Clinical Decision Stability Across Link Choices

Why Decision Stability Is a Separate Question

Several association links can describe how a longitudinal biomarker relates to event risk. A current-value link, a threshold link, or a slope-based link may give different estimates even when each is scientifically plausible. The practical question here is not which link wins a model comparison, nor which link gives the preferred patient ranking. It is whether the same prespecified clinical conclusion follows under the plausible links.

That question requires more than checking whether estimated coefficients have the same sign. A clinical decision is tied to a particular contrast and a threshold. For example, a decision might concern whether a specified difference between two biomarker trajectories corresponds to a hazard ratio above 1.5 at month 12. Changing the contrast, time point, or threshold changes the question being answered.

Define the Contrast and Decision Rule First

In a shared-random-effects joint model, this contrast is commonly interpreted conditionally on the shared random effects and the specified covariates. It is not automatically a population-marginal hazard ratio. If the scientific question instead targets a marginal contrast, that target and its calculation must be specified consistently for every link.

The two trajectories must be defined in a way that makes sense across the candidate links. For instance, a trajectory contrast can specify both the biomarker values and their time pattern. A difference of 5 mg/L at month 12 is not necessarily a common contrast for a current-value link and a slope link: parallel trajectories can differ in current value while having the same slope. If the links do not represent the same contrast, apparent decision instability may simply mean that different questions were compared.

Suppose the prespecified clinical threshold is \(HR^*=1.5\). On the log-hazard scale, this becomes \(c=\log(1.5)\), approximately 0.4055. A point-estimate rule labels the contrast clinically important when \(\hat{\delta}_k>c\). Because estimates are uncertain, a second, more cautious rule uses an interval: label the contrast above threshold if the entire interval lies above \(c\), below threshold if the entire interval lies below \(c\), and indeterminate if it includes \(c\). These are different decision rules; they should not be interchanged after looking at the results.

Uncertainty and a Practical Stability Classification

For a fitted link \(k\), suppose \(\hat{\delta}_k\) estimates the specified contrast and \(SE_k\) is its standard error. Under the usual large-sample approximation that the estimator is approximately normal with a correctly estimated standard error, a 95% Wald interval on the log scale is

The endpoints are symmetric about \(\hat{\delta}_k\). Exponentiating both endpoints gives a 95% interval for the hazard ratio, since the exponential function is increasing. This approximation may be unreliable with weak identification, small samples, poor convergence, or an imprecise standard error; an interval from a different method need not be symmetric on the log scale.

For the interval-based rule, compare each log-scale interval with the same threshold \(c\). Across links, the decision is stable above threshold if every candidate interval lies above \(c\), stable below threshold if every interval lies below \(c\), and not stable if the classifications differ. If intervals straddle \(c\), report the affected link decisions as indeterminate rather than forcing a binary conclusion. A point-estimate comparison can also be reported, but it answers a less cautious question.

Why the Scale and Contrast Must Be Checked

Candidate links often use different functions of the biomarker or its trajectory. A coefficient in a current-value link may describe a one-unit increase in the current biomarker; a slope-link coefficient may describe a unit change in rate per unit time. Those coefficients are not directly comparable simply because they appear in the same position in two model equations. The contrast \(\delta_k\), not the raw association coefficient, is the relevant quantity for the threshold decision.

For a linear current-value association contribution \(A_k(z,t)=\alpha_k m_z(t)\), where \(m_z(t)\) is the biomarker value for trajectory \(z\), the contrast is \(\delta_k=\alpha_k\{m_{z_H}(t^*)-m_{z_R}(t^*)\}\). For a nonlinear link it is the difference of the two transformed values, multiplied or otherwise combined with the model parameters as specified by that link. In either case, the contrast must be computed from the fitted model rather than assumed to equal one reported coefficient.

A threshold link may imply that a small change within one biomarker range has a different association contribution from a change that crosses the threshold. A time-varying association can also make the contrast depend on \(t^*\). Therefore, the clinical time point and trajectory pair should be fixed before comparing results. If the intended decision concerns a time interval rather than a single time, define a suitable time-summary estimand in advance and use it for every link.

Checking Stability Below the Threshold

A stable conclusion need not be a stable positive finding. If every plausible link places the uncertainty interval below the clinical threshold, the evidence is stable against the specified level of clinical importance. This does not necessarily mean the hazard ratio equals 1: it may indicate an association that is positive but smaller than the prespecified threshold.

Diagnostics, Alternatives, and Limitations

Before interpreting a stability classification, verify that the candidate models estimate the same clinical contrast, on the same biomarker scale, at the same time or over the same prespecified time summary. Check that the threshold was chosen for clinical reasons rather than selected to make results agree. Review convergence, standard errors, and interval endpoints; a numerically narrow interval from a poorly fitting or weakly identified model can give false reassurance.

A useful reporting table lists, for each link, the contrast definition, \(\hat{\delta}_k\), its standard error, the log-scale interval, the exponentiated hazard ratio and interval, and the threshold classification. It makes it possible to see whether a conclusion is stable because all estimates are far from the threshold or merely because rounded values obscure a borderline result. Avoid treating overlap between two confidence intervals as a formal test that the link-specific estimates are equal. The purpose here is to assess decisions relative to a clinical threshold, not to test equality between models.

Several alternatives answer different questions. A likelihood comparison assesses relative fit under specified models; it does not by itself determine whether a clinical threshold decision is robust. Model averaging combines information across candidate links according to a stated procedure, whereas this analysis displays the decisions link by link. A single selected link gives a decision conditional on that choice and can conceal sensitivity. None of these approaches replaces the need to define a common contrast.

The classification is limited to the candidate links, fitted data, covariate specification, and uncertainty method considered. It does not account automatically for every source of model uncertainty, and a 95% interval is not a 95% probability that the fixed parameter lies inside that particular interval. Strong stability across several plausible links is useful evidence of robustness, not a guarantee of correctness or a causal conclusion.

Common Pitfall