Interpreting Changes in a Patient’s Predicted Risk

Why a Predicted Risk Can Change

A dynamic prediction is conditional on what is known at a particular time. If a patient has another biomarker measurement, or remains event-free through additional follow-up, the information available for prediction changes. The reported risk may also change simply because the requested prediction horizon changes. These are different reasons for a numerical change, and they should not be interpreted as though they were interchangeable.

The risk is a probability for a specified future interval, conditional on the patient’s observed history and event-free status at the prediction time. It is not a diagnosis, nor does a change in predicted risk by itself show that a biomarker caused a change in the patient’s health. To interpret two predictions sensibly, identify the landmark time, the information included at that landmark, and the endpoint of the interval being predicted.

Definition: Let $T_i$ be the event time for patient $i$, $s$ a landmark time, and $w>0$ a prediction horizon measured from that landmark. Let $Y_i^{(s)}$ denote the patient’s longitudinal measurements observed through $s$, and let $X_i$ denote relevant covariates. The dynamic risk prediction is $$ \pi_i(s,w)=P(s<T_i\leq s+w\mid T_i>s,\;Y_i^{(s)},\;X_i). $$ It refers to the probability of an event in the interval $(s,s+w]$, given that the patient is event-free at $s$ and given the specified history.

How New Information Enters the Prediction

In a joint model, a patient’s latent random effects $b_i$ help describe the underlying biomarker trajectory and may also be associated with the event hazard. Conditional on $b_i$, the event hazard can be written, for example, as

h_i(t\mid b_i,X_i)=h_0(t)\exp\{\gamma^\top X_i+\alpha m_i(t;b_i)\},

where $h_0(t)$ is the baseline hazard, $m_i(t;b_i)$ is the latent marker value, and $\gamma$ and $\alpha$ are model parameters. The conditional survival function is $S_i(t\mid b_i,X_i)=P(T_i>t\mid b_i,X_i)$. This example of a hazard specification is not required for the interpretation below; the key is that the conditional survival function depends on the latent effects.

The patient’s random effects are not observed. Prediction therefore averages over their distribution given the patient’s data. At landmark $s$, the distribution of $b_i$ is informed by both the longitudinal measurements and the fact that the patient has remained event-free through $s$. Under the joint model’s conditional-independence assumptions, its density is proportional to the prior random-effects density multiplied by the longitudinal-data likelihood and the event-free survival likelihood:

p(b_i\mid T_i>s,Y_i^{(s)},X_i) \propto p(Y_i^{(s)}\mid b_i,X_i)\, S_i(s\mid b_i,X_i)\, p(b_i).

Consequently, the dynamic risk is the average of the conditional risk over this updated distribution:

\pi_i(s,w)= \int\left\{1-\frac{S_i(s+w\mid b_i,X_i)} {S_i(s\mid b_i,X_i)}\right\} p(b_i\mid T_i>s,Y_i^{(s)},X_i)\,db_i.

This expression separates two roles. The ratio of survival functions gives the conditional risk for a particular value of $b_i$; the posterior distribution averages that risk over plausible values of $b_i$ for the patient. A new measurement can change the posterior distribution through the longitudinal likelihood. Additional event-free follow-up can change it through the survival likelihood. Changing $w$ changes the interval in the conditional-risk term, even if the patient’s information and posterior distribution are held fixed.

A Short Justification for the Update

For a fixed $b_i$, the probability of an event by $s+w$, conditional on being event-free at $s$, is one minus the conditional probability of surviving from $s$ to $s+w$. By the definition of conditional probability,

P(T_i>s+w\mid T_i>s,b_i,X_i) = \frac{P(T_i>s+w\mid b_i,X_i)} {P(T_i>s\mid b_i,X_i)} = \frac{S_i(s+w\mid b_i,X_i)}{S_i(s\mid b_i,X_i)}.

Subtracting this survival probability from one gives the conditional risk for that $b_i$. Because $b_i$ is unknown, averaging this quantity with respect to $p(b_i\mid T_i>s,Y_i^{(s)},X_i)$ gives the displayed dynamic prediction. The survival term also explains why event-free follow-up is information: values of $b_i$ associated with lower probabilities of surviving to $s$ receive less posterior weight among patients known to have survived to $s$.

This calculation relies on a correctly specified joint model and on the conditional dependence structure it assumes. In particular, the longitudinal-data likelihood and survival likelihood must be combined with the random-effects distribution as specified by the model. The prediction also presumes that the event-free status at the landmark is known. If event status is censored before $s$, one cannot simply treat that patient as known to have survived to $s$.

Worked Example: Separating Biomarker Evidence From an Existing Update

Suppose a hypothetical patient is event-free at a landmark. After incorporating the available history and event-free follow-up, a two-group approximation to the random-effects posterior assigns probability $0.2652$ to a high-risk latent group and $0.7348$ to a low-risk group. A newly observed biomarker has likelihood $0.70$ in the high-risk group and $0.20$ in the low-risk group. Assume these likelihoods are appropriate conditional on group membership and the information already used. Find the updated high-risk-group probability and the predicted risk over the next interval, if the group-specific risks are $0.30$ and $0.08$.

Biomarker update. Bayes’ rule gives the updated high-risk-group probability:

\frac{0.2652(0.70)} {0.2652(0.70)+0.7348(0.20)} = \frac{0.18564}{0.18564+0.14696} = \frac{0.18564}{0.33260} \approx 0.5581.

The corresponding low-risk probability is $1-0.5581=0.4419$, using the unrounded ratio for the calculation. The biomarker evidence has shifted posterior weight toward the high-risk group.

Risk prediction. Average the group-specific risks using the updated weights:

0.5581(0.30)+0.4419(0.08) \approx 0.1674+0.03535 =0.2028.

The predicted event risk for the specified next interval is about $0.2028$, or $20.28\%$. Before this new measurement, but at the same landmark and horizon, the group-mixture risk would have been

$$ 0.2652(0.30)+0.7348(0.08) =0.07956+0.05878 =0.1383. $$

Thus the prediction increased from about $13.83\%$ to $20.28\%$ because the new biomarker changed the group weights, while the group-specific risks and horizon were held fixed. The calculation is illustrative: it does not mean that the biomarker caused the patient’s underlying risk to increase.

Three Sources of Apparent Change

Comparing predictions requires checking whether the same target is being predicted. A comparison can be organized by asking what changed and what was held fixed.

New biomarker evidence: At the same landmark, event-free status, and horizon, incorporating a new measurement changes the longitudinal likelihood and can shift the posterior distribution of $b_i$. It may raise or lower the prediction depending on how the measurement fits the model.
Additional event-free follow-up: Moving to a later landmark after the patient remains event-free changes the survival information in the posterior distribution. This can shift the risk prediction even if no new biomarker is recorded.
Changed horizon: At the same landmark and with the same patient information, asking about a longer or shorter future interval changes the target probability. It does not, by itself, represent new evidence about the patient.

These effects need not be additive or uniquely attributable. For example, the change from a prediction based on biomarker history alone to one based on biomarker history plus event-free follow-up can depend on the order in which the information is incorporated. The posterior distribution is updated jointly, and the size of a “biomarker contribution” can vary if the comparison baseline changes. A clear report specifies the two predictions being compared and which inputs differ.

Worked Example: A Different Horizon, No New Patient Information

Suppose the posterior probability of membership in a high-risk group is $0.60$. At the current landmark, the conditional event risks are $0.12$ for the next three months and $0.32$ for the next nine months in that group. The corresponding low-risk probabilities are $0.03$ and $0.10$. Calculate both predictions using the same posterior weights.

For the three-month horizon:

$$ 0.60(0.12)+0.40(0.03)=0.072+0.012=0.084. $$

For the nine-month horizon:

$$ 0.60(0.32)+0.40(0.10)=0.192+0.040=0.232. $$

The predictions are $8.4\%$ and $23.2\%$, respectively. The patient’s observed history and posterior group weights have not changed; the longer horizon asks about a different interval. The larger probability is not evidence that the patient’s risk “rose” during the calculation.

Worked Example: Event-Free Follow-Up Without a New Measurement

Consider a hypothetical patient whose prior probability of belonging to a high-risk group is $0.50$. Over an interim period, the probability of remaining event-free is $0.80$ for the high-risk group and $0.95$ for the low-risk group. Assume the patient is known to have remained event-free, and no new biomarker measurement arrives. At the new landmark, the next-six-month risks are $0.25$ in the high-risk group and $0.06$ in the low-risk group. For this example, assume these group-specific next-six-month risks are the same at both landmarks, so that the comparison isolates the survival update.

The posterior high-risk probability among those known to have survived the interim period is

\frac{0.50(0.80)} {0.50(0.80)+0.50(0.95)} = \frac{0.40}{0.875} \approx 0.4571.

The prior-weight prediction for a six-month interval is

$$ 0.50(0.25)+0.50(0.06)=0.125+0.030=0.1550. $$

At the new landmark, the prediction is

0.4571(0.25)+0.5429(0.06) \approx 0.1143+0.03257 =0.1469.

Under the stated assumptions, the prediction falls from $15.50\%$ to about $14.69\%$, a difference of about $0.81$ percentage points. This change comes from the event-free follow-up shifting posterior weight toward the group more likely to survive that period. In a model with a changing baseline hazard, the future group-specific risks need not be the same at the two landmarks; then both the posterior update and the changed calendar-time interval can affect the comparison.

Interpreting and Checking Sequential Predictions

A useful comparison states, for each prediction, the landmark, horizon, event-free condition, and longitudinal data included. If the goal is to describe the effect of a new measurement on the model output, hold the landmark and horizon fixed and compare predictions with and without that measurement. If the goal is to describe the prediction after surviving further follow-up, state that the landmark has moved and that survival information has been incorporated. If the horizon changes, report the two horizons explicitly rather than treating the resulting difference as a patient update.

For practical diagnostics, inspect whether the posterior weights or draws change in the expected direction under the fitted model, and verify that the same risk definition and horizon were used in the comparison. Check numerical integration or Monte Carlo error when prediction relies on integration over random effects; the earlier material on likelihood-based integration and Monte Carlo integration is relevant to those computations. If posterior draws are used, verify that the prediction is averaged over the appropriate patient-specific distribution rather than evaluated only at a single estimated random effect.

A large jump is not automatically an error, but it should prompt checks of the measurement scale, timing, missingness treatment, and the likelihood contribution of the new measurement. Predictions can also be unstable when a patient has sparse or noisy measurements or when the fitted model gives little support to the observed history. In such cases, the uncertainty interval for individual risk is important alongside the point estimate. Uncertainty intervals do not, however, explain which information caused a point prediction to change; that requires the controlled comparisons described above.

Common Mistakes and Alternatives

Calling every increase a biomarker effect. The landmark or horizon may have changed as well. Compare predictions with the other inputs held fixed before attributing the difference to the new measurement.
Ignoring event-free follow-up. A patient who reaches a later landmark event-free contributes survival information, even if no biomarker is observed during that period. The posterior distribution at the later landmark should reflect that information under the joint model.
Comparing probabilities for different intervals as if they were the same target. A nine-month risk and a three-month risk refer to different future intervals. State the horizon and interpret each probability for its own interval.
Using a plug-in random effect as though it represented all uncertainty. A prediction based only on one estimated $b_i$ omits variation over plausible random effects. Integrated prediction averages conditional risk over the patient-specific posterior distribution.

A useful alternative to a single unexplained before-and-after comparison is a small set of explicitly defined predictions: one with the old history at the original landmark and horizon, one with the new measurement but the same landmark and horizon when that comparison is meaningful, and one at the updated landmark with the intended horizon. Such comparisons make the target of each number transparent. They do not create a unique causal decomposition: the reported differences remain conditional on the model and on the chosen comparison sequence.

Key takeaway: A dynamic risk is defined by the patient’s information at a landmark and a specified future horizon. New biomarker data and event-free follow-up can update the patient-specific posterior distribution; a different horizon changes the probability being asked for. State these components before interpreting a change.

Check Your Understanding

Use the definitions and calculations above to answer each question. The solutions are included below so that each comparison can be checked.

A patient’s predicted risk changes after a new measurement, but the landmark and horizon are held fixed. What part of the prediction is directly updated by the measurement?
With high-risk posterior weight $0.40$, low-risk weight $0.60$, and six-month group risks $0.20$ and $0.05$, respectively, calculate the mixture risk.
Why can a prediction change after a patient remains event-free even when no new biomarker measurement is made?
A prediction rises when the horizon changes from three to twelve months, with no new observations. Does this show that the patient’s biomarker-related risk increased? Explain.

Solutions. (1) The longitudinal-data likelihood changes, which can change the posterior distribution of the patient’s random effects; the conditional group or random-effect-specific risk is then averaged using updated weights. (2) The risk is $0.40(0.20)+0.60(0.05)=0.080+0.030=0.110$, or $11.0\%$ over six months. (3) Event-free survival contributes information through the survival likelihood, changing the posterior distribution among random effects compatible with surviving to the later landmark. (4) No. The prediction concerns a longer interval, so it is a different target probability; without new patient information, the increase is due to the horizon change rather than a biomarker update.