a substantive and statistical justification.

Quantifying Partial Pooling

Consider a simple normal approximation for two endpoint-specific treatment effects, \(\theta_A\) and \(\theta_B\). Suppose endpoint A has estimate \(\hat{\theta}_A\) with standard error \(s_A\), and endpoint B has estimate \(\hat{\theta}_B\) with standard error \(s_B\). Model the difference between the true endpoint effects as \(\theta_B\mid\theta_A\sim N(\theta_A,\tau^2)\). For this illustration, treat \(\tau^2\) as known. The predictive distribution for endpoint B, conditional on the data for A, has mean \(\hat{\theta}_A\) and variance \(V_A=s_A^2+\tau^2\): it includes uncertainty in A’s estimate as well as the allowed difference between endpoints. If the two endpoint likelihood approximations are independent conditional on their effects, then B’s own estimate contributes variance \(s_B^2\).

The posterior distribution for \(\theta_B\) is normal. Its precision is the sum of the predictive precision from A and the likelihood precision from B. Completing the square gives the posterior mean as a precision-weighted average:

Thus, A’s contribution grows when its predictive variance \(V_A\) is small, while B’s own estimate carries more weight when its standard error is small. As \(\tau^2\) increases, \(V_A\) increases and borrowing weakens. In the limit \(\tau^2\to\infty\), B’s estimate is effectively unpooled. In the limit \(\tau^2\to0\), the model approaches a common effect, though A’s estimate uncertainty still matters.

In this normal model, the posterior variance can be compared directly with \(s_B^2\), the variance from B’s own likelihood approximation. That comparison quantifies model-based precision gain. It should not be treated as a universal conversion to “effective sample size”: outside simple models, the relation between variance and sample size depends on the likelihood and parameterization.

Borrowing Through a Shared Latent Effect

The same principle can operate at the individual level. For a transparent calculation, suppose an individual has latent effect \(U\sim N(0,1)\). Endpoint A measures \(U\) with error variance 0.25, and endpoint B measures \(U\) with error variance 1. The measurement errors are independent conditional on \(U\), and the measurement scales and loadings are assumed known. In a real application, those assumptions and the suitability of a normal latent-effect model need assessment.

This example is deliberately simple. In a nonlinear binary or count endpoint model, integrating over a shared latent effect can make the likelihood and posterior non-normal. The direction and magnitude of information transfer then depend on the link function, the latent-effect distribution, the endpoint-specific loadings, and which observations are available. A shared effect does not mean that one endpoint’s observed value can be substituted directly for the other’s.

When Borrowing Misleads

Borrowing is useful when the endpoint link is scientifically defensible and the data support its strength. It can mislead when the model imposes an overly tight link, when the shared structure is wrong, or when the apparent precision gain hides endpoint-specific differences. Sparse data make this especially important: a small amount of endpoint-specific evidence may be overwhelmed by a strong borrowing assumption.

Diagnostics, Comparisons, and Limits

Interpret borrowing by examining both the estimate and its uncertainty under reasonable alternative assumptions. The following checks help distinguish a stable precision gain from one driven by a fragile link:

• Compare borrowed and unborrowed results. Report how the endpoint estimate changes when its own data are analyzed without cross-endpoint information, and how it changes under weaker and stronger pooling. A large shift signals sensitivity to the borrowing structure.
• Inspect endpoint discrepancies. Compare endpoint-specific estimates and their uncertainty on a common, interpretable scale. Large discrepancies may indicate genuine endpoint differences, an unsuitable shared structure, or simply imprecise estimates; sparse data alone cannot distinguish these explanations.
• Assess the link parameters. If between-endpoint variation or latent-effect loadings are estimated, determine whether the data identify them adequately. Treating a weakly informed variance as known can make the reported gain look more certain than warranted.
• Check model fit and predictions. Examine whether the joint model can reproduce relevant endpoint patterns and whether predictions for the sparse endpoint are plausible. A model that fits one endpoint but systematically misses the other has not earned its precision gain.
• Account for uncertainty in the borrowing structure. The numerical example treated \(\tau^2\) as known. In practice, uncertainty in variance components and other shared parameters must be represented; plugging in an estimated value can understate uncertainty, particularly with few endpoints or sparse data.

The main alternatives are no borrowing, which estimates each endpoint separately; complete pooling, which enforces equality of a parameter; and partial pooling, which allows differences governed by a shared distribution. A joint model with shared latent effects is another option when the scientific mechanism is individual-level dependence rather than equality or exchangeability of endpoint parameters. Separate fits avoid cross-endpoint assumptions but can be imprecise. Complete pooling can be efficient when equality is credible but biased for an endpoint that differs. Partial pooling is more flexible, but its flexibility does not remove the need to validate the assumed distribution of differences.

Even when a borrowing model is appropriate, the gain is specific to the modeled estimand and data structure. It does not establish that changing one endpoint will causally change another, and it does not guarantee improved prediction in a new population. Conclusions should state which endpoint supplied information, through what modeled link, and how sensitive the target estimate was to that link.