Why Missing Biomarker Measurements Matter
A patient’s biomarker history at a prediction time is often incomplete: a scheduled visit may have been missed, a specimen may have failed, or the measurement may not have been recorded. The absence of a value is not itself a biomarker value. In a joint model, the available measurements still inform the patient’s latent trajectory and random effects, but the incomplete history leaves more uncertainty about them. That uncertainty carries through to the predicted event risk.
The calculation below uses the available biomarker values and integrates over the patient-specific quantities that remain uncertain. It does not fill in a missing value with zero, the sample mean, or a guessed value. The key assumption is that the process determining which measurements are missing is ignorable for the model-based inference. That assumption has a precise scope: it does not cover every reason a measurement might be absent.
Model and Missingness Assumption
Write \(b_i\) for the patient-specific random effects and \(\theta\) for the joint-model parameters. A longitudinal submodel describes measurements around a latent trajectory \(m_i(t;b_i,\theta)\); for example, \(Y_{ij}=m_i(t_{ij};b_i,\theta)+\epsilon_{ij}\). A survival submodel specifies the event hazard \(h_i(t\mid b_i,\theta)\). The shared random effects, or the latent trajectory they determine, connect the biomarker and event processes.
For a prediction at time \(s\), the patient is known to be event-free through \(s\). Let \(\mathcal{H}_i(s)\) denote the observed event and censoring history through that time, including that the patient has survived to \(s\), and let \(X_i\) denote observed baseline covariates. The patient-specific prediction uses the distribution of \(b_i\) given the available biomarker history and this survival information. Missing measurements are marginalized rather than assigned invented values.
Ignorability concerns the probability of the missingness pattern given the full data. In this setting, a standard sufficient formulation is that, conditional on observed covariates, observed biomarker values, and observed event history, the probability of the pattern does not further depend on the unobserved biomarker values. In addition, the parameters governing missingness must be distinct from the parameters of the longitudinal and survival models. Under these conditions, the missingness mechanism can be omitted from likelihood-based inference for the joint-model parameters. Missing completely at random is stronger than necessary; missingness may depend on observed information.
For example, a missed visit may be more likely for patients with a previously recorded high biomarker value. That dependence can be compatible with ignorability if the earlier value is observed and is included in the conditioning information and model. By contrast, if a visit is more likely to be missed because the unobserved current biomarker value is high, even after conditioning on the observed history, the missingness mechanism is not ignorable under this formulation.
How the Available Values Enter the Prediction
Under conditionally independent measurement errors, the longitudinal likelihood contribution for a given \(b_i\) is the product of the measurement densities for observed values only. This is because each missing measurement’s density integrates to one when it is marginalized. If measurement errors are correlated over time, the missing values must instead be integrated out of their joint distribution; simply deleting their density terms is not generally valid.
Combining the observed marker data with survival to the landmark gives the patient-specific distribution of the random effects. In a Bayesian analysis, uncertainty in \(\theta\) can also be retained in the conditioning distribution. In a likelihood-based prediction with fitted parameters, the same expression is evaluated at the fitted parameter values and integrated over the random-effects distribution.
For a patient known to be event-free through \(s\), the event-history contribution includes the survival probability through \(s\), conditional on the random effects and covariates. Ignoring this contribution would fail to account for the fact that the patient has survived to the prediction time. Given \(b_i\) and \(\theta\), the risk over the next \(u\) time units is \(1-S_i(s+u\mid b_i,\theta)/S_i(s\mid b_i,\theta)\), where \(S_i(t\mid b_i,\theta)\) is the conditional survival function. The patient-specific prediction averages that conditional risk over the distribution updated by the observed history.
This formula separates two roles of the incomplete history. Observed marker values update the distribution of \(b_i\); missing values provide no direct measurement contribution under the stated model. But the absence of a value does not make the patient’s latent trajectory known, nor does it eliminate uncertainty about that trajectory. The survival information through \(s\) also updates the patient-specific distribution.
Worked Example: Updating a Random Effect With One Missing Visit
Suppose a simplified random-intercept model has \(b_i\sim N(0,1)\), and each scheduled biomarker measurement satisfies \(Y_{ij}\mid b_i\sim N(b_i,1)\), independently conditional on \(b_i\). A patient has a measured value \(Y_{i1}=0.4\); a second scheduled value is missing. Ignore event information for this calculation to isolate the marker-data update.
The likelihood contribution is only the observed measurement density, proportional to \(\exp[-(0.4-b_i)^2/2]\). Combining this with the \(N(0,1)\) prior gives a normal posterior. Its precision is the prior precision plus the observed-data precision, and its mean is the precision-weighted mean:
Thus \(b_i\mid Y_{i1}=0.4\sim N(0.2,0.5)\). The missing second measurement contributes no observed-value likelihood term under the conditional independence and ignorability assumptions. If it were falsely recorded as zero, it would act like a second measured value and yield posterior mean \((0+0.4+0)/3=0.1333\) and variance \(1/3\). That is a different, unjustified update: the patient did not have a measured value of zero.
Worked Patient-Specific Risk Update
The next example shows how the observed biomarker value, survival to the landmark, and a missing scheduled measurement jointly affect a prediction. The calculations use a deliberately small two-point random-effects distribution so that the update can be done by hand.
Worked Example: A Missing Visit and Survival to the Landmark
Consider a hypothetical patient with \(b_i\in\{0,1\}\), each value having prior probability \(0.5\). A measured biomarker value at an earlier visit is \(Y=0\), with \(Y\mid b_i\sim N(b_i,1)\). A later scheduled measurement by landmark \(s=1\) is missing. The patient is known to be event-free at \(s=1\). Suppose the hazard is constant conditional on \(b_i\), with \(h_i(t\mid b_i)=0.1\exp(0.5b_i)\) per year. Assume the missingness pattern is ignorable and the measurement error model is as stated.
First, the observed marker likelihood ratio comparing \(b_i=1\) with \(b_i=0\) is \(\phi(0;1,1)/\phi(0;0,1)=\exp(-0.5)=0.6065\), where \(\phi(y;\mu,\sigma^2)\) is the normal density. The prior odds are one, so after the observed biomarker alone the probabilities are \(P(b_i=1\mid Y=0)=0.6065/(1+0.6065)=0.3775\) and \(P(b_i=0\mid Y=0)=0.6225\). The missing visit adds no marker likelihood term.
Next, survival to \(s=1\) has likelihood \(\exp(-0.1)=0.9048\) for \(b_i=0\). For \(b_i=1\), the hazard is \(0.1\exp(0.5)=0.1649\) per year, so the survival likelihood is \(\exp(-0.1649)=0.8480\). The posterior odds for \(b_i=1\) versus \(b_i=0\), after both the marker and survival information, are \[ 0.6065\times\frac{0.8480}{0.9048}=0.5684. \] Therefore the updated probabilities are \(w_1=0.5684/(1+0.5684)=0.3624\) and \(w_0=1-w_1=0.6376\).
For the next year, the risk conditional on \(b_i=0\) is \(1-\exp(-0.1)=0.09516\). Conditional on \(b_i=1\), it is \(1-\exp(-0.1649)=0.1520\). Averaging these conditional risks over the updated probabilities gives
The predicted one-year event risk is therefore about \(11.58\%\), conditional on being event-free at the landmark and on the observed history. This is not the risk calculated using the prior probabilities alone: both the measured marker and survival to the landmark changed the random-effects weights. The absent measurement was not treated as a zero or as an additional observation.
What Ignorability Excludes
The mathematical convenience of omitting the missingness mechanism is justified only when its assumptions are credible. “The missingness rate is low” does not establish ignorability, and observing that missingness is associated with a recorded covariate does not by itself violate it. The relevant question is whether, after conditioning on the observed information included in the analysis, missingness still depends on unobserved biomarker values or on parameters linked to the outcome model.
Worked Example: When the Missingness Pattern Is Informative
At a prediction time before any future event follow-up, suppose \(b_i\) is equally likely to be 0 or 1. A scheduled biomarker measurement is missing with probability 0.2 when \(b_i=0\) and probability 0.8 when \(b_i=1\). Assume the hazard for the next year is \(0.1\exp(0.5b_i)\) per year. The missingness probability depends directly on the unobserved random effect, so it is not ignorable.
If the pattern is used, the probability of \(b_i=1\) among patients with a missing measurement is \(0.5(0.8)/[0.5(0.8)+0.5(0.2)]=0.8\). The one-year risk is then \(0.2[1-\exp(-0.1)]+0.8[1-\exp\{-0.1\exp(0.5)\}]\) \(=0.2(0.09516)+0.8(0.1520)=0.1406\), or about \(14.06\%\). If the pattern were incorrectly ignored, the equal prior weights would instead give \(0.5(0.09516)+0.5(0.1520)=0.1236\), or about \(12.36\%\). The difference arises because the observed fact that the measurement is missing carries information about \(b_i\) under this mechanism.
In real data, the dependence on an unobserved biomarker cannot generally be tested from the observed values alone. A practical response is to fit sensitivity models that make explicit alternative assumptions about how missingness depends on the unobserved value, and to compare the resulting predictions.
Diagnostics, Alternatives, and Limitations
Diagnostics can identify problems with the observed-data model and reveal patterns that make ignorability questionable, but they cannot prove that missingness is ignorable. Useful checks include tabulating missingness by visit and patient characteristics; comparing missingness rates across recorded covariates and earlier biomarker values; and checking whether the longitudinal model fits the observed measurements adequately across those groups. If missingness is associated with observed information, that information may need to be represented in the analysis. A relationship with observed history does not automatically invalidate ignorability, but omitting relevant observed predictors can make the assumption less plausible.
The main alternatives differ in what they assume and what they retain:
| Approach | What it does | Main concern |
|---|---|---|
| Use observed values in the joint-model likelihood | Marginalizes unobserved measurements and updates patient-specific uncertainty from the data that are available. | Relies on a suitable measurement model and an ignorable missingness mechanism. |
| Complete-case analysis | Restricts analysis to patients with a complete biomarker history. | Can discard substantial information and select a nonrepresentative subset; it does not generally remove bias from informative missingness. |
| Single-value imputation | Substitutes a fixed value, such as a mean or last observation. | Treats a guess as data and generally understates uncertainty; last observation carried forward also imposes an unverified trajectory assumption. |
| Explicit missingness model or sensitivity analysis | Models, or varies assumptions about, the relation between missingness and unobserved values. | Requires assumptions that may not be identified by the observed data and should be reported as sensitivity choices. |
The approach in this tutorial cannot recover information that was never recorded without assumptions. If missingness depends on the unobserved current biomarker after conditioning on the observed history, an ignorable analysis can give a misleading patient-specific update. Also distinguish a missed measurement from a biomarker value that is undefined after an event such as death: post-event absence is not an ordinary missing scheduled measurement. Finally, a model that handles missing biomarker values does not automatically handle a measurement schedule that is itself informative, such as visits triggered by unrecorded clinical deterioration. That requires a suitable model or sensitivity analysis for the observation process.
A Short Derivation of Marginalizing a Missing Measurement
The deletion of a missing measurement from the likelihood follows from integration, not from pretending the value does not exist. For one missing value \(Y_{i2}\), conditional independence of the measurement errors gives a joint density for the two measurements that factors into separate densities given \(b_i\). Integrating over every possible value of the missing measurement leaves the observed measurement density:
The last equality holds because a probability density integrates to one. This argument is conditional on \(b_i\); the observed value still informs \(b_i\), and the distribution of \(b_i\) must still be integrated or updated for prediction. If the measurement errors are correlated, the joint density need not factor in this way, so the integral over the missing value can affect the likelihood for the observed one.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In a model with conditionally independent measurement errors, a patient has one observed marker value and one missing scheduled value. What marker likelihood contribution is used, and why?
- Suppose a visit can be missed more often when the patient’s previously recorded biomarker value is high. Is this necessarily nonignorable? State the relevant condition.
- A patient is event-free at landmark \(s\). Besides the observed biomarker history, what event information enters the update of the patient-specific random-effects distribution?
- In the two-point example, suppose missingness depends on the unobserved random effect. Why can the missingness pattern itself change the prediction?
Solutions. (1) Use the density of the observed value conditional on the random effects and parameters. The missing value is integrated out, and its conditional density integrates to one under conditional independence. (2) No. It can be ignorable if missingness depends only on observed information included in the conditioning set and the missingness parameters are distinct from the model parameters. (3) Survival to \(s\), represented by the event-history likelihood contribution, such as the conditional survival probability through the landmark. (4) The pattern then carries information about the random effect; conditioning on the observed fact that the visit is missing changes the random-effects weights and therefore the averaged future risk.