Questions: Joint Models for Longitudinal and Survival Data

Question 1

A standard mixed-effects model of PSA trajectories in prostate cancer patients ignores the fact that patients with rapidly rising PSA are more likely to die and stop contributing measurements. Why does this create bias?

• A. The model fits fewer data points, reducing power
• B. The dropout is informative — patients with the worst PSA trajectories disappear from the data, making the observed average trajectory appear more favorable than the true population trajectory
• C. Mixed-effects models cannot handle unequal follow-up times
• D. The bias affects only the random effects, not the fixed effects

Answer: (B)

When dropout is related to the biomarker value (informative censoring), the remaining patients at later time points are a biased sample — they are the survivors with better biomarker trajectories. A standard mixed-effects model that treats dropout as ignorable will underestimate the rate of PSA rise in the population because the fastest-rising patients are no longer observed. Joint modeling handles this by explicitly linking the dropout process (survival) to the longitudinal process through shared random effects, so the model 'knows' that missing data are not random.

Question 2

Joint models use the 'true' (unobserved) biomarker value from the longitudinal submodel rather than the observed (measured) value in the survival submodel. Why is this important?

Model answer: Observed biomarker values contain measurement error and are only available at discrete, often irregular, time points. Using raw observed values as time-varying covariates in a Cox model attenuates the association (measurement error biases toward the null) and creates problems at event times that do not coincide with measurement times. The joint model's longitudinal submodel estimates the true underlying trajectory (smooth, continuous, and free of measurement error), and the survival submodel uses this true trajectory to predict the hazard at each moment. This produces unbiased estimates of the biomarker-hazard association.

This is analogous to errors-in-variables bias in regression: using a noisy proxy for the true predictor attenuates the estimated effect. The joint model's longitudinal submodel acts as a denoising filter, estimating the true biomarker level at each time point by borrowing strength across the subject's entire measurement history and the population trajectory. The survival submodel then uses these cleaned values.

Question 3

Dynamic prediction from a joint model updates a patient's survival probability each time a new biomarker measurement is obtained. This is more clinically useful than a single baseline prediction because it incorporates the patient's evolving trajectory.

• T. True
• F. False

Answer: True

A baseline Cox model produces a single survival prediction based on characteristics measured at study entry. A joint model can update this prediction as new longitudinal data arrive — if a patient's biomarker is rising faster than expected, the predicted survival probability decreases accordingly. This dynamic prediction is computed from the posterior distribution of the patient's random effects conditional on all their observed biomarker values, combined with the survival submodel. It is particularly valuable for clinical monitoring, where decisions (continue surveillance, switch treatment, refer for surgery) should reflect the patient's current trajectory, not just their baseline risk.