Why Reporting Link Sensitivity Requires More Than a Coefficient Table
A link-sensitivity analysis asks whether conclusions about the association between a longitudinal process and an event depend on how that process enters the event hazard. A useful report must show more than a list of fitted coefficients: the coefficients may have different units, and a change in their numerical values does not by itself tell a reader whether the scientific conclusion changed. The goal here is a concise comparison that makes the candidate links and their implications visible without presenting one specification as definitive.
The comparison is interpretable only if readers can see what was kept comparable, what each effect contrast means, and how uncertain each estimate is. A confidence interval within one specification describes sampling uncertainty conditional on that specification; it does not measure uncertainty about which association link is appropriate.
Define the Candidate Links and the Comparison
Let \(m_i(t)\) denote the latent longitudinal mean for subject \(i\) at time \(t\), conditional on that subject’s random effects \(b_i\). A shared-random-effects joint model can express the event hazard under candidate specification \(k\) as
where \(h_{0k}(t)\) is the baseline hazard, \(w_i\) is a vector of event-model covariates, and \(g_{ik}(t;b_i)\) is the association quantity specified by link \(k\). Examples include the current value \(m_i(t)\), a cumulative exposure \(A_i(t)=\int_0^t m_i(u)\,du\), or a trajectory slope. The coefficient \(\alpha_k\) describes the association on the scale of that link; it is not automatically comparable across links.
Before interpreting a comparison, state the conditions that make it informative. The candidates should use the same analysis population, outcome and censoring definitions, longitudinal measurements, and core covariate adjustment. Other modelling choices—such as trajectory specification and baseline-hazard flexibility—should be held fixed where feasible, or their differences should be disclosed. Each candidate also requires an adequate fit for its own assumptions. If a model changes several features at once, the reported sensitivity cannot be attributed solely to the link.
These conditions do not make the link specifications identical. They make the source of the comparison clearer. Earlier material on random-effects covariance, trajectory misspecification, baseline-hazard flexibility, sparse visits, and outlying trajectories is relevant because instability from those sources can be mistaken for link sensitivity. Likewise, uncertainty within and between link specifications should not be conflated.
Make the Effect Contrasts Comparable in Meaning
For a current-value link, a change of \(\delta\) units in \(m_i(t)\), holding the other hazard terms fixed, corresponds to a hazard ratio \(\exp(\alpha_V\delta)\). For a cumulative link, the same instantaneous change is not the right contrast: the exposure difference accumulates over time. If one subject’s marker is \(\delta\) units higher throughout a reference interval of length \(\tau\), then the cumulative exposure differs by \(\delta\tau\), giving a hazard ratio \(\exp(\alpha_C\delta\tau)\). A slope link requires a contrast stated in units of marker change per unit time.
This also explains why raw coefficients should not be ranked across links. A current-value coefficient may be per marker unit, a cumulative coefficient per marker-unit-year, and a slope coefficient per marker-unit-per-year. For each candidate, choose and report a clinically or scientifically understandable contrast, with its units and time horizon. Contrasts can then be compared in context, while acknowledging that the underlying links answer distinct questions.
Deriving an Interval for a Reported Contrast
Suppose a fitted link has coefficient estimate \(\hat\alpha_k\) and standard error \(SE(\hat\alpha_k)\). For a fixed contrast multiplier \(c\), the log hazard ratio is \(c\alpha_k\), with estimate \(c\hat\alpha_k\). Its standard error is \(|c|SE(\hat\alpha_k)\). Under the usual large-sample normal approximation, a 95% Wald interval on the log scale is the estimate plus or minus \(1.96\) standard errors. Exponentiating its endpoints gives an interval for the hazard ratio because the exponential function is strictly increasing.
This calculation assumes the coefficient’s estimated sampling distribution is adequately approximated by a normal distribution and treats the contrast multiplier as fixed. For a contrast involving several estimated coefficients, its variance must instead include their covariance. The Wald interval is not a measure of variation across candidate link choices.
Worked Example: Converting Link-Specific Estimates to Hazard Ratios
Consider hypothetical results from three joint models fitted to the same analysis set with the same core adjustment. The selected contrasts are a one-unit higher current value, a one-unit higher marker sustained over one year for the cumulative link, and a one-marker-unit-per-year higher slope. Assume the estimates are on the log-hazard scale, with standard errors shown below.
| Candidate link | Coefficient estimate | Standard error | Contrast multiplier \(c\) |
|---|---|---|---|
| Current value | 0.250 | 0.100 | 1 marker unit |
| Cumulative exposure | 0.180 | 0.090 | 1 marker-unit-year |
| Slope | -0.120 | 0.140 | 1 marker-unit per year |
For current value, the log hazard-ratio estimate is \(0.250(1)=0.250\). The 95% log-scale interval is \(0.250\pm1.96(0.100)=(0.054,0.446)\). Thus the hazard-ratio estimate is \(\exp(0.250)=1.284\), and the interval is \((\exp(0.054),\exp(0.446))=(1.055,1.562)\), using unrounded endpoints for the final rounding.
For cumulative exposure, the one-year sustained contrast has \(c=1\) marker-unit-year. The log-scale interval is \(0.180\pm1.96(0.090)=(0.0036,0.3564)\). Exponentiating gives an estimated hazard ratio of \(\exp(0.180)=1.197\), with 95% interval \((\exp(0.0036),\exp(0.3564))=(1.004,1.428)\).
For slope, the stated contrast is \(c=1\) marker-unit per year. The log-scale interval is \(-0.120\pm1.96(0.140)=(-0.3944,0.1544)\). The hazard-ratio estimate is \(\exp(-0.120)=0.887\), with 95% interval \((\exp(-0.3944),\exp(0.1544))=(0.674,1.167)\). The table would report these contrasts and intervals, not compare \(0.250\), \(0.180\), and \(-0.120\) as though they shared a scale.
Under these fitted models, the first two intervals exclude a hazard ratio of 1, whereas the slope interval includes 1. This is a description of uncertainty within each specification, not a test that selects between the links. The estimates concern conditional associations under the stated contrasts; they do not imply that changing the marker would cause the event hazard to change.
A Compact Table and a Careful Narrative
A practical sensitivity table should make the scale and interpretation visible without overwhelming the reader. A concise version can include the link, the reference contrast, the hazard-ratio estimate and interval, and a short conclusion. State the confidence level and any shared modelling choices in the table note or nearby prose.
Worked Example: Writing a Link-Sensitivity Summary
Using the hypothetical estimates above, a short table could be presented as follows. The contrasts and units belong in the table because the hazard-ratio values alone would not reveal what was compared.
| Association link | Reference contrast | Hazard ratio (95% CI) | Conclusion at the 95% interval level |
|---|---|---|---|
| Current value | 1 marker unit higher at time \(t\) | 1.284 (1.055, 1.562) | Interval excludes 1 |
| Cumulative exposure | 1 marker unit higher sustained for 1 year | 1.197 (1.004, 1.428) | Interval excludes 1 |
| Slope | 1 marker-unit/year higher slope | 0.887 (0.674, 1.167) | Interval includes 1 |
A suitable narrative is: “Across the three prespecified links, the estimated association was positive for the one-unit current-value contrast and the one-year cumulative contrast; both 95% intervals excluded a hazard ratio of 1. The slope contrast had an estimate below 1 and a 95% interval that included 1. The magnitude and direction of the estimate were therefore specification-sensitive, and the intervals remain conditional on their respective links.” This wording reports the differences without declaring a preferred model.
The table’s “excludes 1” wording is a concise description of the intervals, not a claim that the links have been formally compared or that the null is true or false. If a scientific decision depends on a prespecified threshold other than a hazard ratio of 1, report that threshold and evaluate each contrast against it rather than treating statistical significance as the decision rule.
What the Summary Can—and Cannot—Establish
A sensitivity table is a communication device for evidence across plausible specifications. It is not a substitute for checking whether the candidate models are credible. Before summarizing the estimates, examine whether the longitudinal trajectory is adequately represented, whether the event baseline hazard is sufficiently flexible, and whether the observed visit schedule supports the proposed link. Inspect influential trajectories and consider whether association estimates are driven by a small number of subjects. These checks help distinguish genuine link sensitivity from instability due to other modelling choices.
The intervals in the table quantify within-specification estimation uncertainty. They do not automatically include the between-specification variation in point estimates, nor do they account for uncertainty about how plausible each specification is. Model averaging, bootstrap stability, and clinical decision stability are related but distinct analyses. If those analyses have been conducted, report them separately rather than implying that an ordinary confidence interval already incorporates their uncertainty.
Alternative approaches answer different questions. A likelihood comparison evaluates relative fit under specified models and its interpretation depends on the comparison and assumptions; it does not by itself establish scientific robustness. Model averaging combines predictions or estimates according to a stated procedure, rather than displaying each candidate separately. A sensitivity table instead makes the consequences of the candidate links transparent without assigning one a winning status. When candidate links differ in more than their association structure, clearly identify those differences so readers do not attribute all variation to the link.
Common Reporting Mistakes and Diagnostics
- Comparing raw coefficients. Coefficients for current value, cumulative exposure, and slope can have different units. Define an interpretable contrast for each before comparing effects.
- Using a cumulative contrast without its duration. A one-unit difference accumulated over six months is not the same exposure contrast as one sustained for two years. State the time horizon and units explicitly.
- Calling a link the winner because its interval excludes 1. That comparison concerns each specification’s association contrast, not a direct test that one link is preferable. Report the differing conclusions without overstating them.
- Interpreting a confidence interval as between-model uncertainty. Each interval is conditional on its fitted specification. Describe variation in estimates across rows separately.
- Hiding changes in other modelling choices. If trajectory form, covariate adjustment, or baseline hazard also changes, say so. Otherwise the source of the observed sensitivity is ambiguous.
- Rounding before exponentiating. Calculate interval endpoints from unrounded estimates and standard errors, then round the final hazard ratios and endpoints consistently.
Warning signs include substantial changes in estimates after small modelling adjustments, unusually wide intervals, and conclusions that depend on one influential subject or a narrow range of observed visit times. These findings call for careful qualification and further diagnostics, not an automatic choice of the specification with the most favorable estimate.
Worked Example: Reporting a Changed Conclusion Without Choosing a Link
Suppose the analysis team wants to summarize whether the direction of association is stable. From the hypothetical table, the current-value estimate is \(1.284\), the cumulative estimate is \(1.197\), and the slope estimate is \(0.887\). Their 95% intervals are \((1.055,1.562)\), \((1.004,1.428)\), and \((0.674,1.167)\), respectively.
The current-value and cumulative contrasts both have point estimates above 1, while the slope contrast has a point estimate below 1. The slope interval includes 1, so its direction is uncertain at this interval level. A defensible summary is that evidence for a positive association is present under the first two stated contrasts, whereas the slope specification does not provide comparably clear evidence of a non-null association and its point estimate has the opposite direction.
This conclusion is deliberately conditional. The three contrasts have distinct meanings, and none establishes causation. Without a prespecified criterion and a justified direct comparison, the results do not support claiming that the current-value or cumulative link is definitively correct. The changed direction itself is a sensitivity finding to report and investigate.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A cumulative link coefficient is per marker-unit-year. What extra information is needed to state a meaningful hazard-ratio contrast?
- A log hazard-ratio estimate is \(0.20\) with standard error \(0.10\). Calculate the hazard ratio and 95% Wald interval for a one-unit contrast.
- Why should raw coefficients from a current-value and a slope link not be ranked by magnitude?
- One candidate interval excludes 1 and another includes 1. What can a sensitivity report conclude, and what can it not conclude from this fact alone?
- What should be disclosed if the candidate models differ in both link and baseline-hazard specification?
Solutions. (1) State the marker difference and duration over which it accumulates; for example, a one-unit difference sustained for one year gives a one-marker-unit-year contrast. (2) The estimate is \(\exp(0.20)=1.221\). The log interval is \(0.20\pm1.96(0.10)=(0.004,0.396)\); exponentiating gives approximately \((1.004,1.486)\). (3) The coefficients use different units and represent different changes in the latent trajectory, so their numerical magnitudes are not directly comparable. (4) It can report that the within-specification interval conclusions differ; it cannot establish from that fact alone which link is correct or that the intervals test one link against the other. (5) Disclose both the association-link and baseline-hazard differences, since the result cannot be attributed to the link alone.