Why Control the Comparison?
A joint model connects a subject’s longitudinal biomarker trajectory to that subject’s time-to-event outcome. Its association link specifies how the trajectory enters the event hazard. If a researcher changes the link and also changes the covariates, analysis sample, or estimation procedure, a difference in results has several possible explanations. A link-sensitivity analysis is most informative when the link is the deliberate change and the remaining analysis is kept comparable.
“Hold fixed” needs careful interpretation. It usually means keeping the data, the covariate definitions, the longitudinal and survival submodel structures, and the estimation protocol fixed across candidate links. It does not ordinarily mean freezing every fitted parameter at its estimate from the first model. Each candidate joint model is generally refitted, because its association structure can affect the joint likelihood and hence the estimates of other parameters as well.
State the Shared Framework First
Let \(Y_{ij}\) be the biomarker measurement for subject \(i\) at measurement time \(t_{ij}\), let \(m_i(t)\) denote that subject’s latent biomarker trajectory, let \(T_i\) be the event time, and let \(\delta_i\) indicate whether the event was observed rather than censored. Let \(b_i\) denote subject-specific random effects and \(X_i\) denote covariates. One general shared-random-effects model has a longitudinal component such as
and a conditional event hazard of the form
Here \(h_0(t)\) is the baseline hazard, \(\gamma\) is the vector of survival covariate effects, and \(g\) is the association link, which may depend on a specified feature or history of the trajectory. The random effects connect the two outcome processes through the latent trajectory. Candidate links may use different trajectory features or functional forms, but the comparison is only interpretable as a link-sensitivity analysis if the other ingredients are defined consistently.
The model also requires assumptions. For example, the chosen longitudinal model must adequately describe the repeated measurements and their within-subject dependence; the survival submodel and baseline hazard must be appropriate for the event process; and the random-effects distribution and its role in connecting outcomes must be specified. The event and longitudinal processes are modeled jointly conditional on the included covariates and random effects. Any censoring assumptions must also be appropriate for the analysis. Holding an incorrect structure fixed does not make it correct: it only makes the source of differences between candidate fits easier to identify.
What “Fixed” Does and Does Not Mean
A clean comparison holds the specification fixed, not necessarily each estimate. For example, if every candidate model includes the same baseline covariates and random intercept and slope, the estimates of their coefficients and variances may still differ after refitting. Those differences can arise because a changed link alters how the joint likelihood allocates information across the trajectory and event components. They are part of the model’s response to the alternative link, not evidence that the analysis accidentally changed its covariate set.
By contrast, changing the sample, dropping a covariate in one model, using a different baseline-hazard form, or changing the random-effects distribution at the same time as changing the link makes attribution ambiguous. A model fitted to fewer subjects may yield a different association estimate simply because its data differ. A changed longitudinal trajectory model may change the estimated latent biomarker histories, which can in turn change the hazard contribution.
The computational protocol should also be comparable: use the same likelihood or estimation framework, comparable starting-value strategy, convergence criteria, and target numerical precision. The settings need not be mechanically identical if a candidate link requires a different integration strategy, but the resulting approximation accuracy should be comparable and adequate. Otherwise, an apparent link effect could partly be numerical error or failure to find a suitable optimum.
Worked Example: Setting Up a Controlled Link Comparison
Suppose a hypothetical study has 120 subjects, each with repeated biomarker measurements and observed event or censoring information. The analysis compares two candidate links. Both models use the same 120 subjects, the same event and censoring indicators, the same age and treatment covariates, the same longitudinal random-intercept-and-slope structure, and the same baseline-hazard specification.
The first model uses a current-trajectory feature in \(g\), while the second uses a prespecified cumulative trajectory feature. The changed element is the link. Each model is refitted jointly, including its association and other model parameters. Thus, if the estimated treatment coefficient changes, that is a result to report as sensitivity of the fitted model, not a reason to silently freeze the coefficient at its first-model estimate.
Now suppose the second fit also drops 8 subjects with incomplete covariates. The comparison no longer isolates the link: the sample has changed from \(n=120\) to \(n=112\). A defensible repair is to use a common analysis set under a stated missing-data rule, or to acknowledge explicitly that both link and sample changed. Merely labeling the two fits a link comparison would not make their difference attributable to the link alone.
How a Link Change Can Alter a Prediction
For a subject profile, the link affects the hazard through \(g\), and therefore affects cumulative hazard and event risk. Conditional on the model quantities used for prediction, the survival probability through time \(\tau\) is \(S_i(\tau)=\exp\{-H_i(\tau)\}\), where \(H_i(\tau)\) is the cumulative hazard through \(\tau\). The corresponding event risk is \(1-S_i(\tau)\). This identity makes clear why two candidate links can produce different risks even when the rest of the stated model framework is unchanged.
Worked Example: Comparing Predicted Risks Without Rounding Early
For a hypothetical subject profile and a fixed prediction horizon of 12 months, suppose two refitted candidate links give cumulative hazards \(H_A(12)=0.200\) and \(H_B(12)=0.260\). The event, covariates, and prediction horizon are the same; the displayed cumulative hazards are illustrative fitted-model outputs.
Under link A, the 12-month event risk is
This is a predicted risk of about 18.13%. Under link B,
To find the risk difference, calculate from the unrounded expressions rather than subtracting rounded risks:
For this profile, link B gives a fitted 12-month risk about 4.77 percentage points higher than link A. This is a difference between model-based predictions under the two specifications; it does not establish that either link is correct or that changing the link would causally change an individual’s outcome.
Refitting, Diagnostics, and Comparisons
A practical analysis should define its comparison before looking at results. Prespecify the candidate links and the quantities on which conclusions will be compared, such as fitted association patterns, survival predictions for stated profiles, or uncertainty intervals. Keep the prediction profiles and time horizons the same across fits. If conclusions are sensitive to the link, report that dependence rather than selecting one result without explanation.
After fitting, check whether each model actually converged, whether parameter estimates are finite and plausible, and whether numerical integration or optimization warnings occurred. Examine standard errors, uncertainty intervals, and fitted trajectories and survival behavior for signs of instability. A convergence flag alone is not proof of a good fit, while a failed or poorly identified fit should not be treated as a valid candidate simply because its point estimate is convenient. If the candidate models use a likelihood-based criterion such as AIC, its comparison requires fits to the same outcome data under comparable likelihood definitions; it is not a substitute for checking model adequacy.
The principal alternative to changing one modeling element at a time is a broader specification analysis that varies several assumptions together. That may be useful for exploring overall robustness, but it does not isolate link sensitivity. A one-at-a-time link comparison offers clearer attribution, though it cannot reveal every interaction between the link and other model assumptions. If a link appears sensitive to a particular trajectory or random-effects specification, a separately designed analysis can investigate that issue; it should be reported as a distinct change in specification.
A link-sensitivity analysis also has limits. Candidate links may all be misspecified, the data may not contain enough events or longitudinal measurements to distinguish them, and estimates may depend on assumptions about the unobserved biomarker trajectory between measurements. Results can be especially fragile when there is little information over parts of follow-up or when the random-effects structure is weakly identified. Consistency across candidate links supports robustness to those particular choices, not proof that the shared model assumptions hold.
Worked Example: Detecting a Confounded Comparison
Consider two hypothetical fits to the same event outcome. Model A has 10 estimated parameters and maximized log likelihood \(-312.4\). Model B has 10 estimated parameters and maximized log likelihood \(-309.8\). Both use the same subjects, outcome definition, and likelihood framework; only the association link differs. For these conditions, the AIC calculation is \(AIC=-2\ell+2k\), where \(\ell\) is the maximized log likelihood and \(k\) is the number of estimated parameters.
For model A,
For model B,
The AIC difference is \(644.8-639.6=5.2\), favoring model B by this criterion among these two fitted candidates. Because the likelihood framework, data, and parameter count are comparable, the calculation is interpretable as a relative fit-penalty comparison. It does not show that model B’s link is true, nor does it remove the need to inspect convergence, uncertainty, and substantive predictions.
If model B had instead been fitted to a different subject subset or with a different outcome likelihood, this arithmetic would not provide a clean link-only comparison. The computed numbers might still be available, but the contrast would combine differences in link and analysis basis.
Common Mistake and Key Takeaway
- Wrong reasoning: “To isolate the link, I should hold every fitted parameter fixed at its estimate under the first link.”
- Correction: Usually hold the data and non-link model specification fixed, then refit under each link. Freezing estimates instead creates a conditional calculation that omits how the alternative association structure changes joint estimation and its uncertainty.
- Wrong reasoning: “The estimates differ, so I must have changed more than the link.”
- Correction: Other parameter estimates can legitimately move after refitting. Audit the analysis design to ensure the sample, variables, submodels, and estimation protocol were held comparable; then report parameter movement as part of sensitivity.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A candidate model uses a different event definition as well as a different link. Can the difference in results be attributed to the link alone? Why?
- In a controlled comparison, should every coefficient estimate be forced to equal its estimate under the first link?
- For cumulative hazard \(H=0.15\), calculate the event risk through the prediction horizon using \(R=1-\exp(-H)\).
- Two links produce similar predictions, but one fit has an optimizer warning and a non-finite standard error. Is that sufficient evidence of robustness?
- What does a finding that predictions differ by 4.77 percentage points between two candidate links establish, and what does it not establish?
Solutions. (1) No. The event definition changed, so both the link and the outcome analysis changed; the difference cannot be isolated to the link. (2) No. Keep the model specification comparable but refit the parameters under each link; estimates may move as the joint likelihood changes. (3) \(R=1-\exp(-0.15)=1-0.860708=0.139292\), or about 13.93% through the stated horizon. (4) No. The warning and non-finite standard error indicate that the fit may be unreliable, so its prediction cannot be treated as sound evidence of robustness without resolving or appropriately reporting the fitting problem. (5) It establishes that the stated model-based prediction for that profile and horizon is sensitive to those two fitted link specifications. It does not establish which link is true, nor that the link itself causally changes the subject’s risk.