Misclassified Event Causes and Joint Likelihoods

Why Cause Labels Belong in the Likelihood

In competing-risks data, the event time and the event cause answer different questions. A time may be recorded accurately even when the cause label is uncertain—for example, because classification is based on an imperfect diagnostic procedure. Treating that label as certainly correct can distort estimates of cause-specific hazards and their associations with longitudinal measurements. The appropriate likelihood accounts for the uncertainty instead of assigning each event to a single cause with certainty.

This tutorial considers exact event times with right censoring. It combines cause-specific hazards with a probability model for the reported label, then integrates over the shared random effects used to link longitudinal and event processes. The key operation is a sum over possible true causes for each event whose recorded label is uncertain.

Definition: Let \(K_i\in\{1,\ldots,J\}\) be subject \(i\)'s true event cause, and let \(R_i\in\{1,\ldots,L\}\) be the recorded label. The cause-misclassification probability is \(q_{irk}=P(R_i=r\mid K_i=k,\mathcal{C}_i)\), where \(\mathcal{C}_i\) denotes the covariates and other conditioning information specified by the model. For every true cause \(k\), the probabilities satisfy \(q_{irk}\geq 0\) and \(\sum_{r=1}^{L}q_{irk}=1\). Thus, each column of the matrix \(Q_i=(q_{irk})\) gives the distribution of recorded labels for one true cause.

For example, \(q_{i, A, 1}=0.80\) means that an event whose true cause is 1 is recorded as label A with probability 0.80, conditional on the model's stated conditioning information. It is not the probability that the true cause is 1 given label A. That latter probability also depends on the cause-specific hazards and the event history.

Cause-Specific Hazards and the Observed Event Contribution

Write \(\lambda_{ik}(t\mid b_i)\) for subject \(i\)'s cause-specific hazard for true cause \(k\) at time \(t\), conditional on random effects \(b_i\) and the included covariates. The cause-specific cumulative hazard is \(\Lambda_{ik}(t\mid b_i)=\int_0^t\lambda_{ik}(u\mid b_i)\,du\). While the subject remains event-free, the total hazard is the sum over causes, so the conditional event-free survival function is \(S_i(t\mid b_i)=\exp\{-\sum_{k=1}^J\Lambda_{ik}(t\mid b_i)\}\).

Assume the event time is observed exactly, the event process follows these hazards, and censoring is noninformative conditional on the modeled covariates and random effects. Also assume the label mechanism is correctly specified and is conditionally independent of the longitudinal measurements and event time given the true cause and its stated conditioning information. These assumptions permit a factorization into an event-time-and-cause model and a label model. The event-time density for a true cause \(k\) at time \(t\), conditional on \(b_i\), is survival to \(t\) times the cause-specific hazard at \(t\).

f(T_i=t,K_i=k\mid b_i)=S_i(t\mid b_i)\lambda_{ik}(t\mid b_i).

If an event at time \(t\) has recorded label \(r\), its true cause is not observed. Summing the joint density over all possible true causes gives the observed-data event contribution. The label probability weights the hazard for each possible cause; the survival term is common to these alternatives because all causes contribute to the risk of the first event.

L_{i,\mathrm{event}}(t,r\mid b_i) =\sum_{k=1}^{J}S_i(t\mid b_i)\lambda_{ik}(t\mid b_i)q_{irk} =S_i(t\mid b_i)\sum_{k=1}^{J}q_{irk}\lambda_{ik}(t\mid b_i).

For a right-censored subject with censoring time \(C_i\), the event contribution is \(S_i(C_i\mid b_i)\), provided censoring is noninformative under the stated conditioning. There is no cause label to model for a subject who has not had an observed event by that time.

Worked Example: An Event With a Recorded Label

Suppose there are two true causes with constant hazards \(\lambda_1=0.04\) per month and \(\lambda_2=0.06\) per month. A hypothetical subject has an event at \(t=5\) months, recorded as label A. Let \(q_{A1}=0.80\) and \(q_{A2}=0.25\). Assume the label probabilities are constant over time and that the event time and label mechanism satisfy the assumptions just stated.

The cumulative hazards by month 5 are \(0.04(5)=0.20\) and \(0.06(5)=0.30\). Thus the probability of remaining event-free until month 5 is \(S(5)=\exp\{-(0.20+0.30)\}=\exp(-0.50)\approx0.6065\).

The observed event contribution is \(S(5)\{q_{A1}\lambda_1+q_{A2}\lambda_2\} =0.6065\{0.80(0.04)+0.25(0.06)\} =0.6065(0.047)\approx0.02851\) per month.

Conditional on an event at month 5 with label A, and given these hazards and label probabilities, the relative contributions from true causes 1 and 2 are \(0.80(0.04)=0.032\) and \(0.25(0.06)=0.015\). Therefore the model-implied probability of true cause 1 in this event is \(0.032/(0.032+0.015)\approx0.6809\). The label alone does not determine the true cause; both the hazards and misclassification probabilities matter.

Combining the Event and Longitudinal Data

Let \(Y_i\) denote subject \(i\)'s longitudinal measurements, with conditional density or mass function \(f_Y(Y_i\mid b_i)\). In a shared-random-effects joint model, the longitudinal observations and event process are modeled conditionally on the same \(b_i\). For an event at \(T_i\) with recorded label \(R_i=r\), the subject-level observed-data likelihood contribution is obtained by multiplying the conditional longitudinal contribution by the misclassified event contribution and integrating over the random-effects distribution \(p(b_i)\).

L_i =\int f_Y(Y_i\mid b_i) S_i(T_i\mid b_i) \left\{\sum_{k=1}^{J}q_{irk}\lambda_{ik}(T_i\mid b_i)\right\} p(b_i)\,db_i.

For a censored subject, replace the bracketed event-density term by \(S_i(C_i\mid b_i)\). The sum over true causes and the integral over random effects are both essential: the first accounts for an unobserved cause, and the second accounts for unobserved subject-level heterogeneity. In general, the integral of a product is not the product of separate integrals. Longitudinal measurements can therefore affect the event contribution through the information they provide about \(b_i\).

Worked Example: A Joint Contribution at Two Integration Nodes

Consider hypothetical quadrature with two random-effects nodes, each assigned weight 0.50. At node 1, let the cause-specific hazards be 0.04 and 0.02 per month; at node 2, let them be 0.01 and 0.05 per month. An event occurs at month 2 and is recorded as label R, with \(q_{R1}=0.80\) and \(q_{R2}=0.25\). Suppose the longitudinal likelihood contributions at the nodes are 0.60 and 0.20, respectively, in the same density units. The values are hypothetical and are used only to demonstrate the calculation.

At node 1, the total hazard is 0.06 per month, so survival to month 2 is \(\exp\{-2(0.06)\}=\exp(-0.12)\approx0.88692\). The label-weighted hazard is \(0.80(0.04)+0.25(0.02)=0.037\) per month. The conditional event contribution is therefore \(0.88692(0.037)\approx0.032816\) per month.

At node 2, the total hazard is also 0.06 per month, giving the same survival value, \(0.88692\). The label-weighted hazard is \(0.80(0.01)+0.25(0.05)=0.0205\) per month, so the conditional event contribution is \(0.88692(0.0205)\approx0.018182\) per month.

The quadrature approximation to the joint contribution is \(0.50(0.60)(0.032816)+0.50(0.20)(0.018182)\approx0.009845+0.001818=0.011663\), in longitudinal-density units per month. The approximation averages the product of the longitudinal and event contributions at each node; multiplying separately averaged contributions would not generally give the same result.

Why the Sum and Its Assumptions Matter

A short derivation shows why the observed label contribution has the stated form. For a particular true cause \(k\), the joint event-time and label density is the event-time-and-cause density multiplied by the conditional probability of the recorded label. Since the true cause is unobserved, add these mutually exclusive possibilities:

\begin{aligned} f(T_i=t,R_i=r\mid b_i) &=\sum_{k=1}^{J} f(T_i=t,K_i=k\mid b_i) P(R_i=r\mid K_i=k,\mathcal{C}_i)\\ &=\sum_{k=1}^{J}S_i(t\mid b_i)\lambda_{ik}(t\mid b_i)q_{irk}\\ &=S_i(t\mid b_i)\sum_{k=1}^{J}q_{irk}\lambda_{ik}(t\mid b_i). \end{aligned}

This argument depends on the label probabilities being conditional probabilities under the assumed label model. If label accuracy varies with time, covariates, or site, those features should be included in \(q_{irk}\) when supported by the design and data. If the label mechanism also depends on unobserved random effects, that dependence must be modeled rather than silently treated as independent.

When the cause label is completely unavailable, sum the event contribution over all possible recorded labels. Because the probabilities for each true cause sum to one, the result is the usual total event density:

\sum_{r=1}^{L}S_i(t\mid b_i)\sum_{k=1}^{J}q_{irk}\lambda_{ik}(t\mid b_i) =S_i(t\mid b_i)\sum_{k=1}^{J}\lambda_{ik}(t\mid b_i).

Thus, an unobserved label and a misclassified-but-observed label are not interchangeable. A known but noisy label provides information through its probability under each true cause. An entirely missing label contributes no such distinction among causes.

Practical Checks, Alternatives, and Limitations

The model can be fit by maximizing the observed-data likelihood, using numerical integration or an approximation such as quadrature for the random-effects integral. The cause sum is evaluated within that likelihood; it is not replaced by a guessed true cause. Before interpreting estimates, check that the probability matrix is valid, the event and censoring contributions match each subject's status, and the likelihood calculation is numerically stable.

Check the label model. Each entry must be between zero and one, and the probabilities over recorded labels must sum to one for every true cause. Compare estimated or assumed probabilities with validation data, blinded re-adjudication, or information about the classification process when available.
Check information and stability. If some causes or recorded labels are rare, or if the matrix has similar columns, the observed labels may provide little separation between true causes. Large standard errors, strong sensitivity to starting values, or nearly flat likelihood profiles can signal weak identification.
Check model fit. Compare observed and model-implied frequencies of recorded labels across relevant covariate or time groups, and inspect residual or predictive checks for event times and longitudinal outcomes. Poor label-frequency fit can indicate that accuracy varies across groups in a way the model omits.
Check sensitivity to the error assumptions. If the probabilities \(q_{irk}\) are externally specified, assess how reasonable changes in them affect the hazard and association estimates. If they are estimated from the same data, examine whether the data contain enough information to estimate both the hazard structure and label error.

If labels are known to be correct, set \(q_{irk}=1\) when \(r=k\) and zero otherwise; the usual cause-specific event contribution follows. If labels are absent altogether, use the total-hazard event density instead. These alternatives answer different data situations. Simply treating every recorded label as correct is not a neutral shortcut: it imposes the perfect-classification assumption.

A central limitation is identifiability. Without validation information, restrictions, or sufficiently informative structure, different combinations of cause-specific hazards and misclassification probabilities may produce similar observed event and label distributions. The label matrix is therefore not automatically estimable just because a likelihood can be written. Nor does a well-fitting joint likelihood establish that cause labels are accurate or that an estimated association is causal. Interpretations remain conditional on the event, longitudinal, random-effects, censoring, and misclassification assumptions.

Common Pitfall

Common mistake: “The record says label A, so this event contributes \(S_i(t)\lambda_{iA}(t)\).” This treats the observed label as the true cause and assumes perfect classification, even when the model allows errors. Correction: sum over possible true causes, weighting each cause-specific hazard by the probability of the observed label under that cause: \(S_i(t)\sum_k q_{irk}\lambda_{ik}(t)\). If classification is known to be perfect, the probability matrix reduces to the identity case.

Check Your Understanding

Answer each question using the stated model; brief solutions are included below.

For two true causes, \(q_{A1}=0.70\), \(q_{A2}=0.20\), \(\lambda_1(t)=0.03\) per month, \(\lambda_2(t)=0.05\) per month, and \(S(t)=0.80\). What is the event contribution for recorded label A?
Why must the likelihood sum over true causes when the cause is not directly observed?
If labels are entirely unavailable, what event-density term results after summing over all possible labels?
What additional likelihood operation is needed to combine longitudinal measurements with the event contribution when both depend on shared random effects?

Solutions. (1) The weighted hazard is \(0.70(0.03)+0.20(0.05)=0.031\) per month, so the contribution is \(0.80(0.031)=0.0248\) per month. (2) The event could have arisen from any of the possible true causes; their mutually exclusive densities must be added, each weighted by the chance of the observed label under that cause. (3) Since the label probabilities sum to one for each true cause, the term is \(S(t)\sum_k\lambda_k(t)\). (4) Multiply the conditional longitudinal and event contributions at a given random-effects value, then integrate that product over the random-effects distribution.