Patient Risk Rankings Across Association Links

Why Compare Patient Rankings Across Links?

A joint model can connect a patient’s longitudinal biomarker trajectory to the event hazard through different association links. One link may use the current biomarker value, another its rate of change, and another a combination of trajectory features. Even when the same patients and prediction horizon are used, these choices can produce different rankings of predicted event risk. A patient ranked highest under one link need not remain highest under another.

Rankings answer a relative question: which of these patients has the larger predicted risk over a specified horizon? They do not by themselves answer whether any patient’s risk exceeds a clinical threshold, nor do they show how large the risk differences are. The prediction time and horizon must be held fixed for a meaningful comparison.

Definition: Fix a prediction time \(t_0\) and horizon \(\Delta>0\). For patient \(i\) under association-link specification \(k\), the predicted event risk is the probability of an event in \((t_0,t_0+\Delta]\), conditional on being event-free at \(t_0\) and on the information used for prediction. A patient has a higher risk rank than another if this predicted probability is larger. A ranking is the ordering of patients by those probabilities; equal probabilities give a tie.

Risk Is Determined by the Whole Hazard Over the Horizon

For clarity, begin with a conditional shared-random-effects model. Let \(w_i\) denote the patient’s survival covariates, and let \(b_i\) denote the patient-specific random effects governing the biomarker trajectory. Under link \(k\), write the conditional hazard as

h_{ik}(t\mid b_i)=h_{0k}(t)\exp\{\gamma_k^{\mathsf T}w_i+\eta_{ik}(t)\},

where \(h_{0k}(t)\) is the baseline hazard, \(\gamma_k\) is the survival-covariate coefficient vector, and \(\eta_{ik}(t)\) is the association contribution specified by link \(k\), evaluated from the patient’s longitudinal trajectory. For example, a current-value link may use a coefficient times the latent biomarker value; a slope-based link may use a coefficient times its derivative. The link and its parameters together determine \(\eta_{ik}(t)\).

This expression assumes a proportional-hazards form conditional on the random effects and covariates, a nonnegative baseline hazard, and a specified trajectory over the prediction interval. For a plug-in prediction, the random effects are held at chosen patient-specific values. A prediction that accounts for uncertainty in \(b_i\) instead averages the conditional risk over the distribution of \(b_i\) given the patient’s available history. Whichever target is used must be applied consistently across links.

Define the conditional cumulative hazard over the horizon by \(A_{ik}(t_0,\Delta)\). The event risk is a strictly increasing function of this cumulative hazard:

A_{ik}(t_0,\Delta) = \int_{t_0}^{t_0+\Delta} h_{0k}(u)\exp\{\gamma_k^{\mathsf T}w_i+\eta_{ik}(u)\}\,du, \qquad R_{ik}(t_0,\Delta)=1-\exp\{-A_{ik}(t_0,\Delta)\}.

Thus, within one link specification, ranking by predicted risk is equivalent to ranking by cumulative hazard. But cumulative hazard depends on the full trajectory of the association contribution across the horizon, not just its value at \(t_0\). If the score changes over time, a score comparison at a single instant need not give the same ordering as a comparison of integrated risks.

Worked Example: A Reversal Between Two Constant-Score Links

Suppose two hypothetical patients, A and B, have the same survival covariates, and the baseline cumulative hazard over the chosen horizon is \(0.20\) under both link specifications. For this illustration, each patient’s association contribution is constant over that horizon. Under link 1, the contributions are \(1.2\) for A and \(0.8\) for B. Under link 2, they are \(0.3\) for A and \(0.7\) for B. Calculate and compare the risks.

With the shared covariates and constant contributions, the cumulative hazard is \(0.20\exp(\eta)\). Under link 1, A’s cumulative hazard is \(0.20e^{1.2}=0.6640\), so A’s risk is \(1-e^{-0.6640}=0.4852\). B’s cumulative hazard is \(0.20e^{0.8}=0.4451\), giving risk \(1-e^{-0.4451}\approx 0.3592\). Link 1 ranks A above B.

Under link 2, A’s cumulative hazard is \(0.20e^{0.3}=0.2700\), giving risk \(1-e^{-0.2700}=0.2366\). B’s cumulative hazard is \(0.20e^{0.7}=0.4028\), giving risk \(1-e^{-0.4028}=0.3316\). Link 2 ranks B above A. These hypothetical calculations isolate the effect of changing the association contribution; actual fitted links may also have different estimated coefficients or baseline hazards.

When Does Comparing Association Scores Work?

A useful special case gives a direct score-to-risk ordering. Suppose two patients have identical covariates \(w\), the same baseline hazard under a given link, and constant association contributions \(\eta_i\) and \(\eta_j\) over the horizon. Let \(H_0(t_0,t_0+\Delta)\) be the baseline cumulative hazard. Then

A_i=H_0(t_0,t_0+\Delta)\exp(\gamma^{\mathsf T}w+\eta_i), \qquad R_i=1-\exp(-A_i).

The function \(1-\exp(-A)\) is strictly increasing for \(A\geq 0\), since its derivative is \(\exp(-A)>0\). If the baseline cumulative hazard over the horizon is positive, the common multiplier \(H_0\exp(\gamma^{\mathsf T}w)\) is positive, and the exponential function is strictly increasing. Consequently, the risk ordering agrees with the score ordering.

Result: With a common baseline hazard, shared survival-covariate coefficients and covariates, constant association contributions over the horizon, and \(H_0(t_0,t_0+\Delta)>0\), \(R_i>R_j\) if and only if \(\eta_i>\eta_j\). If \(H_0(t_0,t_0+\Delta)=0\), both risks are zero regardless of the scores, so strict score ordering does not imply strict risk ordering.

This result is deliberately conditional. If patients have different covariates, the terms \(\gamma^{\mathsf T}w_i\) and \(\gamma^{\mathsf T}w_j\) also affect their hazards. If the association contributions vary during the horizon, their time-weighted exponentials are integrated. And if alternative links use different fitted baselines or coefficients, comparing \(\eta\) values alone is not a comparison of predicted risks. In all these settings, calculate the risk under each model or compare its cumulative hazard.

Worked Example: A Higher Initial Score Does Not Ensure Higher Risk

Consider two hypothetical patients with identical covariates and a baseline hazard that contributes \(0.10\) of cumulative baseline hazard in each of two equal subintervals. Suppose the association contribution is constant within each subinterval. Patient A has scores \(1\) and \(0\); patient B has score \(0.7\) in both. Calculate the horizon risks under a common baseline.

A’s cumulative hazard is \(0.10e^1+0.10e^0=0.3718\), so A’s risk is \(1-e^{-0.3718}=0.3105\). B’s cumulative hazard is \(0.10e^{0.7}+0.10e^{0.7}=0.20e^{0.7}\approx 0.4028\), so B’s risk is \(1-e^{-0.20e^{0.7}}\approx 0.3315\).

A has the larger score in the first subinterval, but B has the larger total cumulative hazard and predicted risk. The calculation checks the relevant quantity—the integrated hazard over the specified horizon—instead of treating the score at one time as the patient’s risk rank.

Summarising Rank Changes Across Link Specifications

For a set of patients, calculate each patient’s predicted risk separately under every link, using the same prediction time and horizon. Then order patients from highest to lowest risk within each link. A comparison should report more than an impression that the ordering “looks different.” Useful summaries include the patient pairs whose order reverses, whether the highest-ranked or top-\(k\) patients change, and a rank-concordance measure such as Spearman’s rank correlation or Kendall’s tau. These summaries describe ordering, not calibration or clinical value.

Worked Example: Counting Pairwise Reversals

Three hypothetical patients have constant association contributions under two specifications with the same baseline cumulative hazard of \(0.10\) and the same covariates. Under link A, their scores are \(0.9, 0.5, 0.1\) for patients 1, 2, and 3. Under link B, their scores are \(0.2, 0.6, 0.8\), respectively. Find the risks and compare the rankings.

Under link A, the risks are \(1-e^{-0.10e^{0.9}}=0.2180\) for patient 1, \(1-e^{-0.10e^{0.5}}=0.1520\) for patient 2, and \(1-e^{-0.10e^{0.1}}=0.1046\) for patient 3. The ranking is patient 1, then patient 2, then patient 3.

Under link B, the risks are \(1-e^{-0.10e^{0.2}}=0.1150\) for patient 1, \(1-e^{-0.10e^{0.6}}=0.1666\) for patient 2, and \(1-e^{-0.10e^{0.8}}=0.1995\) for patient 3. The ranking is patient 3, then patient 2, then patient 1. All three patient pairs reverse their order. With no ties, Kendall’s tau is \((C-D)/3=(0-3)/3=-1\), where \(C\) and \(D\) are the numbers of concordant and discordant pairs. The perfect reversal says nothing about the size or clinical importance of the risk differences; those require the probabilities themselves and their uncertainty.

Diagnostics, Alternatives, and Limitations

Before interpreting a link-sensitivity comparison, check that both specifications predict risk for the same patients who are event-free at the same \(t_0\), over the same \(\Delta\), and on the same conditioning basis. Confirm that the trajectory is evaluated over the entire horizon, that time units and biomarker scales are consistent, and that any survival covariates are included in the risk calculation. Inspect the fitted trajectories and association contributions for implausible extrapolation; sparse or irregular visits can leave the trajectory during the prediction horizon poorly determined.

Do not infer risk rank from an isolated score. The score ordering result requires its stated common-covariate, constant-score conditions and a positive baseline cumulative hazard.
Distinguish ties from reversals. Equal predicted risks may arise even when scores differ, including when the baseline cumulative hazard over the horizon is zero.
Check the full prediction target. Conditional plug-in risks and risks averaged over random-effects uncertainty are different targets and can yield different orderings.
Report absolute risks as well as ranks. A rank change may occur when two risks are nearly equal, while a high rank alone does not establish a clinically important risk.

Rank comparisons are an alternative to comparing only coefficients or likelihoods: they focus on patient-level ordering over a clinically specified horizon. They complement, rather than replace, absolute-risk comparisons and threshold-based decisions. A threshold classifies patients according to whether their predicted risk crosses a cutoff; a ranking orders them even when none or all cross that cutoff. Model averaging is another distinct approach: it combines predictions across specifications rather than declaring one specification’s ranking definitive.

A plug-in ranking also hides estimation and prediction uncertainty. Patients near one another in rank may exchange positions under small changes in estimated parameters or trajectories. Rank correlation can be insensitive to the magnitude of risk differences, and it does not establish which link is better calibrated or scientifically more credible. Therefore, a ranking should be presented as a feature of the specified models and prediction target, not as a model-free patient truth.

Key takeaway: Compare patients by their predicted risks—or equivalently their cumulative hazards—under each link at the same prediction time and horizon. Association scores alone determine strict risk ordering only under restrictive conditions, including a positive baseline cumulative hazard.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Two patients have identical covariates and constant association contributions \(0.6\) and \(0.2\). The common baseline cumulative hazard over the horizon is \(0.15\). Which patient has higher predicted risk, and why?
What happens to both patients’ risks if the baseline cumulative hazard over the horizon is zero?
Why can a patient with a higher association contribution at \(t_0\) nevertheless have a lower risk over the full horizon?
What is the difference between a risk ranking and a threshold-based classification?

Solutions. 1. The patient with contribution \(0.6\) has higher risk: with common covariates and a constant score, \(0.15\exp(\gamma^{\mathsf T}w)\) is a positive common factor, and the exponential and risk transformations are increasing. 2. Both risks are zero, so they tie regardless of their scores; the positive-baseline condition for strict ordering fails. 3. Risk depends on the integrated hazard over the horizon, so later scores, baseline hazard over time, and covariates can outweigh the score at the prediction instant. 4. A ranking orders patients relative to one another by predicted risk; a threshold classification assigns groups according to whether each predicted risk is above or below a specified cutoff.