Uncertainty Within and Between Link Specifications

Two Different Questions About Uncertainty

In a joint model, the association link connects a subject’s longitudinal biomarker trajectory to the event hazard. Plausible links might use the current biomarker value, a cumulative exposure, or a combination of trajectory features. When each link is fitted to the same data, its association estimate has uncertainty within that fitted specification. The estimates may also differ between specifications. These are distinct quantities, and neither should be mistaken for the other.

Within-specification uncertainty asks how precisely the association parameter is estimated if a particular link and the rest of that model are taken as the analysis specification. Between-specification variation asks how much the fitted association changes across the set of links considered plausible. That variation is informative about sensitivity to the link choice, but it is not automatically a standard error, a confidence interval, or evidence that any one link is correct.

The earlier material on association-link sensitivity considered how different link definitions can change an estimated association. Here the focus is how to describe the uncertainty in those estimates, both within and across specifications, while keeping the two sources separate.

Define a Comparable Quantity Before Comparing Estimates

Let \(m\) index a candidate link specification. A representative shared-random-effects model has a hazard of the form

where \(m_i(t;b_i)\) is the subject-specific longitudinal trajectory, \(g_m\) is the chosen link function, and \(\alpha_m\) is its association coefficient. The estimated coefficient \(\hat{\alpha}_m\) has a standard error \(SE(\hat{\alpha}_m)\) from the fitted model. That standard error describes estimated sampling uncertainty conditional on specification \(m\), under the model and approximation used to calculate it.

Raw coefficients are not necessarily comparable across links. For example, a coefficient multiplying a current value and one multiplying an accumulated exposure can have different units. A useful comparison requires a common scientific scale. One option is to define, for every specification, the log hazard contrast \(\theta_m\) for the same prespecified comparison of biomarker histories over the same time horizon. Then \(\exp(\theta_m)\) is the hazard ratio for that comparison under specification \(m\). If the target is a contrast derived from \(\alpha_m\), its standard error must also be transformed appropriately; for a smooth function, a large-sample delta-method approximation can be used.

Conditions for Interpreting the Two Quantities

A standard error and confidence interval are meaningful only to the extent that the fitted model and its inferential approximation are adequate. For the usual Wald interval, the estimate should be sufficiently regular and its sampling distribution reasonably approximated by a normal distribution. The fitted model should have converged, the parameter should be identifiable, and the reported standard error should correspond to the parameter actually being interpreted. If these conditions are doubtful, a Wald interval can be misleading even when it is computed correctly.

Between-link summaries require their own checks. The candidate links should be stated, scientifically plausible, and fitted to the same target population and event-time framework. The compared estimates must represent the same contrast, with compatible units and direction. If one model’s coefficient represents a one-unit biomarker increase and another’s represents a one-unit increase in cumulative exposure, their raw numerical difference has no direct interpretation. A standardized, prespecified contrast can help, but its definition must be reported.

The fits are usually based on the same individuals and outcomes. Therefore, estimates across links are generally dependent. Their observed spread is still useful as a descriptive sensitivity summary, but it does not by itself quantify repeated-sampling variability of a combined estimate. Do not treat the candidate links as independent replicates of the study.

Summarizing Variation Across Links

For \(K\) candidate specifications, let \(\hat{\theta}_m\) be the estimate of the same common-scale contrast from specification \(m\). A simple descriptive summary is the range, from the smallest to the largest estimate. The sample standard deviation across the \(K\) estimates is another summary:

Here \(s_{\mathrm{spec}}\) describes dispersion within the particular set of specifications that was examined. It is not the standard error of \(\bar{\theta}_{\mathrm{spec}}\), and the displayed mean is not automatically a preferred estimate of the association. The formulas treat the fitted values as a list to summarize; they do not give the specifications probabilities or account for dependence among their estimates.

Why Within-Model Intervals and Between-Link Spread Must Both Be Reported

A collection of point estimates alone hides their precision. Conversely, reporting only one link’s confidence interval hides the sensitivity of the result to the link choice. Reporting link-specific estimates alongside their standard errors or confidence intervals, and then describing the between-link range or standard deviation, makes both features visible.

For example, a narrow interval under each specification can coexist with substantial differences between point estimates. In that case, each fitted model may estimate its own parameter precisely while the scientific conclusion remains sensitive to the association structure. Conversely, estimates may be tightly clustered across links even though each individual interval is wide. This indicates limited link-related spread in the examined set, not necessarily precise estimation overall.

Diagnostics, Alternatives, and Limitations

Before interpreting summaries, check that every fit converged and that estimates and standard errors are finite. Inspect whether any result is unusually influential, weakly identified, or sensitive to a small number of subjects; the earlier material on outlying trajectories is relevant to that assessment. Also check the sign, units, and definition of every reported contrast. A large numerical spread can be an artifact of comparing coefficients with different scales rather than evidence of meaningful link sensitivity.

A direct alternative is to report each specification separately, with its point estimate and interval, without computing an across-link standard deviation. This preserves the full results and avoids implying that a small, hand-selected set of links is a random sample. A compact range or descriptive standard deviation can supplement that presentation, but should not replace it. In either case, state that the intervals are conditional on the corresponding link specification.

These summaries have limitations. The set of candidate links may omit important possibilities, and the spread depends on which links are included. A small spread across a narrow or poorly chosen set does not establish robustness. A large spread does not identify which link is correct. Nor does a descriptive summary account automatically for the dependence among fits or provide a formal uncertainty interval that incorporates link choice. The appropriate conclusion is scoped: describe the results for the candidate specifications examined, and distinguish conditional precision from sensitivity across them.

Common Pitfall

A common mistake is to call the standard deviation of the estimates across links “the standard error” and then combine it with the within-link standard errors as if each fitted link were an independent study. That reasoning is wrong because the models use the same data, the link set is not generally a random sample, and the across-link spread measures sensitivity to specified modeling choices rather than repeated-sampling error. The correction is to report link-specific uncertainty and between-link dispersion as separate summaries, with their meanings and limitations stated explicitly.