Fit and Scientific Robustness Are Different Questions
In a joint model, the association link determines how the longitudinal biomarker trajectory enters the event hazard. Different plausible links can fit the observed data differently and can also yield different estimated hazard contrasts. These are related findings, but they answer different questions: a likelihood comparison assesses relative fit under specified models, whereas scientific robustness asks whether the substantive conclusion persists across plausible ways of representing the association.
For example, a cumulative-exposure link may have a higher maximized likelihood than a current-value link. That does not establish that cumulative exposure is the true scientific mechanism. Nor does it establish that the estimated hazard contrast is insensitive to the link choice. Fit and the stability of the target contrast must be examined separately.
Set Up the Comparison
For subject \(i\), let \(Y_{ij}\) be the longitudinal measurement at visit time \(t_{ij}\), and let \(m_i(t)\) be the subject-specific latent biomarker trajectory. A typical shared-random-effects joint model specifies a longitudinal submodel and an event hazard such as
Here \(b_i\) denotes the subject's random effects, \(w_i\) denotes event-submodel covariates, and \(g_i(t)\) is the chosen association summary of the trajectory. Examples include the current value \(m_i(t)\), a lagged value \(m_i(t-\ell)\), or cumulative exposure \(\int_0^t m_i(s)\,ds\). The coefficient \(\alpha\) multiplies that summary. Because the summaries can have different units and meanings, the numerical value of \(\alpha\) is generally not comparable across links.
A likelihood comparison is interpretable only when the candidate fits use comparable observed data and likelihood definitions. In particular, they should use the same subjects, event and censoring information, longitudinal measurements, covariates, and treatment of the baseline hazard, apart from the intended change in association structure. The random effects must be handled consistently, and the reported likelihoods must refer to the same kind of likelihood. A difference between likelihoods computed under different conventions is not evidence about the link.
Compare Fit Without Overstating It
For a fitted model with maximized log likelihood \(\ell\) and \(k\) estimated parameters, the Akaike information criterion is
Among models fitted to the same data using comparable likelihoods, a smaller AIC indicates a more favorable balance of fit and parameter count under this criterion. AIC differences are relative to the set of models considered; they do not give the probability that a model is true. A lower AIC also does not, by itself, show that a particular hazard contrast is stable.
For nested models, a likelihood-ratio test can be used when its regularity conditions hold. If the smaller model is obtained by fixing \(q\) parameters in the larger model, the statistic is
Under the null restrictions, with regular identifiable parameters that are not on a boundary and the usual large-sample conditions, \(D\) is approximately distributed as a chi-square random variable with \(q\) degrees of freedom. This test asks whether the added parameters improve fit enough to be inconsistent with those restrictions; it does not test whether a scientific conclusion is robust to all plausible links. For nonnested links, this ordinary likelihood-ratio reference distribution does not apply. AIC can be used as a relative fit criterion when its comparability conditions are met, but its ranking still does not settle scientific interpretation.
Worked Examples
Worked Example: AIC Favors One Link, but the Contrast Changes
Suppose two joint models are fitted to the same hypothetical data, with the same longitudinal submodel and covariates. Model A uses a current-value link and has \(\ell_A=-812.4\) with \(k_A=9\). Model B uses a cumulative-exposure link and has \(\ell_B=-808.9\) with \(k_B=10\). Both report a hazard contrast for the same specified trajectory perturbation: a one-unit increase in the latent biomarker maintained over the year preceding time \(t\). For each model, suppose the estimated log hazard contrast and standard error are also available.
Fit comparison. The AIC values are
and
Thus \(\mathrm{AIC}_A-\mathrm{AIC}_B=5.0\): Model B has the lower AIC among these two candidates. This is a relative fit result, not proof that cumulative exposure is the true association.
Hazard contrasts. Suppose Model A estimates a log contrast of \(0.336\), with \(SE=0.120\). Its estimated hazard ratio is \(\exp(0.336)=1.40\). A model-based 95% Wald interval is
Suppose Model B estimates a log contrast of \(0.180\), with \(SE=0.090\). Its estimated hazard ratio is \(\exp(0.180)=1.197\), with interval
The intervals are approximate and describe uncertainty within each fitted model. Both estimates indicate a higher hazard for the specified increase, but their magnitudes differ. The lower AIC for Model B therefore does not make the contrast robust; the contrast changes appreciably with the link. These calculations presume comparable fitted likelihoods, an adequate model-based standard error, and a meaningful, consistently defined trajectory perturbation.
Worked Example: A Likelihood-Ratio Test Does Not Establish Robustness
Consider two hypothetical nested specifications with the same data and all other model components held fixed. The smaller model has a centered current-value association only. The larger adds a slope association, so the smaller model is obtained by setting the added slope coefficient to zero. Suppose their maximized log likelihoods are \(-500.2\) and \(-497.1\), respectively. Assume the added coefficient is identifiable, is not constrained to a boundary, and the large-sample approximation is reasonable.
There is one added parameter, so \(q=1\). The likelihood-ratio statistic is
For a chi-square distribution with 1 degree of freedom, the upper-tail p-value is approximately \(0.0128\). At significance level \(0.05\), the data provide evidence against the restriction that the added slope coefficient is zero, under the stated model and test assumptions.
Now compare the current-value hazard contrast at the centered reference time. In the smaller model, suppose the log contrast is \(0.262\) with \(SE=0.100\). The hazard ratio is \(\exp(0.262)=1.300\), and its approximate 95% interval is
In the larger model, suppose the corresponding centered current-value log contrast is \(0.077\) with \(SE=0.120\). The hazard ratio is \(\exp(0.077)=1.080\), with interval
The likelihood-ratio test supports the added slope term in this comparison, but the estimated current-value contrast is much closer to a hazard ratio of 1 in the larger model, and its interval includes 1. The test does not decide which contrast is scientifically preferable or establish that either specification is adequate. The comparison requires nested models and the regularity conditions stated above; if the added parameter were on a boundary or the models were not nested, this chi-square calculation would not be justified.
Assess Stability on a Common Scientific Scale
A fair robustness assessment compares the same scientific quantity, not raw association coefficients with different units. When a perturbation to the latent trajectory is specified, its effect on each link summary can be calculated, and the corresponding change in the log hazard is \(\alpha\) times that change in \(g_i(t)\). Exponentiating gives the hazard ratio for that perturbation. If no common perturbation is scientifically meaningful, report the link-specific contrasts clearly and do not describe them as directly interchangeable.
Worked Example: Similar Fit Rankings, Different Stability Conclusions
Three candidate links are fitted to the same hypothetical data. For a common contrast, define the trajectory perturbation as adding one biomarker unit throughout the interval from \(t-1\) year to \(t\), with no change earlier. The current-value summary changes by one unit at \(t\); the one-year cumulative exposure summary changes by one unit-year; and a lagged summary evaluated at \(t-0.5\) years changes by one unit. Suppose the estimates below have already been translated to the log hazard contrast for this same perturbation.
| Link | \(\ell\) | \(k\) | Log contrast (SE) |
|---|---|---|---|
| Current value | -420.0 | 8 | 0.278 (0.117) |
| Cumulative exposure | -418.7 | 9 | 0.157 (0.089) |
| Lagged value | -419.1 | 8 | 0.247 (0.116) |
The AIC values are \( -2(-420.0)+2(8)=856.0\) for the current-value link, \( -2(-418.7)+2(9)=855.4\) for the cumulative link, and \( -2(-419.1)+2(8)=854.2\) for the lagged link. The lagged link has the lowest AIC, but the differences from it are only \(1.8\), \(1.2\), and \(0\), respectively. These small differences offer limited separation by this criterion and do not select a scientific truth.
For the current-value link, the hazard ratio is \(\exp(0.278)=1.320\), and the 95% interval is
For the cumulative link, the hazard ratio is \(\exp(0.157)=1.170\), and the interval is
For the lagged link, the hazard ratio is \(\exp(0.247)=1.280\), and the interval is
All three point estimates indicate increased hazard for the defined perturbation, and the estimated direction is stable. The estimated magnitude is less stable: the hazard ratios range from about 1.17 to 1.32, and the cumulative-link interval includes 1. A careful report would distinguish the consistent direction from the uncertainty about magnitude and evidence, rather than claiming either complete robustness or a reversal. This interpretation assumes the trajectory perturbation and its time window are meaningful and are implemented consistently in all three fitted models.
Diagnostics, Alternatives, and Limitations
Before interpreting a comparison, check that the models actually differ in the intended link rather than in data inclusion, covariates, baseline-hazard flexibility, or likelihood calculation. Review convergence, parameter estimates, standard errors, and sensitivity to starting values. Inspect whether the fitted longitudinal trajectories and event patterns are adequately represented; an apparent preference among links can reflect misspecification elsewhere in the joint model. Sparse visits, influential trajectories, or weak information about the event association can make both link rankings and hazard contrasts unstable.
The main alternatives answer different questions. For nested specifications, a likelihood-ratio test assesses restrictions on added parameters under its assumptions. AIC compares relative fit with a penalty for model complexity, including for nonnested candidates when likelihoods are comparable. BIC is another penalized likelihood criterion, with a different penalty; it may rank the same candidates differently. Confidence intervals describe uncertainty within a fitted specification, while comparing estimates across plausible links reveals between-specification sensitivity. None of these alone establishes that a chosen link is scientifically correct.
A common mistake is to say, “This link has the smallest AIC, so its hazard contrast is robust.” The reasoning confuses relative fit with stability: AIC ranks candidates by a fit-and-complexity criterion, while robustness requires examining whether the target contrast remains substantively similar across plausible links. The correction is to report the fit comparison and the common-scale hazard contrasts separately, including their uncertainty and the assumptions behind them.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Two comparable models have \(\ell_1=-300\), \(k_1=7\), and \(\ell_2=-297\), \(k_2=9\). Calculate both AIC values. Which is smaller, and what does that result establish?
- A nested model adds two regular parameters and improves the maximized log likelihood by 2.4. Calculate the likelihood-ratio statistic and its degrees of freedom. What assumptions are needed to obtain a chi-square p-value?
- Why is it usually inappropriate to compare association coefficients directly when one link uses current biomarker value and another uses cumulative exposure?
- Across several plausible links, every estimated hazard ratio exceeds 1, but their intervals and magnitudes differ. Give one feature that may be stable and one that remains link-sensitive.
Solutions. (1) \(\mathrm{AIC}_1=-2(-300)+2(7)=614\), while \(\mathrm{AIC}_2=-2(-297)+2(9)=612\). Model 2 has the smaller AIC among these candidates; this is a relative fit result, not proof that its link is true or that its scientific contrast is robust. (2) \(D=2(2.4)=4.8\), with \(df=2\). The models must be nested, the restrictions regular and identifiable, parameters not on a boundary, and the large-sample chi-square approximation reasonable. (3) The summaries have different units and meanings, so their coefficients do not represent the same contrast. Translate them to a defined common trajectory perturbation when possible. (4) The positive direction may be stable; the magnitude, interval relative to 1, or strength of evidence may remain link-sensitive.